Practical Issues of Using Negative Impedance Circuits As an Antenna Matching Element
Practical Issues of Using Negative Impedance Circuits as an Antenna Matching Element
by
Fu Tian Wong
B.E. (Electrical & Electronic, First Class Honours), The University of Adelaide, Australia, 2011
Thesis submitted for the degree of
Master of Engineering Science
In School of Electrical and Electronic Engineering The University of Adelaide, Australia
June 2011
Copyright © 2011 by Fu Tian Wong All Rights Reserved
Contents
Contents ...... i Abstract ...... iii Statement of Originality ...... v Acknowledgments ...... vii Thesis Conventions ...... ix List of Figures ...... xi List of Tables...... xiii 1 Introduction and Motivation ...... 2 1.1 Introduction ...... 2 1.1.1 Limitations of small antennas ...... 2 1.1.2 Other potential methods ...... 3 1.2 Motivation ...... 4 1.2.1 Broadband matching network ...... 4 1.3 Thesis Overview ...... 7 2 Negative Impedance Converters ...... 10 2.1 Introduction ...... 10 2.2 Linear analysis ...... 10 2.2.1 NIC Realisation ...... 11 2.2.2 Analysis with and without parasitic capacitance ...... 17 2.3 Matching Performance Using a Simple Negating Capacitor ...... 22 2.4 Three element negation network ...... 26 2.5 Chapter summary ...... 30 3 Stability concerns ...... 32 3.1 Introduction ...... 32 3.2 NIC characteristic ...... 32 3.3 Stability Analysis ...... 34 3.4 Circuit Modification ...... 42 3.5 Chapter Summary ...... 56 4 Effects of Non-idealities ...... 58 4.1 Transistor Mismatch and Hfe Variations ...... 58 4.2 Temperature Variations ...... 62 4.3 Supply Voltage Variations ...... 63 4.4 Sensitivity Analysis ...... 66 4.5 Chapter Summary ...... 70 5 Noise Considerations ...... 72 5.1 Introduction ...... 72 5.2 Internal noise ...... 72 5.3 External noise ...... 76 5.4 Chapter summary ...... 82 6 Non-linear Analysis ...... 86 6.1 Introduction ...... 86 6.2 Intermodulation Distortion and Numerical Modelling ...... 87 6.3 Broadcast Stations ...... 96 6.4 Theoretical or Expected IMD Noise Levels ...... 98 6.5 Comparison with External Noise...... 99 6.6 Chapter summary ...... 101 7 Conclusion ...... 104
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7.1 Results and Conclusions ...... 104 Appendix A ...... 109 Appendix B ...... 111 References ...... 113
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Abstract
In the design of antenna systems, it is well known that there are trade-offs between bandwidth and size. As the size of an antenna reduces, in proportion to wavelength, there is a reduction in bandwidth. Wavelength at HF is of the order of tens of meters and so practical HF antennas either have narrow bandwidth or are very large in size. This conclusion holds when passive matching circuits are used, but it is possible that active circuits could provide improved bandwidth. Negative Impedance Converters (NICs) are active circuits that provide a promising avenue for achieving a high bandwidth with electrically small HF antennas. This thesis focuses on tackling the practical issues of using NIC based matching networks for HF reception.
The work presented in this thesis contributes to the research on NICs as HF matching networks in several ways: (i) the interaction of the environment with the non-linearity in the NIC circuit; (ii) a comparison between the external and internal noise effects; and (iii) the stability of the NICs when operated as matching circuits at HF frequencies.
In this thesis, a brief introduction is presented to the previous work to reduce the size of antennas. This includes a short summary on the development of the NIC and its application as a matching network. The thesis then continues with a theoretical analysis of the NIC and its application as an antenna matching circuit.
The thesis also provides an investigation on various practical issues namely the stability of the circuit, device variations, noise and the effects of non-linearity. It was found that device variations, noise and non-linearity did not pose a serious problem. Stability, however, was found to be an important issue and that the NIC circuit had to be carefully loaded to maintain stability. This research is a contribution towards the use of NICs in HF receive systems and could help bring to fruition the dream of small sized HF antennas with high bandwidth. In particular, HF radios for domestic purposes could benefit from such a research outcome.
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Statement of Originality
I, Fu Tian Wong, certify that this work contains no material that has been accepted for the award of any other degree or diploma in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published written by another person, except where due reference has been made in the text.
I give consent to this copy of the thesis, when deposited in the University Library, being available for loan and photocopying, subject to the provisions of the Copyright Act 1968.
I also give permission for the digital version of my thesis to be made available on the web, via the University’s digital research repository, the Library catalogue, and also through web search engines, unless permission has been granted by the University to restrict access for a period of time.
______
Signed Date
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Page vi
Acknowledgments
I am privileged with the opportunity to undertake such a challenging but rewarding Masters Research. It is indeed a gratifying experience to see this research through from the beginning to the end. I extend my sincerest thanks to my family, friends, colleagues and supervisors for their constant support.
I sincerely thank my supervisor, A/Prof Chris Coleman, for his constant guidance and encouragement throughout my Masters candidature. I am immensely grateful for all the time that he has dedicated for discussions on my work. I would also like to deeply thank my research co-supervisor, Dr Said Al-Sarawi, for reviewing my research and providing fruitful insights during our discussions. Further, I would like acknowledge the staff of the School of Electrical and Electronic Engineering at the University of Adelaide, particularly Mr Danny Di Giacomo and Mr Pavel Simcik and the administrative team for their assistance.
For my friends who have seen my ups and downs, encouraged and prayed for me throughout this journey, thank you very much. I thank my family for their unconditional love, support and prayers. This research could not have been completed without them. I would like to thank God for providing me with wisdom and help from all the people listed above, and many others whom have been a great support in many ways.
Fu Tian Wong (June 2010)
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Thesis Conventions
Typesetting
This thesis is typeset using Microsoft Word 2007. The fonts used in this thesis are Times New Roman and Arial.
Referencing
The referencing and citation style adopted in this thesis are based on the Institute of Electrical and Electronics Engineers (IEEE) Transaction style. For electronic references, the last accessed date is shown at the end of a reference.
Units
The units used in this thesis are based on the International System of Units (SI units).
Prefixes
In this thesis, the commonly used numerical prefixes to the SI units are “p” (pico; 10-12), “n” (nano; 10-9), “µ” (micro; 10-6), “m” (milli; 10-3), “k” (kilo; 103), “M” (mega; 106) and “G” (giga; 109).
Spelling
The Australian English spelling is adopted throughout this thesis
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Page x
List of Figures
Figure 1 : Non-Foster circuit reactance cancellation ...... 5 Figure 2 : Time harmonic analysis of antenna system. Antenna represented by a series RC component and a voltage source...... 10 Figure 3 : (a) Basic NIC circuit as proposed by Sussman-Fort [20] (b) Bipolar Junction Transistor equivalent circuit...... 11 Figure 4 : Small signal analysis (excluding rπ) ...... 12 Figure 5 : Small signal analysis (including rπ) ...... 14 Figure 6 : Small signal analysis (Including parasitic capacitance) ...... 18 Figure 7 : A BJT implementation of the negative ‘capacitor’ ...... 24 Figure 8 : Simulation results of a 2 meter dipole by 4NEC2+ program ...... 25 Figure 9 : Equivalent capacitor representing the antenna's reactance (Using a first order approximation) ...... 25 Figure 10 : Equivalent resistor representing the antenna's resistor (Using a first order approximation) ...... 25 Figure 11 : Simple three element network to represent antenna's reactance across 3- 20MHz ...... 27 Figure 12 : The reactance of the three element network (blue curve) reasonably resembles the reactance of the antenna (red curve) ...... 29 Figure 13 : NIC’s two ports terminated by ZL1 and ZL2. One port will be Open Circuit Stable while the other will be Short Circuit Stable (adopted from [24])...... 33 Figure 14 : Circuit setup for Rollet Factor simulation ...... 35 Figure 15 : Rollet Factor simulation result ...... 35 Figure 16 : Reflection coefficient (S11) as seen from the receiver...... 36 Figure 17 : Circuit setup for antenna port reflection coefficient (S11) simulation ...... 37 Figure 18 : Reflection coefficient (S11) as seen from the antenna ...... 38 Figure 19 : Circuit setup for transistor base reflection coefficient (S11) simulation ..... 39 Figure 20 : Reflection coefficient (S11) as seen the receiver side transistor’s base and the antenna side transistor base respectively ...... 40 Figure 21 :(a) Transient Analysis and (b) Fourier Series of the voltage at the receiver (labelled Vout in the circuit) ...... 41 Figure 22 : Modified NIC circuit ...... 43 Figure 23 : Reflection coefficient (S11) as seen from the receiver (a) and antenna (b) respectively ...... 44 Figure 24 : Reflection coefficient (S11) as seen the receiver side transistor’s base (a) and the antenna side transistor base (b) respectively ...... 45 Figure 25 : Transient Analysis (a) and its Fourier Series (b) of the voltage at the receiver ...... 46 Figure 26 : Transient Analysis circuit setup ...... 47 Figure 27 : Transient Analysis (a) and its Fourier Series (b) of the voltage at the receiver ...... 48 Figure 28 : Imaginary part of the input impedance of the NIC with 1:1 transformer .... 49 Figure 29 : Reactance curve of an ideal negative 27pF capacitor...... 50 Figure 30 : Imaginary part of the input impedance of the NIC with a transformer as seen from the receiver ...... 50 Figure 31 : Real part of the input impedance of the NIC as seen from the receiver ...... 51 Figure 32 : Circuit setup for stability sensitivity due to receiver load ...... 52
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Figure 33 : Reflection coefficient as seen from antenna with varying receiver load (Rx) ...... 53 Figure 34 : Reflection coefficient as seen from the base of the antenna-side transistor with varying receiver load (Rx) ...... 53 Figure 35 : Source Follower Circuit Design (note: Vbuff = 7.0 V) ...... 54 Figure 36 : Real part of the input impedance of the overall NIC circuit ...... 55 Figure 37 : (a) Reactance (without buffer circuit) and (b) stability, when both transistor betas are varied ...... 59 Figure 38 : (a) Reactance (without buffer circuit) and (b) stability, when only the antenna side transistor is varied ...... 60 Figure 39 : (a) Reactance (without buffer circuit) and (b) stability, when only the receiver side transistor is varied ...... 61 Figure 40 : (a) Reactance (without buffer circuit) and (b) stability when temperature is varied for military applications ...... 63 Figure 41 : (a) Reactance (without buffer circuit) and (b) stability when the voltage source at the collectors are varied ...... 64 Figure 42 : (a) Reactance (without buffer circuit) and (b) stability when the voltage source at the bases are varied ...... 65 Figure 43 : Sensitivity Analysis for the NIC circuit ...... 67 Figure 44 : Reactance (without buffer circuit) sensitivity analysis across 0.1 to 40 MHz ...... 67 Figure 45 : Reactance sensitivity analysis across (a) 3-10 MHz , (b) 10-12 MHz and (c) 12-40 MHz respectively ...... 68 Figure 46 : Sensitivity of the reflection coefficient as seen from antenna across 0.1 to 40 MHz...... 69 Figure 47 : ADS circuit to analyse internal noise ...... 74 Figure 48 : Noise Figure of the NIC circuit ...... 75 Figure 49 : Minimum Noise Figure of the NIC circuit ...... 75 Figure 50 : Noise voltage at the output of the NIC circuit due to internal noise ...... 76 Figure 51 : Median values for man-made noise power (adopted from [41]) ...... 78 Figure 52 : Noise voltage at the receiver end of the NIC due to environmental noise in rural areas ...... 81 Figure 53 : Noise voltage at the receiver end of the NIC due to environmental noise in the cities ...... 82 Figure 54 : Noise voltage at the receiver end of the NIC circuit due to environmental noise and internal noise...... 82 Figure 55 : NIC two tone harmonic balance analysis ...... 88 Figure 56 : Vout spectrum arising from input frequencies of 15.17 MHz and 15.72 MHz ...... 89 Figure 57 : Surface plots for MATLAB’s least squares quadratic fit to the coefficients 2 0 of (a)α1 and (b) α1 (as described in Equation 6.3)...... 92 Figure 58 : Surface plots for MATLAB’s least squares quadratic fit to the coefficients of (a) 2w2 (i.e. a4) and (b) w2-w1 (i.e. a6) (as described in Equation 6.1)...... 93 Figure 59 : Surface plots for MATLAB’s least squares quadratic fit to the coefficients of (a) w2+w1 (i.e. a5) and (b) 3w1 (i.e. a7) (as described in Equation 6.1)...... 94 Figure 60 : Surface plots for MATLAB’s least squares quadratic fit to the coefficients of (a) 2w1+w2 (i.e. a9) and (b) 2w1-w2 (i.e. a10) (as described in Equation 6.1)...... 95
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List of Tables
Table 1: Input Impedance (Real and Imaginary) and the Voltage Standing Wave Ratio (VSWR) of the NIC output as obtained using ADS ...... 26 Table 2 : Antenna reactance as represented by an equivalent capacitor ...... 27 Table 3 : Reactance of three element network ...... 28 Table 4 : VSWR and impedance characteristics of a three element negated NIC circuit ...... 29 Table 5: Signal level at receiver given a fixed input voltage level ...... 55 Table 6: Temperature durability range according to the context of application ...... 62 Table 7: Man Made Noise according to location. A noise figure of 39.5 dB (bolded) was used for a electric field calculation in Equation 5.7...... 79 Table 8: Least squares quadratic fit for first 6 coefficients (c.f. Equation 6.1; The relationship between coefficients an and bn is described by Equation 6.2) ...... 90 Table 9: Least squares quadratic fit for next 6 coefficients (c.f. Equation 6.1; The relationship between coefficients an and bn is described by Equation 6.2) ...... 91 Table 10: MRF949’s IMD modelled and simulated performance when the 15.17MHz signal at 0.00652V interacts with the 15.72MHz signal at 0.00255V)...... 91 Table 11: Typical shortwave broadcast stations’ signals, as received in Adelaide, Australia...... 96 Table 12: Highest 2w1-w2 or 2w2-w1 components due to the different two tone combinations. (The highest value was bolded) ...... 99 Table 13: Comparison between IMD levels with environmental noise in rural areas and cities...... 100 Table 14: Transistor IMD performance comparison. The 2w1-w2 component represents the most significant IMD problem (as bolded)...... 100 Table 15: MRF949 Die Gummel Poon Parameters ...... 111
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Chapter 1
Introduction and Motivation
HIS chapter presents some brief information on the efforts on minimizing the size of antennas, the limitations in achieving this and the emergent Negative Impedance Converters (NICs) that offer promising results to achieve the previously defined limits on its bandwidth and some claim that NICs have the potential to exceed those limits. The chapter also provides a summary of the organisation of this thesis and its contributions.
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Introduction and Motivation
1 Introduction and Motivation
With the advance of science and technology, electronic components have been miniaturised over the years. However, the size of antennas remains limited by the fact that a traditional design needs a size in an order of a quarter to half a wavelength, thus giving a limit in minimum size. This limitation is most evident in antennas operating on a relatively low frequency band. High Frequency (HF) communications are in the frequency range 3-30 MHz which, contrary to their name, are relatively low frequencies compared with frequencies used for present day communication systems. Such frequencies, however, are still much in use. The size of an antenna is related to its wavelength of operation and thus HF antennas have sizes in the range from 10-100 meters. Therefore, the size of a HF antenna is physically large. This causes spatial inefficiency and would be a hindrance for further development of HF antennas, especially for broadcast reception. For example, spatial inefficiency could be an obstacle for the implementation of Multiple Input and Multiple Output (MIMO) HF antennas. Consequently, this obstacle highlights the need to find methods to reduce the size of HF antennas.
1.1 Introduction This chapter introduces some of the approaches to improve small antenna performance and discusses the motivation and practical issues in utilizing a class of network circuits called non-Foster circuits to achieve this. A non-foster network is a network which contains non-Foster elements. These elements have an imaginary immitance at all real frequencies and the derivative of their reactance is zero or negative [1]. This is elaborated further in section 1.2.1. The structure of this thesis and a concise summary of the novel contributions represented by this work are given.
1.1.1 Limitations of small antennas Wheeler [2] and Chu [3] stated a fundamental limitation on the bandwidth and efficiency (Q-limit) of small antennas. This is given by 11 Q += (1.1) kaka )( 3 where 2π k = (in radians/meter) (1.2) λ
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Chapter 1
and a is defined as the radius of sphere enclosing the maximum dimension of the antenna (in meters). λ is the wavelength of the electromagnetic wave.
This limit implies that for a 2 metre monopole antenna at 10 MHz, the maximum Q is 113.6 and the maximum bandwidth is approximately 88 kHz. This is not sufficient even for a single side band signal and thus the circuit is used with a matching network. Matching networks are said to be able to increase the (half-power) bandwidth by a factor of 3.2 [4]. (It is to be noted that for a real antenna, additional loss is incurred and thus the bandwidth could be increased above the factor of 3.2) Therefore, ever since the formulation of the limitations on bandwidth and efficiency (Q-limit) of small antennas, antenna engineers and scientists have worked on overcoming these limitations. A myriad of methods have been tried over the past few decades to achieve a Q value which is close to the Q-limit. In general, these methods can be categorised into two major categories which are wire engineering and material loading. Firstly, wire engineering involves rearranging the wires, and the structure, of an antenna in order to reduce its length, yet maximise its efficiency and bandwidth. Some examples of wire engineering are folded dipoles [5] and multiple folded arm spherical helix [6], amongst others [7]. The purpose of wire engineering is to reduce the length of the antenna. The second category of methods is material loading which involves adding passive reactive loadings or active devices to help achieve self-resonance for the antenna (so that conjugate matching is not required), improve bandwidth and to control the radiation pattern of the antenna [7]. There is a trade-off between bandwidth and efficiency and one quality is realised at the expense of the other. However, the Q-limit (Q = Quality factor) has not yet been reached [4]. Therefore, further work needs to be done to achieve a closer Q value to the Q-limit.
1.1.2 Other potential methods The attempts in wire engineering and material loading have not been able to bring to fruition an antenna anything close to the Q-limit. To achieve this limit, various innovative ideas have been implemented, such as an antenna in a Negative Index Metamaterial (NIM) [8]. This approach, however, turned out to be flawed as described
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Introduction and Motivation by Hansen [4]. Nevertheless, with the advent of non-Foster circuits, a significant improvement seems possible [4].
These non-Foster circuits give the possibility of circumventing the problem of reactance cancellation for only a single frequency (the case when foster circuits are used [9]). Mayes and Poggio [10] applied non-Foster circuits to antennas, whereby multiple- loading was used with a log-periodic distribution of the active element impedances, hence improving their previous work in which passive periodic loading was used [11].
1.2 Motivation The desire for small antennas, hence making them easily portable, has long been an ideal for antenna engineers. This is due to ecological, aesthetical, and economical reasons. Extensive work had been done in this area, which has significantly reduced the size of some classes of antennas. Nevertheless, size reduction has continued to be a problem for High Frequency (HF) antennas. The Q-limit remains untouched, therefore, it is realistic to expect improvements on the previous work. New dimensions are now open with the advent of non-Foster circuits and efficient broadband matching circuits, are a possibility.
1.2.1 Broadband matching network More often than not, the impedance of the antenna is not matched to the transmission line. Therefore, a matching network is required. Wheeler [2] rightly points out the fact that the associated circuits of the antenna system have a significant effect on the overall bandwidth of operation. Thus, it is vital to have a matching network which is broadband. The topic of broadband matching is not a new one, and many engineers and scientist have attempted for years to improve it. Fano [12] derived the fundamental limitations of a matching circuit, and since then a plethora of techniques and algorithms have been applied and can be found in various textbooks [13], [14]. A notable journal article is by Dedieu [15] where the author describes how the prior art of designing and optimizing a broadband matching circuit has evolved through the years. He later proposed a novel method called ‘Recursive Stochastic Equalization,’ whereby this stratagem circumvents the need to guess the initial equaliser parameter and reduces the computational time. This however, has its drawback, as it requires an equaliser topology to be imposed. Further improvements to overcome this were suggested by Rodríguez et
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Chapter 1 al. [16], their approach bypassing the need of prior knowledge of each network element, and only requiring a definition for the generalised topology of the network. This approach, which includes the utilization of Genetic Algorithms (GAs), also reduces the computational time of the optimization process and may account for the non-idealities of the network elements. Besides this, the prospect of using non-Foster circuits to design broadband matching circuits is promising, as patented by Skahill et al [9]. Armed with a barrage of techniques which have been tried through the years, a potential future project would be to combine some of these methods using non-Foster circuits.
The emphasis of this thesis is on investigating a non-Foster matching network that utilises a Negative Impedance Converter (NIC) and, in particular, its application at HF frequencies (3 – 30 MHz). With NICs, a significantly more broadband matching for small antennas is possible. NICs ‘creates’ negative inductors and negative capacitors which then are able to cancel the reactance of an antenna over a wider range of frequencies, and consequently extend the bandwidth of broadband matching. This is illustrated in Figure 1 where Foster reactance curve describes elements, such as capacitors and inductors, which has a positive reactance slope; as compared with the non-Foster reactance curve produced by NICs.
Figure 1 : Non-Foster circuit reactance cancellation
The concept of Negative Impedance Converters was credited to Marius Latour for his work on negative regenerative systems, though not much actual use was developed at that time [17]. Then in the early 1930s, NICs were designed using vacuum tubes to be
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Introduction and Motivation used in long-line telephony repeaters to provide negative resistance to offset line losses and amplification of the signal. With the advent of transistors, they quickly replaced vacuum tubes in NIC circuits. Linvill [18] designed, built and tested the first transistorised NIC in 1953. From that time, there was appreciable effort to improve and utilise the NIC. Applications include increasing the Q of filters, using the NICs as negative elements at the arms of the antenna, and as a matching network. Of particular relevance to this research is the work done by Harris and Myers [19], as they were the first to apply NICs to match electrically small antennas according to Sussman-Fort [20]. Harris and Myers successfully designed and built a shunt negative capacitor. Op Amps were then used by Perry [17] to realise a negative capacitance for antenna matching and this was used as a series element. Hansen [21] pointed out that, although Op Amps are easier to obtain, it has significant disadvantages due to parasitics, noise, poor efficiency and dynamic range. More information regarding the history of NICs, applied as a matching network, can be found in [20] and [21]. Furthermore, Sussman-Fort [22] provided a catalogue of ten different transistorised NICs, that were designed and published by a number of authors, among which only Linvill [18] and Yanagisawa’s NIC [23] have been built and tested according to Sussman-Fort. Among the NIC circuits published, the paper published in 2006 by Sussman-Fort is particularly relevant. In that research, Sussman-Fort [24] implemented a NIC circuit and managed to produce a broadband, stable, high Q, grounded negative capacitance. These ‘negative’ capacitors and ‘negative’ inductors are non-Foster circuits, i.e. circuits which have a negative dX/df characteristic and traverse the Smith Chart in an anti-clockwise manner. This is the opposite of passive circuits and antennas themselves, thus non- Foster matching is able to cancel the reactance of an antenna over a wider range of frequency. Aberle et. al. [25] followed on that work and simulated the NIC circuit with ADS using a s-parameter model for the NE85630 BJT. His simulation shows that the circuit was stable and achieves broadband matching. However, a perfect bias was used and in [26] he showed that the circuit is not unconditionally stable at frequencies below 31 MHz. The stability of the NIC is a significant issue in its practical implementation and will be discussed thoroughly in this thesis.
Non-Foster circuits involve the usage of active devices like BJTs and MOSFETs. These active devices could contribute significantly to the noise level of the antenna system.
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Chapter 1
Further, a broadband antenna will accept many signals over a wide range of frequency and the inherent nonlinearity of the active devices will cause intermodulation products that could act as artificial noise and further degrade the performance of the antenna. Besides Bahr [27], Krantz and Branner [28], no one has done a thorough noise analysis on Negative Impedance Converters in the application as broadband matching networks for antennas. Bahr concluded in his paper that the NIC contributes an undesirably large amount of noise. However, he did indicate that he used a ‘simplified design approach’, thus there is space for further development via computer aided techniques. Furthermore, developments in transistors have led to a reduction of their noise figure. Krantz and Branner [28] have done a comparative internal noise analysis between various classes of FET filter implementations. These two papers however were not targeted towards NIC antenna matching circuits and do not cover the HF frequency range.
If NICs are to be widely used in matching circuits, it is crucial that a study on the impact of noise and nonlinearity upon a NIC circuit be undertaken to determine the feasibility of NICs as a matching network for small antennas. In particular, it is necessary to determine whether the additional noise caused by the NIC is below that of the natural environmental noise (i.e. externally noise limited system). Non-ideal Negative Impedance Converters are only conditionally stable and sensitive to their loading. Therefore this thesis aims to verify the practicality of NIC implementation of matching networks with regards to noise, non-linearity, stability and other practical considerations. It is also interesting to see how the improvements in transistor technology could contribute to the effectiveness of a non-Foster impedance matching network for small antennas.
1.3 Thesis Overview This research seeks to explore the feasibility of using Negative Impedance Converters as a matching network for High Frequency antenna systems. This chapter has provided a brief literature overview of the work done in order to reduce the size of antennas. From the myriad of strategies attempted in the past, non-Foster matching or NIC matching was chosen as it is an exciting area for improving the performance of small antennas. This chapter has also provided an overview of some of the caveats in pursuing this direction, many of which will be studied in detail in the following chapters.
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Introduction and Motivation
Chapter 2 introduces the mechanics of the negation process by using linear analysis. This analysis also provides a platform to understand the parameters that affect the NIC negation. A major issue in the practical implementation of an NIC is to keep the device stable. Therefore, some simulations and testing regarding the stability characteristics of NIC can be found in chapter 3. Then, the following chapter explores the effects of device variation upon the performance and stability of the NIC circuit.
Chapter 5 considers the internal and external noise of an NIC circuit to ascertain if the device is externally noise limited. As active devices are intrinsic in the functionality of NICs, non-linearity could pose a threat to the practicality of a NIC matching network. Consequently, a thorough non-linear analysis is detailed in chapter 6. A comparison between the different contributors of noise for the NIC circuit can then be made. Finally, chapter 7 concludes the thesis by bringing together the various factors considered in this research and discussing the feasibility of NICs as a matching network for HF antenna systems. Future work is suggested to advance this technology.
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Chapter 2
Chapter 2
Negative Impedance Converters
HIS chapter provides a further introduction to the concept of Negative Impedance Converters. This chapter includes a linear analysis to understand the negating behaviour of an NIC circuit. Finally, the chapter shows how the utilisation of an NIC matching network could improve the bandwidth of an antenna system, and thus allow a greater usage of small antennas.
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Negative Impedance Converters
2 Negative Impedance Converters
2.1 Introduction A negative impedance converter (NIC), as the name suggests, is a device which negates a general impedance. This is done by inverting the polarity of the voltage across its input and output terminals (VNIC) or inverting the current flow into and out of the device (INIC). NICs are active devices, i.e. they need an external power supply. Such devices do not follow Foster’s reactance theorem which states that the reactance monotonically increases with frequency for passive devices. For NICs, the change of reactance with respect to frequency can be negative. Therefore, in the design of a matching network, NICs could greatly increase the frequency range of reactance cancellation and thus they could be used to produce a higher bandwidth in an antenna system. This could release the system from being bounded by Chu-Wheeler’s [2, 3] limit on the bandwidth of small antennas. We will investigate the NIC applied to matching a non-ideal antenna to a receiver.
2.2 Linear analysis
Firstly, a simple linear analysis on the circuit involving the NIC will be considered. This linear analysis can be used as a basis for a non-linear analysis by using the perturbation method (assuming that the NIC is weakly non-linear). Firstly, consider the simplified model of Figure 2.
Figure 2 : Time harmonic analysis of antenna system. Antenna represented by a series RC component and a voltage source.
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Chapter 2
We consider a time harmonic analysis of the above circuit, where s = jω . By applying Kirchhoff Voltage Law, and assuming that the NIC negates a general impedance of Z perfectly, we get 1 S iV (RA iRZi X =−−−+− 0)() (2.1) sC A then, rearranging to obtain the current through the device
V (2.2) i = S 1 RA + RZ −+ X sC A From Ohm’s law,
V R (2.3) iRV == XS 1 X 1 RA +−+ RZ X sC A
By choosing Z to be the capacitor CA, we obtain the maximum voltage possible at the receiver. We have essentially removed the reactance of the antenna (represented by CA) that can cause a massive reduction in receiver voltage at low frequencies.
2.2.1 NIC Realisation
NOTE: (a) This figure is included on page 11 of the print copy of the thesis held in the University of Adelaide Library.
(b)
Figure 3 : (a) Basic NIC circuit as proposed by Sussman-Fort [20] (b) Bipolar Junction Transistor equivalent circuit.
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Negative Impedance Converters
In order to understand the factors that affect a NIC’s negation ability, a small signal analysis was performed on a BJT implementation of a NIC circuit (as shown in Figure 3a). The transistor can be modelled as shown in Figure 3b. This circuit can be simplified by representing the BJTs by current sources alone. This assumption means that the parameter rπ is infinite and Cπ = Cµ=0.
2.2.1.1 rπ is assumed to be infinite and Cπ = Cµ=0
With infinite rπ, the following will represent the circuit of Figure 3a.
Figure 4 : Small signal analysis (excluding rπ)
Each current source is dependent on the voltage drop across its input resistance. Thus, (2.4) 1 ∆= Vfi 1)( where (2.5) −=∆ VvV 121 and (2.6) 2 ∆== Vfii 2 )( where (2.7) −=∆ VvV 212
The voltage drop across the general impedance Z is given by (2.8) − 21 = iZvv
By using Kirchhoff Current Law, we know that current is conserved, thus (2.9) 2 VfVf 1 =∆+∆ 0)()(
The voltage dependent current source (the transistor) can be represented by
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Chapter 2
2 3 (2.10) )( 1 2 3 vgvgvgVf +∆+∆+∆=∆ ...
However, by assuming only weak non-linearity about the bias point, we can use the linear transistor model of (2.11) ∆ )( = 1∆vgVf
The current through the circuit, i.e. (2.6), can be expressed as a sum of two components as given by 1 1 (2.12) )( ∆+∆= VfVfi )( , 2 2 2 2 then by substituting (2.9) into (2.12), we obtain 1 1 (2.13) )( ∆−∆= VfVfi )( , 2 2 2 1 and by substituting (2.5), (2.7) and (2.11) into this equation 1 1 (2.14) )( −−−= VvgVvgi )( 2 211 2 121 which implies 1 1 (2.15) )( −−−= VVgvvgi )( 2 211 2 211
Substituting this into (2.8) gives 1 1 (2.16) =− )( −− − VVZgvvZgvv )( 21 2 211 2 211 from which
1Zg (2.17) vv 21 =− −VV 21 )( . 2 − 1Zg
Assuming that 1Zg ∞→ , (2.18) 21 −−≈− VVvv 21 )( and substituting (2.18) into (2.8) (2.19)
−=− VVZi 21 )( Since the impedance of the NIC is given by
Voltage drop −VV 21 Z NIC = = Current i (2.20)
we have
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Negative Impedance Converters
− Zi ZNIC = −= Z i (2.21)
This shows that, by making the appropriate assumptions, the ideal impedance negation is achieved. The assumptions, however, are ideal and the consequence of real devices will be analysed in the following sections.
2.2.1.2 For the case rπ is not infinite
Figure 5 : Small signal analysis (including rπ)
In any real Bipolar Junction Transistor, the parameter rπ is finite and the small signal model of the transistors needs to include rπ, as shown in Figure 5.
By making the assumption, as in (2.11), we obtain (2.22) 1 = ∆ = −VvgVfi 1211 )()( and (2.23) 2 −=∆= VvgVfi 2112 )()(
A nodal analysis at the nodes labelled in Figure 5, yields at node B − vv (2.24) i = 21 − i b1 Z 2 On noting that
− Vv 12 ib1 = (2.25) rπ and using (2.23), we obtain
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Chapter 2
−Vv − vv 2112 = −− Vvg 211 )( (2.26) rπ Z from which (2.27) 11 1 V1 v2 ( () gv 11 ) +=−++ Vg 21 π Zr Z rπ At node A, − vv (2.28) i = 12 − i b2 Z 1 Then, noting that (2.29) −Vv 21 ib2 = rπ and by using (2.22), we obtain (2.30) − Vv − vv 1221 = −− Vvg 121 )( rπ Z from which (2.31) 11 1 V2 v1( () gv 12 ) +=−++ Vg 11 π Zr Z rπ
By subtracting (2.27) with (2.31), we get 111 1 (2.32) vv 12 )(( g1 VV 21 )(() −−=+−+− g1 ) π Zr Z rπ From this, we can derive a more realistic assessment of the NIC.
2.2.1.3 Finite g1 and rπ
When the parameter g1Z and rπ are finite, we obtain a far more complex behaviour for
ZNIC. By rearranging (2.32), we obtain 1 − g r 1 −−=− VVvv )()( π (2.33) 12 12 21 g1 +− rπ Z
By simplifying the expression to include k2 , we have (2.34) − 12 = − − VVkvv 122 )()( where 1 (2.35) − g r 1 k = π 2 21 g1 +− rπ Z
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Negative Impedance Converters
In order to obtain the input impedance expression, a nodal analysis is performed at nodes A and C
At Node C, (2.36) += iii b11 then, by substituting (2.22) and (2.25) into the equation, we obtain (2.37) −Vv 12 Vgi 11 )( +∆= rπ Further, at Node A (2.38) += iii b213 then, by substituting (2.22) and (2.29) into the equation, we have (2.39) −Vv 21 Vgi 113 )( +∆= rπ By subtracting (2.39) from (2.37), we obtain
− − + VVvv 2112 ii 3 =− (2.40) rπ
− vv 12 Since i = (2.41) 3 Z by substituting i3 into (2.40) yields
11 −VV 12 vvi 12 )(( ) ++−= (2.42) π Zr rπ
Using Equation (2.34) to substitute for − vv 12 )( yields
11 − VV 12 VVki 122 )(( ) ++−−= (2.43) π Zr rπ or
k k22 1 VVi 12 )(( +−−−= ) (2.44) rπ rZ π from which
− Zr Z = π NIC 1 (2.45) ( +− rZg ))( r 1 π π − Z 21 g1 +− rπ Z By applying a number of algebraic manipulations, the following expression is obtained
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Chapter 2
π 1 − − 2)1( rgrZ π Z NIC = (2.46) π gr 1 −− 1
Since current gain factor, β, is given by (2.47)
β = π gr 1 Equation (2.46) is then simplified to the following − β + 2)1( rZ ZZ == π NIC in 1+ β (2.48)
It is to be noted that ZNIC is also the input impedance of the device. From Equation (2.48), it can be seen that for a realistic device, the negation of impedance could be compromised due to device characteristics. Figures 28 and 29 in the next chapter show the difference between a realistic device and an ideal negative 27 pF capacitor. (It is to be noted that the circuit used to obtain the simulations in Figures 28 and 29 has been modified for the purposes of achieving stability). In establishing a realistic analytical model, besides setting a finite rπ and gm, the parasitic effects should be considered to understand its impact on the circuit.
2.2.2 Analysis with and without parasitic capacitance At higher frequencies, Bipolar Junction Transistors tend to have parasitic effects (capacitance in particular). In a BJT’s application as amplifiers, the parasitics reduces the gain at high frequencies. In NIC circuits, these parasitic effects could impact the accuracy of the reactance negation. Therefore, it is important to analyse its impact upon NIC’s performance. Among the parasitic effects, only two of the main parasitic capacitance will be considered, namely the capacitance across the base and emitter (Cπ) and the capacitance across the base and collector of the transistor (Cµ). A circuit which takes into account these parasitic capacitances is shown in Figure 6.
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Negative Impedance Converters
Figure 6 : Small signal analysis (Including parasitic capacitance)
By using the linear transistor model, we obtain (2.49) = ∆ = − VvgVgi 121114 )()( (2.50) −=∆= VvgVgi 211215 )()(
The current passing through the two capacitors Cµ , is given by
(2.51) c −= 211 )( sCvvi µ and (2.52) c −= 122 )( sCvvi µ
From Ohm’s law, − vv (2.53) i = 12 3 Z
Let the parallel impedance of rπ and Cπ be given by
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Chapter 2
/ sCr (2.54) Z = ππ RC 1 rπ + sCπ
Then, by assuming that both transistors are identical, Ohm’s law yields the following equations
(2.55) −Vv 21 ib2 = Z RC (2.56) −Vv 12 ib1 = Z RC
Nodal analysis at the nodes labelled in Figure 6, yields at node A (2.57) c iiii 3214 =−++ 0 Substituting (2.49), (2.51) and (2.53) into the equation gives
− vv 12 )()( isCvvVvg −+−+− = 0 (2.58) 21121 µ 2 Z from which
− vv 12 )()( sCvvVvgi +−+−−= (2.59) 121212 µ Z
A nodal analysis at node C gives (2.60) iii bc 222 =−+ 0
Then, by substituting (2.52) and (2.55) into the equation yields
−Vv 21 122 )( sCvvi µ −−+ = 0 Z RC (2.61) from which
−Vv 21 212 )( sCvvi µ +−= (2.62) Z RC
By equating (2.59) and (2.62),
− vv 12 −Vv 21 121212 )()( sCvvVvgi µ +−+−−= 21 )( sCvv µ +−= Z Z RC (2.63) from which
1 −Vv 21 − 12 2)(( sCvv µ Vvg 121 )() =−−+ Z Z RC (2.64)
At node B, we have (2.65) c iiii 1325 =+++ 0
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Negative Impedance Converters
Substituting (2.50), (2.52) and (2.53) into the equation gives
− vv 12 )()( isCvvVvg ++−+− = 0 (2.66) 12211 µ 1 Z from which 1 )(()( sCvvvVgi +−+−= ) (2.67) 211211 µ Z
Further, nodal analysis at node C gives (2.68) −= iii cb 111
Then, by substituting (2.51) and (2.56) into the equation yields
−Vv 12 i1 = −− 21 )( sCvv µ Z RC (2.69)
By equating (2.67) and (2.69), yields
1 −Vv 12 211211 )(()( sCvvvVgi µ ) =+−+−= −− 21 )( sCvv µ Z Z RC (2.70) from which
1 −Vv 12 − 21 2)(( sCvv µ vVg 121 )() =−++ Z Z RC (2.71) By subtracting (2.64) with (2.71), we obtain
1 − − +VvVv 1221 − 12 2)((2 sCvv µ vVgVvg 121121 )()() =−−−−+ Z Z RC (2.72)
By grouping the voltage terms together and some rearrangements, we have 1 − g Z 1 −−=− VVvv )()( RC 12 12 12 4sCµ g1 +−+ (2.73) Z Z RC This is simplified into (2.74) 12 −−=− VVkvv 124 )()(
1 − g Z 1 where k = RC 4 12 (2.75) 4sCµ g1 +−+ Z Z RC
A nodal analysis at node E gives (2.76) += iii b14
Substituting (2.49) and (2.56) into the equation gives
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Chapter 2
−Vv 12 Vvgi 121 )( +−= Z RC (2.77)
Further, at node F (2.78) −−= iii b25
Substituting (2.50) into the equation gives (2.79) −∆−= iVgi b221
Then by substituting (2.7) and (2.55) into the equation, we obtain
−Vv 21 Vvgi 211 )( −−−= Z RC (2.80)
By adding (2.77) and 2.75, we obtain
− + −VVvv 1212 (2 VVvvgi 12121 ) +−+−= Z RC (2.81) By applying Kirchhoff current law at node C and node D respectively, we obtain (2.82) −= iii cb 222 (2.83) −= iii bc 111
A nodal analysis at node A gives (2.84) c ++= iiii 2143 Then, by substituting (2.49) and (2.82) into the equation yields (2.85) )( −++∆= iiiVgi cbc 221113
Further, at node B (2.86) c −−−= iiii 1253 Then, by substituting (2.50) and (2.83), we obtain (2.87) −+−∆−= iiiVgi bcc 112213
Then, by adding (2.85) and (2.87), we have (2.88) (2 211213 22) −+−++−−= iiiiVvVvgi bbcc 1221
By subtracting (2.81) with (2.88) and by dividing it by half, we obtain
1 − + −VVvv 1212 − ii bb 12 ii 3 =− ii cc 21 +−+ 2 Z RC 2 (2.89)
By substituting (2.51), (2.52), (2.55) and (2.56) into the equation, we obtain
1 − + −VVvv 1212 − − +VvVv 2112 ii 3 =− 21 µ 12 )()( sCvvsCvv µ +−−−+ 2 Z RC 2Z RC (2.90) from which
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Negative Impedance Converters
− + −VVvv 1212 ii 3 =− −− 12 )(2 sCvv µ Z RC (2.91) Then, by substituting (2.53) into the equation 11 1 vvi 12 )(( µ −+−+−= VVsC 12 )(()2 ) RC ZZ Z RC (2.92)
Then by substituting (2.74) into the equation, the following is obtained.
1− k k44 −= VVi 12 )(( −− 4sCk µ )2 Z RC Z (2.93)
Then, by using (2.20), the input impedance is now given by
1− k k44 −1 Zin = ( −− 4 sCk µ )2 (2.94) Z RC Z where k4 and ZRC are given by Equations (2.75) and (2.54), respectively.
Equation (2.94) provides a more thorough description on the input impedance of the NIC, when negating a general impedance, as compared to Equation 2.48. However, the BJTs used in this thesis are MRF 949, which have a maximum parasitic capacitance of 200 fF, thus the effects of these parasitic capacitances are negligible at the HF frequency range. A simulation performed in ADS comparing two transistor models, with and without the parasitic capacitances, yielded a difference in reactance of less than 1.5% at 10 MHz. Thus Equation (2.48) is deemed to be sufficient in describing the NIC’s impedance negation properties.
In this section, the linear analysis performed could provide a better understanding on the non-ideal behaviour of NICs with regard to its negation ability upon a general impedance. Its performance in relation to the transistor parameters and (indirectly) its biasing conditions could be inferred. In the following section, we consider the performance of the NIC when used as a matching network for a HF antenna.
2.3 Matching Performance Using a Simple Negating Capacitor
It is the purpose of this research, to utilise a negative impedance converter circuit, to create a matching network to cancel the reactance of an electrically small antenna. The figure of merit in matching antennas is the Voltage Standing Wave Ratio (VSWR) of the antenna system. This measures the amount of reflection or loss in the system which is a result of how well the circuit is matched. An ideal antenna system’s VSWR is 1:1. However, for practical receive antenna systems, a VSWR of 4:1 or below is an
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Chapter 2
acceptable level for a matched circuit. This section, two programs namely Numerical Electromagnetic Code (NEC) and Agilent’s Advanced Design System (ADS 2009), were used to simulate the antenna system performance. NEC simulates the impedance characteristics of the antenna, while ADS utilises the antenna impedance result, from NEC, to simulate the NIC matching circuit.
Building on the work by Aberle et. al. [25], which used a form of Linvill’s NIC design [18], a circuit (shown in Figure 7) was implemented using a pair of MRF 949 transistors in order to cancel the reactance in an unloaded 2 metres monopole for a large range of frequencies. A 2 meter antenna exhibits capacitive behaviour (over the target frequency range from 3 to 20 MHz) that can be represented by a lumped capacitor in series with a resistor. The antenna’s impedance values (as shown in Figure 8) vary with frequency and these values were simulated using the NEC2 (Numerical Electromagnetics Code) antenna modelling program. The NEC model treated the antenna as a dipole and so calculated values of impedance that were halved in order to obtain those of the intended monopole. These values were then used to obtain equivalent series capacitance and resistance that represent the antenna (Figures 9 and 10 ). A value of 29 pF capacitor was chosen to represent the antenna (internal capacitance of the NIC provided about 2 pF and so the value in the NIC circuit was set at 27 pF). Table 1 shows the resultant VSWR, at the output of the NIC circuit, obtained over the frequency range of 3 – 20 MHz.
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Negative Impedance Converters
um=2 Term Term2 N Ohm Z=50 V_DC SRC18 Vdc=VDC 1 Re f S1P Antenna2 R R37 R=RcOhm top=470 VAR VAR48 Lmid=470 Cb=2 Rtop=6.5 Rbot=2.6 Lbot=220 L Rmid=6.5 C C36 uF C=0.1 Var Eqn =Ltop uH =Ltop =Lbot uH =Lbot L L32 L R=Rtop L L30 L R=Rbot R R39 R=ReOhm BJT_NPN BJT3 Model=MRF949 R R42 R=R2Ohm V_DC SRC20 V Vdc=Vb =Lmid uH =Lmid L L34 L R=Rmid R R41 R=R1Ohm C C34 C=CbnF C C33 pF C=27 N Zin VSWR VSWR2 VSWR1=vswr(S11) Zin Zin2 Zin1=zin(S11,PortZ1) VSWR C C35 C=CbnF V_DC SRC21 V Vdc=Vb =Lmid uH =Lmid L L33 L R=Rmid R R44 R=R1Ohm BJT_Model MRF949 R R43 R=R2Ohm BJT_NPN BJT4 Model=MRF949 =Lbot uH =Lbot L L31 L R=Rbot R R40 R=ReOhm =Ltop uH =Ltop L L29 L R=Rtop C C37 uF C=0.1 R R38 R=RcOhm S-PARAMETERS b=10 VAR VAR49 Rc=200 R1=20000 R2=150000 Re=1600 V VDC=10 top=20 MHz top=20 S_Param SP1 MHz Start=3 S MHz Step=1 Var Eqn um=1 Term Term1 N Ohm Z=50 V_DC SRC19 Vdc=VDC Vout
Figure 7 : A BJT implementation of the negative ‘capacitor’ (Note: A first order approximation of real inductors, given by a series resistor and parallel capacitor, was included in the simulation circuit as a data block labelled Antenna2)
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Chapter 2
Figure 8 : Simulation results of a 2 meter dipole by 4NEC2+ program (Note: the antenna model used in ADS is a monopole and thus its impedance is to be halved)
Figure 9 : Equivalent capacitor representing the antenna's reactance (Using a first order approximation)
Equivalent Radiation Resistance vs Freq
4.5 4 3.5 3 2.5 2
Resistor (R) Resistor 1.5 1 0.5 0 3 5 7 9 11 13 15 Frequency (MHz)
Figure 10 : Equivalent resistor representing the antenna's resistor (Using a first order approximation)
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Negative Impedance Converters
Table 1: Input Impedance (Real and Imaginary) and the Voltage Standing Wave Ratio (VSWR) of the NIC output as obtained using ADS
Freq Real(Zin1) Imag(Zin1) VSWR1 3.000 MHz 1905.612 628.871 42.265 4.000 MHz 483.579 -581.396 23.713 5.000 MHz 114.728 -258.183 14.281 6.000 MHz 47.129 -124.047 8.415 7.000 MHz 27.942 -61.823 4.879 8.000 MHz 21.256 -27.862 3.195 9.000 MHz 18.775 -6.898 2.722 10.00 MHz 17.929 7.370 2.858 11.00 MHz 17.754 17.906 3.222 12.00 MHz 17.840 26.225 3.657 13.00 MHz 18.001 33.215 4.121 14.00 MHz 18.167 39.422 4.610 15.00 MHz 18.335 45.205 5.128 16.00 MHz 18.547 50.780 5.671 17.00 MHz 18.869 56.293 6.225 18.00 MHz 19.378 61.805 6.762 19.00 MHz 20.143 67.333 7.249 20.00 MHz 21.208 72.840 7.655
The VSWR at the output of the NIC is low over a bandwidth of 8 MHz (from 7 to 14 MHz). Traditional methods using passive circuits (i.e. a traditional antenna tuner) would struggle to achieve such results across a similar bandwidth. This was achieved using a fixed capacitor in the NIC despite using a compromised value of 27 pF. The capacitance, from this single capacitor, does not model the reactance of the antenna over the entire frequency range and therefore the reactance cancellation would not be complete. However, by designing a network of elements, which is modelled upon the reactance of the antenna, there could be significant improvements in the reactance cancellation.
2.4 Three element negation network
The NIC in Figure 7 utilised the negation of a single capacitor of 27 pF. However, the reactance of an unloaded short antenna can only be represented well if the capacitance can vary with frequency. Negating the reactance of the antenna with a single capacitor has its limitation. This implies that there could be further improvements in the VSWR performance if we could generate frequency dependent capacitance.
By taking the reactance of the antenna at five spot frequencies, 3, 5, 10, 15 and 20 MHz, it can be represented by equivalent capacitors as shown in Table 2. A simple 3 element
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Chapter 2
LC network, as shown in Figure 11, was designed to match, as closely as feasible, to the reactance of the antenna at the chosen spot frequencies.
Table 2 : Antenna reactance as represented by an equivalent capacitor
Equivalent Frequency Reactance Resistance capacitor (MHz) (F) 3 -1903.4 0.042496 2.7872E-11 5 -1130.1 0.28351 2.8167E-11 10 -536.95 2.2551 2.9641E-11 15 -326.76 3.8279 3.2471E-11 20 -207.85 6.5957 3.8286E-11
L C L C1
C C2
Figure 11 : Simple three element network to represent antenna's reactance across 3-20MHz
The input impedance of the network is given by 1 1 in (sLZ += //) (2.95) 1 sCsC 2 from which sL 1 + (2.96) 2 CCssC Z = 2 21 in 11 sL ++ sCsC 21 Noting that s = jw (2.97) and substituting (2.97) into (2.96) yields
L 1 − (2.98) 2 CCwC Z = 2 21 in j j jwL −− wC1 wC2
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Negative Impedance Converters
By grouping the imaginary terms together, we obtain L 1 − (2.99) 2 CCwC = jZ 2 21 in 11 −+ wL 1 wCwC 2
Equation (2.99) describes the impedance of the LC network. A MATLAB program was then written, utilizing the imaginary part of the input impedance equation, to optimise a choice of L, C1 and C2 in order to match the reactance of the frequency dependent capacitor. The optimised values are: L= 3.5198 µH, C1=12.186 pF, C2=12.473 pF. Table 3 shows the resultant reactance values across the desired frequency range. By comparing Table 3 and Table 2, we see that there is a reasonable match across this frequency range for this simple network. Figure 12 illustrates this match with a plot which compares the antenna reactance and the three element network.
Table 3 : Reactance of three element network
Freq Imag(Zin1) 3.000 MHz -2135.082 4.000 MHz -1591.654 5.000 MHz -1263.251 6.000 MHz -1042.269 7.000 MHz -882.571 8.000 MHz -761.066 9.000 MHz -664.905 10.00 MHz -586.355 11.00 MHz -520.469 12.00 MHz -463.923 13.00 MHz -414.384 14.00 MHz -370.149 15.00 MHz -329.928 16.00 MHz -292.705 17.00 MHz -257.641 18.00 MHz -224.010 19.00 MHz -191.147 20.00 MHz -158.398
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Chapter 2
0 H
-500
-1000
-1500 imag(Zin3) imag(Zin3) (H)
-2000
-2500 2 4 6 8 10 12 14 16 18 20 freq, MHz
Figure 12 : The reactance of the three element network (blue curve) reasonably resembles the reactance of the antenna (red curve)
By using a LC network as the negation element, the VSWR performance of the antenna matching network has improved considerably as shown in Table 4, which lists the input impedance and resultant VSWR, at the output of this NIC circuit.
Table 4 : VSWR and impedance characteristics of a three element negated NIC circuit
Freq Real(Zin1) Imag(Zin1) VSWR1 3.000 MHz 1734.377 583.313 38.614 4.000 MHz 399.116 -409.387 16.445 5.000 MHz 96.246 -152.359 7.128 6.000 MHz 40.772 -56.329 3.295 7.000 MHz 25.379 -18.579 2.318 8.000 MHz 20.245 -2.874 2.480 9.000 MHz 18.486 3.118 2.717 10.00 MHz 17.994 4.292 2.802 11.00 MHz 17.990 2.826 2.789 12.00 MHz 18.148 -0.257 2.755 13.00 MHz 18.322 -4.419 2.754 14.00 MHz 18.465 -9.384 2.818 15.00 MHz 18.586 -15.025 2.968 16.00 MHz 18.731 -21.347 3.220 17.00 MHz 18.970 -28.427 3.588 18.00 MHz 19.382 -36.454 4.094 19.00 MHz 20.035 -45.701 4.772 20.00 MHz 20.969 -56.555 5.679
Whilst a complex network might appear to give an improved performance, resonance within the network could have implications for the stability of the NIC. Consequently, for the purposes of this thesis, a single capacitor of 27 pF is used as the element to be negated. A detailed discussion on the instability of NICs is given in chapter 3.
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Negative Impedance Converters
2.5 Chapter summary The preceding sections have broadly explained the mathematics behind the negation ability of Negative Impedance Converters. It has given an analytical formula, which provides a platform for understanding the linear characteristics of the NIC. It was also shown that, by negating a capacitor, considerable reactance cancellation of the antenna occurs in the circuit. This improves the VSWR of the system across a wide range of frequencies. Less than 4:1 VSWR over a range of 4 MHz was obtained. By designing and negating a network of elements, the bandwidth was increased to about 11 MHz. This is a very large bandwidth for the HF frequency range and would be a good result. This potential, however, is clouded by the possibility of instability in the NIC and this problem will be explored in the next chapter.
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Chapter 3
Stability Concerns
EGATIVE Impedance Converters have significant promise in the application of antenna matching, as shown in the previous chapter. However, concerns arise regarding the stability of the NICs as they are only conditionally stable. This chapter seeks to understand the nature of NIC stability and the methods to achieve stability.
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Stability Concerns
3 Stability concerns
3.1 Introduction
In the design of a matching network it is imperative that the stability of the circuit, i.e. its resistance to oscillate, is thoroughly considered. This is because oscillations would nullify the functionality of the matching network. The stability of a network is frequency dependent and has to be stable across all frequencies, not only at the design frequencies.
There are two types of stability as defined by Pozar [29]:
1) Unconditional stability: “The network is unconditionally stable if |τin|<1 and
|τout|<1 for all passive source and load impedances”
2) Conditional stability: “The network is conditionally stable if |τin|<1 and |τout|<1 only for a certain range of passive source and load impedances. This case is also referred to as potentially unstable.”
Note: τin and τout are the input and output reflection coefficient of the network or circuit.
Unconditional stability is the ideal goal in the design of an active matching network. However, due to the nature of the NIC, it is at best only conditionally stable. This chapter will describe the characteristics of NICs, the methods of predicting stability and investigate the conditions at which the NIC is stable and the modifications that were necessary to achieve stability.
3.2 NIC characteristic
The Negative Impedance Converter design chosen has similarities with an astable multivibrator as the transistors’ bases are coupled to the other transistor’s collector using a capacitor. This cross-couple similarity provides insight into the instability issue that NIC circuits face. A BJT based NIC, which has emitters at the NIC’s input-output ports, has an Open Circuit Stable (OCS) characteristic (OCS means that for any passive load on one port, the device is stable with an open circuit on the other port [24]. Short Circuit Stable (SCS) BJT based NIC circuits, however, have collectors at the input and
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Chapter 3
output terminal pair. (SCS means that for any passive load on one port, the device is stable with a short circuit on the other port [24]). Open Circuit Stable NICs are to be used as series elements, while SCS NICs are to be used as shunt elements. OCS circuits are stable only if they are terminated with relatively large impedances, while SCS circuits are stable only if they are terminated with significantly lower impedance as compared with the input impedance of the NIC.
Sussman-Fort [24] further explains that the inherent conditional stability of an NIC sets a limitation on the magnitude of the impedances at its respective terminations. This limitation is given by the following two equations,
L || > ZZ in11 (3.1) and (3.2)
L2 || < ZZ in2 where ZL1 and ZL2 are the load impedances on the left hand side and right hand side respectively, and Zin1 and Zin2 are the input impedance looking in from the left hand side and right hand side respectively (as shown in Figure 13).
Figure 13 : NIC’s two ports terminated by ZL1 and ZL2. One port will be Open Circuit Stable while the other will be Short Circuit Stable (adopted from [24]).
Sussman-Fort recommends that Inequalities 3.1 and 3.2 be satisfied by at least a factor of two. However, he stated that there is no general answer for this and that it has to be evaluated according to its context. Therefore, the loading of a NIC has to be carefully analysed, according to its context, as its stability is crucial for it to function as a matching network for HF antennas. In the current work, an emitter input / output type NIC will be considered as this is appropriate to the antenna matching problem at HF. A
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Stability Concerns number of stability analysis methods will be used in the next section to analyse the conditions at which the NIC is stable. It is to be noted that other stability prediction methods (aside from those used in this thesis) are applicable, namely Middlebrook’s technique [30] and Rollet’s proviso [31, 32].
3.3 Stability Analysis This research include two main methods of predicting stability in the NIC circuit, namely Reflection Coefficient (or S-parameters) Analysis and Transient Analysis. By setting the conditions for the circuit, the stability of a particular port can be analysed by considering the S11 (S-parameter input port reflection coefficient) of a one port network. If the reflection coefficient of a port (reflection coefficient is shown as S11) is found to be greater than 1, the port may oscillate at that frequency given the appropriate excitations. This analysis will be performed at the input of the receiver and at the output of the antenna (the other port being appropriately load by either the antenna or the receiver). However, as there could be circuit noise and power supply transients, this analysis was found to be necessary at the bases of the transistors also. In order to confirm the results on stability, a transient analysis of the output voltage (at the receiver) was performed as a secondary method in predicting stability of the circuit.
Firstly, the circuit was tested to obtain its Rollet Factor [33] to check for unconditional stability. A requirement for unconditional stability is to have a Rollet Factor greater than one across all frequencies. The circuit setup as shown in Figure 14 was used and the results can be found in Figure 15. The results confirm that unconditional stability was not achieved with this circuit as the Rollet Factor is less than one at frequencies below 3 MHz. This fits in the hypothesis for this chapter as NICs are known to be at best conditionally stable. Thus S11 analysis at the relevant ports (receiver port, antenna port and the transistor bases) were performed to test for conditional stability.
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Chapter 3
R R V_DC L C L V_DC R38 R37 SRC19 L29 C33 L32 SRC18 R=Rc Ohm V_DC R=Rc Ohm Vdc=VDC L=Ltop uH C=27 pF V_DC L=Ltop uH Vdc=VDC R=Rtop SRC21 R=Rtop Vdc=Vb V SRC20 Vdc=Vb V TRANSIENT R S-PARAMETERS R44 R Tran R=R1 Ohm R41 Tran1 S_Param R=R1 Ohm StopTime=100.0 usec SP1 MaxTimeStep=1.0 nsec Start=0.01 MHz R Stop=20 MHz L Var VAR R43 L R Eqn Step=0.01 MHz L33 R42 VAR48 R=R2 Ohm L34 Var VAR L=Lmid uH R=R2 Ohm Lmid=470 Eqn R=Rmid L=Lmid uH Cb=2 VAR49 R=Rmid Rc=200 Rtop=6.5 R1=20000 Rbot=2.6 Lbot=220 R2=150000 C Re=1600 Ltop=470 C35 C Vb=10 Rmid=6.5 C=Cb nF C34 VDC=10 BJT_NPN C=Cb nF BJT_NPN BJT4 BJT3 Model=MRF949 Model=MRF949 Vout L C L 1 Ref L31 VSWR Term C37 StabFact L30 C Term1 C=0.1 uF L=Lbot uH S1P R=Rbot VSWR StabFact L=Lbot uHC36 Num=1 VSWR2 R=Rbot C=0.1 uF Antenna2 Z=50 Ohm StabFact1 Term VSWR1=vswr(S11) StabFact1=stab_fact(S) BJT_Model Term2 R MRF949 N R Num=2 R40 Z=50 Ohm Zin R39 R=Re Ohm StabMeas R=Re Ohm Zin StabMeas Zin2 StabMeas1 Zin1=zin(S11,PortZ1) StabMeas1=stab_meas(S) Figure 14 : Circuit setup for Rollet Factor simulation
1.5
1.0
0.5
0.0
-0.5
-1.0
0 2 4 6 8101 141 1 2 freq, (MHz)
Figure 15 : Rollet Factor simulation result
In order to perform the reflection coefficient analysis at the receiver side, the antenna was directly shorted to ground (i.e. the 50 Ohm S-parameter termination was short circuited). Figure 16 shows the results of the S11 or reflection coefficient simulation as seen from the receiver. It is observed that the circuit is not stable at low frequencies as the S11 is less than one.
Page 35
Stability Concerns
1.2
1.1
1.0
0.9
0.8 mag(S(1,1)) 0.7
0.6
0.5 0 2 4 6 8 10 12 14 16 18 20 freq, MHz Figure 16 : Reflection coefficient (S11) as seen from the receiver
Page 36
Chapter 3
um=1 Term Term1 N Ohm Z=50 V_DC SRC18 Vdc=VDC 1 Ref S1P Antenna2 R R37 R=RcOhm mid=0.086235 VAR VAR48 Lmid=470 Ctop=0.086235 Cb=2 Cbot=0.1124 Rtop=6.5 Rbot=2.6 Lbot=220 Ltop=470 C Rmid=6.5 C C36 uF C=0.1 Var Eqn =Ltop uH =Ltop =Lbot uH =Lbot L L32 L R=Rtop L L30 L R=Rbot R R39 R=ReOhm BJT_NPN BJT3 Model=MRF949 R R42 R=R2Ohm V_DC SRC20 V Vdc=Vb =Lmid uH =Lmid L L34 L R=Rmid R R41 R=R1Ohm N C C34 C=CbnF Zin VSWR VSWR2 VSWR1=vswr(S11) Zin Zin2 Zin1=zin(S11,PortZ1) VSWR C C33 pF C=27 C C35 C=CbnF V_DC SRC21 V Vdc=Vb =Lmid uH =Lmid BJT_Model MRF949 L L33 L R=Rmid R R44 R=R1Ohm R R43 R=R2Ohm BJT_NPN BJT4 Model=MRF949 =Lbot uH =Lbot L L31 L R=Rbot R R40 R=ReOhm =Ltop uH =Ltop L L29 L R=Rtop C C37 uF C=0.1 R R38 R=RcOhm S-PARAMETERS b=10 VAR VAR49 Rc=200 R1=20000 R2=150000 Re=1600 V VDC=10 top=20 MHz top=20 S_Param SP1 MHz Start=0.01 S MHz Step=0.01 Var Eqn V_DC SRC19 Vdc=VDC R R45 Ohm R=50 Vout
Figure 17 : Circuit setup for antenna port reflection coefficient (S11) simulation
Next, the same analysis was done to check the reflection coefficient as seen from the Antenna. In this analysis, the receiver was replaced by a 50 Ohm resistor, as shown in Figure 17. The simulation results are given in Figure 18, and show that the antenna port is stable.
Page 37
Stability Concerns
1.0
0.8
0.6
0.4
0.2
0.0
0 2 4 6 8 1 12 1 16 18 20
freq, MHz Figure 18 : Reflection coefficient (S11) as seen from the antenna
Next, the reflection coefficient analysis was performed at the bases of the transistors. Figure 19 shows the circuit setup for the reflection coefficient analysis at the base of the antenna-side transistor. Similarly, the other transistor base is tested by moving the S- parameter termination along with a DC blocking capacitor in Figure 19, to the base of the receiver side transistor. This blocking capacitor is required so that it does not affect the DC bias of the circuit. The results of these two simulations are shown in Figure 20. The result of Figure 20, where the reflection coefficient is greater than 1, indicates that the circuit could oscillate at low frequencies.
Page 38
Chapter 3
V_DC SRC18 Vdc=VDC Ref 1 S1P Antenna2 R R37 R=RcOhm mid=0.086235 VAR VAR48 Lmid=470 Ctop=0.086235 Cb=2 Cbot=0.1124 Rtop=6.5 Rbot=2.6 Lbot=220 Ltop=470 C Rmid=6.5 C C36 uF C=0.1 Var Eqn =Ltop uH =Ltop =Lbot uH =Lbot L L32 L R=Rtop L L30 L R=Rbot R R39 R=ReOhm BJT_NPN BJT3 Model=MRF949 um=1 Term Term1 N Ohm Z=50 C C38 uF C=0.1 R R42 R=R2Ohm V_DC SRC20 V Vdc=Vb =Lmid uH =Lmid L L34 L R=Rmid R R41 R=R1Ohm N C C34 C=CbnF Zin VSWR VSWR2 VSWR1=vswr(S11) Zin Zin2 Zin1=zin(S11,PortZ1) VSWR C C33 pF C=27 C C35 C=CbnF V_DC SRC21 V Vdc=Vb =Lmid uH =Lmid BJT_Model MRF949 L L33 L R=Rmid R R44 R=R1Ohm R R43 R=R2Ohm BJT_NPN BJT4 Model=MRF949 =Lbot uH =Lbot L L31 L R=Rbot R R40 R=ReOhm =Ltop uH =Ltop L L29 L R=Rtop C C37 uF C=0.1 R R38 R=RcOhm S-PARAMETERS b=10 VAR VAR49 Rc=200 R1=20000 R2=150000 Re=1600 V VDC=10 top=20 MHz top=20 S_Param SP1 MHz Start=0.01 S MHz Step=0.01 Var Eqn V_DC SRC19 Vdc=VDC R R45 Ohm R=50 Vout
Figure 19 : Circuit setup for transistor base reflection coefficient (S11) simulation
Page 39
Stability Concerns
5
4
3
2 (a)
1
0 0 2 4 6 8 10 12 14 16 18 20
freq, MHz
2.0
1.8
1.6
1.4
1.2
(b) 1.0
0.8
0.6
0.4
0 2 4 6 8 10 12 14 16 18 20
freq, MHz
Figure 20 : Reflection coefficient (S11) as seen the receiver side transistor’s base and the antenna side transistor base respectively
Lastly, a transient analysis is performed as a secondary test for stability and to understand the characteristics of instability. A transient analysis using a maximum step size of 1 ns and a stop time of 100 us were found to be sufficient for this case. The results in Figure 21 show that the circuit is unstable and its oscillations grow with time. The Fourier series of the waveform shows that oscillations occur at a number of frequencies. Therefore, if this circuit is to be utilised as a matching network for HF antennas, some major changes would be required to stabilise this circuit without
Page 40
Chapter 3
nullifying its ability to negate impedances. This will be discussed in the following section.
300
200
100
0 (a) Vout, mV Vout, -100
-200
-300 0 10 20 30 40 50 60 70 80 90 100 time, usec
140
120
100
80
(b) 60 fs(Vout), mV 40
20
0 1E5 1E6 1E7 freq, Hz
Figure 21 :(a) Transient Analysis and (b) Fourier Series of the voltage at the receiver (labelled Vout in the circuit)
Page 41
Stability Concerns
3.4 Circuit Modification
The simulation results in section 3.3 confirm that the circuit is not stable. Therefore, modifications to the circuit are necessary. The circuit was biased using standard amplifier biasing methodology. Since stability is the main problem, the circuit was re- biased to produce better stability factors by designing the NIC such that the voltage at the base of the transistor is 1/10th that of the voltage of the supply as suggested by Gonzales [34].
Literature review of the work done by Linvill and Sussman-Fort, at the beginning of this chapter, explained that this circuit is only open circuit stable (as the input and output of the NIC are at the emitter ends of the transistors). For this circuit setup, the impedance of the receiver (50 Ohms) was found to be not sufficiently large as compared to the input impedance of the NIC. Thus a transformer with a turns ratio of 1:10 was placed before the receiver, such that the receiver impedance would appear to be 100 times larger to the NIC. It is to be noted that this solution is not ideal, as it reduces the already small antenna voltage.
The reflection coefficient results (Figures 16 and 20) suggest that a large proportion of the instability occurs at low frequencies. This could be circumvented by adding filters into the circuit. However, a careful selection and placement of filters were necessary in order to avoid grounding a large proportion of the signal and thus nullify the functionality of the circuit. After much experimentation and analysis, it was found that by implementing a band-stop filter and two resistively loaded notch filters, the circuit could be stabilised. The modified circuit is shown in Figure 22. By following the same steps in section 3.3, the circuit was tested for its reflection coefficient across a number of points and a transient analysis was performed. The results are shown in Figures 23 to 25.
Page 42
Chapter 3
Ref V_DC SRC18 Vdc=VDC mid=0.086235 1 VAR VAR48 Lmid=470 Ctop=0.086235 Cb=1 Cbot=0.1124 Rtop=6.5 Rbot=2.6 Lbot=470 Ltop=200 C Rmid=6.5 S1P Antenna2 Var Eqn R R37 R=RcOhm b=20 C C36 C=0.1 uF VAR VAR51 Rc=2900 R1=59800 R2=4200 Re=100 V VDC=20 VAR VAR61 Rstab=1 Var Var Eqn Eqn =Ltop uH =LbotuH L L39 L R= L L30 L R=Rbot R R39 R=Re Ohm Vin BJT_NPN BJT3 Model=MRF949 C C66 C=0.001 uF =5 nH R R76 R=5 Ohm L L47 L R= R R69 R=RstabOhm =1 uH C C63 C=5 uF L L43 L R= R R68 R=Rstab Ohm R R42 R=R2 Ohm V_DC SRC20 Vdc=VbV R R41 R=R1 Ohm N Zin Zin Zin2 Zin1=zin(S11,PortZ1) C C105 C=1.0 nF C C33 C=27pF BJT_Model MRF949 VSWR VSWR2 VSWR1=vswr(S11) VSWR C C106 C=1.0 nF V_DC SRC21 Vdc=Vb V R R44 Ohm R=R1 =10 uH C C77 C=0.1055 nF L L56 L R= R R75 R=95 Ohm R R43 R=R2 Ohm =Lbot uH BJT_NPN BJT4 Model=MRF949 L L31 L R=Rbot R R40 R=Re Ohm =Ltop uH L L38 L R= C C101 C=0.1uF R R38 R=Rc Ohm TF TF2 T=0.10 V_DC SRC19 Vdc=VDC S-PARAMETERS um=1 Term Term1 N Ohm Z=50 top=40 MHz S_Param SP1 Start=0.1 MHz S Step=0.1 MHz Vout
Figure 22 : Modified NIC circuit
Page 43
Stability Concerns
m8 freq=29.320000000MHz mag(S(1,1))=0.9972207755 Max 1.00 m8
0.99
(a) 0.98 mag(S(1,1))
0.97
0.96 0 5 10 15 20 25 30 35 40 freq, MHz m8 freq=30.000000000kHz mag(S(1,1))=0.9999998369 Max m8 1.00
0.95
0.90
0.85 (b) 0.80
0.75 mag(S(1,1))
0.70
0.65
0.60 0 5 10 15 20 25 30 35 40 freq, MHz
Figure 23 : Reflection coefficient (S11) as seen from the receiver (a) and antenna (b) respectively
Page 44
Chapter 3
m8 freq=33.940000000MHz mag(S(1,1))=0.9990352243 Max m8 1.0
0.9
0.8
(a) 0.7
0.6 mag(S(1,1)) 0.5
0.4
0.3 0 10 20 30 40 50 60 freq, MHz m8 freq=10.000000000kHz mag(S(1,1))=0.9963402964 Max m8 1.0
0.9
(b) 0.8
0.7 mag(S(1,1))
0.6
0.5 0 10 20 30 40 50 60 freq, MHz
Figure 24 : Reflection coefficient (S11) as seen the receiver side transistor’s base (a) and the antenna side transistor base (b) respectively
Page 45
Stability Concerns
15
10
5
0
-5
(a) fV Vout, -10
-15
-20
-25 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 time, msec
(b)
Figure 25 : Transient Analysis (a) and its Fourier Series (b) of the voltage at the receiver
Figures 23 to 25 show that the circuit is stable under this set of conditions. Transient analysis, adding a transient voltage step at the power supply and some initial conditions (+0.1 V at receiver-side transistor base) confirmed this result. The circuit setup used is shown in Figure 26.
Page 46
Chapter 3
Ref 1 S1P Antenna2 C C36 C=0.1 uF =Lbot =Lbot uH L L30 L R=Rbot R R39 Ohm R=Re Vin BJT_NPN BJT3 Model=MRF949 V_DC SRC18 Vdc=VDC TRANSIENT R R37 R=Rc Ohm C C66 uF C=0.001 =5 nH R R78 R=1.5 Ohm L L47 L R= topTime=100 usec topTime=100 Tran Tran1 S nsec MaxTimeStep=0.5 R R69 Ohm R=Rstab =Ltop =Ltop uH L L39 L R= =1 uH C C63 uF C=C1 L L43 L R= R R68 Ohm R=Rstab R R42 Ohm R=R2 high=-1V =Lbot =Lbot uH VtStep SRC3 Vlow=0V V Delay=10nsec Rise=1nsec L L41 L R=Rbot R R41 Ohm R=R1 t Vsup 1=5 VAR VAR61 Rstab=1 VAR VAR60 C C3=0.47 ar ar V V Eqn Eqn C C78 C=1 nF V_DC SRC20 Vdc=Vb V BJT_Model MRF949 C C33 pF C=27 C C79 C=1 nF =Lbot =Lbot uH V_DC SRC21 Vdc=Vb V R R44 Ohm R=R1 L L40 L R=Rbot NodeSet NodeSet1 V V=0.1 InitCond InitCond1 V V=0.1 In it Se t Co nd No de R R43 Ohm R=R2 =10 =10 uH C C77 nF C=0.1055 L L56 L R= R R75 R=5 Ohm =Ltop =Ltop uH L L38 L R= R R38 R=Rc Ohm S-PARAMETERS BJT_NPN BJT4 Model=MRF949 top=40 MHz top=40 S_Param SP1 Start=0.1 MHz S MHz Step=0.01 =Lbot =Lbot uH L L31 L R=Rbot R R40 Ohm R=Re V_DC SRC19 Vdc=VDC mid=0.086235 VAR VAR48 Lmid=470 Ctop=0.086235 Cb=1 Cbot=0.1124 Rtop=6.5 Rbot=2.6 Lbot=470 Ltop=200 C Rmid=6.5 C C37 C=0.1 uF VSWR VSWR2 VSWR1=vswr(S11) ar VSWR V Eqn b=20 VAR VAR51 Rc=2900 R1=59800 R2=4200 Re=100 V VDC=20 TF TF1 T=0.1 N ar V Eqn Zin Zin Zin2 Zin1=zin(S11,PortZ1) Vout um=1 Term Term1 N Z=50 Ohm
Figure 26 : Transient Analysis circuit setup
Page 47
Stability Concerns
The additional parameters were necessary as to simulate a perturbation that could trigger oscillation. In the case, when the circuit is unstable, a spurious signal that grows with time will be produced by the transient analysis. The transient analysis was run for 1 ms with a maximum step size of 0.5 ns. The result in Figure 27 shows a spike at the start due to the voltage step input for the transient analysis, but the spurious signal does not grow with time and the Fourier Series of the signal does not indicate oscillation. This transient and reflection coefficient analysis confirms that the circuit is stable.
50
40
30
20
(a) mV Vout, 10
0
-10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 time, msec
(b)
Figure 27 : Transient Analysis (a) and its Fourier Series (b) of the voltage at the receiver
Page 48
Chapter 3
However, the solution applied to achieve a stable circuit has come at cost of a slight degradation in performance. This was expected because a number of filters were introduced into the NIC itself and a 0.1 turns ratio transformer was used at the receiver end.
With regards to the negation ability of the NIC, the inclusion of the filters has reduced the performance of reactance negation. This is because a notch filter designed for 4.9 MHz was used due to instability at that frequency. This notch filter, however, includes a 95 Ohm resistor to reduce the effects of the attenuation to a sufficient level for stability. The reactance negation of the NIC circuit upon a 27 pF capacitor is shown in Figure 28. For simplicity in reading the simulation, the result was obtained using a 1:1 turns ratio transformer and the antenna was shorted. This result is to be compared with the reactance curve of an ideal negative 27 pF capacitor as simulated by ADS in Figure 29. The overall imaginary impedance of the circuit is shown in Figure 30.
m1 m7 freq=6.450MHz m2 m3 freq=5.000MHz imag(Zin1)=665.259 freq=10.00MHz freq=15.00MHz imag(Zin1)=607.443 Max imag(Zin1)=532.369 imag(Zin1)=306.330
1000 m4 m6 m7m1 freq=20.00MHz freq=3.000MHz m6 m2 imag(Zin1)=188.007 imag(Zin1)=471.490 500 m3 m4m5 m5 0 freq=22.00MHz imag(Zin1)=159.671
-500 imag(Zin1)
-1000
-1500 0 5 10 15 20 25 30 35 40 freq, MHz Figure 28 : Imaginary part of the input impedance of the NIC with 1:1 transformer and without the antenna
Page 49
Stability Concerns
m1 m2 m3 freq=3.000MHz freq=6.150MHz freq=10.00MHz imag(Zin1)=1964.876 imag(Zin1)=958.476 imag(Zin1)=589.463
6000 m4 m5 freq=15.00MHz freq=20.00MHz 5000 imag(Zin1)=392.975 imag(Zin1)=294.731
4000
3000
imag(Zin1) m1 2000 m2 1000 m3 m4 m5
0 0 2 4 6 8 10 12 14 16 18 20 freq, MHz Figure 29 : Reactance curve of an ideal negative 27pF capacitor
By comparing Figures 28 and 29, it is observed that at lower frequencies, below 6.15 MHz, the reactance curve of the NIC has been severely affected by the filters. However, at higher frequencies the accuracy improves. Some discrepancies in the reactance negation are to be expected, but the overall circuit reactance cancellation, has a moderately good performance as can be seen in Figure 30. It will be noted, however, that the transformer has reduced the reactance to 1/100 of its value at the output of the NIC.
m6 m7 m1 freq=3.000MHz freq=5.000MHz freq=10.50MHz imag(Zin1)=-0.091 imag(Zin1)=0.005 imag(Zin1)=0.449 Max m2 m2m1 freq=10.00MHz 0.5 imag(Zin1)=0.387 m7 m6 m4m5 0.0 m3 m3 freq=15.00MHz imag(Zin1)=-0.213 -0.5 m4 -1.0 freq=20.00MHz imag(Zin1) imag(Zin1)=-0.088
-1.5 m5 freq=22.00MHz -2.0 imag(Zin1)=-0.061 0 10 20 30 40 50 60 freq, MHz Figure 30 : Imaginary part of the input impedance of the NIC with a transformer as seen from the receiver
Page 50
Chapter 3
Another disadvantage with this solution, as briefly mentioned earlier in this section, is that the introduction of the 1:10 transformer also reduces the already small antenna voltage. In addition, the real resistance becomes even smaller in magnitude and thus causing the impedance that is seen by the receiver to be further from the ideal of 50 Ohm. The real part of the input impedance of the matching network, and antenna, as seen from the receiver is shown in Figure 31.
1.2 m10 m10 1.0 freq=11.25MHz real(Zin1)=1.016 0.8 Max m11 0.6 freq=29.32MHz real(Zin1)=0.070
real(Zin1) Min 0.4
0.2 m11
0.0 0 10 20 30 40 50 60 freq, MHz Figure 31 : Real part of the input impedance of the NIC as seen from the receiver
In order to improve on this solution, an understanding on what impedance values are necessary for stability would be helpful. Thus a stability analysis on the effect of impedance loading on the receiver side is performed by using a batch analysis (parameter sweep) of the receiver impedance and its effect on stability. The circuit setup is shown in Figure 32. The reflection coefficient of the antenna port and transistor base port is shown in Figure 33 and Figure 34.
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Stability Concerns
Term Term1 Num=1 Z=50 Ohm Ref 1 S1P Antenna2 C C36 C=0.1 uF BATCH SIMULATION =Lbot =Lbot uH L L30 L R=Rbot R R39 Ohm R=Re ergeDatasets=no Vin BatchSimController BatchSim1 Log= Dec=1.0 Start=50.0Var="Rx" Stop=50000.0 UseSweepPlan=yes UseSweepModule=no SweepModule="" SweepArgument= UseSeparateProcess=no M RemoveDatasets=no Analysis[1]="SP1" BJT_NPN BJT3 Model=MRF949 C C66 uF C=0.001 =5 nH R R76 R=5 Ohm L L47 L R= V_DC SRC18 Vdc=VDC VAR VAR62 Rx=500 R R69 Ohm R=Rstab Var Eqn R R37 R=Rc Ohm =1 uH C C63 uF C=C1 L L43 L R= =Ltop =Ltop uH L L39 L R= R R68 Ohm R=Rstab R R42 Ohm R=R2 V_DC SRC20 Vdc=Vb V =Lbot =Lbot uH L L41 L R=Rbot R R41 Ohm R=R1 1=5 VAR VAR61 Rstab=1 VAR VAR60 C C3=0.47 Var Var Eqn Eqn C C78 C=1 nF BJT_Model MRF949 C C33 pF C=27 C C79 C=1 nF V_DC SRC21 Vdc=Vb V =Lbot =Lbot uH L L40 L R=Rbot R R44 Ohm R=R1 R R43 Ohm R=R2 =10 =10 uH C C77 nF C=0.1055 L L56 L R= R R75 Ohm R=95 =Ltop =Ltop uH L L38 L R= R R38 R=Rc Ohm S-PARAMETERS BJT_NPN BJT4 Model=MRF949 top=20 MHz top=20 S_Param SP1 Start=0.1 MHz S MHz Step=0.01 =Lbot =Lbot uH L L31 L R=Rbot R R40 Ohm R=Re V_DC SRC19 Vdc=VDC VSWR VSWR2 VSWR1=vswr(S11) VSWR mid=0.086235 VAR VAR48 Lmid=470 Ctop=0.086235 Cb=1 Cbot=0.1124 Rtop=6.5 Rbot=2.6 Lbot=470 Ltop=200 C Rmid=6.5 C C37 C=0.1 uF Var Eqn b=20 VAR VAR51 Rc=2900 R1=59800 R2=4200 Re=100 V VDC=20 Vout N Var Eqn Zin Zin Zin2 Zin1=zin(S11,PortZ1) R R77 R=Rx Ohm
Figure 32 : Circuit setup for stability sensitivity due to receiver load
Page 52
Chapter 3
1.08
1.06
1.04 mag(S(1,1))
1.02 Rx=50.000000
1.00 Rx=500.000000
0.98 Rx=5000.000000 Rx=50000.000000 0.96 mag(S(1,1)) 0.94
0.92
0.90
0.88
0.86 0 2 4 6 8 10 12 14 16 18 20 freq, MHz Figure 33 : Reflection coefficient as seen from antenna with varying receiver load (Rx)
1.00
0.95 mag(S(1,1)) 0.90 Rx=50.000000 0.85 Rx=500.000000 0.80 Rx=5000.000000 0.75 Rx=50000.000000
mag(S(1,1)) 0.70
0.65
0.60
0.55
0.50 0 2 4 6 8 10 12 14 16 18 20 freq, MHz Figure 34 : Reflection coefficient as seen from the base of the antenna-side transistor with varying receiver load (Rx)
From Figures 33 and 34, we can conclude that the NIC requires a loading of at least 5000 Ohms at the receiver side for the circuit to be stable. This re-affirms the number of turns chosen for the transformer used. However, as previously discussed, the transformer does not match the circuit to 50 Ohms. An alternate solution is to incorporate a buffer amplifier between the NIC and the receiver. A buffer circuit has a
Page 53
Stability Concerns significant advantage over the 1:10 transformer solution, as it does not cause a drop in antenna voltage. Its disadvantage, however, is that it could add additional noise. There are two main choices for a buffer circuit design for this application, namely an emitter follower and a source follower. A source follower was chosen as it provides larger input impedance as compared to an emitter follower.
A simple source follower circuit was designed and shown in Figure 35. The source follower was designed to provide an input impedance of 500k Ohms and an output impedance of 50 Ohms. The high input impedance would cause a greater proportion of the signal voltage to occur across the receiver impedance and yet fulfil the stability requirements. The result of including the buffer into the circuit is shown in Figure 36, where the real part of the input impedance was successfully matched to 50 Ohms.
V_DC SRC34 Vdc=Vbuff
R R118 R=Rbuf kOhm Var Eqn VAR VAR63 Rbuf=1000.0 NIC_Output
C ap_nms_2N6660_19930601 C102 M2 C=0.1 uF Vout Term Term1 C R Num=1 C101 R R117 Z=50 Ohm C=0.1 uF R115 R=Rbuf kOhm R=665 Ohm
Figure 35 : Source Follower Circuit Design (note: Vbuff = 7.0 V)
Page 54
Chapter 3
60
50
40
30 real(Zin3) 20
10
0 0 2 4 6 8 10 12 14 16 18 20 freq, MHz
Figure 36 : Real part of the input impedance of the overall NIC circuit
In order to compare the performance of the different circuits, a transient analysis was performed using a 1 mV voltage source, placed within the antenna circuit and its frequency was varied. The resultant voltage across the receiver input is listed in Table 5.
Table 5: Signal level at receiver given a fixed input voltage level
Buffer NIC & 1:10 Freq Buffer (2N6660) (2N6660) & transformer (MHz) (V) NIC (V) (V) 3.00 6.975E-5 4.623E-4 1.508E-5 5.00 5.847E-5 4.624E-4 1.274E-5 10.00 0.002 4.624E-4 4.328E-4 11.24 0.004 4.320E-4 6.331E-4 15.00 0.002 4.624E-4 3.383E-4 20.00 0.001 4.624E-4 2.596E-4
From Table 5 we can observe that the NIC circuit with the buffer (column 2) shows a marked improvement in performance for frequencies above 10 MHz over standard active antenna matching that uses the buffer alone (column 3). Sussman-Fort, in his recent work [35], confirm that a buffer circuit enhances the performance of the NIC and surpasses traditional active antenna matching. This circuit also has a substantial
Page 55
Stability Concerns decrease in signal loss as compared with the NIC circuit containing the transformer (column 4).
3.5 Chapter Summary
NICs are, at best, conditionally stable. By applying a reflection coefficient analysis at various points within the circuit and by utilizing transient analysis, the circuit’s stability can be tested. The NIC’s stability is particularly dependent on its loading. For emitter based NICs, it is Open Circuit Stable and require a high impedance load. Besides that, for an NIC to be stable, it was found that the DC bias of the transistors needs to be carefully chosen, such that the voltage at the base is relatively low (1/10th or smaller) as compared with the supply voltage. Last but not least, there were inherent instabilities at low frequencies in this circuit which could be removed by band-stop filters and notch filters. It is to be noted that the design of the filters is a complicated matter as the filters themselves could interact with the circuit and cause a different set of instabilities. The solutions applied to stabilise the NIC circuit have caused deterioration in the bandwidth of the NIC. Further work could be explored in order to minimise this loss in performance.
By introducing a source follower at the output of the NIC, substantial improvements were made in the matching of the real resistance of the antenna to the 50 Ohms of the receiver. This combination has improved the voltage level received at the receiver while maintaining circuit stability. Over the appropriate bandwidth of the NIC, its performance is better than traditional active matching. Future work could include other methods of stability prediction that may provide a better understanding of the conditions for stability. This could improve the efficiency of the design process of NICs. In the practical implementation of the NIC circuit, device variation could have a detrimental effect on the circuit. The following chapter explores some of the common manufacturing variations and its consequence upon the NIC circuit.
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Effects of Non-idealities
N this chapter, the effects of non-idealities upon the performance of the NIC will be analysed. Some of the non-idealities considered are parameter variations, environmental changes and supply voltage fluctuations. A sensitivity analysis was also performed to understand the sensitivity of the parameters chosen and its impact on the NIC performance.
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Effects of Non-idealities
4 Effects of Non-idealities
In the preceding chapter, we have seen that stability is a significant issue to be considered for NIC circuits. We have established, through reflection coefficient analysis and transient analysis, that the NIC with the circuit conditions and loading in section 3.4, is stable. However, in the practical implementation of NIC circuits, device variations and other non-idealities could affect stability and other aspects of performance. The parameters that will be considered are transistor beta value, temperature, supply voltage and some of the lumped components used in the circuit. In particular, asymmetry in transistor properties could be a problem. The effect of variations upon factors such as stability and impedance negation performance will be analysed.
4.1 Transistor Mismatch and Hfe Variations The NIC circuit, in this research, has a degree of symmetry and includes two identical transistors. However, in the manufacturing of these transistors, there could be differences in their beta or hfe values. Hence, it is important to consider the impact of a mismatch in the two BJTs.
The BJTs’ (MRF949) datasheet specifies that it has a minimum beta value of 50. This implies that there could be a fair degree of variance about this value. Thus a batch simulation in ADS2009 was performed, by varying the beta values for one and/or both of the transistors. In Figure 37, both of the transistors’ beta values were varied simultaneously and its reactance and the stability are simulated with a parameter sweep of beta starting from 50 to 200. The stability parameter is taken by looking into the antenna end of the NIC. It is to be noted that the reactance simulation, shown in this chapter, does not include the buffer circuit and is seen from the 50 Ohm receiver. Similarly, asymmetric beta is analysed by varying only one transistor’s beta value at a time, fixing the other at 175 (a typical beta value for this transistor). The results are given in Figures 38 and 39. This shows the effect of transistor mismatch upon the important performance parameters of the circuit.
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80
60 beta=50.000000 beta=75.000000 40 beta=100.000000 beta=125.000000 20 beta=150.000000
0 beta=175.000000 (a) beta=200.000000 imag(Zin1) -20
-40
-60
-80 0 2 4 6 8 10 12 14 16 18 20 freq, MHz 1.00
beta=50.000000 0.99 beta=75.000000 beta=100.000000 beta=125.000000 0.98 beta=150.000000 beta=175.000000 0.97 beta=200.000000 (b) S(1,1)
0.96
0.95
0.94 0 2 4 6 8 10 12 14 16 18 20 freq, MHz
Figure 37 : (a) Reactance (without buffer circuit) and (b) stability, when both transistor betas are varied
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50
40 beta=50.000000 30 beta=75.000000
20 beta=100.000000 beta=125.000000 10 beta=150.000000 0 beta=175.000000 (a) beta=200.000000 imag(Zin1) -10
-20
-30
-40
-50 0 2 4 6 8 10 12 14 16 18 20 freq, MHz 1.00
beta=50.000000 0.99 beta=75.000000 beta=100.000000 0.98 beta=125.000000 beta=150.000000 beta=175.000000 0.97 beta=200.000000 (b) S(1,1)
0.96
0.95
0.94 0 2 4 6 8 10 12 14 16 18 20 freq, MHz
Figure 38 : (a) Reactance (without buffer circuit) and (b) stability, when only the antenna side transistor is varied
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80
60 beta=50.000000 beta=75.000000 40 beta=100.000000 beta=125.000000 20 beta=150.000000 0 beta=175.000000 (a) beta=200.000000 imag(Zin1) -20
-40
-60
-80 0 2 4 6 8 10 12 14 16 18 20 freq, MHz 1.00
beta=50.000000 0.99 beta=75.000000 beta=100.000000 0.98 beta=125.000000 beta=150.000000 beta=175.000000 0.97 beta=200.000000 (b) S(1,1)
0.96
0.95
0.94 0 2 4 6 8 10 12 14 16 18 20 freq, MHz
Figure 39 : (a) Reactance (without buffer circuit) and (b) stability, when only the receiver side transistor is varied
The results of Figures 37 to 39 show that beta variation has little effect on the performance of the NIC. However, it is interesting to note that the performance of the NIC is more sensitive to the beta value of the receiver end transistor. The following section will explore the impact of temperature variations on the performance of the NIC.
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4.2 Temperature Variations
HF systems are used in a number of different environmental conditions. The durability of the NIC circuit is important to be able to withstand non-ideal environmental conditions. One environmental variation that could have a significant impact is temperature changes. In this section, we will test the NIC’s performance under a range of temperatures. There are standards which specify the temperature durability of a circuit, depending on the application context of the circuit. One of them is the United States Military Standard, MIL-STD-810, which details the required circuit durability under a wide range of temperature variation. Table 6, as obtained from Altera Corporation [36] , shows the temperature variations that the circuit design should take into account, such that the circuit fulfils the specified performances.
Table 6: Temperature durability range according to the context of application
Application Temperature Range Civilian 0 to 85 ˚C Industrial -44 to 100 ˚C Military -55 to 125 ˚C
The most stringent temperature durability range, namely the one for military applications, was chosen and an ADS simulation was performed, by varying the circuit’s temperature, in order to simulate the performance of the NIC circuit under these harsher conditions. This temperature variation test was applied to the lumped components and transistors of the NIC. It is to be noted that high temperature stability capacitors with a temperature coefficient of 30ppm/˚C were used. The results, as shown in Figure 40, lead to an understanding that the stability and negation performance of the circuit is not significantly affected by temperature variation of the transistors.
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50
40
30 T1=-55.000000
20 T1=-10.000000 T1=35.000000 10 T1=80.000000 0 T1=125.000000
(a) imag(Zin1) -10
-20
-30
-40
-50 0 2 4 6 8 10 12 14 16 18 20 freq, MHz
1.00
0.99 T1=-55.000000
0.98 T1=-10.000000 T1=35.000000 0.97 T1=80.000000 (b) S(1,1) T1=125.000000 0.96
0.95
0.94 0 2 4 6 8 10 12 14 16 18 20 freq, MHz
Figure 40 : (a) Reactance (without buffer circuit) and (b) stability when temperature is varied for military applications
4.3 Supply Voltage Variations Variations in the power supply for the NIC circuit, could directly affect the collector current and hence affect NIC’s performance. There are three DC voltage sources in the
NIC circuit namely the voltage at the transistors’ collector (VCC), bases (VBB) and at the
MOSFET’s drain (labelled as Vbuff in the circuit). Variations at the Voltage source of the Buffer would not analysed as they do not affect the negation ability and stability of the circuit. ADS simulations were performed to analyse the impact of minor supply voltage variations (less than ±10%) for the other two voltage sources.
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Effects of Non-idealities
Firstly, the voltage at the transistors’ collector is varied from 18 V to 22 V, and its impact upon reactance (without buffer circuit), and the stability as seen from the antenna are shown in Figure 41. The same simulations were performed by varying the base voltage source instead and its results are shown in Figure 42. The results obtained show that minor supply voltage variations, for both collector and base voltage, do not affect the performance of the NIC.
50
40
30 Vcc=18.000000 20 Vcc=18.500000 10 Vcc=19.000000
0 Vcc=19.500000 (a) Vcc=20.000000 imag(Zin1) -10 Vcc=20.500000 -20 Vcc=21.000000 -30 Vcc=21.500000
-40 Vcc=22.000000
-50 0 5 10 15 20 25 30 35 40 freq, MHz 1.00
0.99 Vcc=18.000000 0.98 Vcc=18.500000 Vcc=19.000000
0.97 Vcc=19.500000 (b) S(1,1) Vcc=20.000000
0.96 Vcc=20.500000 Vcc=21.000000
0.95 Vcc=21.500000 Vcc=22.000000
0.94 0 2 4 6 8 10 12 14 16 18 20 freq, MHz Figure 41 : (a) Reactance (without buffer circuit) and (b) stability when the voltage source at the collectors are varied
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50
40
30 Vbb=18.000000 20 Vbb=18.500000 10 Vbb=19.000000
0 Vbb=19.500000 (a) Vbb=20.000000 imag(Zin1) -10 Vbb=20.500000 -20 Vbb=21.000000 -30 Vbb=21.500000 Vbb=22.000000 -40
-50 0 5 10 15 20 25 30 35 40 freq, MHz m1 freq=100.00000000kHz S(1,1)=0.9999997587 / -0.0932602033 Vbb=22.000000 Max m1 1.00
Vbb=18.000000 0.99 Vbb=18.500000 Vbb=19.000000 0.98 Vbb=19.500000 (b) Vbb=20.000000 0.97 Vbb=20.500000
S(1,1) Vbb=21.000000 Vbb=21.500000 0.96 Vbb=22.000000
0.95
0.94 0 2 4 6 8 10 12 14 16 18 20 freq, MHz
Figure 42 : (a) Reactance (without buffer circuit) and (b) stability when the voltage source at the bases are varied
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Effects of Non-idealities
4.4 Sensitivity Analysis Besides the variation found in section 4.1, there are other process variations that occur for the other practical elements used in the NIC circuit. A simple example is the tolerance in resistor values and other lumped components. Little variations in the values, or parasitics, of these elements could contribute to a poorer performance for the NIC. ADS provides a method for sensitivity analysis, in order to understand the impact that is contributed by each of these elements.
A sensitivity analysis using ADS optimisation simulation was performed upon some of the major elements in the circuit (as shown in Figure 43), namely the bias resistors, chokes, capacitors and filters within the NIC circuit. This analysis was carried across the frequency range 0.1 to 40 MHz. The result of the sensitivity analysis upon the reactance of the NIC circuit (excluding the buffer circuit) is shown in Figure 44. ADS’ normalised sensitivities use the approximate gradient (single-point sensitivity) to predict the percentage change in the response due to a 1 % change in the design variable. Figure 44 would then imply that, for a 1 % increase in inductor value at the left hand side notch (Lnotch1), an increase of about 150 % in the magnitude of the reactance would occur. This, however, is the maximum value recorded across the entire frequency range.
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Var VAR Var VAR S-PARAMETERS Eqn Eqn PARAMETER SWEEP OPTIM VAR64 VAR51 Lmid=470 {o} Rc=2900 {o} ParamSw eep Optim S_Param Ctop=0.086235 R1=59800 {o} Sw eep1 Optim1 SP1 Cb=1 {o} R2=4200 {o} SimInstanceName[1]="Optim1" OptimType=Sensitivity Start=0.1 MHz Cbot=0.1124 Re=100 {o} SimInstanceName[2]="SP1" StatusLevel=4 Stop=40 MHz Rtop=6.5 Vb=20 SimInstanceName[3]= UseAllOptVars=yes Step=0.01 MHz Rbot=2.6 VDC=20 SimInstanceName[4]= UseAllGoals=yes Lbot=220 {o} SimInstanceName[5]= Ltop=470 {o} SimInstanceName[6]= Var Cmid=0.086235 Eqn VAR GOAL GOAL VAR61 Rmid=6.5 Rnotch1=95 {o} Var VAR Var VAR Cnotch1=0.1055 {o} Eqn Eqn Goal Goal VAR66 VAR50 Lnotch1=10 {o} sensS11 sensAbsImagZin Rnotch2=5 {o} beta=175 Expr="mag(S(1,1))" Expr="mag(imag(Zin1))" Var Eqn VAR Lnotch2=5 {o} T1=25 SimInstanceName="SP1" SimInstanceName="SP1" VAR65 Cnotch2=0.001 {o} Min=0 Min=0 Cbsf=5 {o} Max=10 Max=10 Lbsf=1 {o} Weight= Weight= Rbsf=1 {o}
V_DC R L C L R L38 L39 V_DC SRC19 R38 C33 R37 L=Ltop uH V_DC V_DC L=Ltop uH SRC18 Vdc=VDC R=Rc Ohm C=27 pF R=Rc Ohm R= SRC21 SRC20 R= Vdc=VDC Vdc=Vb V Vdc=Vb V
R R R44 R41 R=R1 Ohm R=R1 Ohm
V_DC R L SRC34 R43 R L40 L Vdc=7 R42 R=R2 Ohm L=Lbot uH L41 L=LbotR=R2 uH Ohm R=Rbot R=Rbot
R R118 C C R=Rbuf kOhm C79 C R R R C63 R C78 R68 R69 BJT_NPNR75 C=1 nF C=Cbsf uF R76 C=1 nF R=Rbsf Ohm R=Rbsf Ohm BJT4 R=Rnotch1 Ohm R=Rnotch2BJT_NPN Ohm Vin Model=MRF950a BJT3 Model=MRF949 R L L
C 1 Ref R117 L31 L43 L ap_nms_2N6660_19930601 C77 R=Rbuf kOhmL=Lbot uH L=Lbsf uH L L30 C M2 C=Cnotch1 nF R=Rbot BJT_Model R= L47 L=Lbot uHC36 S1P Var VAR L=Lnotch2 nH Antenna2 Vout MRF949 Eqn R=Rbot C=0.1 uF Term Bf=beta VAR60 R= Term1C R C1=5 C3=0.47 Num=1C101 R R40 L R Z=50 OhmC=0.1 uF Var VAR C R115 Eqn R=Re Ohm L56 R39 R=665 Ohm VAR63 L=Lnotch1 uH C66 R=Re Ohm Rbuf=1000.0 R= C=Cnotch2 uF BJT_Model MRF950a Bf=175
Figure 43 : Sensitivity Analysis for the NIC circuit
200
150
100
50
Reactance Sensitivity Reactance 0
-50 Re R2 R1 Rc Ltop Lbot Cb Lmid Lnotch1 Cnotch1 Rnotch1 Rbsf Lbsf Cbsf Cnotch2 Lnotch2 Rnotch2
sensVariables
Figure 44 : Reactance (without buffer circuit) sensitivity analysis across 0.1 to 40 MHz
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Effects of Non-idealities
The simulation was repeated for the range 3-10 MHz, 10-12 MHz and 12-40MHz in order to identify the frequency for which the filter was problematic. Figure 45 (c) shows the range 12-40 MHz for which the filter does not represent a problem. Figure 45 (a) and Figure 45 (b) show that in the range 3 – 12 MHz, the tolerance of filter coefficients must be as low as possible to achieve the expected negation performance.
40
30
20
(a) 10
Reactance Sensitivity Reactance 0
-10 Re R2 R1 Rc Ltop Lbot Cb Lmid Lnotch1 Cnotch1 Rnotch1 Rbsf Lbsf Cbsf Cnotch2 Lnotch2 Rnotch2
sensVariables 200
150
100
50 (b)
Reactance Sensitivity Reactance 0
-50 Re R2 R1 Rc Ltop Lbot Cb Lmid Lnotch1 Cnotch1 Rnotch1 Rbsf Lbsf Cbsf Cnotch2 Lnotch2 Rnotch2
sensVariables 2.0
1.5
1.0
0.5
(c) 0.0 Reactance Sensitivity Reactance -0.5
-1.0 Re R2 R1 Rc Ltop Lbot Cb Lmid Lnotch1 Cnotch1 Rnotch1 Rbsf Lbsf Cbsf Cnotch2 Lnotch2 Rnotch2
sensVariables Figure 45 : Reactance sensitivity analysis across (a) 3-10 MHz , (b) 10-12 MHz and (c) 12-40 MHz respectively
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A deviation of 10 % in Lnotch1 was found to cause a deviation of 30 Ohms in the overall imaginary impedance of the circuit. Thus a minimum tolerance of 10% is recommended for each the filter elements used. It is to be noted that the introduction of the filters affected the input impedance across the 27pF capacitor. As a result, the sensitivity of the negating capacitor might be affected. The sensitivity effect of the filters in relation to the negating capacitor could be investigated further as a future research.
Next, we analyse the sensitivity of the reflection coefficient (as seen from the antenna) due to variations in the lumped components. The sensitivity analysis result (shown in Figure 46) shows the positive impact of the filters in improving stability. However, it also suggests that there is a trade-off involved in the choice of filters, due to interaction between the filters. This complicates the search to design the appropriate filters.
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20 Reflection Coefficient Coefficient Sensitivity Reflection
-0.25 Re R2 R1 Rc Ltop Lbot Cb Lmid Lnotch1 Cnotch1 Rnotch1 Rbsf Lbsf Cbsf Cnotch2 Lnotch2 Rnotch2
sensVariables
Figure 46 : Sensitivity of the reflection coefficient as seen from antenna across 0.1 to 40 MHz.
Figure 46 show that variations in the lumped components do not significantly impact the stability performance of the NIC circuit.
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Effects of Non-idealities
4.5 Chapter Summary The impact of process variation, temperature changes and supply voltage variation upon the performance of the NIC circuit was discussed and simulated in this chapter. It was found that the NIC performance was not heavily affected by these variations. The exception, however, is the variations in the filter component values. A sensitivity analysis indicates that the reactance cancellation ability is severely affected by variation in the filter components. The stability of the circuit, however, is not sensitive to these variations. For the present, however, the filter solution is a workable option, providing filter components are carefully chosen. Since the NIC consists of active circuits, the intrinsic noise in the circuit could pose threats, as it may be at a higher voltage level than the signals received, and thus diminish the usefulness of the NIC matching network. The following chapter explores some of these issues.
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Noise Considerations
OISE is an important consideration in the design of an antenna system. In this chapter, a noise analysis will be performed for a receive antenna system utilizing the NIC circuit in this thesis. Two main categories of noise, namely environmental noise and internal noise will be considered and compared, in order to understand if the device is externally or internally noise limited. The ideal goal for an antenna matching network designer is that the circuit be externally noise limited.
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5 Noise Considerations 5.1 Introduction Noise is an important parameter to consider in the design of an active matching network for an antenna system. Noise generated by active matching circuits can be at a relatively high level and hence ‘drown’ the signals of interest, and thus diminish the benefits of having a NIC circuit which successfully negates the reactance of the antenna. Noise is, essentially, undesired signals. Noise could be segmented into two major categories, namely internal and external noise. External noise arises from sources such as atmospheric noise, man-made noise, etc. From the circuit designer’s perspective, not much could be done about the environmental noise. Internal noise, however, has its origin in circuit imperfections, some components creating more noise than others. Thus, in designing a matching network, it is a goal of the circuit design to mitigate the internal noise of the circuit such that the device is environmentally noise limited. This chapter seeks to understand the impact of the different sources of internal, and external noise and to estimate the typical noise levels contributed by these sources. We can thus conclude whether typical NIC circuits are externally noise limited or whether the internal noise dominates.
5.2 Internal noise Circuits create noise internally. This noise originates in the electronic components themselves. There are three types of internal noise namely Thermal noise, Shot noise and Flicker. Thermal noise (or Johnson-Nyquist noise, as it is sometimes known) is generated from the motion of charge carriers inside an electrical conductor at equilibrium. With or without any applied voltage, there is a random movement of electrons which will produce a noise voltage. This is independent of frequency and thus it is also known as ‘white noise’. Shot noise is produced by active devices. This occurs when the current flow is discontinuous, or when there is a ‘jump’ in current. Shot noise particularly occurs across the junctions of the BJT. It increases when the bias current is increased. It should be noted that flicker noise also appears in active devices, but its effect is negligible at HF frequencies (its amplitude is inversely proportional to frequency).
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As the NIC implementation in this research consists of 2 Bipolar Junction Transistors, a MOSFET and numerous lumped components, only thermal noise and shot noise are to be considered. Advanced Design System 2009’s Harmonic Balance simulation was used to perform a non-linear noise simulation to analyse the internal noise of the circuit. The circuit was setup as in Figure 47. It is to be noted that the temperature was set to 16.85 ˚C in order to achieve the highest accuracy for noise analysis as suggested by ADS’ documentations [37]. Figures 48 and 49 give the specified noise figure and the minimum noise figure of the circuit respectively. The voltage at the receiver (as shown in Figure 50) shows a maximum voltage level of 4.40 nV. This is significantly smaller than typical signals generated by the receiver antenna (c.f. Table 5 in section 3.4). This level, however, must be compared with the environmental noise in order to understand which noise source is limiting its Signal to Noise ratio (SNR).
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req[1]=freq1MHz P_nTone PORT1 Num=1 Ohm Z=50 F P[1]=polar(dbmtow(-117.65),0) V_DC SRC35 Vdc=VDC Ref 1 S1P Antenna3 R R122 R=RcOhm HARMONICBALANCE rder[1]=order =LtopuH =Lbot uH HarmonicBalance HB2 MaxOrder=3 Freq[1]=freq1 MHz O Other= L L60 L R= C C105 C=0.1uF L L30 L R=Rbot R R123 R=Re Ohm Vin BJT_NPN BJT3 Model=MRF949 =5nH C C104 C=0.001 uF R R119 R=5Ohm L L58 L R= OPTIONS R R120 R=RstabOhm emp=16.85 Options Options1 T Tnom=25 =1 =1 uH C C103 uF C=5 HB NOISE CONTROLLER NOISE HB L L59 L R= oiseNode[1]=Vout NoiseCon NC1 N NoisyTwoPort=yes R R121 R=Rstab Ohm R R131 Ohm R=R2 mid=0.086235 V_DC SRC38 Vdc=VbV VAR VAR66 Lmid=470 Ctop=0.086235 Cb=1 Cbot=0.1124 Rtop=6.5 Rbot=2.6 Lbot=470 Ltop=200 C Rmid=6.5 R R132 Ohm R=R1 ar V Eqn b=20 VAR VAR65 Rc=2900 R1=59800 R2=4200 Re=100 V VDC=20 ar C C108 C=1.0nF V Eqn C C111 C=27 pF VAR VAR64 Rstab=1 ar C C107 C=1.0nF V Eqn BJT_Model MRF949 V_DC SRC37 Vdc=VbV R R129 Ohm R=R1 =10uH C C106 C=0.1055nF L L61 L R= R R124 R=95Ohm R R130 R=R2Ohm =LbotuH BJT_NPN BJT4 Model=MRF949 L L62 L R=Rbot R R125 R=Re Ohm =Ltop uH L L63 L R= C C110 C=0.1 uF VAR VAR63 Rbuf=1000.0 R R127 R=Rbuf kOhm R R126 R=Rbuf kOhm ar V Eqn R R133 R=Rc Ohm SNR SNR1 SNR1=snr(Vout,Vout.noise,{1}) R R128 R=665 Ohm ap_nms_2N6660_19930601 M3 V_DC SRC39 Vdc=VDC rder=3 VAR VAR3 o freq1=15 V_DC SRC36 Vdc=7 C C109 C=0.1 uF um=2 ar V Eqn Term Term2 N Ohm Z=50 Vout
Figure 47 : ADS circuit to analyse internal noise
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30
25
20
15 nf
10
5
0 2 4 6 8 10 12 14 16 18 20
noisefreq, MHz Figure 48 : Noise Figure of the NIC circuit
8
7
6
5
NFmin 4
3
2
1 2 4 6 8 10 12 14 16 18 20 noisefreq, MHz
Figure 49 : Minimum Noise Figure of the NIC circuit
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5 m1
4
3
2 Vout.noise, nV Vout.noise, m1 1 noisefreq=11.00MHz Vout.noise=4.403E-9 Max 0 2 4 6 8 10 12 14 16 18 20
noisefreq, MHz Figure 50 : Noise voltage at the output of the NIC circuit due to internal noise
5.3 External noise In the design and planning of any RF communication system, it is important to compare the wanted signal levels with the background signal level, i.e. its external noise. This is because the natural background noise could set a limit on the sensitivity of receiver system (antenna plus radio receiver). A good receiver system is one which its internal noise is below the external noise. i.e. the system is externally noise limited. In the preceding section, we have considered the internal noise sources and its overall noise contribution to the NIC circuit. In this section, we seek to understand the sources of external noise and its impact on the noise level of the NIC. The sources of external noise consist of atmospheric, galactic and man-made noise. The significance or magnitude of these noise sources varies according to the frequency range of the communications system.
Atmospheric noise is generated by a number of sources that are distributed worldwide. They are impulsive in nature and thus span a large frequency and are caused by radiation from lightning discharges, emissions from atmospheric gases and the earth’s surface. Their impact is especially strong at frequencies at and below HF (less than 30 MHz). Lightning across the globe can be ‘heard’ as it travels across continents using the
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Earth’s ionosphere as a waveguide. Furthermore, there is almost continuous lightning activity around the world, approximately 100 lightning strokes happen per second [38]. Thus the impact of atmospheric noise is significant in raising the external noise floor and will thus set a limit on the sensitivity of radio systems.
In the design of an antenna system, there is no need to achieve sensitivity of the system that is better than the external noise. It is therefore important to quantify the external noise according to the operating frequency range targeted. As atmospheric noise is a random process and has variation across localization, time and season, a comprehensive probabilistic model is required. With these consideration, CCIR, or Comité Consultatif International pour la Radio (currently known as ITU) has produced several atmospheric noise models, taken from 16 measurement stations worldwide across a period of 4 years [39]. This model can be used by radio engineers when designing a receive system.
Another source of external noise is galactic or cosmic noise which arises from the sun and other stars. Its impact on earth is dependent on the ionospheric shielding and varies with frequency. In the HF band, its impact is considerably lower than atmospheric and man-made noise and thus can be neglected.
Last but not least, mankind has contributed to an increasing amount of external noise. Man-made radio noise is caused by a variety of sources of which the most significant noise contributor is from electrical equipment. These consist of electrical machinery, spark ignition systems, switching transients, discharge lighting and etc. Since man-made noise originate from man-made technologies, it is dependent on the distance of the noise sources to the antenna system and it occurs at random time with a short duration and random magnitude. Bianchi [40] states that distance, frequency, emitted power, continuous or impulsive nature of the emitted waves, its polarization and modulation are important characteristics that describe man made noise.
Bianchi proceeds further and claims that man made noise has been increasing steadily for the past century or so. This makes sense as devices which utilises Electromagnetics are becoming increasing available, affordable and in a wider range of applications.
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International Telecommunication Union Radiocommunication sector (ITU-R), formerly known as CCIR, has produced numerous recommendations regarding radio noise level. The ITU recommendations include information on noise figures from the different noise sources in order for radio designers to estimate system performance. This information is obtained empirically in different locations in the world at different times. ITU Radio Noise Recommendation P.372-9 [41] provides a noise figure versus frequency graph for the Noise Figure estimation of man-made noise (as shown in Figure 51)
NOTE: This figure is included on page 78 of the print copy of the thesis held in the University of Adelaide Library.
Figure 51 : Median values for man-made noise power (adopted from [41])
From Figure 51, we see that the noise figure of man-made noise increases as frequency decreases. In the HF region, the level is significant and varies depending on localization.
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In comparison with atmospheric noise level, man-made noise’s impact varies with time. At night time, natural noise level is considerably higher than the noise contributed by artificial noise sources. However, during daytime, the atmospheric noise level is significantly lower than man-made noise (25 dB or more). In terms of probability, the external noise of the antenna is dominated by man-made noise most of the time (99.5%).
In the design of an antenna system, the aim is to limit the level of internal noise to below that of its external noise. Here, according to the figure above, a probable estimate for the spot frequency of 10 MHz in rural areas, gives approximately an external noise figure of 40 dB. The precise values are given in Table 7.
Table 7: Man Made Noise according to location. A noise figure of 39.5 dB (bolded) was used for a electric field calculation in Equation 5.7.
Freq (MHz) City (dB) Residential (dB) Rural (dB) Quiet Rural (dB) 3 63.58 59.28 53.98 39.95 4 60.12 55.82 50.52 36.38 5 57.44 53.14 47.84 33.61 6 55.25 50.95 45.65 31.34 7 53.39 49.09 43.79 29.43 8 51.78 47.48 42.18 27.77 9 50.37 46.07 40.77 26.31 10 49.10 44.80 39.50 25.00 11 47.95 43.65 38.35 23.82 12 46.91 42.61 37.31 22.74 13 45.94 41.64 36.34 21.74 14 45.05 40.75 35.45 20.82 15 44.22 39.92 34.62 19.96 16 43.45 39.15 33.85 19.16 17 42.72 38.42 33.12 18.41 18 42.03 37.73 32.43 17.70 19 41.38 37.08 31.78 17.03 20 40.76 36.46 31.16 16.39
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Noise Considerations
Then, a calculation is required to obtain the voltage induced on the antenna by the external noise, which then could be compared against its internal noise and Intermodulation effects. The following formulas as obtained from [41] applies (5.1) pn f a = 0bkt where fa : external noise factor
pn : available noise power from an equivalent lossless antenna k : Boltzmann’s constant = 1.38 X 10-23 J/K
t0 : reference temperature (K) taken as 290 K b : noise power bandwidth of the receiving system (Hz) Equation (5.1) can also be written as: (5.2) an BFP −+= 204 dBW where (5.3) n = log10 pP n
= log10 bB (5.4) and (5.5) log10 tk 0 = −204 Note that Equation (5.2) is based on the assumption that noise is incident on the antenna uniformly in all directions. (This is a reasonable assumption for a small antenna.)
For the case where a short monopole is used along with a perfecting conducting ground plane, the vertical electric field strength is given as below [41]: (5.6) an += log20 MHz BfFE −+ 5.95 dB (µV/m) where
En : field strength in bandwidth b, and
fMHz : centre frequency (MHz).
For the HF frequency range, Fa = 39.5 dB (rural area), f MHz = 10 , and B = 10 log (3000 Hz) for the receiver noise bandwidth. Substituting these into Equation (5.6) gives (5.7) En −= 23.1 dB (µV/m) The voltage induced at the antenna is given by the following formula
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Chapter 5
(5.8) = .hEV eff Substituting (5.7) into (5.8), and assuming that the 2 meters length antenna is perfectly aligned with the noise signals (to obtain the worst case estimate).
− 23.1 (5.9) 20 hEV eff =×== µV736.1210. The voltage induced then would be 1.736 µV.
By placing this voltage as a noise source (with the same magnitude across the 3 – 20 MHz range) at the antenna end, and ‘switching’ off internal noise effects, the voltage at the receiver end of the NIC was simulated and found to have a voltage of 1.797 µV at 10MHz. This calculation and simulation is repeated for every 1 MHz and the noise at the receiver end of the NIC is shown in Figure 52. The results show that man made noise could range from 85.12 nV to 2767 nV depending on the frequency. This value could then be compared with the internal noise simulated in Figure 50. It is to be noted that these values are relevant to rural area application. In cities, however, the man made noise is different and is shown in Figure 53. A comparison would lead to a conclusion that the noise contribution due to environmental noise sources, for both rural or city areas, is higher than the internal noise of the circuit. It is to be noted that the shape of the curve was affected by the filters introduced to achieve stability (Chapter 3).
Figure 52 : Noise voltage at the receiver end of the NIC due to environmental noise in rural areas
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Noise Considerations
Figure 53 : Noise voltage at the receiver end of the NIC due to environmental noise in the cities
5.4 Chapter summary This chapter has described the different noise sources for the NIC circuit, namely its intrinsic internal noise and the environmental noise. This is crucial in order to understand the limitation of the NIC’s noise performance. The results obtained in this chapter leads to an understanding that the circuit’s noise performance is limited mainly by the environmental noise. Figure 54 confirms that conclusion by comparing the internal noise of the circuit with the voltage at the receiver caused by external sources (man-made noise as measured from rural areas), across 3 to 20 MHz.
Figure 54 : Noise voltage at the receiver end of the NIC circuit due to environmental noise and internal noise.
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Chapter 5
The simulated internal noise gives a maximum of 4.40 nV at the receiver which is significantly smaller as compared to the voltage induced by environmental sources which gives a minimum voltage of 85.12 nV at the receiver. However, in order to confirm that the circuit is externally noise limited, the remaining noise component needs to be considered, namely noise from the Intermodulation distortion (IMD) of the signals that enter the antenna system. This is the effect of non-linearity due to the presence of active devices in the circuit. The non-linear behaviour of the NIC and its IMD effects are discussed in the next chapter.
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Noise Considerations
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Chapter 6
Non-linear Analysis
HE ultimate goal of the NIC matching network is to increase the bandwidth of the system and to enable a higher voltage level for the received signals of interest compared to the noise floor. However, a higher bandwidth implies that more signals are received. Active devices which comprise the NIC circuit would interact with these signals non-linearly and thus increase the noise floor or block the signals of interest. In this chapter, a non-linear analysis was performed to model the non-linear behaviour of the NIC and typical signals were used to understand the IMD performance of the NIC.
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Non-linear Analysis
6 Non-linear Analysis
6.1 Introduction “In these circuits (small signal amplifiers), non-linearities are responsible for phenomena that degrade system performance and must be minimised” – Stephen A. Maas [42]
In the preceding chapters, the NIC was shown to be able to provide a broadband match in the HF frequency range and that its noise performance is only limited by the environmental noise. However, NIC circuits utilise active devices in order to achieve positive feedback to negate reactance. Active devices are inherently non-linear and hence this introduces a whole set of non-linear problems which needs to be carefully considered in determining the practicality and usefulness of the NIC matching circuit. In particular, we need to analyse whether Intermodulation products have the potential to mask the signals that we seek to receive.
Non-linear circuits are circuits whereby the superposition principle does not apply. This is due to the non-linear terms in the transfer characteristics of BJTs. This gives rise to problems like Intermodulation distortion, co-channel distortion, desensitization, harmonics and etc. Consequently, frequency domain techniques are unable to provide an acceptable analysis of the circuit behaviour. Agilent Advanced Design System 2009 Harmonic Balance simulation was chosen as the tool to analyse the non-linear behaviour of the NIC circuit.
This chapter examines the non-linear behaviour of the NIC circuit and its consequences. It provides the reader with a numerical model to approximate the NIC’s non-linear response towards incoming signals. This numerical model can then be utilised to estimate the intermodulation products received from broadcast stations. Typical levels of broadcast signals are derived and then used to estimate the expected magnitude of Intermodulation products. These products are then compared with the other unwanted signals described in chapter 5 (i.e. noise).
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Chapter 6
6.2 Intermodulation Distortion and Numerical Modelling The NIC, as shown in chapters 2 and 3, successfully matches an antenna over a wide range of frequencies in the HF frequency region. A larger bandwidth, however, implies that more undesired signals are received into the system. This does not cause problems for a passive circuit. However, NICs consists of active devices such as BJTs, FETS and Op-amps and these active devices introduce non-linear behaviour to the circuit. The non-linearity of the circuit will create new frequencies due to the interaction between the input signals. These signals modulate each other and could significantly increase the noise floor of the system. The noise floor, if raised too high, would reduce the signal to noise ratio significantly. A widely used figure of merit for the non-linear performance of an active circuit is a circuit’s third order intercept point (IIP3). A two tone test was simulated in order to analyse the Intermodulation Distortion (IMD) of the NIC circuit.
The two tone test was used in order to build a numerical model describing the non- linear behaviour of the NIC. Two tones with an input voltage of 3 mV were used. The first tone consists of frequencies 3, 5, 10, 15 and 20 MHz, while the second tone consists of 3.01, 5.01, 10.01, 15.01 and 20.01 MHz. These two tones were used in combination with each other. It is to be noted that the frequencies differ by 0.01 MHz such that the frequency components could be distinguished. These values are representative of typical values of broadcast channels in the HF frequency range. These simulations were performed by using the harmonic balance analysis provided by the ADS software simulation tool (Figure 55 shows the ADS simulation setup). The harmonic balance analysis only required three orders of harmonics in order to achieve similar results, with a maximum deviation of less than 0.8 %, with a seven order analysis. Thus the three order analysis was deemed to be sufficient. Figure 56 shows an example spectrum output a three order analysis is performed on an two tone test with at 15.17 MHz and 15.72 MHz.
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Non-linear Analysis
[1]=polar(0.00625,0) V [1]=polar(0.00625,0) V_nTone SRC1 Freq[1]=freq1 MHz Freq[2]=freq2 MHz V V V[2]=polar(0.00255,0) rder=3 VAR VAR2 freq2=15.72 o freq1=15.17 V_DC SRC18 Vdc=VDC ar V Eqn Ref 1 S1P Antenna2 R R37 R=Rc Ohm HARMONIC BALANCE HARMONIC rder[2]=order HarmonicBalance HB1 MaxOrder=3 Freq[1]=freq1MHz Freq[2]=freq2MHz Order[1]=order O Other= C C36 C=0.1 uF =Ltop =Ltop uH =Lbot =Lbot uH L L39 L R= L L30 L R=Rbot R R39 Ohm R=Re Vin BJT3 Model=MRF949 C C66 uF C=0.001 =5 nH R R76 R=5 Ohm L L47 L R= R R69 Ohm R=Rstab =1 uH C C63 C=5 uF L L43 L R= R R42 Ohm R=R2 R68 Ohm R=Rstab V_DC SRC20 Vdc=Vb V mid=0.086235 R R41 Ohm R=R1 VAR VAR48 Lmid=470 Ctop=0.086235 Cb=1 Cbot=0.1124 Rtop=6.5 Rbot=2.6 Lbot=470 Ltop=200 C Rmid=6.5 ar V Eqn b=20 VAR VAR51 Rc=2900 R1=59800 R2=4200 Re=100 V VDC=20 VAR VAR61 Rstab=1 ar ar C C105 C=1.0 nF V V Eqn Eqn C C33 pF C=27 C C106 C=1.0 nF BJT_Model MRF949 V_DC SRC21 Vdc=Vb V R R44 Ohm R=R1 =10 =10 uH C C77 nF C=0.1055 L L56 L R= R R75 Ohm R=95 R R43 Ohm R=R2 =Lbot =Lbot uH BJT4 L L31 L R= R R40 Ohm R=Re =Ltop =Ltop uH L L38 L R= C108 C=0.1 uF R R118 kOhm R=Rbuf R117 kOhm R=Rbuf VAR VAR62 Rbuf=1000.0 R R38 R=Rc Ohm ar V Eqn R R115 Ohm R=665 M2 V_DC SRC19 Vdc=VDC V_DC SRC34 Vdc=7 C C101 C=0.1 uF Vout um=1 Term Term1 N Z=50 Ohm
Figure 55 : NIC two tone harmonic balance analysis
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Chapter 6
m3 m4 freq=15.17MHz freq=15.72MHz dBm(Vout)=-33.853 dBm(Vout)=-42.025 -20 m3 m4 -40 m5 freq=14.62MHz -60 dBm(Vout)=-135.337
-80
-100
dBm(Vout) -120 m5 -140
-160
-180 0 5 10 15 20 25 30 35 40 45 50 freq, MHz
Figure 56 : Vout spectrum arising from input frequencies of 15.17 MHz and 15.72 MHz
The results of the two tone simulation were used to create a model that characterised the nonlinear behaviour of the NIC based active antenna. The theory of numerical analysis is as follows. For a two-tone test, the voltage V measured at the output of the NIC can be described by the following expression.
= cos + cos + 2 2cos + 2 2cos + cos( + )twwVVatwVatwVatwVatwVaV 13222111 241 212152 3 3 2 + cos( ) +− 1721216 )3cos( + 281 )3cos( + 192 2 + 21 )2cos( twwVVatwVatwVatwwVVa (6.1) 2 2 2 + 110 2 21 )2cos( +− 2111 )2cos( ++ 211212 − 12 )2cos( twwVVatwwVVatwwVVa
V1 and V2 are the amplitude of the two signals at the input to the NIC and w1 and w2 are their frequencies respectively.
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Non-linear Analysis
By using the results of the ADS simulations, the data for the spectra of each combination of input frequencies was calculated. By applying a least squares quadratic fit to this data, the coefficients an (n>2) can be found for a frequency dependent model of the form:
2 2 n 1521423121 +++++= wbwbwwbwbwbba 26 (6.2)
Coefficients a1 and a2 are different from the other coefficients as they have voltage dependence of the form:
0 1 2 2 2 a αα ++= VV )( (6.3) 11 2 1 1 2
0 1 2 2 2 a αα ++= VV )( (6.4) 22 2 2 1 2 A MATLAB program was written to apply the least squares fit to all these coefficients according to Equations 6.2, 6.3 and 6.4. The results of the MATLAB computations are tabulated in Tables 8 and 9. In addition, surface plots were drawn to graphically describe the variation of coefficients with frequency. It is to be noted that the black dots shown in Figures 58 to 60 represents the coefficients of the respective frequency components given in Equation 6.1, while 57 refers to Equation 6.3.
Table 8: Least squares quadratic fit for first 6 coefficients (c.f. Equation 6.1; The relationship between coefficients an and bn is described by Equation 6.2)
a1 a2 a3 a4 a5 a6 2 0 2 0 α1 α1 α 2 α 2 2w1 2w2 w1+w2 w2-w1 -7.24E- -7.24E- - -6.41E- -4.31E- b1 -2.91E-01 -2.91E-01 -5.56E-01 01 01 8.29E+00 01 01 2.41E- -1.92E- b2 2.64E+00 2.14E-01 9.02E-02 5.33E-06 7.06E-02 5.24E-02 01 16 -1.07E- -1.09E- b3 2.41E-01 -1.86E-15 -4.49E-17 9.02E-02 7.06E-02 5.24E-02 16 16 3.19E- b4 3.30E-18 4.23E-17 3.48E-18 1.25E-18 -2.26E-08 2.20E-03 7.85E-04 18 -8.05E- -6.64E- -1.98E- b5 4.51E-18 -1.41E-01 -3.15E-03 -1.58E-07 -3.13E-03 03 03 03 2.85E- -8.05E- -1.98E- b6 5.47E-17 2.66E-18 1.24E-18 -3.15E-03 -3.13E-03 18 03 03
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Chapter 6
Table 9: Least squares quadratic fit for next 6 coefficients (c.f. Equation 6.1; The relationship between coefficients an and bn is described by Equation 6.2)
a7 a8 a9 a10 a11 a12 3w1 3w2 2w1+w2 2w1-w2 2w2+w1 2w2-w1 b1 -1.77E-01 -1.77E-01 -5.14E-01 -5.66E-01 -5.14E-01 -5.68E-01 b2 5.49E-02 1.57E-05 7.39E-02 6.81E-02 5.65E-02 8.03E-02 b3 4.30E-06 5.48E-02 5.65E-02 8.10E-02 7.39E-02 6.93E-02 b4 8.85E-09 -3.24E-06 1.69E-03 2.53E-03 1.69E-03 2.53E-03 b5 -2.01E-03 -1.03E-06 -3.25E-03 -3.20E-03 -2.51E-03 -3.95E-03 b6 -1.65E-07 -2.01E-03 -2.51E-03 -3.99E-03 -3.25E-03 -3.25E-03
This model was then validated by comparing its modelled IMD results with the ADS simulated results (as shown in Table 10). It can be seen that the model and simulation are sufficiently close, in most cases, to provide a model that is representative of the intermodulation introduced by device non-linearity. It is to be noted that the quadratic fit used to model the w2-w1 coefficients (Figure 58b) show some discrepancies between simulated and curve fitted results. Hence a higher order polynomial could be used to curve fit these coefficients. However, as this frequency component can be filtered out in typical HF applications, these discrepancies should not pose any practical problems.
Table 10: MRF949’s IMD modelled and simulated performance when the 15.17MHz signal at 0.00652V interacts with the 15.72MHz signal at 0.00255V).
Freq 2w1 2w2 w1+w2 w2-w1 2w1+w2 2w1-w2 2w2+w1 2w2-w1 Modelled 1.50E- 2.27E- 1.09E- 7.16E- 5.77E- 6.73E- 2.24E- 2.67E- Mag(V) 05 06 05 06 08 08 08 08 ADS 1.32E- 2.00E- 1.03E- 1.03E- 5.36E- 5.41E- 2.09E- 2.11E- Mag(V) 05 06 05 05 08 08 08 08
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Non-linear Analysis
ααα21
1 1 0.9
0.8 0.8
0.7 0.6 0.6
(a) Coef 0.4 0.5 Coef 0.4 0.2 0.3
0 0.2
20 0.1 15 0 10 20 15 5 10 5 w2 w1 ααα01
1
0.9
1 0.8
0.8 0.7
0.6 0.6
(b) Coef 0.4 0.5 Coef
0.2 0.4 0.3 0 0.2 20 15 20 0.1 15 10 0 10 5 5 w2 w1
2 Figure 57 : Surface plots for MATLAB’s least squares quadratic fit to the coefficients of (a)α1 and 0 (b) α1 (as described in Equation 6.3).
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Chapter 6
2w2 0.35
0.3
0.3 0.25
0.2 0.2
(a) Coef 0.1 0.15 Coef
0 0.1
0.05 5
10 20 0 15 15 10
20 5 w1 w2 w2-w1
0.4
0.35
0.6 0.3
0.25 0.4 0.2
0.15 Coef 0.2
(b) Coef 0.1 0 0.05
0 20 -0.05 15 20 15 10 -0.1 10 5 5 w2 w1
Figure 58 : Surface plots for MATLAB’s least squares quadratic fit to the coefficients of (a) 2w2 (i.e. a4) and (b) w2-w1 (i.e. a6) (as described in Equation 6.1).
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Non-linear Analysis
w1+w2
0.6
0.5 0.6 0.4 0.4 0.3
(a) Coef 0.2 0.2 Coef 0 0.1
20 0 15 20 15 10 -0.1 10 5 5 w2 w1
3w1
0.18
0.16
0.14 0.15 0.12
0.1 0.1
Coef 0.08 (b) 0.05 Coef 0.06 0 0.04
20 0.02 20 15 0 15 10 10 -0.02 5 5 w2 w1
Figure 59 : Surface plots for MATLAB’s least squares quadratic fit to the coefficients of (a) w2+w1 (i.e. a5) and (b) 3w1 (i.e. a7) (as described in Equation 6.1).
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Chapter 6
2w1+w2
0.5
0.4 0.5
0.4 0.3 0.3
0.2 0.2 Coef
(a) 0.1 Coef
0 0.1 -0.1
20 0 15 20 15 10 -0.1 10 5 5 w2 w1
2w1-w2
0.6
0.5
0.8 0.4
0.6 0.3 0.4
(b) Coef 0.2 0.2 Coef
0 0.1
-0.2 0 20 20 15 -0.1 15 10 10 -0.2 5 5 w2 w1
Figure 60 : Surface plots for MATLAB’s least squares quadratic fit to the coefficients of (a) 2w1+w2 (i.e. a9) and (b) 2w1-w2 (i.e. a10) (as described in Equation 6.1).
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Non-linear Analysis
6.3 Broadcast Stations
In the preceding section, we have developed a numerical model describing the non- linear effects and the Intermodulation distortion of the NIC circuit. The model from Tables 8 and 9 could be utilised to estimate the intermodulation products received from broadcast stations. The level of possible input signals can be derived from the World Radio TV Handbook [43] and a number of internet sources [44, 45]. Some typical, but relevant, broadcast stations from Australia and internationally is detailed in Table 11. The voltage induced at the receiving antenna was calculated and is shown in Table 11.
Table 11: Typical shortwave broadcast stations’ signals, as received in Adelaide, Australia.
Voltage (V) at Distance EIRP Station Frequency (kHz) receiving (km) (kW) antenna 5995, 7150, 9580, 2300km Radio Australia 11660, 13605, 300 0.00652
15170 2310, 4835 1300km 50 0.00471 National ABC 2485, 5025 2400km 50 0.00255 2325, 4910 1800km 50 0.0034 6170, 7440, 9765, Radio New Zealand 11725, 13660, 3400km 100 0.00255 International 15720 Trans World Radio 9870, 11580 5361km 100 0.00162 Pacific KWHR World Harvest 6120, 9930 9174km 100 0.00094 Radio (Hawaii)
Far East Broadcasting 9670, 11650, Company (Northern 5590 100 0.00155 15380 Mariana Islands)
National Broadcasting Commission of Papua 4890, 9675 2970 100 0.00292 New Guinea 5995, 7405, 9840, Voice of America 11580, 13775, 16778 500 0.00115 15120 4830, 7245, 9655, Radio Thailand 6937 500 0.00279 15370
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Chapter 6
Voice of America (Relay 7110, 9770, 6943 500 0.00279 station – Thailand) 11785, 15265 Radio Singapore 9530 5418 250 0.00253 International (RSI) BBC Far Eastern Relay 3915, 5975 5424 250 0.00252 station (Singapore)
The voltage induced at the antenna (column 5 in Table 11) was calculated as follows. It is to be noted that the distance calculations were made for a path from Adelaide (which has the geographical coordinates 34°55’S 138°35’E) to the respective transmitting base stations. We will assume a receiver with a 2m monopole antenna (typical of short wave reception).
The Electric Field Strength received is given by
EIRP30 E = (6.5) R where E : field strength (V/m) EIRP : Effective Isotropic Radiated Power (W) R : Distance (m)
The voltage induced at an antenna is given by (6.6) = .hEV eff where heff is the effective height of the antenna.
Assuming that the 2 meter short monopole antenna is perfectly aligned with the electric field for a maximum voltage condition (worse case) = EV (6.7) Since (6.8) 1 = lh eff 2 where l = physical length of the antenna (assuming that the antenna is short).
Thus the voltage induced, at the receiving antenna, by the broadcast stations can be calculated from Equation (6.5) and has been listed in Table 11. With these values, and
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Non-linear Analysis the numerical model produced in section 6.2, an estimate of the IMD products can be obtained and is shown in the following section.
6.4 Theoretical or Expected IMD Noise Levels
Intermodulation between signals received can cause major problems by generating artificial signals that interfere with desired weak signals. In the preceding section, typical broadcast signals were calculated to assist in predicting some typical IMD noise to be expected in the real world environment. Amongst frequencies created due to non- linearity, the 2w1-w2 and 2w2-w1 components are the most significant concern. This is because such IMD components are sometimes extremely difficult to filter out due to their proximity, in frequency, to the desired signal frequency.
A wide range of typical broadcast signals were given in section 6.3 and Table 11. From these signals, a matrix was tabulated to discover the worse case combination, such that the strongest 2w1-w2 or 2w2-w1 component that is realised. This value is then compared to other sources of noise, such as the environmental noise and internal noise in chapter 5.
From Table 11 the signals broadcasted by Radio Australia, National ABC, Radio New Zealand International and National Broadcasting Commission of Papua New Guinea will be considered due to their signal strength and frequency. Since Radio Australia’s HF signals are the largest in magnitude, it will be one of the two signals which could give the worse case IMD combination.
The signals broadcasted by Radio Australia are received with a voltage of 6.52 mV on frequencies 9.58, 11.66, 13.6 and 15.17 MHz. These are labelled as signal (a), (b), (c) and (d) respectively. Similarly, the signal by National ABC will be labelled as signal (e). Radio New Zealand International broadcasts at 9.77 MHz is labelled by (f), 11.73 MHz by (g), 13.66 MHz by (h) and 15.72 MHz by (i). The signal from the broadcast station at Papua New Guinea is labelled as signal (j). A matrix, using these notations, is shown in Table 12 along with the highest calculated level of 2w1-w2 or 2w2-w1 component. This is necessary to discover the worse case combination and the IMD behaviour of the NIC.
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Chapter 6
Table 12: Highest 2w1-w2 or 2w2-w1 components due to the different two tone combinations. (The highest value was bolded)
Highest 2w1-w2 (a) (b) (c) (d) or 2w2-w1 (dBm) (e) -137.60 -136.59 -136.63 -137.35 (f) -136.43 -135.47 -135.07 -135.04 (g) -135.70 -134.65 -134.14 -134.00 (h) -135.57 -134.37 -133.73 -133.50 (i) -136.03 -134.59 -133.79 -133.43 (j) -135.30 -134.34 -133.95 -133.94
From Table 12, we can see that there is a tendency for the IMD to become worse as frequency increases. This, however, is not always true as can be seen in row (e), i.e. the combinations involving National ABC signal. The highest IMD signal was the 2w1-w2 component caused by the combination of signal (i) and (d), i.e. the 15.17 MHz and 15.72 MHz signals. In a crowded broadcast spectrum, this is hard to remove with filtering, as the IMD is at 14.62 MHz. This would be closer if the two input signals were nearer in frequency. This IMD level is -133.43 dBm, or 67.34 nV. This, however, is just an estimate value. The simulated value (obtained through ADS) is 54.1 nV (as also shown in Figure 56 and Table 10). In the next section, a comparison will be made between some typical IMD levels with the external noise that was discussed in section 5.3.
6.5 Comparison with External Noise
As discussed in section 5.3, the external noise of the NIC in the HF frequency range varies with frequency. Thus the frequency range of the IMD components is an important consideration in comparing the IMD with the external noise. (Man-made noise was obtained from Figures 52 and Figure 53.) Only the signal interactions from Radio Australia and Radio New Zealand International are considered here as they produce the highest levels of IMD.
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Non-linear Analysis
Table 13: Comparison between IMD levels with environmental noise in rural areas and cities.
IMD Frequency (MHz) IMD Voltage (nV) Rural Noise (nV) City Noise (nV) (2w1-w2 component) 14.62 54.1 1839.4 5463.7 11.59 58.54 2683.5 8216.3 5.82 18.3 85.1 523.4
From Table 13, we see that environmental noise is significantly higher across the entire frequency range for both rural and city areas. This difference increases with frequency and this implies that the IMD effects are lesser of a concern at high frequencies. Therefore, for applications in the frequency range above 10 MHz, IMD’s effect is manageable as the environmental noise is the limiting factor for the NIC matching circuit’s noise performance. Thus overall, the device is noise limited according to its context of application.
In addition, improvements in transistor technology could further improve the non-linear performance of the NIC. The transistor used in this thesis is MRF949. It is published in MRF949’s datasheet that its IP3 level is at +29 dBm at 1 GHz with a collector current of 20 mA. This explains the good non-linear performance for the NIC circuit. Another low noise and highly linear BJT, namely BFP460, was used to check for incremental improvement. The simulation results of both transistors’ Intermodulation products are listed in Table 14 as a means of comparison.
Table 14: Transistor IMD performance comparison. The 2w1-w2 component represents the most significant IMD problem (as bolded).
Freq 2w1 2w2 w1+w2 w2-w1 2w1+w2 2w1-w2 2w2+w1 2w2-w1 MRF949 1.32E-05 2.00E-06 1.03E-05 1.03E-05 5.36E-08 5.41E-08 2.09E-08 2.11E-08 Mag(V) BFP460 1.30E-05 1.99E-06 1.02E-05 1.01E-05 5.22E-08 5.27E-08 2.04E-08 2.06E-08 Mag(V)
From Table 14, we see that there is some reduction in IMD noise level. This is a positive result and implies that as transistors improve in their non-linear performance,
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Chapter 6
IMD will be less of an issue even at lower frequencies. However, some caution should be taken in the choice of the transistor and the NIC’s context of application.
6.6 Chapter summary
In this chapter, a non-linear analysis was performed on the NIC circuit. This has not been done in prior work, but it is necessary because NICs utilise active devices to achieve positive feedback to negate impedance. A numerical model was produced to model the non-linear behaviour of the NIC, which was then used to predict the Intermodulation Distortion products produced due to the input of typical broadcast signals. It was found that the device, at the HF range, is externally noise limited as the IMD’s 2w1-w2 and 2w2-w1 components are at a level at least 4 times lower than the environmental noise. It was also found that by choosing a transistor with a better non- linear performance, the IMD level can be reduced. This implies that improvements in transistor technology could extend the scope of application for future NIC circuits to be used at lower frequencies.
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Non-linear Analysis
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Chapter 7
Conclusion
HIS chapter draws together the conclusions from the work described in this thesis and provides recommendations for further research. It summarises all the factors that are important in the design of an NIC matching network for HF receive antenna systems. In addition, a summary of the original contributions produced from this research is given.
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Conclusion
7 Conclusion
7.1 Results and Conclusions
The concept of Negative Impedance Converters (NIC), since its inception over half a century ago, has attracted the attention of scientists and engineers. Such circuits have the potential to cancel impedance and hence overcome certain limitations with passive circuits. In the case of antennas, the ability to cancel impedance over a wide range of frequency can overcome the limitations of passive matching. This research aims to analyse the feasibility of NICs as matching networks for HF receiver antenna systems, in particular, to be able to provide broadband matching for small (length << wavelength) antenna.
Chapter 1 contains a literature review on the efforts to reduce the size of antennas and, in particular, the use of NICs. In chapter 2, we focus on the definition of Negative Impedance Converters and carry out a linear analysis in order to understand their operation. Through simulation, it is shown that an NIC is able to achieve an effective match over the frequency range 8 – 12 MHz for a 2 meter monopole. This is achieved by negating a single capacitor in order to cancel the capacitive reactance of the 2 meter monopole. By replacing the capacitor with a network of elements, the bandwidth was improved to 11 MHz. However, due to the parallel resonance of the network, this modification had implications for the instability of the NIC circuit.
As NICs are known to be only conditionally stable, this is an issue that has to be carefully analysed. Chapter 3 analysed stability by means of two methods, namely a reflection coefficient analysis and a transient analysis. By applying these methods to the circuit, it was found that the circuit oscillates due to low frequency instability. Steps were then taken to stabilise the circuit and, in particular, the loading of the NIC had to be carefully chosen. In addition, it was found that the DC bias of the transistors needs to be designed, such that the voltage at the base is relatively low (1/10th or smaller) as compared with the supply voltage. Finally, a band-stop filter and a notch filter were introduced to remove frequencies at which the instability occurred. This solution, however, has the drawback that it causes deterioration in the NIC’s bandwidth.
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Chapter 7
It was also found, in chapter 3, that the matching of the NIC circuit to the 50 Ohms of the receiver could be significantly improved by introducing a source follower buffer amplifier at the output of the NIC. Such an approach leverages the fact that an NIC is open circuit stable. The use of a buffer is similar to the methods used in traditional active antennas. In this case, however, the NIC significantly improves performance as the reactance of the monopole is cancelled, which increases output voltage.
The objective of this thesis had been to understand the practical issues of using NICs as antenna matching elements. Therefore, it is important to understand the effects of device variations upon the performance of the circuit. Chapter 4 explores the impact of device variation, temperature changes and supply voltage variation upon the performance of the NIC circuit. It was found that the NIC performance is not heavily affected by these variations. Variations in the filter values, however, have a significant effect on the performance of the NIC. Consequently, careful selection of filter values is necessary for the NIC to perform as simulated. A sensitivity analysis was performed and this confirmed the conclusions made in chapter 4.
In any antenna system, noise can pose a problem if the level is higher than that of the signals to be received. Chapter 5 explores the effects of NIC circuit noise and external environmental noise. The results obtained in this chapter lead to the conclusion that noise performance is limited mainly by the external environmental noise and not by internal noise. Figure 54 confirms the conclusion by comparing the internal noise of the circuit with the voltage at the receiver caused by external sources (man-made noise as measured from rural areas), across the band 3 to 20MHz.
Another possible source of ‘noise’ is the artificial signals produced by the Intermodulation Distortion (IMD) of the signals that enter the antenna system. This is the due to the presence of active devices within the NIC and the large bandwidth. In chapter 6, a non-linear analysis was performed on the NIC circuit. As a result, a numerical model was produced which could then be used to predict the Intermodulation Distortion products due to the input of typical broadcast signals. Among the IMD components, typically the 2w1-w2 and 2w2-w1 components are the ones that cause the
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Conclusion most problems as they can be hard to filter out without removing the desired signals. However, the IMD levels of these components (given a set of typical broadcast stations’ signals as received in Adelaide), were found to be at least 4 times lower than the environmental noise. Therefore, the effect of IMD does not change the conclusion that the NIC in this research is externally noise limited. Indeed, it was also found that by choosing a transistor with a better non-linear performance, the IMD level can be further reduced.
The worked performed in this research could be improved in a number of ways. For instance, other methods of predicting stability, namely Middlebrook’s technique [30] and Rollet’s proviso [31, 32], could be explored. Such an approach yields a better understanding of where in the circuit the instability arise and hence provides some insight into how the instability might be eliminated. It is clear that the use of filtering within the NIC is not an ideal solution and some improved techniques for reducing instability are required if NIC based matching is to be viable.
This thesis has demonstrated that, in a short antenna, an NIC can provide effective antenna matching over a large range of frequencies by eliminating the capacitive reactance of the antenna. The limitations of the NIC, however, mean that it is probably best employed as part of an active antenna and not the total device. Explicitly, it is best combined with a traditional active antenna such as a source follower buffer amplifier. Further, we have demonstrated that possible concerns, such as internal noise and non- linearity, are not an issue for the target application, i.e. a short antenna for HF communications reception.
The major contribution of this work has been the analysis of the application of NICs to HF communication reception. This includes the following: 1) The interaction of the environment with the non-linearity in the NIC circuit. 2) A comparison between the external and internal noise effects 3) The stability of the NICs when operated as matching circuits for these frequencies.
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Chapter 7
HF antennas can be very large and impractical. It is hoped that the work of this thesis has provided some progress towards HF antennas of a manageable size. In particular, such antennas would be useful for HF radios for domestic purposes where antenna size is a major concern.
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Conclusion
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Appendix A
Software Implementation
A Matlab program was written in order to create a numerical model for a non-linear analysis of the NIC circuit, as given in chapter 6. If this model is to be reproduced, this program is to be used concurrently with ADS to obtain data. Then, the data is to be processed according to Equation 6.1, 6.3 and 6.4. Matlab 7.10.0 (R2010a) was used to implement this program.
% Surface model of the voltage independent coefficient of (2w1+w2) term. % BJT used = MRF949, 2 meter monopole antenna % Points chosen = 3, 5, 10, 15, 20 MHz close all;
%% Instructions: % Manually load the excel file (OrderedList) % which consists of columns data for the different signal combinations % File>Importdata>ChooseFile>Sheet1>Next> % Then choose CreateVectorsFromEachColumnUsingColumnNames y=x2w1pw2; % Add the alphabet 'x' to the labels which start with a number % p - represents 'plus' % m - represents 'minus'
% Other examples: % y=x2w1; % 2w1's coefficient
% Declaring the frequency points which relates to the data x2=[3; 5; 10; 15; 20; 3; 5; 10; 15; 20; 3; 5; 10; 15; 20; 3; 5; 10; 15; 20; 3; 5; 10; 15; 20];
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Appendix x1=[3; 3; 3; 3; 3; 5; 5; 5; 5; 5; 10; 10; 10; 10; 10; 15; 15; 15; 15; 15; 20; 20; 20; 20; 20]; x=[x1 x2];
% A surface plot using quadratic fit is implemented in the following: stats = regstats(y,x,'quadratic','beta'); b = stats.beta; % Model coefficients xx1 = linspace(min(x1),max(x1),32); xx2 = linspace(min(x2),max(x2),32); yy=linspace(min(y), max(y), 32); [X1,X2] = meshgrid(xx1,xx2); Y = b(1) + b(2)*X1 + b(3)*X2 + b(4)*X1.*X2 + b(5)*X1.^2 + b(6)*X2.^2; % a= b(1) + b(2)*X11 + b(3)*X22 + b(4)*X11.*X22 + b(5)*X11.^2 + b(6)*X22.^2
% Plotting the surface plot hmodel = scatter3(X1(:),X2(:), Y(:), 5, Y(:), 'filled'); hold on hdata = scatter3(x1,x2,y,'ko','filled'); axis tight xlabel('w1'); ylabel('w2'); zlabel('Coef'); hbar = colorbar; ylabel(hbar, 'Coef'); title('{\bf Coefficient of (2w1+w2) }')
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Appendix
Appendix B
SPICE MODELS
Bipolar Junction Transistors 1) MRF949
Table 15: MRF949 Die Gummel Poon Parameters
Name Value Name Value Name Value IS 4.598E-16 IRB 8.00E-05 TF 1.00E-11 BF 175 RBM 3 XTF 50 NF 0.9904 RE 0.45 VTF 1.2 VAF 22 RC 6 ITF 0.32 IKF 0.08 XTB 0 PTF 32 ISE 1.548E-14 EG 1.11 TR 1.00E-09 NE 1.703 XTI 3 FC 0.9 BR 76.1 CJE 8.70E-13 NR 0.9952 VJE 0.905 VAR 2.1 MJE 0.389 IKR 0.02059 CJC 3.60E-13 ISC 3.395E-16 VJC 0.4907 NC 1.13 MJC 0.2198 RB 8 XCJC 0.43
2) BFP460 .OPTION TNOM=25, GMIN= 1.00e-12 *BFP460 C B E .SUBCKT BFP460 1 2 3
CBEPAR 22 33 1.875E-013 CBCPAR 22 11 1.48E-013 CCEPAR 11 33 8.007E-016 LB 22 2 9.312E-010 LE 33 3 3.981E-010 LC 11 1 4.664E-010 CBEPCK 2 3 1.154E-016 CBCPCK 2 1 1.617E-014 CCEPCK 1 3 2.749E-015 Q1 11 22 33 4 M_BFP460
.MODEL M_BFP460 NPN(
+ IS = 1.221E-016
+ BF = 187.3
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Appendix
+ NF = 1.005 + VAF = 37.95
+ IKF = 0.5364
+ ISE = 6.757E-014
+ NE = 2.312
+ BR = 14.19 + NR = 1.004 + VAR = 2.455 + IKR = 0.0866 + ISC = 1.335E-015 + NC = 1.5 + RB = 5.708 + IRB = 0 + RBM = 1.968 + RE = 0.2919 + RC = 1.067 + XTB = -0.001 + EG = 1.11 + XTI = 5 + CJE = 3.967E-013 + VJE = 0.4605 + MJE = 0.4485 + TF = 4.702E-012 + XTF = 18.02 + VTF = 3.248 + ITF = 0.8641 + PTF = 0.1 + CJC = 2.777E-013 + VJC = 0.6477 + MJC = 0.2943 + XCJC = 0.7031 + TR = 2.703E-006 + CJS = 3.01E-013 + MJS = 0.08335 + VJS = 0.1506 + FC = 0.5 + KF = 0 + AF = 1) ***************************************************************
.ENDS BFP460
Field Effect Transistors 1) 2N6660
*2N6660 MODEL * .MODEL 2N6660 NMOS (LEVEL=3 RS=0.36 NSUB=1.0E15 +DELTA=0.1 KAPPA=0.0506 TPG=1 CGDO=6.343E-10 +RD=0.43 VTO=1.600 VMAX=1.0E7 ETA=0.0223089 +NFS=6.6E10 TOX=1.0E-7 LD=1.698E-9 UO=862.425 +XJ=6.4666E-7 THETA=1.0E-5 CGSO=9.09E-9 L=2.5E-6 +W=5.0E-3) .ENDS
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