Martijn Huynen Injection Locking and Pulling EMC-Aware Design of A

EMC-Aware Design of a Low-Cost Receiver Circuit under Injection Locking and Pulling

Martijn Huynen

Supervisors: Prof. dr. ir. Dries Vande Ginste, Prof. dr. ir. Johan Bauwelinck Counsellors: Ir. Gert-Jan Stockman, Dr. ir. Frederick Declercq, Dr. Guy Torfs

Master's dissertation submitted in order to obtain the academic degree of Master of Science in Electrical Engineering

Department of Information Technology Chairman: Prof. dr. ir. Daniël De Zutter Faculty of Engineering and Architecture Academic year 2013-2014

EMC-Aware Design of a Low-Cost Receiver Circuit under Injection Locking and Pulling

Martijn Huynen

Supervisors: Prof. dr. ir. Dries Vande Ginste, Prof. dr. ir. Johan Bauwelinck Counsellors: Ir. Gert-Jan Stockman, Dr. ir. Frederick Declercq, Dr. Guy Torfs

Master's dissertation submitted in order to obtain the academic degree of Master of Science in Electrical Engineering

Department of Information Technology Chairman: Prof. dr. ir. Daniël De Zutter Faculty of Engineering and Architecture Academic year 2013-2014

Preface

This master’s dissertation is the conclusion to a lustrum at the faculty of Engineering and Architecture at Ghent University. Writing this preface is one of the ﬁnal feats in this one-year work and makes you contemplate about the work that has been done and the help you received. This is why I want to seize the opportunity to express my sincerest gratitude to all those who helped bring this work to a successful ending. First and foremost, I would like to thank the chairman, prof. dr. ir. D. De Zutter, as well as prof. dr. ir. D. Vande Ginste and prof. dr. ir. J. Bauwelinck for giving me the opportunity to carry out this thesis research at the Electromagnetics Group and at the INTEC design group of the Department of Information Technology and providing me with the necessary facilities and materials to successfully complete this work. In particular, I thank prof. Vande Ginste for giving splendid advice, putting forward great suggestions and for his general support. I am very appreciative as well of the excellent guidance and cheerful positivism of prof. Bauwelinck. If I would have to name one person without whom this dissertation would not have come to a good end, it would be, without any doubt, ir. Gert-Jan Stockman. His dedication and determination kept me from not seeing the wood for the trees. For his inexhaustible patience and his thorough feedback, I thank him from the bottom of my heart. Special thanks goes to dr. ir. Frederick Declercq and dr. ir. Guy Torfs for aiding me in the design of the oscillator. There is no doubt about it that without their expertise, the design would not have worked properly in the end. I am also grateful for the suggestions and assistance, however small, from various people of both research groups. They may not have always realised it at the time, but every contribution to this work was highly appreciated. I would also like to thank my fellow thesis students with whom I shared the computer room of the research group for the past year. Irven Aelbrecht, Olivier Caytan, Erica Debels, Niels Lambrecht and our Italian friend Lorenzo Silvestri always backed me up when I needed some encouragement and created a fun atmosphere to work on the thesis. My roommate, Niels Van den Putte, deserves special thanks as well since we shared the fun and not so fun moments of student life for the past ﬁve years. We are going our separate ways now, but I wish him the best of luck in his last year of veterinary medicine. For going through this book and correcting my numerous writing errors, my sincerest gratitude goes towards my cousin, Sacha. Last but certainly not least, I wish to express my deepest gratitude to my friends and family. In particular, I thank my parents and my sister without whose unconditional support, not only during my studies, I would have never made it this far.

Martijn Huynen, June 2014 Admission to Loan

The author gives permission to make this master’s dissertation available for consultation and to copy parts of this master’s dissertation for personal use. In the case of any other use, the limitations of the copyright have to be respected, in particular with regard to the obligation to state expressly the source when quoting results from this master dissertation. De auteur geeft de toelating deze masterproef voor consultatie beschikbaar te stellen en delen van de masterproef te kopi¨erenvoor persoonlijk gebruik. Elk ander gebruik valt onder de beperkingen van het auteursrecht, in het bijzonder met betrekking tot de verplichting de bron uitdrukkelijk te vermelden bij het aanhalen van resultaten uit deze masterproef.

Martijn Huynen, June 2014 EMC-Aware Design of a Low-Cost Receiver Circuit under Injection Locking and Pulling by

Martijn HUYNEN

Master’s Dissertation submitted to obtain the academic degree of Master of Science in Electrical Engineering

Academic 2013–2014

Promoters: Prof. dr. ir. Dries VANDE GINSTE, Prof. dr. ir. Johan BAUWELINCK Supervisors: Ir. Gert-Jan STOCKMAN, Dr. ir. Frederick DECLERCQ, Dr. ir. Guy TORFS

Faculty of Engineering and Architecture Ghent University

Departement of Information Technology Chairman: Prof. Dr. Ir. Dani¨elDE ZUTTER

Summary

This master’s dissertation encompasses the design of a low-cost radio frequency (RF) receiver front-end, intended for application in the 2.45 GHz industrial, scientific and medical (ISM) radio band. In order to reduce the cost, the RF preselect filter is omitted, which affects the immunity of the total system. Out-of-band signals can leak into the oscillator and cause non-linear effects such as injection locking and injection pulling. The designed oscillator is a negative resistance oscillator that produces a fundamental tone at 2.344 GHz with an output power of 9 dBm. The employed mixer is a commercially available double balanced diode mixer. For the first time in literature, this work demonstrates the critical necessity to take into account the potentially detrimental effects caused by injection locking and pulling during Electromagnetic Compatibility (EMC)-aware design. Throughout the design cycle, it is advised to adopt rigorous theoretical analysis, thorough simulation, and careful prototyping and measurements.

After an introduction to EMC-aware design in Chapter 1, a theoretical approach to the exam- ined phenomena of injection locking and pulling is given in Chapter 2. Chapter 3 continues with a summary of the various oscillator and mixer topologies and holds a discussion of non-linear simulation tools as well. The design of the negative resistance oscillator and the mixer is explained in Chapter 4. Chapter 5 is devoted to the measurements of both the functional and the EMC-behaviour of the individual building blocks and the total front-end under injection locking and pulling. Finally, conclusions and an outline for future research are formulated in Chapter 6.

Keywords superheterodyne receiver; low-cost circuit; negative resistance oscillator; electromagnetic compatibility (EMC); injection locking and pulling EMC-Aware Design of a Low-Cost Receiver Circuit under Injection Locking and Pulling Martijn Huynen Supervisors: prof. dr. ir. D. Vande Ginste, prof. dr. ir. J. Bauwelinck, ir. G.-J. Stockman, dr. ir. F. Declercq and dr. ir. G. Torfs

Abstract— This master’s dissertation focusses on the design of a low- the natural frequency, the so-called lock range, the oscillator’s cost radio frequency (RF) receiver front-end, intended for operation in the frequency will shift to the frequency of the external signal, 2.45 GHz industrial, scientific and medical (ISM) radio band. The effects of co-located sources, in particular injection locking and pulling, are observed called injection locking. The width of this range is proportional and characterised in both the local oscillator (LO) and the complete sys- to the amplitude of the injected signal, the natural frequency tem. The consequences on the receiver’s performance are evaluated as well. of the oscillator and inversely proportional to the quality factor For the first time in literature, this work demonstrates the critical necessity (Q) of the oscillator, a measure for the frequency selectivity [2]. to take the potentially detrimental effects caused by injection locking and pulling into account during Electromagnetic Compatibility (EMC)-aware When the frequency is brought just outside of the lock range af- design. terwards, the frequency of the oscillator does not return to its Keywords— superheterodyne receiver; low-cost circuit; negative resis- original state. Quite the opposite, the oscillator will still attempt tance oscillator; electromagnetic compatibility (EMC); injection locking to lock to the external signal. However, as it cannot keep up, its and pulling frequency undergoes a quick transition back towards its natural frequency, overshoots and tries to lock once again to the external I.INTRODUCTION signal. To put it briefly, the oscillator’s frequency response has VER the last few decades, an explosive growth in wireless become time-dependent because it is being pulled towards the Ocommunication has been witnessed. Together with the rise external frequency, hence the name injection pulling. This effect of smarthphones and other multimedia applications, the demand manifests itself in the frequency domain by the appearance of a for low-cost communication circuits has increased drastically. lot of tightly spaced frequency components close to the external In an attempt to reduce the cost of a standard superheterodyne frequency. The further one departs from the lock range, the less receiver, the radio frequency (RF) preselect filter is sometimes pronounced the effect becomes whereby the natural frequency omitted. This measure should not be taken lightly, however, as it will start prevailing once again. can affect the immunity of the receiver system to interfering signals. When out-of-band signals (for the intended application), III.DESIGNOFA RECEIVER CIRCUIT having a frequency in the vicinity of the local oscillator (LO) A. Negative resistance oscillator frequency, leak into the oscillator, they can cause injection locking and/or pulling. These non-linear effects have the potential to One possible view on oscillators that is convenient for high jeopardise the proper operation of the total receiving chain [1]. operating frequencies is that of the negative resistance oscillator. In this master’s dissertation, the goal is to design a low-cost An unstable amplifier can be represented by a one-port network radio frequency (RF) receiver front-end, intended for use in the that delivers a complex input impedance at its port with the real 2.45 GHz industrial, scientific and medical (ISM) radio band. part being negative. The frequency can then be fixed by com- Subsequently, the system is subjected to co-located sources to pensating the input impedance with a matching network to the induce injection locking and pulling and evaluate the receiver’s load. A first design was made using a bipolar junction transistor performance under these conditions. It is shown that EMC- (BJT), i.e. the BFP640 by Infineon Technologies, together with aware design needs to encompass rigorous theoretical analy- discrete components to provide the necessary matching. After sis, thorough simulation, and careful prototyping and measure- manufacturing, the oscillator turned out to oscillate at 6 GHz in- ments. stead of the wanted 2.38 GHz. The lack of a strong frequency This abstract starts off with a short introduction to injection dependence was the main cause. A redesign was made on a locking and pulling in order to get a solid grasp of their ma- DE104 substrate by Elprinta (thickness of 800 µm, r of 4.75 jor properties (Section II). The design of the different building and tan δ of 0.02) with the BFP183 by Infineon Technologies blocks is discussed in Section III. Measurements on the oscil- and using transmission lines to improve frequency selectivity. lator and the total system are described and interpreted in Sec- Table I together with Figure 1 depicts the oscillator that pro- tion IV. Conclusion and future research are outlined in Sec- duces a ground tone at 2.344 GHz with a power of 9 dBm. The tion V. first two overtones are substantially suppressed by 13.5 dB and 17.5 dB. II.INJECTION LOCKINGAND PULLING B. Diode mixer Injecting a signal with a frequency close to the oscillator’s natural frequency can cause injection locking or pulling. When A mixer from a previous project was re-used. A Mini Circuits the injected frequency is within a certain frequency range around ADE-35+ double balanced diode mixer with a BP2U+ power

3.3 V 50 Measurement L2 Simulation C2 l3 R1 Out 40

l1 30 [MHz]

L1 l2 L R2 f 20 C1

R3 C3 10

0 40 30 20 10 0 Fig. 1. Schematic of the ﬁnal oscillator design. − − − − Pinj /P0 [dB]

TABLE I Fig. 2. Lock range of the stand-alone oscillator. COMPONENTVALUESANDTRANSMISSIONLINEDIMENSIONSOFTHE FINAL OSCILLATOR. B. Receiver front-end Component Value Component Value R1 1500 Ω l1 / w1 35 mm / 0.9 mm During the measurements on the total system, two tones were R2 820 Ω l2 / w2 19.6 mm / 0.9 mm inserted into the mixer via the RF port. The useful signal (f1) at R3 27 Ω l3 / w3 10.2 mm / 1.416 mm 2.45 GHz, and an interfering signal at a variable frequency (f2). L 37 nH L 37 nH Both had an input power of 20 dBm. Figures 3 and 4 show the 1 2 − C1 52.1 pF C3 52.1 pF IF spectrum for different values of f2. C2 100 pF C4 27 pF As the exact LO frequency in the total system is 2.3317 GHz, the nominal IF frequency for the information signal is 118.3 MHz, indicated in both ﬁgures by the dashed vertical line. Note that the lock range of the oscillator is much smaller in this splitter (with one terminated port) at its LO input was used to set-up, compared to Figure 2, as the effective injected power is perform the downconversion from the RF signal to an interme- substantially lower. At the edge of the lock range (Figure 3), diate frequency (IF). The circuit exhibits a conversion loss of the spectrum consists of a single frequency component which 7.5 dB and an RF to LO isolation of about 25 dB. This isolation has clearly shifted. Depending on the building blocks after the lowers the undesired effective power of co-located sources that front-end, this can cause problems if not dealt with properly. reaches the oscillator.

30 118.315 MHz IV. MEASUREMENTS − 40 Measurements are performed using an N5242A PNA-X from − 50 Agilent Technologies to generate one single tone or a two-tone − signal, depending on the exact measurement. Frequency spectra 60 are captured using a Rhode & Schwarz FSV40 signal analyser. − 70 In order to measure the LO spectrum while it was embedded in − the system, a quadrature hybrid was used. 80 − Magnitude [dBm] 90 − A. Oscillator 100 − 110 Figure 2 depicts the one-sided lock range (fL) of the oscillator − 118 118.2 118.4 118.6 by itself as a function of the ratio of the injected power (Pinj) Frequency [MHz] to the oscillator’s output power (P0). Power was injected into the output and results for both simulation and measurements are Fig. 3. IF spectrum for f2 = 2.331685 GHz, i.e. the edge of the lock range. plotted. One clearly sees that the lock range starts off very small for low injected powers but grows drastically for larger injected The spectrum in the injection pulling region (Figure 4) differs powers. Simulation results predict a smaller lock range which is completely from the one in the injection locking situation [3]; almost certainly caused by the higher Q factor of the simulated the spectrum consists of a lot of closely spaced components, circuit, as not all parasitic effects can be simulated. making the extraction of information from this IF spectrum impossible. The time-dependent oscillator has ruined the proper operation of the receiver structure. For stronger interference signals, the frequency range in which the operation is disturbed increases and within this range, the pulling effects become even more prominent.

30 − 40 − 50 − 60 − 70 − 80 − Magnitude [dBm] 90 − 100 − 110 − 118 118.2 118.4 118.6 Frequency [MHz]

Fig. 4. IF spectrum for f2 = 2.33166 GHz, causing strong injection pulling.

V. CONCLUSIONSAND FUTURE RESEARCH A low-cost RF front-end for the 2.45 GHz ISM band is designed and manufactured. The occurrence of injection locking and pulling is demonstrated in both the stand-alone oscillator and in a realistic scenario on the total system. As such, it is clearly shown that EMC-aware design critically depends on a design strategy that can take these non-linear effects into account. In future work, a co-design between the oscillator and the mixer could provide some interesting opportunities for further cost and footprint reduction. Adding an (active) antenna to the receiver circuit and performing additional injection measurements via radiation, will yield new insights into non-linear phenomena described in this dissertation. Moreover, besides being an ideal platform for EMC-evaluation, such a complete system also constitutes a fully functional prototype for the further development of low-cost and wearable electronic products.

REFERENCES [1] R. Adler, “A Study of Locking Phenomena in Oscillators”, Proceedings of the IRE, vol. 34, no. 6, pp. 351-357, June 1946. [2] B. Razavi, “A study of Injection Locking and Pulling in Oscillators”, IEEE Journal of Solid-State Circuits, vol. 39, no. 9, pp. 1415-1424, Septem- ber 2004. [3] I. Ali, A. Banerjee, A. Mukherjee, and B. N. Biswas, “Study of Injection Locking With Amplitude Perturbation and Its Effect on Pulling of Oscilla- tor”, IEEE Transactions on Circuits and Systems I - Regular Papers, vol. 59, no. 1, pp. 137-147, January 2012. EMC-bewust ontwerp van een lagekost ontvangstcircuit onder invloed van storende synchronisatiefenomenen Martijn Huynen Begeleiders: prof. dr. ir. D. Vande Ginste, prof. dr. ir. J. Bauwelinck, ir. G.-J. Stockman, dr. ir. F. Declercq and dr. ir. G. Torfs

Abstract—Deze masterproef bespreekt het ontwerp van een lagekost ra- II.INJECTIONLOCKINGENPULLING diofrequent (RF) ontvangstcircuit voor gebruik in de industrial, scientific and medical (ISM) radioband. Effecten die veroorzaakt worden door on- Injection locking en pulling worden veroorzaakt door een sig- gewenste ingangssignalen, met in het bijzonder injection locking en pul- naal in de oscillator te sturen met een frequentie in de buurt van ling, werden waargenomen en gekarakteriseerd zowel in de lokale oscil- de eigenfrequentie van de schakeling. Als deze externe frequen- lator (LO) als in het totale systeem. De gevolgen op de prestaties van de ontvanger worden tevens onderzocht. Dit werkt toont voor het eerst de kri- tie binnen een bepaalde frequentieband valt, de zogenaamde tische noodzaak aan van EMC-bewust ontwerp waarbij de potentieel zeer lock range, zal de oscillatiefrequentie verschuiven naar deze schadelijke effecten veroorzaakt door injection locking en pulling in rekening stoorfrequentie. Dit noemt men injection locking. De breedte worden gebracht. van deze lock range is evenredig met de amplitude van het stoor- Trefwoorden— superheterodyne ontvanger; lagekost circuit; negatieve signaal, de eigenfrequentie van de oscillator en omgekeerd even- weerstand oscillator; elektromagnetische compatibiliteit; injection locking en pulling redig met de kwaliteitsfactor (Q), een maat voor de frequentie- selectiviteit van de schakeling [2]. Als de frequentie van het externe signaal vervolgens juist buiten de lock range wordt ge- I.INLEIDING bracht, zal de oscillator niet terugkeren naar zijn normale werking. Integendeel, de oscillator zal proberen te synchroniseren N de afgelopen decennia hebben we een explosieve groei met het externe signaal. Aangezien dit niet vol te houden is, zal Imeegemaakt van draadloze communicatie. Daarnaast heeft er een snelle terugval naar de eigenfrequentie plaatsvinden na de opkomst van smartphones en andere multimediatoepassingen een bepaalde tijd. Door de snelheid wordt deze frequentie ech- de vraag naar lagekost communicatiecircuits fors doen stijgen. ter voorbij geschoten waarna er opnieuw een poging wordt on- Om de kost van een standaard heterodyne ontvanger te drukken, dernomen om te vergrendelen met het stoorsignaal. Kortom, de wordt er soms geopteerd om de radiofrequent (RF) voorselec- ogenblikkelijke oscillatiefrequentie wordt tijdsafhankelijk om- tiefilter achterwege te laten. Deze maatregel moet echter goed dat deze wordt aangetrokken richting het externe signaal, der- overwogen worden aangezien de ongevoeligheid aan stoorsig- halve de naam injection pulling. In het frequentiedomein wordt nalen hieronder zou kunnen lijden. Als signalen buiten de voor dit zichtbaar door het ontstaan van vele, dicht opeengepakte fre- de applicatie gekozen band vallen, maar een frequentie hebben quentiecomponenten in de buurt van de stoorfrequentie. Als in de buurt van de lokale oscillator (LO) frequentie, kunnen zij men de externe frequentie verder verwijdert van de lock range, injection locking en/of pulling veroorzaken. Deze niet-lineaire zal dit tijdsafhankelijk effect geleidelijk aan afnemen en keert fenomenen kunnen mogelijks het correct functioneren van het de oscillator terug naar zijn eigenfrequentie. volledige systeem in gevaar brengen [1]. III.ONTWERP VAN EEN ONTVANGSTCIRCUIT Het doel van deze thesis is een lagekost ontvangstcircuit te ontwerpen dat functioneert voor ingangsfrequenties die in de A. Negatieve weerstand oscillator 2.45 GHz industrial, scientific and medical (ISM) radioband Een mogelijke invalshoek om oscillatorontwerp aan te vatten, vallen. Daarna wordt het systeem onderworpen aan stoorsig- passend voor hoge werkfrequenties, is via het concept van de nalen om injection locking en pulling uit te lokken. De presta- negatieve weerstand oscillatoren. Een onstabiele versterker kan tie van de ontvanger onder deze omstandigheden wordt vervol- voorgesteld worden door een eenpoortnetwerk dat een ingangs- gens geevalueerd.¨ Dit werkt toont aan dat EMC-bewust ontwerp impedantie aanlevert met een negatief reeel¨ deel. De frequentie een rigoureuze, theoretische voorstudie, grondige simulaties, en wordt vervolgens vastgelegd door een geschikt aanpassingsnet- nauwkeurige metingen van prototypes dient te omvatten. werk naar de last te voorzien. In een eerste ontwerp werden De korte inleiding tot injection locking en pulling van deze een bipolaire transistor, een BFP640 van Infineon Technologies, abstract leidt tot het nodige inzicht in de belangrijkste principes. en discrete componenten gebruikt om de oscillator te realiseren. Vervolgens wordt er ingegaan op het ontwerp van de verschil- Na vervaardiging bleek de oscillator een frequentie van 6 GHz lende bouwblokken in sectie III. Metingen op de alleenstaande te produceren in plaats van de gewenste 2.38 GHz. De oorzaak oscillator en op het totale systeem worden besproken in sec- voor deze afwijking werd gevonden bij het gebrek aan een sterk tie IV. Sectie V ten slotte, beschrijft de voornaamste conclusies frequentieafhankelijk element in het circuit. Een herontwerp en beschrijft de mogelijkheden tot verder onderzoek. werd vervolgens gemaakt op het DE104 substraat van Elprinta (dikte van 800 µm, r van 4.75 en tan δ van 0.02) waarbij een A. Oscillator andere transistor, de BFP183 vann Infineon Technologies, werd Figuur 2 toont de enkelzijdige lock range (fL) van de au- gebruikt samen met transmissielijnen om de frequentiestabiliteit tonome oscillator als functie van de verhouding tussen het te bevorderen. In tabel I worden de componentwaarden weerge- ge¨ınjecteerde vermogen (Pinj) en het uitgangsvermogen van de geven die de oscillator van figuur 1 bepalen. Deze oscillator oscillator (P0). Het stoorsignaal werd ingebracht via de uit- produceert een grondtoon van 2.344 GHz met een uitgangsver- gang en zowel meet- als simulatieresultaten zijn weergegeven. mogen van 9 dBm. De twee eerste boventonen worden sterk De lock range is duidelijk zeer klein voor lage stoorvermogens onderdrukt met 13.5 dB en 17.5 dB. maar groeit spectaculair voor hogere ge¨ınjecteerde vermogens. Simulatieresultaten voorspellen een kleinere lock range dan de 3.3 V metingen uitwijzen. Dit is te wijten aan de hogere Q-factor van de oscillator in simulaties aangezien niet alle parasitaire effecten L2 C2 l3 in rekening kunnen worden gebracht. R1 Uit

C4 50 Meting l1 Simulatie

L1 l2 40 R2 C1

R3 C3 30 [MHz] L f 20

Fig. 1. Schema van het uiteindelijke oscillator ontwerp. 10 TABEL I COMPONENTWAARDENENTRANSMISSIELIJNDIMENSIESVOORHET 0 40 30 20 10 0 UITEINDELIJKEONTWERP. − − − − Pinj /P0 [dB] Component Waarde Component Waarde R1 1500 Ω l1 / w1 35 mm / 0.9 mm Fig. 2. Lock range van de autonome oscillator. R2 820 Ω l2 / w2 19.6 mm / 0.9 mm R 27 Ω l / w 10.2 mm / 1.416 mm 3 3 3 B. Ontvangstcircuit L1 37 nH L2 37 nH C1 52.1 pF C3 52.1 pF Twee tonen werden via de RF poort in het systeem gebracht. C2 100 pF C4 27 pF Het eerste signaal (f1) is het nuttige signaal op 2.45 GHz terwijl de tweede toon (f2) het stoorsignaal voorstelt dat in frequentie verandert gedurende de metingen. Beide tonen hebben een in- gangsvermogen van 20 dBm. Figuren 3 en 4 tonen het MF B. Diode mengtrap − spectrum voor verschillende frequenties van het stoorsignaal. Een mixerschakeling uit een vorig project werd hergebruikt. Aangezien de exacte LO frequentie in het totale systeem De ADE-35+, een dubbel gebalanceerde diode mixer van Mini 2.3317 GHz bedraagt, is de nominale MF frequentie voor het Circuits met een BP2U+ vermogensverdeler (met een getermi- nuttige signaal 118.3 MHz. Deze frequentie wordt in beide fi- neerde derde poort) aan zijn LO ingang, werd gebruikt om de guren aangeduid met een verticale streeplijn. Merk op dat, in omzetting van het RF signaal naar de middenfrequentie (MF) te vergelijking met figuur 2, de lock range in deze situatie gevoelig realiseren. Het circuit wordt gekenmerkt door een omzettings- kleiner is door de eerder vernoemde mixerisolatie. Aan de rand verlies van 7.5 dB en een RF-LO isolatie van circa 25 dB. Deze van de lock range (figuur 3) bestaat het spectrum uit eén´ enkele isolatie verlaagt het ongewenste, effectieve vermogen van het frequentiecomponent die duidelijk verschoven is. Afhankelijk stoorsignaal dat de oscillator bereikt. van de verdere bouwblokken in het totale ontvangstsysteem kan dit voor problemen zorgen als dit niet correct wordt opgevangen. IV. METINGEN Metingen werden uitgevoerd met behulp van een N5242A Het frequentiespectrum in het injection pulling gebied (fi- PNA-X van Agilent Technologies om eén-´ en tweetoonmetin- guur 4) verschilt volledig van het spectrum in de injection loc- gen uit te kunnen voeren. Frequentiespectra werden opgemen- king situatie [3]; in plaats van eén´ enkele component bestaat het ten aan de hand van een Rhode & Schwarz FSV40 signaal ana- spectrum nu uit vele, dicht opeengepakte componenten. Zij ma- lysator. Om het spectrum van de LO te kunnen opmeten tijdens ken het onmogelijk om nog correcte informatie uit dit signaal gebruik in het ontvangstcircuit, werd een kwadratuur hybride te halen. Het is duidelijk dat de tijdsafhankelijkheid van de os- koppelaar gebruikt. cillatorfrequentie de correcte werking van de ontvanger dwars- bijkomende inzichten kunnen opleveren in de niet-lineaire ef- 30 118.315 MHz fecten die hier besproken werden. Een dergelijk compact cir- − cuit is bovendien niet enkel een geschikt platform voor verdere 40 EMC-evaluatie, maar vormt eveneens een prototype dat dient tot − de verdere ontwikkeling van lage-kost, wearable elektronische 50 − producten. 60 − REFERENTIES 70 − [1] R. Adler, “A Study of Locking Phenomena in Oscillators”, Proceedings of the IRE, vol. 34, no. 6, pp. 351-357, June 1946. 80 − [2] B. Razavi, “A study of Injection Locking and Pulling in Oscillators”, IEEE Magnitude [dBm] Journal of Solid-State Circuits, vol. 39, no. 9, pp. 1415-1424, Septem- 90 − ber 2004. [3] I. Ali, A. Banerjee, A. Mukherjee, and B. N. Biswas, “Study of Injection 100 − Locking With Amplitude Perturbation and Its Effect on Pulling of Oscilla- tor”, IEEE Transactions on Circuits and Systems I - Regular Papers, vol. 110 59, no. 1, pp. 137-147, January 2012. − 118 118.2 118.4 118.6 Frequentie [MHz]

Fig. 3. MF spectrum voor f2 = 2.331685 GHz, i.e. de rand van de lock range. boomt. Voor sterkere stoorsignalen wordt het frequentiegebied waar ontvangst onmogelijk wordt alleen maar groter. Bovendien worden deze ongewenste effecten alleen maar sterker in het ge- troffen frequentiegebied.

30 − 40 − 50 − 60 − 70 − 80 − Magnitude [dBm] 90 − 100 − 110 − 118 118.2 118.4 118.6 Frequentie [MHz]

Fig. 4. MF spectrum voor f2 = 2.33166 GHz, wat leidt tot sterke injection pulling.

V. CONCLUSIEENVERDERONDERZOEK Een lagekost RF ontvangstcircuit voor de 2.45 GHz ISM radioband werd ontworpen en gefabriceerd. Het optreden van injection locking en pulling werd aangetoond in zowel de oscillator op zich als in de totale ontvangststructuur. Dit beves- tigt de noodzaak tot een EMC-bewuste ontwerpstrategie waarbij de negatieve gevolgen van deze niet-lineaire fenomenen ter- dege in rekening wordt gebracht. Bij verder onderzoek zou een co-ontwerp tussen de oscillator en de mengschakeling enkele interessante mogelijkheden kunnen bieden om de kost en opper- vlakte verder te reduceren. Door het toevoegen van een (actieve) antenne aan het ontvangstcircuit kunnen aanvullende injectie- metingen via invallende straling worden uitgevoerd, wat leift tot CONTENTS i

Contents

List of Abbreviations iii

1 Introduction 1 1.1 EMC-aware Design ...... 1 1.2 Low-Cost Receiver ...... 1 1.3 Goal and Outline ...... 3

2 Injection Locking and Pulling 4 2.1 Introduction ...... 4 2.2 Intuitive Approach ...... 4 2.3 Formal Approach ...... 6 2.3.1 Derivation of the Lock Range ...... 6 2.3.2 Derivation of Injection Pulling ...... 8

3 Receiver Circuit Topologies 17 3.1 Oscillator and Mixer Topologies ...... 17 3.1.1 Oscillator topologies ...... 17 3.1.2 Mixer Topologies ...... 20 3.2 Design Strategy and CAD Software ...... 21 3.3 Non-linear Simulation ...... 22 3.3.1 Transient Simulation ...... 22 3.3.2 Harmonic Balance Simulation ...... 23

4 Design of a Low-Cost Receiver Circuit 26 4.1 Realisation of Transmission Lines ...... 26 4.2 Oscillator Design ...... 28 4.2.1 First Oscillator Design ...... 28 4.2.2 Second Oscillator Design ...... 39 4.3 Mixer Choice ...... 47

5 Measurements 48 5.1 Quadrature Hybrid ...... 48 5.2 Oscillator Measurements ...... 49 5.3 Mixer Measurements ...... 50 CONTENTS ii

5.4 Injection Locking Measurements ...... 53 5.4.1 Oscillator ...... 53 5.4.2 Complete System ...... 59

6 Conclusions and Future Work 65

A Calculations 67 A.1 Injection Locking Calculations ...... 67 A.1.1 Derivation of (2.3) ...... 67 A.1.2 Derivation Phase Characteristic RLC Tank ...... 68 A.2 Injection Pulling Calculations ...... 68

A.2.1 Derivation of Single Sinusoid Form of Ssum ...... 68 A.2.2 Approximation for dψ/dt ...... 69 A.2.3 Approximation for tan (θ ψ)...... 69 − A.2.4 Behaviour of θ in Injection Pulling ...... 70 A.2.5 Expression for dθ/dt ...... 71 A.2.6 Series Expansion of (A.31) ...... 71 A.2.7 Expression for cos θ ...... 72

Bibliography 73

List of Figures 77

List of Tables 79 CONTENTS iii

List of Abbreviations

AC alternating current

ADS Advanced Design System

BJT Bipolar Juction Transistor

CAD computer-aided design

CPU central processing unit

CPW coplanar waveguide

DC direct current

DUT device under test

EM electromagnetic

EMC electromagnetic compatibility

FR ﬂame retardant

FSE Fourier Series Expansion

FSE−1 inverse Fourier Series Expansion

GCPW grounded coplanar waveguide

HB Harmonic Balance

IF intermediate frequency

IoT Internet of Things

ISM industrial, scientiﬁc and medical

LNA low-noise ampliﬁer

LO local oscillator

LTCC low temperature co-ﬁred ceramic

MoM Method of Moments CONTENTS iv

MOSFET metal-oxide-semiconductor ﬁeld-eﬀect transistor

NF noise ﬁgure

PCB printed circuit board

PLL phase-locked loop

PNA programmable network analyser

RF radio frequency

SDR software-deﬁned radio

SMA SubMiniature version A

SOM self-oscillating mixer

SRF self-resonant frequency

TL transmission line

VCO voltage-controlled oscillator INTRODUCTION 1

Chapter 1

Introduction

1.1 EMC-aware Design

In the modern era of rapidly advancing technology, wireless communication has become ubiqui- tous. Almost every electronic device, ranging from computers over smartphones and tablets to washing machines and even coffee machines, is interconnected in the Internet of Things (IoT). This widespread use of radio waves to create short and long range wireless communications, cre- ates a polluted and harsh electromagnetic environment. At the same time, sensitive and critical electronic equipment should operate in this environment without disturbance of its proper functionality. The ability of impinging signals to cause havoc in certain devices is called interference. The objective of a contemporary design engineer is to create appliances that function properly in this environment, in other words devices that are immune to the effects of the environment, without causing interference in neighbouring systems themselves [1]. Electromagnetic compatibility (EMC) aware design of receivers, as considered in this dissertation, applies techniques to avoid susceptibility and strengthen the immunity in the design phase in order to avoid costly redesigns at the end of the cycle. Other evolutions in electronics such as rising operation frequencies, increasing bit rates and miniaturisation of the components, further complicate the design of an EMC compliant device as parasitic effects grow and interconnections start to exhibit wave phenomena. These trends have lead to a growing understanding that in order to avoid EMC problems at the end of the design cycle during testing, it is crucial to estimate immunity and susceptibility during the early design stages of the receiver and take appropriate measures. In this dissertation, only signals received by the system will be considered while the emission of possible interference signals by the circuit itself will not be discussed.

1.2 Low-Cost Receiver

In order to communicate via radio waves, the information has to be placed on a carrier with a certain frequency. At the receiver side, the incoming information signal has to be separated from the other signals and down-converted to a suitable frequency for further processing. A common architecture to achieve this operation is the superheterodyne front-end as shown in INTRODUCTION 2

Figure 1.1 [2]. The information signal enters the receiver at the radio frequency (RF) via the antenna. The signal is then subjected to a (variable) RF preselect filter to remove unwanted signals that could jeopardise further operation and to reduce the overall received noise in the system. Further down the line, the filtered signal is sent through a low-noise amplifier (LNA) to enhance sensitivity as future steps, in particular the mixer, have a substantial noise figure (NF). After amplification, the signal is applied to the mixer. The RF signal is converted to a lower, intermediate frequency (IF) by multiplying it with a signal produced by the local oscillator (LO). The IF component is filtered out of the other mixing components by the IF filter and after amplification by the IF amplifier, the signal is demodulated to extract the information. Note that these last steps can also be performed in the digital domain.

Preselect IF Demo- LNA Mixer IF filter filter amplifier dulator RF Out

Figure 1.1: Structure of a superheterodyne receiver.

In recent times, interest in the fabrication of compact, low-cost receivers has grown. Together with active and flexible antennas [3], they clear the way for compact wearable electronics that have a wide variety of applications. One way to reduce the cost and footprint of the receiver’s front-end is to omit the RF preselect filter, which typically is a steep analogue filter and thus an expensive component. Besides, it also requires quite a bit of real-estate on e.g., a printed circuit board (PCB). In some applications, with the most notable example being software-defined radio (SDR) [4], the received signal’s nature is so broadband that using an RF preselect filter is not even possible. The signal is down-converted directly by a (variable-frequency) LO signal and further filtering and image rejection is performed in the digital domain. An issue that can arise from this omission, is the existence of co-located sources; besides the useful signal, undesired signals outside the application’s frequency band enter the system and get down-converted to an IF frequency as well. Only when these signals are sufficiently spaced apart, the IF filter will suppress these signals and remove the problem. Unfortunately, co-located sources can induce another severe problem; when an undesired signal (fco) has a frequency close to the frequency of the LO, leakage through the mixer into the oscillator can alter its operation considerably (see Figure 1.2). A shift of the oscillator frequency, called injection locking, may occur as well as the creation of a time-dependent oscillation frequency (injection pulling) can occur. INTRODUCTION 3

fRF = 2.45 Ghz +fco Preselect LNA Mixer ﬁlter

fIF = 70 MHz

fco fLO = 2.38 GHz

Figure 1.2: Structure of a low-cost RF front-end.

1.3 Goal and Outline

In this master’s dissertation, an RF front-end will be designed to operate in the industrial, scientific and medical (ISM) radio band centred around 2.45 GHz. This band contains some widely used applications such as Bluetooth, Wi-Fi and cordless phones. In order to achieve an IF frequency of 70 MHz, a local oscillator is required that produces a sinusoidal signal at 2.38 GHz. The goal of this work is to design a compact, low-cost receiver front-end and to observe and analyse injection locking and pulling phenomena in the constructed system. To this end, a simple oscillator and mixer will be designed to make up the core of the receiver. Injection locking and pulling will be treated theoretically first in order to define the main characteristics of these non-linear effects. Next, measurements and simulations induce the phenomena and evaluate their impact on the performance of the receiver circuit. After the background information provided in this chapter, a theoretical approach to injection locking and pulling will be given in Chapter 2. Continuing with Chapter 3, the possible receiver circuit topologies will be explored and compared to make the best choice for this particular application. This chapter also contains a section in which the most common non-linear simulation methods are discussed (Section 3.3). The design of the receiver’s building blocks is described in Chapter 4. Injection locking and pulling experiments together with various other measurements are outlined in Chapter 5. Lastly, conclusions and future work are presented in Chapter 6. INJECTION LOCKING AND PULLING 4

Chapter 2

Injection Locking and Pulling

2.1 Introduction

When two systems with independent oscillating frequencies are brought into close contact, the mutual influence can change the oscillating behaviour of both systems. With nominal frequencies that are very similar but differ by a small percentage, the mutual influence can cause both systems to oscillate at a certain frequency which lies somewhere in between the nominal frequencies of the two systems. This situation is called injection locking. When the deviation between the nominal frequencies is too large for injection locking but still small enough to cause deviation, injection pulling can occur. In this scenario, the oscillation frequency of both systems will be pulled towards each other and become time-dependent but the two systems will never oscillate at exactly the same frequency. Injection pulling is always considered to be an undesirable phenomenon due to the time- dependent frequency of the influenced system. Injection locking, on the other hand, can be very useful in some applications; e.g. a strong oscillator can be set more precisely by injecting the signal of a lower power, but more stable oscillator and by relying on injection locking to achieve a well-defined frequency. In the system discussed here, however, both injection locking and pulling are considered undesirable. This chapter starts with an intuitive approach (Section 2.2) to these non-linear phenomena where simple examples and illustrations from nature provide a basic insight. In Section 2.3, a more rigorous and mathematically profound approach is presented which concentrates more on the occurrence of injection locking and pulling in electronic circuits.

2.2 Intuitive Approach

The first scientific observation of injection locking was done by the famous Dutch scientist Christiaan Huygens [5]. While laying ill in bed, Huygens noticed that two clocks mounted on his mantelpiece synchronised over time while having slightly different intrinsic oscillation frequencies (due to different lengths of the pendula). He presumed that tiny vibrations in the mantelpiece caused both pendulums to interact and, eventually, to swing at the same frequency but with a phase shift of 180◦. Modern day re-examinations (like the experiment in [6]) have INJECTION LOCKING AND PULLING 5

shown that Huygens’ assumption was indeed correct. Another reproduction of the experiment showed that both in-phase and anti-phase relations can occur [7]. A very similar experiment to the Huygens’ experiment can be performed at home and is even featured in an episode of the popular science entertainment program MythBusters on Discovery Channel [8]. The experiment synchronises two or more mechanical metronomes with the same frequency but with initial phase shifts. In this situation, the term entrainment is used rather than injection locking since the latter normally refers to the situation with different frequencies for the systems involved. This experiment should work too when the metronomes have slightly different periods but since locking takes longer in this situation, they tend to unwind before total synchronisation occurs. Figure 2.1a shows the initial set-up. Three metronomes are placed on a thin wooden board which is in turn placed on cardboard cylinders to create a smoothly movable base. The colourful stickers aid to determine the phase shift between the three metronomes. After a minute, the three metronomes synchronise and their phase difference fades away as shown in Figure 2.1b. At this point, the wooden board is rocking back and forth noticeably, revealing a strong constructive influence amongst the metronomes.

(a) Initial set-up of the metronomes. (b) Synchronised result.

Entrainment/injection locking is observed not only in man-made systems but in biological systems as well. The most basic example of injection locking in nature is the synchronisation of the daily rhythm of plants and animals to the day-night cycle on earth. This is called a circadian rhythm and is the most important phenomenon in the biological field of chronobiology. Some organisms, like mammals, even have multiple internal clocks of approximately 24 hours that are injection locked to one master pacemaker located in the brain [9]. This master clock in turn is locked to the period of the Earth’s rotation. Several biological markers, like core body temperature [10] or melatonin levels [11], are intrinsically linked with the master clock and thus with the day-night cycle. The lack of a proper entrainment causes sleep derivation problems and is shown to cause health problems like obesity and polyphagia [11]. Injection locking/entrainment can occur between different animals as well; e.g. mosquitoes of both genders have different wing-beat frequencies but opposite-sex pairs synchronise this frequency in response to each other [12]. Another interesting example is the synchronous flashing of large groups of certain species of fireflies in Southeast Asia [13]. Animals can also lock on INJECTION LOCKING AND PULLING 6

VDD H | |

CT RT LT

1 Vout − Itot

log ω ω0 ω1 Iosc 6 H M1

Iinj log ω φ0

(a) Simple oscillator with injected current. (b) Bode plot of the tank characteristics. to other cycles present in nature. Most types of corals present in the same colony/place, for example, tend to spawn simultaneously at certain lunar phases (mostly full moon) and at a certain position of the Sun (e.g. sunset) to increase chances of reproduction [14]. Corals are thus locked onto the cycle of both the Moon and the Sun.

2.3 Formal Approach

The mathematical equations that govern injection locking and pulling will be derived in this section using a simple oscillator, as shown in Figure 2.2a, as model. This derivation is mainly based on the work of Razavi [15], Ali [16] and Adler [17]. In Subsection 2.3.1, an expression for the range of frequencies where locking occurs will be derived. The next subsection, Subsection 2.3.2 will be on the equations that govern the system in the region of injection pulling.

2.3.1 Derivation of the Lock Range

The resonance tank does not contribute to the phase shift at the free running frequency ω0 (Figure 2.2b) so the inverting buffer and the transistor make up the required total phase shift ◦ of 360 . The injection of an alternating current of frequency ωinj and magnitude Iinj causes an extra phase shift (φ0) in the loop and oscillation must thus occur at another frequency (ω1) since the tank has to compensate this extra phase shift. As a result, all three currents (Iosc, Itot and Iinj) will differ in phase as shown in Figure 2.3. The phase difference between Iosc and Iinj is denoted as θ and φ0 is the phase difference between Itot and Iosc. The angle between the latter pair is such that Itot has the same phase as Vout (and Iosc) after being shifted over φ0 by the tank at frequency ω1. This angle reaches a maximum and will thus limit the lock range. An expression for φ0 can be derived from Figure 2.3a via the law of sines (and the law of INJECTION LOCKING AND PULLING 7

I I tot tot

I osc I osc φ0 φ0 θ θ

Iinj Iinj

(a) Arbitrary φ0 (b) φ0,max

Figure 2.3: Vector diagrams depicting the phase of the diﬀerent currents. cosines):

Iinj sin φ0 = sin θ (2.1) Itot I sin θ = inj . (2.2) 2 2 Iosc + Iinj + 2IoscIinj cos θ q By diﬀerentiating (see Appendix A.1.1 for details), one ﬁnds that the maximum equals

Iinj sin φ0,max = (2.3) Iosc and occurs for I cos θ = inj . (2.4) −Iosc This situation is shown in Figure 2.3b and reveals that at the edge of the lock range, the injected current and the current through the tank have a phase difference of 90◦. To convert this phase difference to a maximum frequency deviation from the natural frequency, an expression for the phase shift exerted by a tank is needed. This expression is given by (details see Appendix A.1.2): 2Q tan φ0 (ω0 ωinj) (2.5) ≈ ω0 − where Q is the quality factor, a.k.a. Q-factor, of the tank. Realising from Figure 2.3b that tan φ0 = Iinj/Itot and using Pythagoras’ theorem for Itot, one finds an expression for the one- sided lock range: ω0 Iinj 1 ωL = (ω0 ωinj)max = (2.6) − 2Q · Iosc · 2 Iinj 1 2 s − Iosc

ω0 Iinj which reduces to ωL for Iinj Iosc. Under the same assumption one ﬁnds a linear ≈ 2Q · Iosc relation between sin θ and (ω ω ): 0 − inj ω0 Iosc sin θ (ω0 ωinj) (2.7) ≈ 2Q · Iinj − INJECTION LOCKING AND PULLING 8

Iinj Iosc

ω ω0

Figure 2.4: Region of injection locking also known as Arnold tongue.

Ssum Sinj + Sout = Ainj cos ωinjt = Aosc cos (ωinjt + θ(t))

Figure 2.5: Block diagram model of an oscillator under injection pulling. because I I and sin φ tan φ . tot ≈ osc 0 ≈ 0 One sees the dependency of the lock range on Iinj and Iosc as well as the linear approximation for small values of Iinj/Iosc. Furthermore, it is clear that the lock range shrinks as the strength of the injected signal lowers. The region of injection locking is called an Arnold tongue [18] and is shown in Figure 2.4.

2.3.2 Derivation of Injection Pulling

For the derivation of the oscillator in the injection pulling region, the oscillator will be modelled by the system in Figure 2.5. The injected signal is modelled as a signal with frequency ωinj while the output is represented by a cosine with frequency ωinj and a possibly time-varying phase θ(t). Hereafter, explicit mentioning of the time dependence of θ is dropped not to clutter the calculations. The choice to represent the output with a frequency of ωinj instead of ω0 might seem peculiar but is done because it facilitates the calculation. The goal is to ﬁnd an expression for θ since it holds information about the instantaneous frequency of the oscillator.

This expression will be found by calculating Ssum, sending it through the RLC tank and equating the result to the expression for Sout. In order to subject Ssum to the phase characteristic of the tank (2.5), an expression of the form A cos (ωt + φ) is needed seeing that superposition is not valid in phase equations. The INJECTION LOCKING AND PULLING 9

simple form of Ssum, however, is a sum of two sinusoids, so some calculations are required to transform this expression to a useful form. Details can be found in Appendix A.2.1 where the resulting expression is shown to be:

S A cos (ω t + ψ) (2.8) sum ≈ osc inj with A sin θ tan ψ = osc (2.9) Ainj + Aosc cos θ assuming A A . inj osc Sending this signal through the RLC tank will add a phase shift proportional to the instan- dψ taneous frequency ω + : inj dt 2Q dψ S = A cos ω t + ψ + arctan ω ω . (2.10) out osc inj ω 0 − inj − dt 0 Equating this expression to the original one for Sout as shown in Figure 2.5, results in the following equation: 2Q dψ ψ + arctan ω ω = θ (2.11) ω 0 − inj − dt 0 dψ By using the approximations for and tan (θ ψ) as derived in Appendix A.2.2 and A.2.3, a dt − diﬀerential equation for θ is obtained:

dθ ω0 Ainj = ω0 ωinj sin θ (2.12) dt − − 2Q · Aosc = ω ω ω sin θ (2.13) 0 − inj − L with ω equal to (2.6) under the assumption that A A . By setting dθ/dt = 0, the L inj osc solution for sin θ corresponds to a steady state solution (region of injection locking) and equals (2.7) derived in Subsection 2.3.1. When the derivative of θ is not zero, however, the system suﬀers from injection pulling and the time-dependent nature of θ and its derivative will cause a time-dependent instantaneous frequency of the output signal. In order to obtain an expression for θ as a function of t, the derivative of θ to t has to be integrated out of (2.13). Using some clever substitutions and basic integral identities (see Appendix A.2.4), one gets: θ ω ω ω t tan = L + b tan b (2.14) 2 ω ω ω ω 2 0 − inj 0 − inj with ω = (ω ω )2 ω2 . b 0 − inj − L q d(ωinjt + θ) The instantaneous frequency of the system equals = ω + dθ/dt. Using some dt inj basic trigonometric properties (details in Appendix A.2.5), dθ/dt is found to be

dθ ω (β2 1) = L − (2.15) dt β + cos (2α α ) − 0 with β = (ω ω )/ω , 2α = ω t = ω β2 1t and tan α = β2 1. As dθ/dt is periodic 0 − inj L b L − 0 − in ωb, so is the instantaneous frequency. Inp order to ﬁnd an expressionp for the average frequency, INJECTION LOCKING AND PULLING 10

2.05 2.04 2.03 2.02 2.01

[GHz] 2 osc

ω 1.99 1.98 1.97 1.96

1.95 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 ωinj [GHz]

Figure 2.6: Diagram showing the average oscillation frequency as a function of ωinj. it is advantageous to ﬁnd an alternative expression for the derivative of θ. This formula is based on the proof given in Appendix A.2.6 and results in ∞ dθ = sgn (β)ω β2 1 1 + 2 ( 1)nrn cos n(2α α ) (2.16) dt L − − − 0 " n=1 # p X with r = β sgn (β) β2 1 and sgn ( ) the sign/signum function that returns the sign of − − · the argument. The averagep value of the phase derivative is now simply found to be the factor in front of (2.16): sgn (β)ω β2 1 = sgn (β)ω . The average frequency is thus ω + L − b inj 2 2 sgn (ω0 ωinj) (ω0 ωinj) ωp. This average frequency ωosc as a function of ωinj is shown − − − L in Figure 2.6 forq ω0 = 2 GHz and ωL = 50 MHz. The grey zone depicts the injection locking region where the oscillation frequency is constant and equal to the injected frequency. The frequency becomes time-dependent outside this area and its average evolves back to the natural frequency as the injected signal’s frequency drifts away from the natural frequency. It should also be noted that the average frequency is always lower than ω0 when ωinj < ω0 and always higher than ω0 in the other situation. Not only the average frequency is inﬂuenced by the injected signal. Therefore, the remainder of this section is devoted to deriving and describing the interesting behaviour of the spectrum and time domain signal in the injection pulling domain. To this end, the equation for Sout will be rearranged to be a sum of sinusoids:

Sout = Aosc cos (ωinjt + θ) (2.17) = A cos (ω t) cos (θ) A sin (ω t) sin (θ) (2.18) osc inj − osc inj An expression for sin θ can easily be found from (2.13) and (2.16): ∞ sin (θ) = r sgn (β)2 β2 1 ( 1)nrn cos n(2α α ) . (2.19) − − − − 0 "n=1 # p X INJECTION LOCKING AND PULLING 11

An expression for cos θ proves to require more trigonometric manipulations (see Appendix A.2.7) and a proof very similar to A.2.6 in order to get:

∞ cos (θ) = 2 β2 1 ( 1)nrn sin n(2α α ) . (2.20) − − − 0 "n=1 # p X Plugging this back into (2.17) one gets

∞ S = A r sin (ω t) + 2A β2 1 ( 1)n+1rn sin (ω t + n sgn (β)(2α α )) out − osc inj osc − − inj − 0 "n=1 # p X (2.21) ∞ = A r sin (ω t) + 2A β2 1 ( 1)n+1rn sin ([ω + n sgn (β)ω ]t φ ) − osc inj osc − − inj b − n "n=1 # p X (2.22) with φn = n sgn (β)α0. One notices that the signal has a component at the injected frequency and a series of sidebands spaced by ωb that are only present at one side of the injected signal. If sgn β = 1, thus ωinj < ω0, the sidebands only appear at frequencies higher than ωinj, explaining the fact that the average frequency is higher than ω (Figure 2.6). For sgn β = 1 the opposite inj − is true.

In order to fully understand the effect of injection pulling, the spectrum of Sout as given by (2.22) will be studied for various values of ωinj (and thus β). Only examples with ωinj < ω0 will be discussed since the other situation is completely analogous. Starting from ωinj < ω0, the injected signal’s frequency will be increased until locking occurs. The different spectra can be found in the subsequent subfigures of Figure 2.7. For (ω ω ) ω β 1, (2.22) can be approximated by making first order Taylor 0 − inj L ⇒ approximations of the various factors:

1 1 1 r = β 1 1 β 1 1 + = − − β2 ≈ − 2β2 2β r 1 1 1 1 r β2 1 = βr 1 − − β2 ≈ 2 − 4β2 ≈ 2 r p 1 1 1 1 r2 β2 1 = βr2 1 − − β2 ≈ 4β − 8β3 ≈ 4β r p 1 1 ω = ω β2 1 = (ω ω ) 1 (ω ω ) 1 (ω ω ). b L − 0 − inj − β2 ≈ 0 − inj − 2β2 ≈ 0 − inj r p This results in the following expression for Sout: A A S = osc sin (ω t) + A sin (ω t + φ ) osc sin ([2ω ω ]t + φ ). (2.23) out − 2β inj osc 0 1 − 2β 0 − inj 2 As β 1, one sees (Figure 2.7a) that the spectrum has a quasi-symmetric shape with a strong component very close to the natural frequency and two small sidebands spaced by ω (ω b ≈ 0 − ωinj). The strongest component is not exactly at ω0 because β = 5 for ωinj = 1.75 GHz and this is not large enough for the component to be at the center frequency. A smaller ωinj, however, INJECTION LOCKING AND PULLING 12

would make the ﬁgure absurdly wide if kept to scale with the others. The two sidebands are well below the central frequency component so in the time domain, the signal consist of a sine with the natural frequency perturbated by a fast but very small beat.

Raising ωinj but still making sure that β is substantially larger than one, the crude approximations made above should be replaced by their more precise, second order expressions resulting in a more complex expression for Sout: A 1 1 ω2 S = osc sin (ω t) + A 1 sin ω L t + φ out − 2β inj osc − 2β2 0 − 2 ω ω 1 0 − inj 1 1 1 ω2 A sin 2ω ω L t + φ . (2.24) − osc 2β − 4 β3 0 − inj − ω ω 2 0 − inj The main component has now shifted from the natural frequency towards the injected frequency and lost a bit of its strength (as shown in Figure 2.7b). The spacing of the sidebands has decreased since ωb lowered and they no longer have the same amplitude. The amplitude of the third component has increased less than the signal at ωinj. The beat frequency has thus decreased but is nonetheless still quite high which is why this region (and the ﬁrst one) is called fast beat operation. Since the amplitude of the main component lowers while the second component grows stronger, one expects a value of ωinj where both components will be equal in magnitude. Re- ferring to expression (equation (2.22)) for S , this occurs for r = r β2 1 β = √5/2 out − ⇒ ⇒ ω ω = 1.118 ω ω = 1.9441 GHz (see Figure 2.7c). At this point both frequency 0 − inj · L ⇒ inj p components will have an amplitude of (ϕ 1)A = 0.618 A with ϕ = (1 + √5)/2 the golden − osc osc ratio. When ωinj is brought even closer to ω0, the component at ωinj will become dominant, ωb will become small and we no longer speak of fast beat but quasi-lock. As β 1, the approximations used above are no longer valid and for a correct representation ' of the signal the original formula (equation (2.22)) for Sout should be used. One notices (Fig- ure 2.7d) that there are still no components below ωinj but a large number of closely spaced, progressively decreasing spectral components. If β eventually reaches one, locking occurs and the spectrum consist of one and only one component (Figure 2.7e). All other spectral components have coincided with the injected frequency since the beat frequency steadily went towards zero and their amplitude steadily decreased as well. While the most interesting phenomena occur in the frequency domain, the time domain holds the origin of the terms fast beat and quasi-locking. Because of this, two speciﬁc values for

ωinj will be scrutinised and special detail will be directed towards the time domain. Figures 2.8 and 2.9 show θ, the instantaneous frequency, the time-domain signal and the spectrum for two diﬀerent injected frequencies ω1 = 1.34 GHz and ω2 = 1.49 GHz with ωL enlarged to 500 MHz in order to make the eﬀect in the time domain easier to see. ω1 lies far away from the lock range and is thus an example of a fast beat signal. ω2 lies just a bit below the lower bound of the lock range and will thus experience quasi-lock.

Although ω1 is clearly far enough removed from the central frequency to be placed in the fast beat region, the strongest component is pulled towards ωinj because β = 1.6 is not large enough to make sure the pulling is negligibly small (Figure 2.8d). In spite of this, the non-linear phase INJECTION LOCKING AND PULLING 13

ω ω ω b ≈ 0 − inj

ω ωinj ω0

(a) ωinj = 1.75 GHz.

ωb

ω ωinj ω0

(b) ωinj = 1.925 GHz.

ωb

ω ωinj ω0

ωb

ω ωinj ω0

(d) ωinj = 1.948 GHz.

ωb = 0

ω ωinj ω0

(e) ωinj = 1.95 GHz.

Figure 2.7: Spectrum for various values of ωinj (amplitude in log scale). INJECTION LOCKING AND PULLING 14

π 2

0 [radians] θ

π − 2

π − 0 0.4 2π/ωb 0.8 2π/ωb 1.2 2π/ωb 1.6 2π/ωb 2 2π/ωb · · t [s] · · ·

(a) θ(t).

2.5

2.25

1.75 Instantaneous frequency [GHz]

1.5 0 0.4 2π/ωb 0.8 2π/ωb 1.2 2π/ωb 1.6 2π/ωb 2 2π/ωb · · t [s] · · ·

(b) ωinst(t).

1.5

0.5

0 Magnitude 0.5 −

1 −

1.5 − 0 0.4 2π/ωb 0.8 2π/ωb 1.2 2π/ωb 1.6 2π/ωb 2 2π/ωb · · t [s] · · ·

ω ωinj 1.79 ω0 2.28 2.77

(d) Spectrum (all quantities in GHz).

Figure 2.8: θ, the instantaneous frequency, the waveform and the spectrum (log scale) for

ωinj = 1.3 GHz. INJECTION LOCKING AND PULLING 15

relationship causes the instantaneous frequency to ﬂuctuate around ω0 with a preference for low frequencies (Figures 2.8b and 2.8d). This results in a waveform that is sometimes lagging behind the natural frequency, sometimes leading the same signal (Figure 2.8c). With ωb = 0.5 GHz, a beat cycle takes about three periods of the output signal, so the beat phenomenon is of the same magnitude as the injected signal and the natural frequency causing a fast beat as compared to their period.

A totally different story can be told for ω2 as can be seen on Figure 2.9. The phase is clearly more non-linear (Figure 2.9a) with an almost flat section in the middle signifying that the instantaneous frequency lies very close to the injected frequency at that time. This flat section is followed by a steep, short very non-linear transition causing a swift change in the frequency (Figure 2.9b). The waveform (Figure 2.9c) reflects this behaviour. Please note the different time axis of this figure as not to get confused with the above subfigures. For the first 40 percent of the interval, the phase is relatively flat and the output signal has almost the same frequency as the injected signal. This region is called quasi-lock as the output seems locked to the input signal despite being outside the lock range. After some time, however, the signal cannot hang on any longer and experiences the rapid phase shift, so called phase slip, perceptible as a region of higher frequency. For this example, this region spans from 40 percent to approximately 65 percent of the beat period. One also notices that more periods of the output signal fit into one beat period as ωb = 0.1 GHz, signifying that this period is an order of magnitude larger than the period of the injected signal.

If ωinj would be brought even closer to the edge lock range (1.5 GHz), the quasi-lock region would become more and more dominant and the phase slips would become swifter and take up an increasingly smaller percentage of the beat period. In the end, when ωinj = 1.5 GHz, the phase slips would vanish, ωb = 0 GHz and locking would be achieved. To conclude this chapter, it is requisite to mention that the above analysis assumes that the injected signal only influences the phase of the oscillator and that the amplitude is undisturbed because the injected power is supposed to be small as compared to the oscillator output. When the injected power increases other phenomena come into play and a more complete analysis is required that takes amplitude modulation into account, as given in Ali [16]. The effect is, however, small even for larger input power and the effect is mainly limited to a smaller lock range and the rise of small frequency components below or above ωinj for ωinj < ω0 and ωinj > ωinj, respectively. INJECTION LOCKING AND PULLING 16

π 2

0 [radians] θ

π − 2

π − 0 0.4 2π/ωb 0.8 2π/ωb 1.2 2π/ωb 1.6 2π/ωb 2 2π/ωb · · t [s] · · ·

(a) θ(t).

2.5

2.25

1.75 Instantaneous frequency [GHz]

1.5 0 0.4 2π/ωb 0.8 2π/ωb 1.2 2π/ωb 1.6 2π/ωb 2 2π/ωb · · t [s] · · ·

(b) ωinst(t).

1.5

0.5

0 Magnitude 0.5 −

1 −

1.5 − 0 0.2 2π/ωb 0.4 2π/ωb 0.6 2π/ωb 0.8 2π/ωb 1 2π/ωb · · t [s] · · ·

ω ωinj 1.59 1.69 1.79 1.89 1.99 2.09 2.19

(d) Spectrum (all quantities in GHz).

Figure 2.9: θ, the instantaneous frequency, the waveform and the spectrum (log scale) for

ωinj = 1.49 GHz. RECEIVER CIRCUIT TOPOLOGIES 17

Chapter 3

Receiver Circuit Topologies

As discussed in Section 1.2, the front-end of a low-cost receiver consists solely of an LNA, a mixer and an LO. The influence of the LNA is neglected in this thesis. This approximation is valid as long as the bandwidth of the LNA is large enough, such that it does not significantly affects the injecting signal which will cause the injection locking or injection pulling behaviour. As the LO signal (2.38 GHz) lies close to the RF signal (2.45 GHz), the injected signals with the largest influence will most likely fall in the passband of the LNA. Hence, our approach is of practical importance. Both for the oscillator and the mixer a lot of possible topologies can be used. However, the focus lies on low-cost, so the topologies chosen for the mixer and oscillator are small and simple. This choice will obviously have its repercussion on the performance and will result in a sub-optimal receiver. This emphasises the importance and the goal of this thesis, which is to observe and study injection locking and injection pulling in low-cost receivers, rather then designing the “perfect” receiver. This chapter starts with Section 3.1, listing the different topologies for both oscillator and mixer that were suitable for this application. The chapter then continues with a short overview of the design strategy and the applied computer-aided design(CAD) software in Section 3.2. To conclude this chapter, Section 3.3 describes the main non-linear simulation tools that were used extensively throughout the design.

3.1 Oscillator and Mixer Topologies

3.1.1 Oscillator topologies

To keep the oscillator small, simple and cheap, no quartz crystal oscillators like the Pierce crystal oscillator [19] are considered here, nor a voltage controlled oscillator (VCO) embedded in a phase-locked loop (PLL). The remaining group of oscillators can be divided into two groups based on the output waveform. Harmonic or sinusoidal oscillators produce a sinusoid of a single frequency whilst relaxation oscillators produce non-sinusoidal waveforms like a square or a sawtooth wave [20]. As the mixer generally requires a single frequency signal to perform frequency conversion, the oscillator must be of the former type and only this group will be considered hereafter. RECEIVER CIRCUIT TOPOLOGIES 18

All harmonic oscillators have the same operating principle: a passive frequency-selective element is used to establish the frequency of the output signal and an active element is used to compensate for the losses of the passive element to ensure a non-zero amplitude. The compensation of the losses can be achieved by applying external positive feedback, resulting in a so-called feedback oscillator while loss compensation can be obtained by means of internal positive feedback as well, leading to a negative resistance oscillator. Both classes of oscillators are depicted in Figure 3.1. It should be noted that the terms internal and external feedback are not absolute so the division into classes is not simply black-and-white. The main diﬀerence actually lies in the design tactic and not so much in the operating principle.

Sin + A(s) Sout Matching Destabilised Z Z load network ampliﬁer S

β(s) ZOUT ZIN

(a) Schematic representation of a feed- (b) Schematic representation of a negative resistance oscil- back oscillator. lator.

Figure 3.1: Two classes of harmonic oscillators.

Feedback Oscillators

The operation of a feedback oscillator is based on a (frequency-selective) ampliﬁer A(s) with feedback β(s), containing the passive frequency-selective element (Figure 3.1a). This block diagram has the familiar transfer function: S A(s) out = . (3.1) S 1 β(s)A(s) in − One sees that the denominator becomes zero when A(s)β(s) = 1. This condition is called the Barkhausen criterion [21] and is more commonly written as two separate conditions:

A(s)β(s) = 1 (3.2) | | A(s)β(s) = 2nπ, n N. (3.3) ∠ ∈ A denominator equal to zero signifies that an output can exist without the need of an input signal, exactly what one expects of an oscillator. It should be noted that the Barkhausen criterion is a necessary but not a sufficient condition for oscillation. Figure 3.2 depicts two examples of common harmonic feedback oscillators. Figure 3.2a shows the simplified schematic of a common collector Colpitts oscillator. The bipolar junction transistor (BJT) acts as the amplifier while the two capacitors and the coil form the feedback factor β. The feedback path itself is formed by the capacitive voltage divider C1 and C2. Colpitts oscillators or similar topologies as the Clapp or the Hartley oscillator are widely used for both fixed frequency oscillators and VCOs. Therefore, their design procedure is well-known and the simple topology is another advantage. These topologies, however, tend to suffer from parasitic RECEIVER CIRCUIT TOPOLOGIES 19

feedback paths with rising frequencies, lowering the control over the oscillating frequency during the design. Another widely used feedback oscillator is the differential cross-coupled oscillator. Fig- ure 3.2b shows an example of this oscillator with metal-oxide-semiconductor field-effect transistors (MOSFETs) as the active elements. The main advantage of this circuit is the presence of two outputs (at the drain of both transistors) with a phase difference of 180◦. These two signals can be very useful in differential circuits or in mixers who need both signals, e.g. an active Gilbert cell. Like the Colpitts oscillator, however, the cross-coupled oscillator suffers from parasitic feedback when the frequency rises.

Vcc

C2 Ic

(a) Colpitts oscillator (simpliﬁed biasing (b) Cross-coupled oscillator. circuit).

Figure 3.2: Two examples of harmonic feedback oscillators.

Negative Resistance Oscillators

Figure 3.1b shows the principle of operation of a negative resistance oscillator. An active, non- linear element such as a tunnel diode [22] or a transistor is used to compensate the losses. This compensation manifests itself as an input impedance with a negative real part, a so- called negative resistance [23]. In some conﬁgurations, the impedance ZS is added to the active element to narrow the region of negative impedance, i.e. to make the characteristic more frequency-dependent. As to acquire a stable oscillation at a favoured frequency, this complex input impedance ZIN must be compensated by the load at the desired frequency of oscillation, f0. Since the load generally does not meet these requirements, a matching network transforms Zload to the correct ZOUT such that

R = R OUT IN f=f0 ZIN + ZOUT = 0 => − | (3.4) f=f0  | XOUT = XIN  − |f=f0 with R the resistance and X the reactance of the impedance Z. It should be noted that these condition are necessary but not suﬃcient just like the Barkhausen criterion. This matching RECEIVER CIRCUIT TOPOLOGIES 20

network is ideally strongly frequency-selective (as the rest of the circuit should be) since this improves the stability of the oscillation; a small deviation from the natural frequency leads to a signiﬁcant violation of (3.4) when there is a strong frequency dependency. The downside of this approach is the relative bigger inﬂuence of deviating component values, e.g. due to manufacturing tolerances. To conclude, it should be noted that the output can be chosen at the location of ZS as well, since oscillation occurs globally in an oscillator circuit.

3.1.2 Mixer Topologies

The main decision in mixer topologies consists of choosing between a passive or an active mixer design. Passive mixers like diode mixers [24] have superior noise performance and linearity but since no ampliﬁcation takes place, the conversion gain is always negative making the design of the rest of the receiver structure more challenging. Active mixers like the Gilbert cell [19] on the other hand, have a positive conversion gain, thus opening the possibility for a simpler IF ampliﬁer, but taste defeat as compared to the passive mixers when talking about noise or linearity. Since passive mixers, especially diode mixers, having only a few components, are easy to use and can be made very compact, they are given preference in this work.

Passive Diode Mixers

The simplest diode mixer possible is the single diode mixer (Figure 3.3a). This circuit solely relies on the non-linear nature of a single Schottky diode [25] (for low LO power) or the switching behaviour of this diode (for high LO power) to generate the mixing components of LO and RF. The desired IF frequency is filtered out of all the other mixing terms at the other side of the diode. Since the LO and RF signal are injected in the same circuit node, their inherent isolation from one another is zero. Some form of isolation can be achieved by inserting filters but seeing that the frequencies of both signals tend to lie close together, this would require steep bandpass filters with the accompanying extra cost, real-estate on the PCB, and design effort. Isolation between the two input ports and the IF output is poor as well due to the lack of symmetry in this one-sided circuit. To accommodate this lack of isolation, the single balanced diode mixer (Figure 3.3b) uses a centre-tapped balun to achieve isolation between LO and IF. This can be understood by noticing the symmetric nature of the circuit when there is no signal present at the RF port. Isolation between RF and IF is still not achieved (as they are injected on the same node) but as their frequencies are generally spaced far apart, some form of isolation can be obtained by applying a relatively simple cross-over filter. The need for a balun can cause problems at the frequencies considered in this thesis (above 1 GHz) since coupling between the coils destroys the symmetry and thus the proper operation of this component. Especially the need for a centre tap is problematic. Luckily, a solution exists in the form of a Marchand balun [26]. This balun uses transmission lines instead of coils and has a centre tap to ground (Figure 3.4) which suits perfectly for this mixer. Modern day commercial Marchand baluns are produced in low temperature co-fired ceramic (LTCC) technology and operate at frequencies up to several GHz [27]. RECEIVER CIRCUIT TOPOLOGIES 21

VLO

VRF VLO VRF

(a) Single diode mixer. (b) Single balanced diode mixer.

IF VLO VRF

Figure 3.3: Three types of passive diode mixers.

λ/2

λ/4 λ/4

Figure 3.4: Marchand balun in microstrip technology.

In order to achieve isolation between all three signal ports, a second balun is needed, resulting in the double balanced diode mixer (Figure 3.3c). This mixer, with its typical quad diode ring, acquires isolation by adding an extra layer of symmetry. Commercially available diode mixers are typically double balanced due to the superior isolation between all ports.

3.2 Design Strategy and CAD Software

Both the design of the oscillator and the mixer require accurate models of the employed components due to their strong non-linear operation in the receiver circuit and a good simulation tool is thus indispensable. To this end, the Advanced Design System (ADS) by Agilent Technologies [28] is used since it incorporates the planar 3D simulator (a.k.a. a 2.5D solver) Momentum and a highly versatile circuit simulation environment with both linear simulations, like S-parameters, and non-linear simulators, like Harmonic Balance (HB) or Transient. Momentum is a 3D planar electromagnetic (EM) simulator originally developed by academic RECEIVER CIRCUIT TOPOLOGIES 22

researchers of the EM group at Ghent University [29]. This simulator uses frequency-domain Method of Moments (MoM) to simulate complete board structures (including multi-layer) accurately taking effects like coupling or losses into account. An extra advantage and powerful tool in Agilent ADS is the co-simulation between Momentum and the circuit simulator. A design will typically start in the circuit simulator since it provides fast simulations and parameter sweeps and thus a quick way to gain insight in the operation of the circuit. When the design is finalised, including good component values, the PCB layout is drawn in Momentum as to take the effect of the substrate and the metal lines into account. Afterwards, the S-parameters of this layout are exported to the circuit simulator to simulate the entire circuit including a good model for the interconnection, providing an accurate approximation of the real behaviour of the system.

3.3 Non-linear Simulation

Agilent ADS incorporates a number of tools for non-linear circuit analysis. In this section, two of these simulations methods, leveraged throughout this thesis, are outlined.

3.3.1 Transient Simulation

The Transient simulation computes the response of the circuit in the time domain by solving the integral and/or diﬀerential equations governing the currents and voltages in the circuit. As solving this set of possibly non-linear equations analytically is complex or impossible for non- trivial circuits, numerical methods are applied to approximate the exact results. A summary of how this is done in Agilent ADS is given in this section (more details can be found in the online help section of Agilent Technologies [30]). A pure transient simulation represent all elements of the circuit in the time domain. This can be problematic, however, since common elements in RF circuits like distributed elements (e.g. microstrip lines) have an intrinsic frequency-dependent behaviour and can often not be easily represented in the time-domain. Therefore, these kinds of elements have to be approximated by lumped equivalents or some assumptions like frequency-independent loss are made to solve this problem. While being valid at low frequencies, these assumptions can result in discrepancies between simulation and reality at higher frequencies. This is why Agilent ADS includes a second type of time-domain simulation, viz. the Con- volution simulation, which represents distributed elements in the frequency domain. In order to calculate the response of these elements to certain stimuli, the input signals are convolved (hence the name) with the impulse response of the element. The Transient simulation will auto- matically apply the convolution method to elements that cannot be represented exactly in the time domain so both simulators are actually used at the same time. The main disadvantage of the convolution simulation is the substantially increased amount of calculation, resulting in an increase in CPU (central processing unit) time. The actual simulation starts by calculating the DC solution since a transient simulation always starts at t = 0 and furthermore assumes that all non-DC sources are switched oﬀ at this starting point in time. After computing the starting condition, the simulator moves a certain time interval forwards to t1. After evaluating the independent sources at t1, an attempt is made RECEIVER CIRCUIT TOPOLOGIES 23

to solve the differential equations using a numerical integration method (see next paragraph) and a limited number of Newton-Raphson method steps. If no convergence is reached after this calculation, the time step is reduced and the algorithm tries again to reach convergence. This process continues as long as the time step stays above a certain threshold; going below this threshold results in an error. After convergence the error on the obtained values is estimated and if this error is acceptable, the simulator will advance to the next time point. The initial step taken from ti to ti+1 is estimated based on the results for the voltages/current at time ti: a rapidly varying system at ti will result in a small initial step. The numerical integration can be performed using two different algorithms. The traditional algorithm is called the trapezoidal integration method and estimates the derivative at instant n as 2 ∆x = (x x − ) ∆x − . (3.5) n ∆t n − n 1 − n 1 In general, the use of this numerical integration method works fine and provides accurate results. However, the algorithm can result in numerical oscillation, meaning oscillation that is not actually there but is just a result of the algorithm itself. This oscillation can cause extremely long simulation times. For oscillator circuits, the trapezoidal integration method is known to stumble upon convergence problems very easily, indicating that another numerical integration algorithm is more suitable for these circuits. The second provided method is the so-called Gear’s backward difference method. This algorithm approximates the derivative at instant n as a weighted sum of values at previous time points:

∆xn = a0xn + a1xn−1 + ... + akxn−k. (3.6) where k determines the order of the Gear’s method. This order will be adapted throughout the calculations to speed up the calculation time if possible. This method is known to be very stable in most circuits but may be less accurate than the trapezoidal integration method.

3.3.2 Harmonic Balance Simulation

Harmonic balance is a complementary non-linear simulation which describes the circuit in the frequency domain [31]. The main advantage is that distributed elements can be represented elegantly and there is no need to fiddle around with convolutions in the time domain. Since calculations are performed in the frequency domain, transient effects are not simulated but in most cases one is mainly interested in the steady state response so this causes no problems. The description below is based on the summary of harmonic balance given in [18] and the more mathematically profound analysis in [25]. The harmonic balance simulator assumes that all current and voltages are described sufficiently accurate by means of a Fourier series with a limited number of unrelated fundamentals and a limited amount of harmonics. All circuit variables can therefore be represented by a finite vector of amplitudes and phases. In order to calculate all variables, the simulator divides the circuit into the independent sources, the linear subcircuit and non-linear subcircuit (shown in Figure 3.5). The contribution of the independent sources and the linear circuit to the currents

IL,i can be calculated straightforwardly using linear manipulations of the frequency domain RECEIVER CIRCUIT TOPOLOGIES 24

IL,1 INL,1

V1 VS,1 IL,2 INL,2

VS,2 Linear V2 Non-linear . Subcircuit Subcircuit . . . IL,N INL,N VS,K

Figure 3.5: Schematic representation of an HB circuit.

equations. The computation of the currents resulting from the non-linear circuit (INL,i) is very hard in the frequency domain as these elements are intrinsically defined in the time domain. The goal is to compute the current such that Kirchhoff’s current law is satisfied at all intermediate nodes: IL,i + INL,i = 0. Considering the voltages Vi as state variables and realising that the non-linear subcircuit’s current contribution is a function of these state variables, an error function for the optimisation algorithm is formulated as

E(V) = ASVS + ALV + INL(V) (3.7) with V the matrix containing all voltages Vi, INL the matrix grouping the currents INL,i and the matrices AS and AL denoting the transformation from the voltages VS,i and Vi to the currents IL,i respectively. It should be noted that the individual voltages and currents are vectors containing the amplitude and phase at the selected frequencies; explaining why the collection of all voltages and currents are represented in matrices. As mentioned before, the dependency of INL on V is generally not analytically solvable, certainly not in the frequency domain. To circumvent this inconvenience, the harmonic balance simulation applies an iterative process (Figure 3.6). Starting with a reasonable guess for the voltages V, the time domain representation of these state variables are computed using the inverse Fourier Series Expansion (FSE−1). The voltages are then plugged into the non-linear time domain equations of the non- linear subcircuit. The resulting time domain currents are transformed back to the frequency domain using the forward FSE. This enables the algorithm to evaluate the error function. An algorithm, usually Newton-Raphson, is then applied to evolve towards a negligible small norm RECEIVER CIRCUIT TOPOLOGIES 25

External sources Linear subcircuit Newton-Raphson

E(V) = ASVS + ALV + INL(V) 0 z }| { z }| { | {z } V v(t) i(t) INL 1 FSE− i(t) = f(v(t)) FSE

Figure 3.6: Solving equation (3.7). of the error function: Vn+1 = Vn [JE]−1 En(Vn) (3.8) − Vn with [JE]−1 the Jacobian matrix of the error function evaluated using the voltages at the Vn previous iteration, Vn. The Newton-Raphson method ends when the variation between subsequent values of Vn are within tolerance bounds or when the maximum number of iteration is reached, resulting in an error. The last issue that remains with harmonic balance is the selection of the fundamental frequencies. During simulation of non-autonomous circuits, i.e. circuits that do not generate frequencies themselves, e.g. an LNA, this selection is trivial and involves the frequency of the injected signals, e.g. one-tone or multi-tone measurements. For autonomous circuits, e.g. oscillators, however, the exact fundamental frequency is unknown, although a good estimation might be available. Agilent ADS provides two ways to solve this problem. The first solution consists of placing the OscPort component in the feedback loop of the oscillator and providing an estimate of the fundamental frequency. This component will actually turn the closed-loop system into an open- loop circuit and finds the oscillation frequency by searching the point of zero phase shift and a loop gain greater than one. The second method is called the OscProbe method and works as described in Section 5.5 in [32]. The simulator inserts a special voltage source at a specified circuit node. The voltage source is special since it is only connected at the fundamental frequency but not at DC or any harmonics. The frequency of the injected voltage is the fundamental frequency if and only if the voltage at this node is non-zero and the current through the source is zero. This way, the fundamental frequency can be found and the simulation is forced to find a non-trivial solution. DESIGN OF A LOW-COST RECEIVER CIRCUIT 26

Chapter 4

Design of a Low-Cost Receiver Circuit

Based on the discussion of the possible oscillator and mixer topologies in Section 3.1, it was decided to design a negative resistance oscillator to function as oscillator and a single balanced diode mixer serve to as frequency mixer. Feedback oscillators suﬀer too much from parasitic feedback paths at the operating frequencies of the system in this thesis. Modelling all these parasitics during the design is tedious so it makes a good, low-cost design impossible. The design procedure for negative resistance oscillators facilitates the inclusion of parasitics from the beginning, provided that good models for the components are used, a considerable advantage to the classical feedback oscillator. The single balanced diode mixer was preferred to the double balanced diode mixer or the single diode mixer since it makes a good compromise between simplicity and performance. While providing some form of isolation between ports as compared to the single diode mixer, the use of a single balun is a cost advantage to the double balanced variant. As the use of transmission lines (TLs) is inevitable in RF design, various ways to implement transmission lines on a board structure will be discussed Section 4.1. Afterwards, the design and redesign of the negative resistance oscillator will be treated in Section 4.2. To conclude this chapter, the implementation of the frequency mixer will described in Section 4.3.

4.1 Realisation of Transmission Lines

Transmission line components are commonly found in RF circuits and are realised in various ways. The most common ones used in PCB technology are shown in Figure 4.1 [33]. The ﬁrst realisation, the stripline (Figure 4.1a), consists of a signal line sandwiched between two ground planes inside the substrate. However, since the goal is to design a simple, low-cost circuit, the oscillator is realised on a PCB with only two layers, where the backplane functions as ground. This low impedance solid metal plane provides a good reference plane for the entire circuit. Moreover, connections to ground can be made as short as possible by placing vias through the substrate wherever a ground connection is needed. That way, current loops are kept as small as possible, minimising radiation and susceptibility to impinging noise signals. DESIGN OF A LOW-COST RECEIVER CIRCUIT 27

W W W h t t s s h h εr, µr εr, µr εr, µr

(a) Stripline. (b) Microstrip. (c) GCPW.

Figure 4.1: Transmission line realisations.

The first possible realisation on a double-sided PCB is the microstrip line shown in Fig- ure 4.1b. It consists of a signal line on top of a dielectric substrate with a ground plane on the bottom. It is very popular due to the easy fabrication and simplicity, the simple integration with components and of course its planar topology. The dielectric material in the microstrip is made up of both the substrate and the air above the signal line, implying that EM fields are present in both of these media. As such, the microstrip radiates which is not the case with the stripline where the fields are confined to the region between the two outer conductors. Another consequence of the discontinuous medium is the fact that a microstrip is inherently dispersive; with rising frequency, fields will be more and more confined to the substrate and thus experience a decreasing phase velocity even when the dielectric substrate itself is non-dispersive. Another important characteristic is the loss of the line. The two main contributions to the losses are the finite conductivity of the metal conductors and the radiation losses. The ohmic losses increase with the impedance which is in its turn inversely proportional to W/h. As it happens, the current density is higher for smaller strips and as such, the losses are higher. The radiation losses are roughly proportional to W/h as a wider signal line forces more field lines to travel (partially) through the air and thus radiate. The second appropriate choice for this application is the grounded coplanar waveguide (GCPW) (Figure 4.1c). The structure consist of one central metal signal line placed in between two metal planes functioning as ground planes. In the grounded coplanar waveguide the other side of the substrate is covered with a ground plane while the standard CPW lacks this ground plane. No matter what topology is chosen, it is crucial that all ground planes are at the same potential. In the CPW structure, this can be obtained by placing air bridges (bond wires) over the signal line to connect both ground planes. The GCPW provides the opportunity for a more elegant solution in the form of vias between the upper ground planes and the back plane. Note that both the air bridges and the vias should be spaced closely together, e.g. closer than λ/20, in order to ensure a constant potential on all ground planes [34]. The need for so many vias can be problematic in a large circuit at higher frequencies, as this drastically increases simulation time in an EM solver and can even form a risk for the structural integrity of the board. The main advantage of a CPW topology is the proximity of the ground planes. Not only is this very convenient for connecting components to ground as extra vias are not needed, the ground planes also attract the field lines, resulting in substantially lower radiation losses. What is more, the usually small gap s forces the fields to be mainly confined to the dielectric substrate, DESIGN OF A LOW-COST RECEIVER CIRCUIT 28

resulting in less dispersion. When vias are placed close to the gap in the GCPW topology, one gets even better results as the fields are concentrated almost entirely inside the substrate. The ground planes have the advantage of reducing crosstalk as well. The design of a (G)CPW for a certain impedance Z0 and substrate thickness is highly different from the design of a microstrip with the same requirements as the microstrip has only one degree of freedom, viz. W , whereas a CPW has two, viz. W and s, meaning that an infinite amount of CPWs meet the requirements. This property can be used to avoid discontinuities and tapering in a circuit. However, due to its lower effective dielectric constant, the width of the signal line tends to be smaller for a CPW, thus increasing the ohmic losses. Further in the design, both the microstrip and the GCPW are employed to realise transmission lines. The reasons for one of both in a particular situation are explained at that point.

4.2 Oscillator Design

The design of the oscillator that will function as the LO in the receiver circuit will be discussed in this section. The ﬁrst design for the oscillator, including the design steps, the result and the problems will be discussed in Section 4.2.1. The redesign, during which the mistakes from the ﬁrst design are corrected in order to achieve a correctly functioning oscillator, will be described in Section 4.2.2.

4.2.1 First Oscillator Design

The general goal in the design of a negative resistance oscillator is to achieve an active circuit, producing a ﬁxed input impedance with a negative real part, which is compensated at the desired oscillating frequency by a load matching circuit (see Section 3.1.1). This design was done using the following step-by-step plan:

1. Design an ampliﬁer circuit.

2. Make the circuit unstable.

3. Select a load in the unstable region.

4. Determine ZIN and compensate with a correct ZS.

5. Perform simulations to ensure start-up and correct frequency behaviour.

This plan is mainly based on [35]. It should be noted that this approach slightly diﬀers from

Figure 3.1b. While the impedance ZS and the amplifier provide the negative resistance to the output matching in the figure, the output matching together with the amplifier fulfil this role in the first design. Every step will be discussed in more detail in the following sections based on the original design. DESIGN OF A LOW-COST RECEIVER CIRCUIT 29

Design an ampliﬁer circuit

The most important choices in an amplifier design, and so by extension in an oscillator design, are the topology and the active element, i.e. the transistor. A common base topology was chosen as it provides an elegant way to apply local feedback (see next section). For the transistor, a (NPN) bipolar junction transistor BFP640 from Infineon Technologies AG [36] was selected. A BJT was preferred because they generally have superior noise performance in oscillator circuits as compared to MOSFETs [37]. This particular transistor, BFP640, was picked from a sample kit (ensuring the availability) based on the sufficiently high transition frequency (around 40 GHz), appropriate maximum voltage levels (higher than the power supply, Vcc = 3.3 V) and the high maximum power dissipation (200 mW).

Vcc

L2 R1

port 2

Component Value C1 port 1 R1 1500 Ω

L1 R2 820 Ω R 2 R3 27 Ω

C1 10 pF R3 C2 100 pF

L1 100 nH L2 100 nH

Figure 4.2: Common base circuit. Table 4.1: Component values of Figure 4.2.

Figure 4.2, and the accompanying Table 4.1, describe the design of the common base ampli-

ﬁer. The base voltage is set by the voltage divider R1 and R2 while the DC emitter current is set by the resistor R and by the voltage V V across its terminals with V = V 0.7 V 3 B − BE BE d ≈ and VB = 1.17 V. Furthermore, resistor R3 introduces negative feedback in the biasing circuit: an increase in emitter current (for example due to a higher temperature) will result in a lower value for Vbe which, in turn, causes the emitter current to decrease. The claim that R1 and R2 behave as a voltage divider is not entirely true as some current ﬂows into the base of the BJT. The approximation is still valid, however, since the current through the voltage divider is designed such that it is at least ten times IB. The coils in the circuit function as RF chokes, indicating that they behave as short circuits at

DC and block currents at the operating frequency due their substantial impedance. C1 connects the base of the transistor to ground at 2.38 GHz, hence the name common base and C2 fulﬁls the role of decoupling capacitor. The two unnamed capacitors act as DC blocks and are used for DESIGN OF A LOW-COST RECEIVER CIRCUIT 30

the S-parameters simulations. A DC simulation computes the DC currents and voltages which determine the operation point of the circuit. Results can be found in Table 4.2.

Quantity Value

VC 3.3 V VB 1.14 V VE 0.33 V

IE 12.2 mA IB 51.8 µA

IR2 1.39 mA

Table 4.2: DC characteristics of the ﬁrst oscillator design.

The magnitude of the S11 and S22 parameters for this ampliﬁer are shown in Figure 4.3a. Regions where the magnitude of the small signal S-parameters is larger than one, indicate that the circuit is potentially unstable, while a magnitude smaller than one indicates a stable region. These conditions only hold if the circuit is terminated with 50 Ω, the standard reference impedance for our S-parameters. Hence, the ampliﬁer is potentially unstable in the region spanning from 4.36 Ghz to 12.8 GHz.

6 4 S S | 11| 8.79 GHz | 11| S22 3.5 S22 5 | | 2.46 GHz | | 3 4 2.5

3 2

Magnitude Magnitude 1.5 2 1 1 0.5

00 2.5 5 7.5 10 12.5 15 00 2.5 5 7.5 10 12.5 15 Frequency [GHz] Frequency [GHz]

(a) S-parameters of Figure 4.2. (b) S-parameters of Figure 4.4.

Figure 4.3: S-parameters of the common base circuit with and without base coil.

Make the circuit unstable

Despite not being a suﬃcient condition, it is a good practice to start the design by making the magnitude of the reﬂection parameters maximal at the desired frequency of operation. Since it lies in the stable region, however, positive feedback needs to be added to move the region of potential instability to lower frequencies. In a common base topology, local positive feedback can be achieved by adding a coil to the base of the transistor. The coil forms an LC resonator together with the (parasitic) base-collector capacitor Cbc of the BJT and C1 [38]. At resonance, there exists a low impedance path between the output (collector) and the input (base), paving the way DESIGN OF A LOW-COST RECEIVER CIRCUIT 31

Vcc

L2 R1

port 2 Component Value

R1 1500 Ω C1 port 1 R2 820 Ω

L1 R3 27 Ω R2 L3 C1 2.2 pF C2 100 pF

R3 L1 100 nH L2 100 nH L3 15 nH

Figure 4.4: Common base circuit with base Table 4.3: Component coil. values of Figure 4.4.

for (positive) feedback. Tuning the value of the coil and changing capacitor C1 as well, enables one to achieve maximum magnitude for the S-parameters at 2.38 GHz. The resulting circuit and components values are shown in Figure 4.4 and Table 4.3, respectively. Figure 4.3b depicts the magnitude of the S-parameters as a function of frequency. One sees that the magnitude of the

S11 and S22 parameters is maximal in the neighbourhood of 2.38 GHz. The range of potential instability for S11 spans from 1.11 GHz to 3.95 Ghz, a narrower region than the original ampliﬁer.

Select a load in the unstable region

One is obviously not limited to a load of 50 Ω, so another method is needed to check the stability (or potential instability) of the system with a different load. To this end, stability circles can be used. Stability is defined by the condition Γ < 1 with | in| S S Γ Γ = S + 12 21 L , (4.1) in 11 1 S Γ − 22 L hence Γin is the input reflection coefficient. The potentially unstable region, on the other hand, is characterised by the condition Γ > 1. The border is thus defined by the equality Γ = 1. | in| | in| By substituting the expression for Γin in this equality, squaring both sides and completing the square (for details, see Section 12.2 in [19]), one realises that this border forms a circle on the

Smith chart, which is a Cartesian coordinate system for the load reﬂection coeﬃcient ΓL. The centre of the circle and the radius are given by (S ∆S∗ )∗ C = 22 − 11 (4.2a) L S 2 ∆ 2 | 22| − | | S S R = 12 21 , (4.2b) L S 2 ∆ 2 22 | | − | |

DESIGN OF A LOW-COST RECEIVER CIRCUIT 32

where ∗ denotes the complex conjugate and ∆ = S S S S is the determinant of the 11 22 − 12 21 scattering parameter matrix. An example of such a stability circle is (partly) displayed in Figure 4.5 as the black circle. It is unclear, however, if the stable region lies outside or inside of this circle. Nevertheless, the stability for a load of 50 Ω is known by inspecting S and | 11| the location of this load is the centre point of the Smith chart. For Figure 4.5, the potentially unstable region is located inside the black circle as S = 3.1 at 2.38 GHz. | 11|

+j1.0

+j0.5 +j2.0

+j0.2 +j5.0

0.0 0.2 0.5 1.0 2.0 5.0 ∞

ΓL

−j0.2 −j5.0

−j0.5 −j2.0

−j1.0

Figure 4.5: Smith chart with unity stability circle (black), ﬁxed Γ circle (grey) and Γ . | in| L

In order to make an unstable amplifier, a load with reflection coefficient ΓL must be chosen inside the potentially unstable region. It is prudent to choose a load that is not located too close to the stability circle; a deviation of the desired load impedance due to parasitics, temperature variations, etc. could make the circuit stable instead of unstable, making oscillations impossible straight away. Therefore, it is useful to pick a load such that Γ = ρ, with ρ a number bigger | in| than one. It can be proven in a very similar way that for all values of ρ, the equation results in a circle on the Smith chart. The derivation is completely analogous to the one for ρ = 1 as found in Pozar [19] and results in the centre and radius: (ρ2S ∆S∗ )∗ C = 22 − 11 (4.3a) L ρ2 S 2 ∆ 2 | 22| − | | S S R = ρ 12 21 . (4.3b) L ρ2 S 2 ∆ 2 22 | | − | | The grey circle in Figure 4.5 depicts a circle with ρ = 2.5. The load for the oscillator is chosen on this circle at the marked point. This particular point, Γ = 0.127 83.41◦, was chosen L ∠ − because it lies deep in the unstable region and its position close to the center on the circle of normalised unity resistance provides easy matching to the load of 50 Ω. The matching network consists of a single series capacitor of 5.22 pF and has the extra functionality of a DC block. DESIGN OF A LOW-COST RECEIVER CIRCUIT 33

Compensate the negative resistance

With the load chosen in the unstable region, the input impedance of the transistor circuit, looking from port two, should have a negative resistance. The expression for this impedance can be calculated as follows: Γin is obtained by substituting the chosen ΓL into (4.1); the resulting reflection coefficient is then transformed to an impedance using the equation 1 + Γ Z = Z in , (4.4) in 0 1 Γ − in with Z0 the reference impedance for the S-parameters, viz. 50 Ω. For the value of ΓL chosen in the previous section, this results in an input impedance of ( 28.56 + j25.06) Ω. The negative − real part is apparent and indicates the ability of the circuit to overcome the inevitable losses at the desired frequency. The surrounding frequencies are, however, possible candidates for oscillation as well since they also exhibit negative resistance. In other words, the frequency of oscillation itself has yet to be fixed. This is done by offering a complex impedance at port two that compensates the complex part of the input impedance at the required frequency, the second condition in (3.4). The real part of the input impedance is compensated as well by the real part of impedance ZS (see Figure 3.1b) in order to stabilise the oscillation. The magnitude of the negative resistance is proportional to the gain of the unstable amplifier. At small signal levels, this gain is constant and dictated by the operation point and the small signal properties of the transistor and the surrounding circuitry. As the signal level rises, however, the transistor’s properties start to change and the small signal approximations are not valid any longer. The gain decreases and, consequently, the magnitude of the negative resistance lowers as well. Accordingly, the oscillation will not build up if the small signal negative input resistance is compensated completely. If little or no compensation is applied, however, the oscillator circuit will be driven in a very strong non-linear regime and strong harmonics will be generated, resulting in a strongly distorted output waveform. Luckily, a good rule of thumb exists that forms a compromise between guaranteed start up and small distortion; R = 1/3 R . S | in| This guideline does not come out of the blue; it is based on a strongly simplified behaviour of the negative resistance as the amplitude changes [39]. The input resistance equals R for zero − 0 amplitude, in other words, it corresponds to the small signal value of the negative resistance. The resistance is assumed to rise linearly, up to the magnitude Am for which the negative resistance reaches zero. The power available from the network is thus: 1 P = [VI∗] (4.5) avn 2< 1 = I2 [Z ] 2 < in = 1/2A2 (R [1 A/A ]) . (4.6) − 0 − m By setting the derivative of the above expression with respect to A to zero, one ensures maximum power transfer from the oscillator to the load. This occurs for an amplitude A of 2/3Am, corresponding to an input impedance of 1/3R . So if R = 1/3 R , the net resistance for small − 0 S | 0| amplitudes is negative, enabling start up but the net resistance vanishes before the amplitude becomes too big which would cause distortion. DESIGN OF A LOW-COST RECEIVER CIRCUIT 34

Vcc

L2 C3 R1 Component Value Out R1 1500 Ω 50 Ω R2 820 Ω

C1 R3 27 Ω R4 9.5 Ω L1 C 2.2 pF C4 1 R2 L3 C2 100 pF C3 5.22 pF R R 3 4 C4 2.67 pF

L1 100 nH L2 100 nH L3 15 nH

Figure 4.6: Initial oscillator design. Table 4.4: Component values of Figure 4.6

Thus the impedance Z was chosen as (9.52 j25.06) Ω, compensating the imaginary part S − exactly at 2.38 GHz and compensating the small signal real part partially to enable the build up of the oscillation. The impedance ZS is simply realised using a resistor of 9.5 Ω in series with a capacitor of 2.67 pF. With all components values designed, the circuit looks like depicted in Figure 4.6 and the corresponding Table 4.4.

Non-linear simulations

The design so far is based on small signal S-parameters. In the final operation mode, however, the oscillator will not be working in its small signal regime; serious gain compression and strong non- linear effects will come into play. This is why the above results are only a first approximation of the true operation of the circuit. Moreover, start up of an oscillator can only be checked easily using the non-linear transient simulator. Together with the harmonic balance simulation, it forms the main validation of operation in simulation. Both the harmonic balance and the transient simulation (with Gear’s method, Section 3.3.1) predict an oscillation at 1.51 GHz with 11.2 dBm output power at the fundamental. Figure 4.7 shows the spectrum (computed using a HB simulation) and the waveform (computed using a transient simulation). The waveform is not sinusoidal at all, a phenomenon that manifests in strong harmonics in the output spectrum. The second harmonic is only 4 dB weaker than the fundamental tone and the third and fourth harmonic are significant as well with powers around 2 dBm. Worst of all is of course the fundamental frequency, which deviates a great deal from the desired 2.38 GHz. Substantially tuning components enables one to shift the frequency towards 2.38 GHz and lower the overtones at the cost of lower output power. Opting for a smaller coil DESIGN OF A LOW-COST RECEIVER CIRCUIT 35

15 2.5 2 10 1.5 5 1 0.5 0 0 0.5 − -5

Magnitude [dBm] 1 Ouput voltage [V] − 1.5 -10 − 2 − 2.5 − -15 1.51 3.02 4.54 6.05 7.56 9.07 10.59 12.1 13.61 0 0.2 0.4 0.6 0.8 1 Frequency [GHz] Time [ns] (a) Spectrum computed by HB simulation. (b) Waveform computed by transient simulation.

Figure 4.7: Non-linear simulation results of the small signal design. and capacitor at the base, raises the frequency a lot and increasing the capacitance at the emitter aids in getting a higher frequency as well. Increasing the losses in the circuit by increasing the resistor at the emitter lowers the power of the overtones, resulting in an output waveform with a more sinusoidal nature. The new component values can be found in Table 4.5. The resulting spectrum and waveform are shown in Figure 4.8. The frequency of oscillation is now 2.388 GHz, almost exactly the frequency needed. Even though the power of the fundamental tone has dropped to 6.5 dBm, the improvement in spectral purity is remarkable. The first overtone is almost 20 dB lower than the first harmonic and all subsequent harmonics are even smaller. The output waveform reflects these smaller harmonics; it is almost perfectly sine-like with a small imbalance due to the harmonics.

Component Value

R1 1500 Ω R2 820 Ω R3 27 Ω R4 22.5 Ω

C1 1.2 pF C2 100 pF C3 5 pF C4 3.9 pF

L1 100 nH L2 100 nH L3 13 nH

Table 4.5: Component values of the improved initial design. DESIGN OF A LOW-COST RECEIVER CIRCUIT 36

10 1 0 3/4

-10 1/2 1/4 -20 0 -30 -1/4 Magnitude [dBm] -40 Ouput voltage-1/2 [V]

-3/4 -50 -1 -60 2.39 4.78 7.16 9.55 11.94 60 60.2 60.4 60.6 60.8 61 Frequency [GHz] Time [ns] (a) Spectrum computed by HB simulation. (b) Waveform computed by transient simulation.

Figure 4.8: Non-linear simulation results of the improved initial design.

Drawing the layout

The oscillator designed in the previous sections is made up of ideal components and is as such not realisable and measurable. Furthermore, the inﬂuence of the layout must not be neglected as the length of the interconnections and the extra capacitance and inductance can have its repercussions on the operation of the oscillator. This is why a layout was drawn in Momentum (Figure 4.9) which is just a straightforward implementation of the oscillator circuit described above (Figure 4.6) with room for some extra components at the base and emitter to enable ﬁne-tuning of critical values.

Figure 4.9: Layout of the ﬁrst oscillator.

A design choice that should be noted is the small trace between two pins of the transistor. It so happens that the package of the BFP640 has four pins and two diametrically opposed pins are emitter pins. Connecting only one of them to the bias circuit and the resonator and leaving DESIGN OF A LOW-COST RECEIVER CIRCUIT 37

the other one open would make the circuit very sensitive to impinging noise signals and vice versa, it would lead to a strongly radiating structure. Connecting these two pins avoids these risks. Unfortunately, the only two options to obtain this is, on the one hand, by connecting both pins underneath the transistor and, on the other hand, by making the connection through vias and an aperture in the ground plane. The first solution is the least intrusive one and causes the least troubles, so it is implemented here. The substrate used is an FR4 (flame retardant) variant, viz. DE104, by Elprinta with a thickness of 800 µm, a dielectric constant r of 4.75 and a loss tangent tan δ of 0.02. As the influence of the layout and the real component models is significant, the design cycle described above was actually repeated starting from the layout with the same bias circuit and choosing/tuning the other components step by step. This results in the component values displayed in Table 4.6. Simulation results are shown in Figure 4.10. A power of 6.6 dBm is predicted for the fundamental component at 2.385 GHz. The first overtone is 16 dB weaker and the second overtone is about 10 dB lower than the fundamental tone with a power of 3.5 dBm. − The waveform shows a periodic signal with the general shape of a sine but with quite strong higher harmonic effects clearly visible. Note that both spectrum and waveform are results from the harmonic balance simulation as the transient simulation converged but predicted that the oscillation would die out immediately. Measurements have shown, however, that the oscillation does not die out.

Component Value Series

R1 1500 Ω CR0603-FX Bourns [40] R2 820 Ω CR0603-FX Bourns R3 27 Ω CR0603-FX Bourns R4,a 1.2 Ω CR0603-FX Bourns R4,b 20 Ω CR0603-FX Bourns C1,a 1.2 pF Murata GQM18 [41] C1,b 1.1 pF Murata GQM18 C2 10 nF Murata GRM18 [42] C3 1.8 pF Murata GQM18 C4 1.8 pF Murata GQM18

L1 100 nH Coilcraft 0603CS [43] L2 100 nH Coilcraft 0603CS L3,a 1.6 nH Coilcraft 0603CS L3,b 4.3 nH Coilcraft 0603CS

Table 4.6: Component values of the ﬁnal ﬁrst design. DESIGN OF A LOW-COST RECEIVER CIRCUIT 38

10 1 5 3/4 0 1/2

-5 1/4

-10 0

-15 -1/4 Magnitude [dBm] Ouput voltage-1/2 [V] -20 -3/4 -25 -1 -30 2.39 4.77 7.16 9.54 11.93 14.31 16.7 0 0.2 0.4 0.6 0.8 1 Frequency [GHz] Time [ns] (a) Spectrum computed by HB simulation. (b) Waveform computed by HB simulation.

Figure 4.10: Non-linear simulation results of the ﬁnal ﬁrst design.

Validation of the ﬁrst design

After soldering the components on the PCB, the power source was connected and the oscillator was attached to a signal analyser (Rhode & Schwarz FSV40 [44]) in order to measure the spectrum. As shown in Figure 4.11, the oscillator produces a fundamental tone of 11.35 dBm but at completely the wrong frequency, namely 5.966 GHz. The ﬁrst two overtones have a power of 4.06 dBm and 10.69 dBm, respectively, which is a good suppression as compared to the − − fundamental tone.

10 5.966 GHz 11.35 dBm 0 -4.06 dBm 10 − 20 − 30 − Magnitude [dBm] 40 − 50 − 0 5 10 15 20 25 Frequency [GHz]

Figure 4.11: Measured spectral response of the ﬁrst oscillator circuit.

While being a major setback at the time, in retrospect the oscillator was doomed to oscil- DESIGN OF A LOW-COST RECEIVER CIRCUIT 39

late at a different frequency. Figure 4.12 depicts the real (black) and imaginary (grey) input impedance at the emitter of the oscillator before the components at the resonator where added. This is exactly the place in the circuit where a negative input resistance is desired at the chosen frequency. The real part is negative in three different regions, all potential oscillation frequencies. As this is not the only condition for oscillation, a full non-linear simulation must be performed to check whether other frequencies are possible. By tinkering with the settings of the harmonic balance, simulations show that oscillation is possible in the third region at 7.33 GHz. While it is clearly not the oscillation frequency that was actually measured, it shows that the large extra area of negative resistance is problematic as it can support additional oscillation frequencies. Due to this fundamental flaw in the design, it was decided to redesign the oscillator using a slightly different approach.

150 Rin Xin 100

0 Impedance [Ω]

50 −

100 − 0 1 2 3 4 5 6 7 8 9 Frequency [GHz]

Figure 4.12: Real (black) and imaginary (grey) part of the input impedance at the emitter of the ﬁnal ﬁrst design.

4.2.2 Second Oscillator Design

The main cause of the wrong behaviour of the ﬁrst oscillator design is the use of small discrete components. Beside their large sensitivity to parasitic elements and uncertainty of the exact component value, the lack of a narrow region of negative impedance has its origin in these discrete components. Low component values lead to low Q-factors, indicating weak frequency dependent behaviour. Under these circumstances, it may be advantageous to use transmission lines. Not only do they have a decent Q-factor, but they also have an intrinsic periodic frequency behaviour; resonance does not occur at a single frequency but at the harmonics of this frequency as well. This results in a strong frequency dependence over a large frequency range. DESIGN OF A LOW-COST RECEIVER CIRCUIT 40

In order to benefit from these profitable properties, it was decided to incorporate transmission lines into the second design to obtain a narrower area of negative resistance. Moreover, a transmission line in the resonator and/or the output matching network results in a more strongly varying imaginary part, improving the frequency stability. The main downside to the use of transmission lines is the fact that, once made, they cannot be changed whereas discrete components can easily be replaced by other components. Therefore, the choice was made to provide a mix of transmission lines and discrete components to preserve some way of tuning the behaviour. As for the type of transmission line realisation, the microstrip was the best option for this circuit. The dimensions of the circuit are actually not small so proper grounding on the top layer to achieve a GCPW topology, would require a huge amount of vias. Although the parasitics of the microstrip are more pronounced, Momentum simulations were used to make a good approximation of these effects. Another measure was taken to ensure oscillation in the proper frequency range. Another transistor, BFP183 from Infineon Technologies AG [45], was chosen because it has a much lower transition frequency, fT (8 GHz). As this frequency is an indication of the maximal frequency at which the transistor can amplify signals, the lower fT should make it harder for the circuit to oscillate at higher frequencies, hopefully avoiding the situation that was experienced with the first oscillator.

Small signal design

The same biasing circuit as the first oscillator is used, where only two coils, L1 and L2 that function as RF chokes, are lowered in value to raise their self-resonant frequency (SRF). Be- cause of the different transistor, the DC voltages and currents differ but only by a fraction (see Table 4.7). The magnitude of the small signal S-parameters for the circuit without added instability are depicted in Figure 4.13a. As was the case with the first oscillator design, the inherent instability is located too high, although only slightly too high, and must be lowered by some form of positive feedback. In order to get maximum instability at 2.38 GHz, a microstrip stub is added at the base instead of a coil and a larger capacitance C1 = 27 pF was selected. The width of the strip was chosen to be 0.9 mm which is actually the width of the pad for the base pin of the BJT and the landing pad of the capacitor at the end of the line. On the same substrate as before, DE104 by Elprinta, this yields a transmission line of 63.7 Ω. The length was tuned starting from 34.4 mm, corresponding to 180◦ electrical length, until maximum instability was achieved around 2.38 GHz with a length of 36 mm (Figure 4.13b). Not only has the frequency shifted, the region of possible instability has narrowed significantly, which limits possible regions of negative resistance further in the design. DESIGN OF A LOW-COST RECEIVER CIRCUIT 41

Quantity Value

VC 3.3 V VB 1.08 V VE 0.42 V

IE 15.4 mA IB 162.9 µA

IR2 1.32 mA

Table 4.7: DC values for the second design.

3 3 S S | 11| 2.39 GHz | 11| S22 S22 2.5 3.05 GHz | | 2.5 | |

2 2

1.5 1.5 Magnitude Magnitude 1 1

0.5 0.5

00 2.5 5 7.5 10 12.5 15 00 2.5 5 7.5 10 12.5 15 Frequency [GHz] Frequency [GHz]

(a) Common base circuit with BFP183. (b) Common base circuit with microstrip at the base.

Figure 4.13: S-parameters of the common base circuit with and without base microstrip.

At this point, the decision was made to follow a different approach; the resonator at the emitter was designed first, resulting in a negative resistance at the collector. Subsequently, the load matching was picked such that it compensates the input impedance at the desired frequency. The emitter resonator consists of a stub with a capacitor (C3) to ground in series, just like the circuit at the base of the transistor. The resulting circuit is shown in Figure 4.14. A value of 27 pF was chosen for the capacitor, the width of the line fixed to 0.9 mm, easing the placement of the capacitor and an initial value for the line of 23 mm. This length (l2) as well as the length of the microstrip at the base (l1) were tuned to the values found in Table 4.8, resulting in the input impedance at the collector, depicted in Figure 4.15. The impedance varies very abruptly, indicating strong resonant behaviour of certain circuit elements. There is now only one main region of negative resistance spanning from 1.69 GHz to 2.49 GHz. Around three other frequencies, 320 MHz, 4.66 GHz and 5.6 GHz respectively, the circuit produces a small negative resistance as well but these are much less distinct than the main region. At 2.38 GHz, the input impedance is ( 78.23 j176.8)Ω. This implies that the output matching network has − − to transform the load impedance of 50 Ω to an impedance of (26.08+j176.832) Ω or equivalently ◦ a reflection coefficient (ΓL) of 0.927∠31 . The matching network to achieve this wanted reflection coefficient is depicted in Figure 4.16. DESIGN OF A LOW-COST RECEIVER CIRCUIT 42

Vcc

L2 R1

port 2 Component Value

R1 1500 Ω

l1 R2 820 Ω R 27 Ω l 3 L 2 1 C1 27 pF R2 C1 C2 100 pF C3 C3 27 pF R 3 L1 37 nH L2 37 nH

l1 37 mm l2 19.5 mm

Figure 4.14: Circuit generating a negative re- Table 4.8: Component val- sistance at the collector. ues resulting in negative resistance.

Capacitor C4 acts as DC block and will be connected to the collector. The long transmission line (l3) rotates the impedance on the Smith chart while the short stub, l4 (shorted to ground in AC), moves the impedance from the center to the edge of the Smith chart. The width of l3 is chosen to be 1.416 mm,yielding a 50 Ω microstrip, while the width of l4 equals 0.9 mm which adapts nicely to the capacitor C5. The load will be connected to the second 50 Ω line at the far right. With l3 = 9 mm and l4 = 1 mm and both capacitors having a value of 27 pF, the matching network produces the required reﬂection coeﬃcient. DESIGN OF A LOW-COST RECEIVER CIRCUIT 43

300 Rin Xin 200

100

Impedance [Ω] 100 −

200 −

300 − 0 1 2 3 4 5 6 7 8 9 Frequency [GHz]

Figure 4.15: Real (black) and imaginary (grey) part of the input impedance of the second design.

C5 C 4 l3 l4

Figure 4.16: Output matching network. DESIGN OF A LOW-COST RECEIVER CIRCUIT 44

Non-linear simulations

Combining the transistor and the matching network should result in oscillation at 2.38 GHz or at least in the vicinity of this frequency. However, one has to remember that the design so far is based on small signal simulations of the components and that the real behaviour of the oscillator will diﬀer from these simulations. An harmonic balance simulation of the current circuit consequently shows that no oscillation occurs. Tuning the lengths of the four transmission lines in the circuit, however, yields oscillation at 2.38 GHz. As the design so far was already made with real component models and using a Momentum simulation to model the interconnection and the transmission lines, this design should result in a correctly functioning real circuit after fabrication. A representation of the ﬁnal circuit’s layout can be found in Figure 4.17 and component values are listed in Table 4.10.

Component Value Series

R1 1500 Ω CR0603-FX Bourns R2 820 Ω CR0603-FX Bourns R3 27 Ω CR0603-FX Bourns

C1 27 pF Murata GQM18 C2 100 pF Murata GRM18 C3 27 pF Murata GQM18 C4 27 pF Murata GQM18 C5 27 pF Murata GQM18

L1 37 nH Coilcraft 0603CS L2 37 nH Coilcraft 0603CS

l1 35 mm l2 19.6 mm l3 10.2 mm l4 1.45 mm

Table 4.9: Component values of the second design.

Although the footprint of the BFP183 diﬀers from that of the BFP640, the transistor used in the second design has two diametrically opposed emitter pins as well. That is why in this design a trace is placed underneath the transistor in this realisation too. The connection for the power source is brought to the top of the board using a bend to ease the placement of a connector. DESIGN OF A LOW-COST RECEIVER CIRCUIT 45

Figure 4.17: Final layout of the second oscillator design. DESIGN OF A LOW-COST RECEIVER CIRCUIT 46

Validation and fabrication

After fabrication of the PCB, the components as listed in Table 4.10 were soldered and the output of the board was connected to the signal analyser (Rhode & Schwarz FSV40). Unfortunately, no oscillation was measured. Only after adding a capacitor of 0.5 pF in parallel with coil L2, oscillation was measured at 2.6 GHz with a power of 0 dBm. This relatively low output power is caused by power leaking away to ground via the small capacitance that was added and the capacitance to ground in the output matching network. In an attempt to raise this output power, a copy of the board was soldered with the same components but omitting capacitor

C5 to reduce the loss of RF power to ground. For this configuration, oscillation did occur in this circuit at 2.56 GHz with a fundamental tone output power of 5.7 dBm. Further research is required to determine the exact operation of this oscillator. Changing both capacitors C1 and C3 to 5.1 pF 47 pF shifted the frequency response to 2.35 GHz with an output power of 6.7 dBm. k It is thus demonstrated that fine-tuning the capacitors enables one to achieve oscillation at exactly 2.38 GHz. Consequently, this oscillator was kept as the final LO for the injection locking measurements described in Chapter 5. The final component values are summarised in Table 4.10.

Component Value Series

R1 1500 Ω CR0603-FX Bourns R2 820 Ω CR0603-FX Bourns R3 27 Ω CR0603-FX Bourns C 5.1 pF 47 pF Murata GQM18 1 k C2 100 pF Murata GRM18 C 5.1 pF 47 pF Murata GQM18 3 k C4 27 pF Murata GQM18 C5 omitted

L1 37 nH Coilcraft 0603CS L2 37 nH Coilcraft 0603CS

l1 35 mm l2 19.6 mm l3 10.2 mm l4 1.45 mm

Table 4.10: Component values of the functioning oscillator. DESIGN OF A LOW-COST RECEIVER CIRCUIT 47

4.3 Mixer Choice

Originally, the idea was to design a single balanced diode mixer with a Marchand balun to function as the frequency mixer in the receiving circuit. However, as little time remained after the two oscillator designs, the decision was taken to purchase an existing mixer instead of designing it from scratch. As the mixer is not so crucial in the occurrence of injection locking/pulling, the use of a commercially available mixer has the advantage of proving that this non-linear phenomenon applies to real world applications, relaying on commercial off-the-shelf components, as well. In order to stay as close as possible to the original decisions, the type of mixer was preserved although a double balanced mixer was chosen as single balanced diode mixers are a rare find. The choice of the mixer itself is, in the first place, mainly based on its applicability in the frequency range of interest and secondly, in the isolation between RF and LO as very high isolation between these two ports would limit the effects of injection locking/pulling. Based on these two requirements, a handful mixers from Mini-Circuits were an option with the two top candidates the ADE-35+ [46] and the MBA-25L+ [47]. A third specification that has to be taken into account is the power of the oscillator, as every mixer performs best for a limited range of LO powers. The first mixer works best at +7 dBm LO power while the second one operates optimal at +4 dBm LO power. At the moment of purchase, however, the exact power of the designed oscillator circuit was yet unknown. Luckily, a board from a previous project with an ADE35+ on it and a splitter at the LO side, BP2U+ from Mini-Circuits [48] was available in the research group so this could be reused. In case the output power would be lower (around +4 dBm or lower), an MBA-25L+ was ordered.

Figure 4.18: Layout of the MBA-25L+ mixer.

A separate board, fabricated by Eurocircuits, was designed for this mixer to facilitate measurements. The substrate is the I-Tera substrate from Isola [49] with a thickness of 508 µm, a dielectric constant r of 3.3 and a loss tangent tan δ of 0.0036. Because of the small dimensions of the board, the connections were made using GCPWs instead of microstrips. The layout is shown in Figure 4.18. MEASUREMENTS 48

Chapter 5

Measurements

With all the necessary building blocks designed and fabricated, validation through measurements is crucial to verify theory and simulations. Moreover, measurements of the total system mimic the real application in which this low-cost front-end of an RF receiver will be used. The experiments mainly focus on the occurrence of injection locking/pulling and their influence on the performance of the receiver circuitry. The quadrature hybrid, an RF component that is used extensively during the measurements, will be discussed in Section 5.1. The spectral quality of the final oscillator design is measured in Section 5.2. Next, the mixer performance will be verified in Section 5.3. Injection locking/pulling measurements of both the oscillator on its own and of the total system will be performed and interpreted in Section 5.4.

5.1 Quadrature Hybrid

Directional couplers are four-port RF components that can be used for splitting power flowing in different directions. A quadrature hybrid is a special type of directional coupler that produces a 90◦ phase shift between the output signals (Section 7.5 in [19]). The microstrip implementation of this component is called a branch-line hybrid and was designed on the I-Tera substrate from Isola that was used for the mixer board as well (see Section 4.3). The layout is shown in Figure 5.1 and will be used to explain the operation of the quadrature hybrid. Power entering the component through the input port (port 1) will be diverted to ports 2 and 3 with a phase difference of 90◦ and 3 dB loss due to the split. No power will be seen at the fourth port, therefore called the isolated port. Due to the symmetrical nature of the component, the same reasoning can be made for power entering at any of the other ports. The branch-line hybrid relies on the resonant behaviour of the central transmission lines and is, as such, a narrowband implementation of the ideal quadrature hybrid. The bandwidth is in practice limited to about 10% 20% of the central frequency [19]. In order to evaluate the − performance of the hybrid coupler a four-port S-parameter measurement was performed with a N5242A PNA-X (Programmable Network Analyser) by Agilent Technologies [50] using the same port numbering as used in Figure 5.1. The resulting S-parameters are shown in Figure 5.2. The centre frequency is approximately 2.5 GHz at which the isolation between port 1 and 4 MEASUREMENTS 49

Input Output Port 1 Port 2

Port 4 Port 3 Isolated Output

Figure 5.1: Branch-line hybrid implementation. is 26 dB and the attenuation to the outputs (port 2 and 3) 3.4 and 3.3 dB, respectively. The performance is clearly narrowband as isolation has dropped to around 14 dB at 0.25 GHz away from the center frequency.

5 −

10 −

15 −

Magnitude [dB] 20 −

S11 25 S − 12 S13 S14 30 − 1 1.5 2 2.5 3 3.5 4 Frequency [GHz]

Figure 5.2: Magnitude of the S-parameters of the branch-line hybrid.

5.2 Oscillator Measurements

The spectral properties of the ﬁnal oscillator are measured using a Rhode & Schwarz FSV 40 signal analyser. The measured spectrum is shown in Figure 5.3. The full spectrum (Figure 5.3a) depicts a fundamental tone at 2.344 GHz with an output power of 9.04 dBm. The second and third harmonic are suppressed by 13.5 dB and 17.5 dB, respectively; a decent suppression that leads to a relatively pure sine in the time domain. However, no oscilloscope was available that could measure up to at least a couple harmonics so the exact waveform could not be inspected. A detailed look at the fundamental tone in Figure 5.3b indicates a more precise oscillation frequency of 2.3439 GHz with a slightly lower output power of 8.94 dBm. Note that there are two symmetrical sidebands to the fundamental tone at 900 kHz. These spurious signals exist due to ± parasitic feedback paths or weak secondary oscillations that produce more or less discrete spikes in the spectrum as well as intermodulation products with the harmonics. As these sidebands are suppressed by more than 50 dB, they can be ignored in practice except for situations in which the phase noise is important, as they will impact this characteristic. At this point it MEASUREMENTS 50

should be noted that the oscillation frequency is sensitive to the exact power supply voltage and the operating temperature of the circuit. The exact fundamental frequency thus tends to differ between different measurements or can even drift during a single measurement. The difference is mostly limited to a few MHz.

10 2.344 GHz 2.3439 GHz 5 9.04 dBm 0 8.94 dBm 5 -4.51 dBm 10 − − -8.46 dBm 15 20 − − 30 25 − − 40 35 − − 50

Magnitude [dBm] Magnitude [dBm] − 45 − 60 − 55 70 − − 65 80 − 0 2.5 5 7.5 10 12.5 15 17.5 − 2.335 2.34 2.345 2.35 Frequency [GHz] Frequency [GHz] (a) Full spectrum. (b) Detail of the fundamental tone.

Figure 5.3: Spectrum of the ﬁnal oscillator design.

5.3 Mixer Measurements

The performance of a mixer is mainly determined by the conversion gain (or loss, in case of a passive mixer) and the isolation between the three ports. As injection locking will occur by inserting a tone through the RF port, the main interest lies in the RF to LO isolation. As mentioned before, an ADE-35+ was available at the research group from a previous project. The mixer was already placed on a board but together with a power splitter in the conﬁguration as depicted in Figure 5.4. With a proper load of 50 Ω attached to the third port of the splitter, it behaves like a two-port element with a loss of approximately 3.5 dB between the two functional ports. During the isolation and conversion loss measurements, the combination of the splitter and the mixer will be considered as one three-port element.

BP2U+ ADE-35+ LO

IF 50 Ω

Figure 5.4: Block diagram of the ADE-35+ set-up. MEASUREMENTS 51

The second mixer, a MBA-25L+ was soldered on the board as designed in Section 4.3. Sub- Miniature version A (SMA) connectors were used to connect the mixer board to test equipment or other boards containing building blocks for the receiver circuit.

Agilent Technologies Agilent Technologies Rhode & Schwarz N9310A RF Signal Generator N5242A PNA-X FSV40

RF LO DUT Agilent Technologies Rhode & Schwarz N5242A PNA-X FSV40 IF 50 Ω RF LO IF DUT

(a) Measurement of the conversion loss. (b) Measurement of the RF to LO isolation.

Figure 5.5: Mixer measurement set-ups.

Conversion loss

The conversion gain in a mixer is defined as the ratio of the IF power to the inserted RF power. In passive mixers, this ratio is always smaller than one (or negative in dB) and is therefore redefined as the conversion loss, which is the inverse of the conversion gain. The set-up to measure the conversion loss is shown in Figure 5.5a. The device under test (DUT) is either the board with the ADE-35+ on it as shown in Figure 5.4 or the board with the MBA-25L+. An Agilent Technologies N5242A PNA-X is used as the LO that drives the mixer. An N9310A RF Signal Generator [51] is used to inject a tone into the RF input. The third device is a Rhode & Schwarz FSV40 signal analyser that measures the IF spectrum. In order to compare the measured results to the data presented in the datasheet, a fixed IF frequency of 30 MHz was chosen. The RF power was fixed to 0 dBm for both mixers and the LO power set to 7 dBm for the ADE-35+ and to 4 dBm for the MBA-25L+.

10 ADE-35+ MBA-25L+ 9

7 Conversion loss [dB] 6

52 2.2 2.4 2.6 2.8 3 Frequency [GHz]

Figure 5.6: Conversion loss measurements for both mixers. MEASUREMENTS 52

Figure 5.6 depicts the conversion loss for both mixers as a function of the LO frequency. The conversion loss of the ADE-35+ is strongly dependent on the LO frequency due to the frequency dependence of the internal baluns. This behaviour is also clearly visible in the datasheet, certainly if one takes into account that the eﬀective LO power at the mixer is 3.5 dBm. The general trend closely resembles the data that is reported for an LO power of 4 dBm. The trend in the conversion loss of the MBA-25L+ closely resembles the reported data as well. The MBA-25L+ clearly outperforms the ADE-35+ in terms of conversion loss but it should be noted that the conversion loss performance of the ADE-35+ improves with higher LO power, although it will still be worse than the MBA-25L+.

Isolation

The isolation from the RF input to the LO port was measured using the set-up represented in Figure 5.5b. An Agilent Technologies N5242A PNA-X acts as the RF input signal. The signals leaking into the LO port are measured using a Rhode & Schwarz FSV40 signal analyser. As no signals are measured at the IF port, it is terminated using a precision load of 50 Ω. Both cables connecting the DUT to the equipment have a loss of 1 dB at the operation frequencies in question. However, this loss is compensated during the processing of the results.

ADE-35+, PRF = 0 dBm 40 ADE-35+, PRF = 10 dBm MBA-25L+, PRF = 0 dBm MBA-25L+, PRF = 10 dBm

30 Isolation [dB]

20 2 2.2 2.4 2.6 2.8 3 Frequency [GHz]

Figure 5.7: RF to LO isolation measurements for both mixers.

Figure 5.7 shows the isolation for both mixers and for two diﬀerent input powers. One notices a strong dependence of the isolation on the RF input power for both mixers. This phenomenon is most distinct for the MBA-25L+; for a strong input signal the isolation is almost constant across the measured frequency range while a smaller input signal shows a better isolation that is, however, strongly frequency-dependent. The internal mixing diodes behave more frequency dependent when operated at lower input powers. The board with the ADE-35+ as mixer has the general tendency to provide better isolation for rising frequencies. The isolation is less dependent MEASUREMENTS 53

on the input power than with the other mixer. Since a strong dependence on both frequency and input power would complicate the evaluation of future measurements, it was decided to use the ADE-35+ as the mixer for the measurements on the total receiver structure. Moreover, the oscillator output power of 9 dBm would overdrive the MBA-25L+, giving rise to stronger intermodulation products caused by higher non-linearity. Taking the loss caused by the splitter into account, the eﬀective LO power that reaches the ADE-35+ will be approximately 5.5 dBm, which is lower than the optimal 7 dBm. This will result in a higher conversion loss but a better linearity.

5.4 Injection Locking Measurements

With all the individual components properly characterised, measurements can be performed to determine whether injection locking and pulling occurs. If it does, specifying the lock range is the main characteristic that defines the severity of the phenomenon. Measurements will be performed on the oscillator itself, first in order to observe the phenomenon and to check whether the theory from Chapter 2 accurately predicts the real situation. These measurements, together with simulation results, will be presented in Subsection 5.4.1. Second, in Subsection 5.4.2, measurements are carried out on the total system to investigate the influence of the injection locking and pulling on the receiver’s performance.

5.4.1 Oscillator

In order to induce injection locking in the oscillator, a disturbance has to be injected into the circuit. In a low-cost receiver, this will happen via the output of the oscillator. However, in order to measure the spectrum of the oscillator in this situation, a quadrature hybrid is needed to insert the disturbance while still being able to measure the spectrum. An interfering signal can actually also enter the oscillator circuit via a second coupling path, i.e. the signal enter via the power supply line. The lock range will be measured in both situations to investigate the inﬂuence of the point of injection. Afterwards, the shape of the spectrum, as predicted by the theory, will be veriﬁed experimentally and by means of simulation results.

Lock range

The measurement set-up for determining the lock range is drawn in Figure 5.8a. The injected signal is once again provided by an Agilent Technologies N5242A PNA-X and the spectrum measured by a Rhode & Schwarz FSV40 signal analyser. The spectrum is separated from the injected signal by the quadrature hybrid discussed in Section 5.1. Simulation results are obtained by means of transient simulations on the ﬁnal design with capacitance C5 present (see Subsection 4.2.2). It should be noted that the circuit concerned is simulated without a model for the PCB as the transient simulation does not converge if this model is included. The circuit model, however, does include real component models and lossy transmission lines so it still yields an appropriate approximation of the real circuit. MEASUREMENTS 54

Agilent Technologies Rhode & Schwarz N5242A PNA-X FSV40 Agilent Technologies N5242A PNA-X Power supply Power supply 3.3 V 3.3 V Rhode & Schwarz Picosecond FSV40 Oscillator 5541A

50 Ω

Oscillator

(a) Injection via the output. (b) Injection via the power supply.

Figure 5.8: Lock range measurement set-ups.

The natural frequency of the real oscillator is 2.3702 GHz with an output power of 8.8 dBm. The fundamental tone of the simulation lies at 2.4395 GHz and has a power of 5.02 dBm. In both cases, the lock range was determined by choosing a fixed injection power and sweeping the frequency until locking occurred. Next, the injection power was raised and the new boundaries of the lock range were found. Figure 5.9a depicts the measured and simulated frequency fL, i.e. half the lock range, as a function of the ratio of the normalised input power, i.e. compensated for the cable and coupler losses, and the output power of the fundamental tone. Rising injected power causes a rapid increase of the lock range, as was to be expected. One also notices that the simulated oscillator is more robust than the real oscillator, as the lock range is substantially smaller for the same power ratio. Figure 5.9b shows a detail of the same graph with theoretical curves added. These curves are based on (2.6) (see Section 2.3.1). As the Q-factor of both oscillators is unknown, the curve is fitted using the three lowest measured points of each oscillator. One notices that the theoretical curve lies above the measured results at all time, indicating that it behaves as a worst-case result. Including amplitude modulation into the theoretical derivations, results is a modified, smaller lock range that lies closer to the measured results [16]. For larger values of Pinj/P0 the theoretical curve grows extremely quickly as (2.6) becomes singular for a ratio of 0 dB which is clearly not a physical result. Figure 5.10 depicts the Arnold tongues (see Section 2.3.1) for the real and simulated oscillator. Both regions of injection locking are very narrow and symmetrical for small injected powers. When the injected power grows, the boundaries of injection locking drift apart and become asymmetrical. The effect is more pronounced for the simulated oscillator but it should be noted that a stronger signal is injected as well. These higher powers are not injected into the real oscillator during measurements in order to avoid damaging the circuit. Figure 5.8b shows the set-up for injection locking by inserting the disturbance in the power supply. An Agilent Technologies N5242A PNA-X provides the RF signal that is combined with the power supply using a 5514A bias tee by Picosecond Pulse Labs [52]. The oscillator spectrum is measured using a Rhode & Schwarz FSV40 signal analyser. The measured lock range is displayed in Figure 5.11. One notices that the lock range is significantly smaller (approximately MEASUREMENTS 55

15 50 Measurement Measurement Simulation Theory (measurement) Simulation Theory (simulation) 40

10 30 [MHz] [MHz] L L f 20 f

5 10

0 40 30 20 10 0 25 20 15 10 − − − − − − − − Pinj /P0 [dB] Pinj /P0 [dB] (a) Full range. (b) Detail including ﬁtted theoretical curves.

Figure 5.9: Lock range measurements/simulations.

0 10 5 − 5 10 − 0 15 − 5 [dB] [dB] −

0 20 0 −

/P /P 10 −

inj 25 inj

P − P 15 − 30 − 20 35 − − 25 40 − − 30 2.325 2.34 2.355 2.37 2.385 2.4 2.415 − 2.365 2.39 2.415 2.44 2.465 2.49 2.515 Frequency [GHz] Frequency [GHz] (a) Fabricated oscillator. (b) Simulated oscillator.

Figure 5.10: Measured and simulated Arnold tongues. MEASUREMENTS 56

twenty times) but follows the same trend as measured in the case of injection in the output. This experiment shows that the oscillator is clearly more robust to locking phenomena when power is injected via the power supply. This is clearly caused by the lower eﬀective power that is injected into the transistor as the decoupling capacitor C2 leads most power to ground.

2.5

1.5 [MHz] L f 1

0.5

0 40 30 20 10 0 − − − − Pinj /P0 [dB]

Figure 5.11: Lock range with injection in the power supply.

Injection pulling

Now that the typical characteristic of the oscillator under inﬂuence of an external signal, viz. the lock range, is measured, the spectrum in the injection pulling region can be inspected. From the theoretical analysis in Section 2.3.2, one expects to see a central frequency that moves to the edge of the lock range as the injected frequency approaches this edge and the rise of many, tightly spaced frequency components as locking draws nearer. In case of the measurements, the interfering tone is injected via the power supply to avoid spectral deforming by the coupler. During the simulations, however, the power is injected in the output. The otherwise narrow lock range would require extremely long simulation time in order to achieve mediocre spectral resolution. Figure 5.12 depicts several spectra for both the measurements (left column) and simulations (right column). Starting from the top with Figures 5.12a and 5.12b, the injected tone’s frequency is increased beginning at the upper edge of the lock range. In both situations, the injected power is 20 dBm. The natural frequency of the real oscillator is 2.34645 GHz with − a measured lock range of 77.5 kHz. The simulated oscillator produces a fundamental tone at 2.43978 GHz with a lock range of 1.935 MHz. With an injected frequency slightly above the lock range, one sees the typical asymmetrical series of closely spaced frequency components. The frequency component at the injected frequency is clearly the strongest in Figures 5.12c and 5.12d indicating that the circuit is in quasi-lock. As the frequency increases, the frequency components start to drift apart. More- over, the fundamental tone loses its dominance and becomes equal to the second tone, which is the shifted oscillator frequency (Figures 5.12e and 5.12f). This frequency, that denotes the border between quasi-lock and fast beat, lies at 1.97 f above the natural frequency for the real · L MEASUREMENTS 57

oscillator and 1.147 f for the simulation. This is different from the theory, as calculations · L show that this point should occur at 1.118 f above the natural frequency (see Section 2.3.2). · L A possible explanation is the leakage of injected power to the output of the oscillator, causing a higher power at this frequency than actually generated by the oscillator. A second reason could be that the assumptions made during the theoretical derivations lead to results that can only be used as estimations for the real behaviour. Note the weak frequency components at the other side of the injected tone as well. These components are not predicted by the theory either. These extra components arise from amplitude modulation that is caused by the injected tone, but amplitude modulation was not taken into account in the theoretical derivations [16]. Leading the injected frequency further away, keeps shifting the oscillator component back to its natural frequency. The other frequency components drift further apart and diminish in strength (Figures 5.12g and 5.12h). Sufficiently far away from the lock range, the oscillator returns to its natural frequency and only two extra tones remain as shown in Fig- ures 5.12i and 5.12j. The fact that their power is not the same as one would theoretically expect, is once again caused by the leakage of the injected tone to the output. Although the lock range of the oscillators is rather small, both in simulations and measurements, the region where the effects of injection pulling are considerable is substantially larger. From the measurements, for example, the range of influence is approximately 20 times the lock range. The difference in this range between theory and measurements is concerning as well. Circuit designers should be very aware of this! MEASUREMENTS 58

0 0

20 20 − − 40 − 40 − 60 − Magnitude [dBm] Magnitude [dBm] 60 80 − − 100 80 − 2.346 2.3462 2.3464 2.3466 2.3468 − 2.43 2.435 2.44 2.445 2.45 Frequency [GHz] Frequency [GHz] (a) finj = f0 + fL = 2.346525 GHz. (b) finj = f0 + fL = 2.44172 GHz.

0 0

20 20 − − 40 − 40 − 60 − Magnitude [dBm]

Magnitude [dBm] 60 − 80 − 80 100 − − 2.3455 2.346 2.3465 2.347 2.3475 2.42 2.43 2.44 2.45 2.46 Frequency [GHz] Frequency [GHz] (c) f = f + 1.322 f = 2.34655 GHz. (d) f = f + 1.106 f = 2.44192 GHz. inj 0 · L inj 0 · L 0 0

20 20 − − 40 − 40 − 60 −

Magnitude [dBm] 60 Magnitude [dBm] − 80 − 80 100 − − 2.3455 2.346 2.3465 2.347 2.3475 2.42 2.43 2.44 2.45 2.46 Frequency [GHz] Frequency [GHz] (e) f = f + 1.97 f = 2.3466 GHz. (f) f = f + 1.147 f = 2.442 GHz. inj 0 · L inj 0 · L 0 0

20 20 − −

40 40 − − 60 − 60

Magnitude [dBm]− Magnitude [dBm] 80 − 80 − 100 − 2.345 2.346 2.347 2.348 2.42 2.43 2.44 2.45 2.46 Frequency [GHz] Frequency [GHz] (g) f = f + 8.42 f = 2.3471 GHz. (h) f = f + 4.765 f = 2.449 GHz. inj 0 · L inj 0 · L 0 0

20 20 − −

40 40 − − 60 − 60 Magnitude [dBm] Magnitude [dBm] − 80 − 80 − 100 − 2.344 2.346 2.348 2.35 2.4 2.42 2.44 2.46 2.48 Frequency [GHz] Frequency [GHz] (i) f = f + 31.65 f = 2.3489 GHz. (j) f = f + 15.618 f = 2.47 GHz. inj 0 · L inj 0 · L

Figure 5.12: Spectrum of injection pulled oscillator for various injected frequencies and Pinj = 20 dBm. Left column are measurements results and the right column simulation results. − MEASUREMENTS 59

5.4.2 Complete System

An oscillator does not function separately in a receiver circuit and, as such, it is important to research the susceptibility of the complete system, as this could diﬀer strongly from the individual performance of the oscillator. Firstly, the lock range is measured once again for two major reasons. This measurement checks whether the inspected phenomena actually occur in the total receiver circuit. Moreover, it enables one to compare the robustness of the oscillator in its intended application. Secondly, two-tone measurements provide additional insights in the eﬀects of injection locking/pulling on the receiver’s performance.

Agilent Technologies Rhode & Schwarz N5242A PNA-X FSV40

Power supply Power supply 3.3 V 3.3 V IF ADE-35+ ADE-35+ RF RF IF 50 Ω Oscillator LO Oscillator LO

50 Ω 50 Ω

Rhode & Schwarz Agilent Technologies FSV40 N5242A PNA-X

(a) Measurement of the lock range. (b) Measurement of the IF and LO spectrum.

Figure 5.13: Set-ups for evaluating the total system.

Lock range

The lock range is measured using the set-up depicted in Figure 5.13a. All the devices have the same function as before, with the only difference being the mixer that is inserted this time. As the IF spectrum is not of importance during this measurement, it is terminated with a precision load of 50 Ω. The oscillator is connected to the quadrature hybrid that couples approximately half of this power to the mixer and half to the signal analyser. With this circuit loading the output of the oscillator, oscillation occurs at 2.332 GHz with an output power of 10.45 dBm. Using the same power supply voltage, the oscillator on its own produces a tone of 2.344 GHz with a power of 9 dBm, hence the load caused by the coupler and the mixer clearly shifts the frequency and increases the output power. Injected RF powers are varied from 30 dBm up to 10 dBm. However, the mixer isolation and − quadrature hybrid losses have to be taken into account to estimate the actual power injected into the oscillator. The measured lock range versus the ratio of this effective injected power and the oscillator power is plotted in Figure 5.14. One notices the very small effective injected powers, resulting in a substantially smaller lock range than measured for the standalone oscillator, when subjected to disturbances at the output (Figure 5.9a). At higher power ratios, approximately MEASUREMENTS 60

1.25

0.75 [MHz] L

f 0.5

0.25

0 70 60 50 40 30 − − − − − Pinj /P0 [dB]

Figure 5.14: Lock range of the total system.

40 dB, the lock range curve deviates from the one measured for the oscillator on its own. − The explanation for this phenomenon is the change in the load that is experienced by the oscillator. When the mixer has to handle stronger signals, its impedance at the LO port begins to change, thus inﬂuencing the oscillator. The same phenomenon has a second consequence: in the injection pulling region, for injected frequencies smaller than the natural frequency, the oscillation frequency initially moves away from the injected frequency as the latter is brought closer. This is in complete contrast with the behaviour observed in the free running oscillator.

Two-tone measurements

The first two-tone measurement imitates the real application in which the circuit will be used. The first tone is located at 2.45 GHz, the centre frequency of the ISM band, and represents the useful signal that carries the information. The second tone represents an undesired, interfering signal that reaches the input of the mixer as well. The strength of the interfering signal as compared to the information signal will be varied and its impact on the receiver performance will be evaluated. The set-up for these measurements is depicted in Figure 5.13b. While the set-up is nearly identical to previous ones, there are some important changes. The IF spectrum is measured by a Rhode & Schwarz FSV40 signal analyser as it is more accurate for lower frequencies than an Agilent Technologies N5242A PNA-X. The latter is used to measure the LO spectrum as well as to generate the two tones. As a first test, the useful signal is sent into the RF port by itself in order to evaluate the mixer performance in an ideal environment. With an LO frequency of 2.33172 GHz, the IF frequency of the information signal lies at 118.3 MHz. Figures 5.15a and 5.15b show the IF spectrum for input powers of 20 dBm and 5 dBm, respectively. On the first spectrum, one − notices four components: the wanted IF component, leakage from both the RF and LO port and an intermodulation product 2f f . For stronger input signals, a lot of extra intermodulation 0 − RF products raise above the noise floor, cluttering the spectrum. The power of the intermodulation MEASUREMENTS 61

products close to the IF frequency, however, is at least 30 dB smaller than the IF frequency itself. Taking into account that an IF ﬁlter will suppress these components even more, one can conclude that the receiver works properly, even for larger input signals.

0 0 f f RF 10 10 IF − f0 − f 20 20 0 − − fRF 30 fIF 30 − − 40 40 − 2f0 fRF − − 50 50 − − Magnitude [dBm] Magnitude [dBm] 60 60 − − 70 70 − − 80 80 − 0 0.5 1 1.5 2 2.5 3 − 0 0.5 1 1.5 2 2.5 3 Frequency [GHz] Frequency [GHz] (a) Weak input signal (P = 20 dBm). (b) Strong input signal (P = 5 dBm). RF − RF Figure 5.15: Normal operation of the total circuit with information signal at 2.45 GHz.

Bringing the second signal into play, the ﬁrst situation considered consists of two tones that have the same power. The information signal (f1) and the interfering signal (f2) are injected at a power of 20 dBm. The second tone is swept from 2 GHz towards the oscillation frequency of − 2.33172 GHz and both the IF and the LO spectra are observed on their respective measurement devices. For a large spacing between the two components (f2 = 2 GHz), the mixer just transforms both signals to their respective IF frequencies of 118.3 MHz and 331.7 MHz (see Figure 5.16a).

The undesired component at 331.7 MHz can, however, be ﬁltered out easily. Raising f2 to 2.33 GHz starts to introduce results of fast beat operation into the IF spectrum as shown in Figure 5.16b. Small symmetric components start to appear around the IF frequency of the information signal. As these components are spaced not far enough from fIF,1, they will pass through the IF ﬁlter. Fortunately, they are very small compared to the desired component, and hence, they do not yet cause problems. Approaching the fundamental frequency a step further, gives rise to more and stronger components that start to hinder the operation of the receiver (see Figure 5.16c). Driving the oscillator into quasi-lock totally ruins the IF spectrum and makes it impossible to extract the information out of the mixing components as is evident from Figure 5.16d. Locking the oscillator to the interfering signal results in a nice pure IF spectrum (Figure 5.16e) but as the LO frequency has changed, so has the IF frequency. Depending on the circuit behind this front-end, this locked situation may also be problematic. MEASUREMENTS 62

0 30 − 10 − f0 40 − 20 50 − − fIF,1 f2 f1 30 fIF,2 60 − − 40 70 − − 80 50 − Magnitude [dBm] − Magnitude [dBm] 90 60 − − 100 70 − − 110 0 0.5 1 1.5 2 2.5 3 − 116 117 118 119 120 Frequency [GHz] Frequency [MHz]

(a) f2 = 2 GHz. (b) f2 = 2.33 GHz.

30 30 − − 40 40 − − 50 50 − − 60 60 − − 70 70 − − 80 80 − − Magnitude [dBm] 90 Magnitude [dBm] 90 − − 100 100 − − 110 110 − 116 117 118 119 120 − 118 118.2 118.4 118.6 Frequency [MHz] Frequency [MHz]

30 118.315 MHz − 40 − 50 − 60 − 70 − 80 − Magnitude [dBm] 90 − 100 − 110 − 118 118.2 118.4 118.6 Frequency [MHz]

(e) f2 = 2.331685 GHz.

Figure 5.16: IF spectra for various f and P = 20 dBm. 2 2 − MEASUREMENTS 63

The second scenario keeps the same information signal but raises the power of the second signal to 10 dBm, representing a strong interfering/jamming signal. Starting from a lower frequency of 1.8 GHz, the same procedure is applied. At this starting frequency, the spectrum contains a lot more components than in the previous situation, as shown in Figure 5.17a. The strong IF component at 531.7 MHz is the downconverted version of the jamming signal. Pro- vided that a strong IF ﬁlter is applied after the front-end, extracting the information should still be possible. Nonetheless, loss of quality is to be expected. Further raising the frequency beyond the point where both IF frequencies coincide, i.e. at f2 = 2.2134 GHz, the spectrum for f2 = 2.3 GHz seems very problematic (see Figure 5.17b). Both IF frequencies are accompanied by evenly spaced components that are actually downconverted injection pulling components of the LO. Moreover, these two groups of unwanted frequency components overlap and make extraction of the useful IF component impossible.

A higher f2 gives rise to extra components that are spaced closer together, cluttering the IF spectrum even more. The resulting spectrum is depicted in Figure 5.17c. Entering the region of quasi-beat, two separate groups of tightly spaced injection pulling components are observed with the one resulting from the jamming signal being located at very low frequencies (shown in Figure 5.17d). Locking the signal clears the spectrum but, once again, shifts the IF frequency as seen in the detailed spectrum of Figure 5.17e. A third experiment is performed for another purpose. The oscillator was locked to a signal of magnitude 0 dBm at the RF port and a frequency of 2.35271 GHz. A second signal was then injected through the RF port as well and tests for injection pulling and locking on this second signal were performed for increasing power. However, raising the power of the second tone to 10 dBm still caused no visible locking or pulling, regardless of how close f2 was brought to the locked frequency! One can therefore conclude that the oscillator becomes very robust to injection pulling and locking if already locked on a ﬁrst tone, even if the original locking signal has lower power. This may be used to make the receiver circuit more robust and immune, albeit at the cost of a second oscillator or maybe even a PLL. MEASUREMENTS 64

10 10 − fIF,2 f2 fIF,2 f0 − 20 20 − − fIF,1 f1 30 30 fIF,1 − − 40 40 − − 50 50 − − 60 Magnitude [dBm] 60 Magnitude [dBm] − − 70 70 − − 80 80 − 0 0.5 1 1.5 2 2.5 3 − 0 50 100 150 200 Frequency [GHz] Frequency [MHz]

(a) f2 = 1.8 GHz. (b) f2 = 2.3 GHz.

10 10 f − fIF,2 − IF,2 20 20 − − 30 fIF,1 30 fIF,1 − − 40 40 − − 50 50 − − 60 60 Magnitude [dBm] Magnitude [dBm] − − 70 70 − − 80 80 − 0 50 100 150 200 − 0 50 100 150 200 Frequency [MHz] Frequency [MHz]

30 − f 40 IF,1 − 50 − 60 − 70 − 80 − Magnitude [dBm] 90 − 100 − 110 − 117 117.5 118 118.5 119 119.5 Frequency [MHz]

(e) f2 = 2.3312 GHz.

Figure 5.17: IF spectra for various f2 and P2 = +10 dBm. CONCLUSIONS AND FUTURE WORK 65

Chapter 6

Conclusions and Future Work

The goal of this master’s dissertation was to design an RF front-end of a low-cost receiver that operates in the 2.45 GHz ISM band and to analyse its EMC behaviour. A typical characteristic of low-cost receivers, viz. the omission of the RF preselect filter, was to be evaluated. The resulting susceptibility of the receiver circuit to non-linear phenomena that could jeopardise the proper functioning of the receiver were to be inspected. The specific effects that were to be studied and characterised were injection locking and pulling. The local oscillator, a critical building block of the RF front-end, was designed and implemented as a negative resistance oscillator. The final design, operating at a power supply voltage of 3.3 V, produces an output signal at a fundamental frequency of 2.344 GHz with a power of 9 dBm for the first harmonic. An impressive suppression of 13.5 dB and 17.5 dB for the first and second overtone, respectively, is achieved. The mixer used to down-convert RF signals to the IF frequency, is a Mini Circuits ADE-35+. Injection locking and pulling were theoretically studied, resulting in a better understanding of the nature and (potentially adverse) effects of these phenomena. Simulations on the designed oscillator predicted the occurrence of injection locking and pulling and confirmed the typical spectral properties of the various stages. Measurements of the manufactured oscillator showed that both injection locking and pulling are induced in the real circuit as well. Moreover, it was observed that the span in which injection pulling occurs is substantially larger than the one predicted by the theoretical derivation or the simulations. This is caused by the more realistic loading conditions that were present during measurements, but that are absent during simulations and extremely hard to take into account from a theoretical viewpoint. The influence on the circuit’s immunity of different points of injection was evaluated as well. Although the lock range for injection via the power supply connection is considerably smaller than injection via the output, its effects are found not to be negligible. Measurements on the total front-end (mixer and oscillator) showed that injection locking and pulling are not avoided by the isolation of the mixer. Although the lock range is obviously smaller in normal operation than for the oscillator on its own, the presence of a co-located source still has the potential to ruin the operation of the receiver by shifting the IF frequency or by introducing a time-dependency. The stronger the co-located source, the larger its influence and the bigger the frequency range in which it can cause havoc. CONCLUSIONS AND FUTURE WORK 66

In summary, from this work, it is concluded that both injection locking and pulling can have detrimental eﬀects on the EMC-performance of low-cost receivers. To the best of the author’s knowledge, these observations and investigations have so far not been reported in literature within this context. Although predicted by theory and simulations, the full extent of potential EMC problems becomes clear after prototyping. Any RF front-end designed should be aware of the described non-linear phenomena and their consequences. Therefore, it is considered opportune to turn this newly acquired knowledge into EMC-design guidelines that are to be applied and integrated in the design cycle of novel electronic products.

It is now also clear that further investigations concerning EMC-effects of injection locking and pulling are of critical importance for state-of-the-art RF circuit design. Therefore, some suggestions for future work are presented below. In the first place, the circuit itself could be improved in various ways. The existing oscillator can be tuned to oscillate at 2.38 GHz, the precise intended LO frequency. The exact cause of the current operation (omitting a certain capacitor) should also be verified by means of simulation in order to evaluate the cause of its correct functioning. Other oscillator topologies are interesting possibilities as well. The use of a quartz crystal oscillator or a VCO embedded in a PLL would most likely increase the immunity of the oscillator substantially but the trade-off with the low- cost requirement should be taken into account at all times. Designing a novel mixer, opens the possibility to co-design which could tackle the problem of the changing load. One could, for example, opt for a passive self-oscillating mixer (SOM) to further reduce the footprint and cost, if this proves to be feasible with commercially available components. Injection locking and pulling themselves could be further explored as well. Including amplitude modulation into the calculations explains extra frequency components that are observed during injection pulling, especially for large injected signals. More developed theory should also be able to explain other large signal effects better, including a more precise expression for the lock range. In terms of measurements and simulations, room for improvement is present as well. Time domain measurements could provide additional insights into the various stages of injection pulling and provide an easier way to compare transient simulation results with measurements, resulting in drastically decreased simulation times. A complete functioning front-end, also encompassing an antenna, LNA and IF filter, makes additional interesting experiments possible. Instead of injecting signals into the system via cables, both the useful signal and the co-located sources could be transmitted to the circuit via EM radiation, mimicking the real application of the receiver front-end and enabling evaluation of the system’s performance in another EMC context. Besides being an ideal platform for further EMC evaluation, such a complete system may also immediately constitute a fully functional prototype for further development of low-cost and wearable systems. CALCULATIONS 67

Appendix A

Calculations

A.1 Injection Locking Calculations

A.1.1 Derivation of (2.3)

Starting with a rewritten form of (2.1):

1 cos2 θ sin φ0 = − (A.1) v I2 I u osc + 1 + 2 osc cos θ u 2 u Iinj Iinj u t the derivative with respect to the cosine of θ yields:

1 I I 2 osc + inj + 2 cos θ d sin φ0 1 Iosc Iinj Iosc =  2  d cos θ 2 Iinj 1 cos θ  −     I2 I  I osc + 1 + 2 osc cos θ ( 2 cos θ) 1 cos2 θ 2 osc I2 I − − − I inj inj ! inj 2 . (A.2) · 2 Iosc Iosc 2 + 1 + 2 cos θ Iinj Iinj ! To ﬁnd the maximum, this derivative has to be zero. The ﬁrst term in (A.2) does not vanish so possible solutions have to be found in the second term. The numerator of this term is a quadratic equation in cos θ:

2 2 2 2IoscIinj cos θ + 2 Iosc + Iinj cos θ + 2IoscIinj = 0 (A.3) The two possible roots of this equation are:

Iinj Iosc cos θ1 = , cos θ2 = (A.4) −Iosc −Iinj

The second solution, however, results in sin φ0 = 1 which means the tank should provide a phase shift of 90◦. An RLC tank can only provide this shift for ω so the second solution does → ∞ Iinj Iinj not correspond to a real situation. The only solution is thus sin φ0,max = for cos θ = . Iosc −Iosc CALCULATIONS 68

A.1.2 Derivation Phase Characteristic RLC Tank

The transfer function of an RLC tank is given by jωL Z(s) = . (A.5) jωL (1 ω2LC) + − R The phase characteristic thus equals:

2 ωL ωL π ωL ω0 ∠Z(ω) = arctan arctan = arctan . (A.6) 0 − ω2 2 − R · ω2 ω2 R 1 0 − − ω2 0 This expression can be rewritten using the following property of inverse trigonometric functions: π arctan x = arctan x−1 for positive x. Furthermore, one can say that R R = Q since 2 − ωL ≈ ω0L the interest lies in the phase behaviour close to ω0. Consequently, we get

2 2 ω0 ω ∠Z(ω) arctan Q −2 . (A.7) ≈ ω0

The nominator of the argument can be approximated via ω2 ω = ω2 [ω + (ω ω )]2 = 0 − 0 − 0 − 0 2(ω ω )ω (ω ω )2 2ω (ω ω). The approximate expression for the tangent of the − − 0 0 − − 0 ≈ 0 0 − phase eventually becomes: 2Q tan ∠Z(ω) (ω0 ω). (A.8) ≈ ω0 −

A.2 Injection Pulling Calculations

A.2.1 Derivation of Single Sinusoid Form of Ssum

The simple expression for Ssum is simply the sum of Sin and Sout:

Ssum = Ainj cos ωinjt + Aosc cos (ωinjt + θ) (A.9) = (A + A cos θ) cos (ω t) A sin θ sin (ω t). (A.10) inj osc inj − osc inj In order to achieve a single sinusoid form, the goal is to get an expression on which an inverse trigonometric angle sum identity can be used:

A sin θ S = (A + A cos θ) cos ω t osc sin ω t sum inj osc inj − A + A cos θ inj inj osc cos ψ sin ψ = (A + A cos θ) cos ω t sin ω t inj osc cos ψ inj − cos ψ inj A + A cos θ = inj osc cos (ω t + ψ) (A.11) cos ψ inj with A sin θ tan ψ = osc . (A.12) Ainj + Aosc cos θ CALCULATIONS 69

Assuming that A A , an approximation of cos ψ can be obtained: inj osc −1 cos ψ = 1 + tan2 ψ p −1 A sin θ 2 = 1 + osc  A + A cos θ  s inj osc   −1 2 2 = (Ainj + Aosc cos θ) Ainj + Aosc + 2AoscAinj cos θ q −1 (A + A cos θ) A2 + 2A A cos θ (A.13) ≈ inj osc osc osc inj q resulting in a simpliﬁed expression for Ssum:

S A2 + 2A A cos θ cos (ω t + ψ) (A.14) sum ≈ osc osc inj inj qA cos (ω t + ψ) (A.15) ≈ osc inj A.2.2 Approximation for dψ/dt

Applying arctan ( ) to (A.12), gives an expression for ψ as a function of θ. Taking into account · dψ that the derivative of arctan u equals 1/(1 + u2), the expression for is dt dψ 1 d tan ψ = dt A sin θ 2 · dt 1 + osc A + A cos θ inj osc 2 2 2 (Ainj + Aosc cos θ) Aosc cos θ(Ainj + Aosc cos θ) + Aosc cos θ dθ = 2 2 2 Aosc + Ainj + 2AoscAinj cos θ · (Ainj + Aosc sin θ) · dt 2 Aosc + AoscAinj cos θ dθ = 2 2 . (A.16) Aosc + 2AoscAinj cos θ + Ainj · dt

Assuming that Ainj Aosc, the nominator and denominator of the factor in the above expression dψ dθ can be approximated by the term in A2 , so eventually one gets that . osc dt ≈ dt

A.2.3 Approximation for tan (θ ψ) − An expression for tan (θ ψ) is easily found by using the trigonometric angle sum identities and − substituting tan ψ by formula (A.12): sin θ cos ψ sin ψ cos θ tan (θ ψ) = − (A.17) − sin θ sin ψ + cos θ cos ψ sin θ tan ψ cos θ = − sin θ tan ψ + cos θ sin θ(Ainj + Aosc cos θ) Aosc sin θ cos θ = 2 − sin θ + cos θ(Ainj + Aosc cos θ) A sin θ = inj (A.18) Aosc + Ainj cos θ A inj sin θ (A.19) ≈ Aosc CALCULATIONS 70

with the last approximation, again, made by assuming A A . inj osc A.2.4 Behaviour of θ in Injection Pulling

The differential equation (2.13) is a separable equation so can be solved by integrating the following equation dθ = dt. (A.20) ω ω ω sin θ 0 − inj − L In order to integrate this equation, a quadratic form will be introduced in the denominator by performing some trigonometric substitutions: θ θ sin θ = 2 sin cos 2 2 θ θ = 2 tan cos2 2 2 θ tan = 2 2 . (A.21) θ 1 + tan2 2 This transforms (A.20) to 2dθ θ 2 cos2 2 = dt. (A.22) θ θ ω ω + (ω ω ) tan2 2ω tan 0 − inj 0 − inj 2 − L 2 θ To ease the following steps, the substitution u = tan is performed here: 2 2du = dt (A.23) (ω ω ) + (ω ω )u2 2ω u 0 − inj 0 − inj − L The goal is now to transform the denominator to a purely quadratic form a2 + v2 since the primitive function of this expression is arctan ( ). This can be done by completing the square · 2(ω ω )du 0 − inj = dt (A.24) (ω ω )2 ω2 + ((ω ω )u ω )2 0 − inj − L 0 − inj − L and performing the substitution v = (ω ω )u ω . Defining ω as (ω ω )2 ω2 and 0 − inj − L b 0 − inj − L taking the integral of both sides result in q 2 v arctan = t (A.25) ω ω b b ignoring the integration constant since the exact phase is not as important as the overall time- θ dependent phase behaviour. Rearranging and substituting v for u and u for tan , finally gives 2 an expression for θ as a function of t: θ ω ω ω t tan = L + b tan b . (A.26) 2 ω ω ω ω 2 0 − inj 0 − inj CALCULATIONS 71

A.2.5 Expression for dθ/dt

Starting from (A.26), one ﬁnds the equation 1 β2 1 θ = 2 arctan + − tan (α) (A.27) "β p β # where β = (ω ω )/ω and 2α = ω t = ω β2 1t. Taking the derivative of this equation 0 − inj L b L − with respect to t results in p dθ ω (β2 1)β = L − (A.28) dt 2 cos2 α 1 + β2 1 tan α + β2 − p ω (β2 1)β = L − (A.29) cos2 α sin2 α + 2 β2 1 sin α cos α + β2 sin2 α + β2 cos2 α − − ω (β2 1) = L − p . (A.30) cos 2α β2 1 + − sin 2α + β β p β Deﬁning α as arctan β2 1 enables one to invoke the inverse trigonometric angle sum 0 − identity for the cosine, leadingp to dθ ω (β2 1) = L − . (A.31) dt β + cos (2α α ) − 0 A.2.6 Series Expansion of (A.31)

In order to express (A.31) as a sum of sinusoids, the complex exponential form of the cosine will be used as well as the properties of a geometric series. For this proof, it will be assumed that sgn β = 1 and since β = (ω ω )/ω this means that β 1 in the region of injection − 0 − inj L | | ≥ pulling. Defining r as β + β2 1 gives a number whose magnitude is always smaller than one − and with a negative sign (thep same sign as β). With these definitions, one finds that dθ ω (β2 1) = L − dt β + cos (2α α ) − 0 1 r2 = ω β2 1 − − L − 1 + r2 + 2r cos η p 1 1 = ω β2 1 1 rejη re−jθ − L − − 1 + rejη − 1 + re−jη p ∞ ∞ = ω β2 1 1 rejη ( 1)mrmejmη re−jθ ( 1)mrme−jmη − L − − − − − " m=0 m=0 # p ∞ X X = ω β2 1 1 + 2 ( 1)mrm cos n(2α α ) (A.32) − L − − − 0 " n=1 # p X where η was defined as 2α α to simplify notation. For sgn β = 1 and r = β β2 1 a − 0 − − completely analogous proof leads to p ∞ dθ = ω β2 1 1 + 2 ( 1)mrm cos n(2α α ) . (A.33) dt L − − − 0 " n=1 # p X CALCULATIONS 72

An expression valid for all signs of β is thus found to be

∞ dθ = sgn (β)ω β2 1 1 + 2 ( 1)nrn cos n(2α α ) (A.34) dt L − − − 0 " n=1 # p X with r = β sgn β β2 1. − − p A.2.7 Expression for cos θ

The derivation starts from the trigonometric identity: cos θ = 1 sin θ tan (θ/2) because an − expression for sin θ ((A.31) and (A.20)) and tan (θ/2) (A.26) has already been found.

1 β2 1 β2 1 cos θ = 1 + − tan α β − − β β − β + cos (2α α ) " p # − 0 β2 1 β2 1 + (β2 1) tan α tan α[β2 + cos(2α) + β2 1 sin (2α)] = − − − − − β · β + cos (2α α ) p p − 0 p β2 1 sin (2α) β2 1 cos (2α) = − − − −p β · β + cosp (2α α0) ∞ − = 2 β2 1 ( 1)nrn sin n(2α α ) (A.35) − − − 0 "n=1 # p X The last step requires a proof very similar to the one in Section A.2.6 and is thus not repeated here. BIBLIOGRAPHY 73

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List of Figures

1.1 Structure of a superheterodyne receiver...... 2 1.2 Structure of a low-cost RF front-end...... 3

2.3 Vector diagrams depicting the phase of the diﬀerent currents...... 7 2.4 Region of injection locking also known as Arnold tongue...... 8 2.5 Block diagram model of an oscillator under injection pulling...... 8

2.6 Diagram showing the average oscillation frequency as a function of ωinj...... 10 2.7 Spectrum for various values of ωinj (amplitude in log scale)...... 13 2.8 θ, the instantaneous frequency, the waveform and the spectrum (log scale) for

ωinj = 1.3GHz...... 14 2.9 θ, the instantaneous frequency, the waveform and the spectrum (log scale) for

ωinj = 1.49 GHz...... 16

3.1 Two classes of harmonic oscillators...... 18 3.2 Two examples of harmonic feedback oscillators...... 19 3.3 Three types of passive diode mixers...... 21 3.4 Marchand balun in microstrip technology...... 21 3.5 Schematic representation of an HB circuit...... 24 3.6 Solving equation (3.7)...... 25

4.1 Transmission line realisations...... 27 4.2 Common base circuit...... 29 4.3 S-parameters of the common base circuit with and without base coil...... 30 4.4 Common base circuit with base coil...... 31 4.5 Smith chart with unity stability circle (black), fixed Γ circle (grey) and Γ .. 32 | in| L 4.6 Initial oscillator design...... 34 4.7 Non-linear simulation results of the small signal design...... 35 4.8 Non-linear simulation results of the improved initial design...... 36 4.9 Layout of the first oscillator...... 36 4.10 Non-linear simulation results of the final first design...... 38 4.11 Measured spectral response of the first oscillator circuit...... 38 4.12 Real (black) and imaginary (grey) part of the input impedance at the emitter of the final first design...... 39 4.13 S-parameters of the common base circuit with and without base microstrip. . . . 41 LIST OF FIGURES 78

4.14 Circuit generating a negative resistance at the collector...... 42 4.15 Real (black) and imaginary (grey) part of the input impedance of the second design. 43 4.16 Output matching network...... 43 4.17 Final layout of the second oscillator design...... 45 4.18 Layout of the MBA-25L+ mixer...... 47

5.1 Branch-line hybrid implementation...... 49 5.2 Magnitude of the S-parameters of the branch-line hybrid...... 49 5.3 Spectrum of the ﬁnal oscillator design...... 50 5.4 Block diagram of the ADE-35+ set-up...... 50 5.5 Mixer measurement set-ups...... 51 5.6 Conversion loss measurements for both mixers...... 51 5.7 RF to LO isolation measurements for both mixers...... 52 5.8 Lock range measurement set-ups...... 54 5.9 Lock range measurements/simulations...... 55 5.10 Measured and simulated Arnold tongues...... 55 5.11 Lock range with injection in the power supply...... 56

5.12 Spectrum of injection pulled oscillator for various injected frequencies and Pinj = 20 dBm. Left column are measurements results and the right column simulation − results...... 58 5.13 Set-ups for evaluating the total system...... 59 5.14 Lock range of the total system...... 60 5.15 Normal operation of the total circuit with information signal at 2.45 GHz. . . . . 61 5.16 IF spectra for various f and P = 20 dBm...... 62 2 2 − 5.17 IF spectra for various f2 and P2 = +10 dBm...... 64 LIST OF TABLES 79

List of Tables

4.1 Component values of Figure 4.2...... 29 4.2 DC characteristics of the first oscillator design...... 30 4.3 Component values of Figure 4.4...... 31 4.4 Component values of Figure 4.6 ...... 34 4.5 Component values of the improved initial design...... 35 4.6 Component values of the final first design...... 37 4.7 DC values for the second design...... 41 4.8 Component values resulting in negative resistance...... 42 4.9 Component values of the second design...... 44 4.10 Component values of the functioning oscillator...... 46