Power Combiners, Impedance Transformers and Directional Couplers

Home , Balun, Electrical length, Voltage transformer

By Andrei Grebennikov Infineon/DICE

any RF applica- Transmission-Line Transformers This is the first of a multi- tions require and Combiners part article that provides a Mpower combiners The transmission-line transformers and textbook-style review of an or dividers, impedance combiners can provide very wide operating important group of RF transformers and direc- bandwidths and operate up to frequencies of 3 circuits used in applications tional couplers. In the GHz and higher [1, 2]. They are widely used in such as power amplifiers, case of combiners, it is matching networks for antennas and power antenna systems and critical, particularly at amplifiers in the HF and VHF bands, in mixer measurement systems higher frequencies, that circuits, and their low losses make them espe- the correct types are used cially useful in high power circuits [3, 4]. to achieve the desired power performance Typical structures for transmission-line trans- when combing individual active devices to formers consist of parallel wires, coaxial achieve higher power. cables or bifilar twisted wire pairs. In the lat- The methods for configuration of the com- ter case, the characteristic impedance can eas- biners or dividers differ, depending on the ily be determined by the wire diameter, the operating frequency, frequency bandwidth, insulation thickness, and, to some extent, the output power, and size requirements. Coaxial twisting pitch [5, 6]. For coaxial cable trans- cable combiners with ferrite cores are often formers with correctly chosen characteristic used to combine the output powers of power impedance, the theoretical high frequency amplifiers intended for wideband applica- bandwidth limit is reached when the cable tions. The device output impedance is usually length comes in order of a half wavelength, low at high power levels; so, to match this with the overall achievable bandwidth being impedance with a standard 50-ohm load, coax- about a decade. By introducing the low-loss ial-line transformers with specified high permeability ferrites alongside a good impedance transformation are used. For nar- quality semi-rigid coaxial or symmetrical strip row-band applications, the N-way Wilkinson cable, the low frequency limit can be signifi- combiners are widely used due to their simple cantly improved providing bandwidths of sev- practical realization. For microwaves, the size eral or more decades. of combiners should be very small and, there- The concept of a broadband impedance fore, the hybrid microstrip combiners (includ- transformer consisting of a pair of intercon- ing different types of the microwaves hybrids nected transmission lines was first disclosed and directional couplers) are commonly used and described by Guanella [7, 8]. Figure 1(a) to combine output powers of power amplifiers shows a Guanella transformer system with or oscillators. In this paper, a variety of differ- transmission line character achieved by an ent combiners, impedance transformers and arrangement comprising one pair of cylindri- directional couplers for application in RF and cal coils that are wound in the same sense and microwave transmitters is given with descrip- are spaced a certain distance apart by an tions of their schematics and operational prin- intervening dielectric. In this case, one cylin- ciples. drical coil is located inside the insulating

20 High Frequency Electronics High Frequency Design COMBINERS & COUPLERS

Figure 1 · Schematic configurations of Guanella 1:1 and 4:1 transformers. cylinder and the other coil is located Figure 2 · Schematic configurations of a coaxial cable transformer. on the outside of this cylinder. For the currents flowing through both windings in opposite directions, the corre- in the two times higher impedance currents—flowing in both transmis- sponding flux in the coil axis is negli- 2Z0 at the input and two times lower sion line inner and outer conductors gibly small. However, for the currents impedance Z0/2 at the output. By in phase, and in the same direction— flowing in the same direction through grounding terminal 4, such a 4:1 are suppressed, and the load may be both coils (common-mode), the latter impedance transformer provides balanced and floating above ground may be assumed to be connected in impedance matching of the balanced or balanced with a center tap ground- parallel, and a coil pair represents a source to the unbalanced load. In this ed load, thus operating as a balun [9, considerable inductance for such cur- case, when terminal 2 is grounded, it 10]. If the characteristic impedance of rents and acts like a choke coil. With performs as a 4:1 unun (unbalanced- the transmission line is equal to the terminal 4 being grounded, such a 1:1 to-unbalanced transformer). With a terminating impedances, the trans- transformer provides matching of the series-parallel connection of n coil mission is inherently broadband. If balanced source to unbalanced load pairs, each having the characteristic not, there will be a dip in the and is called a balun (balanced-to- impedance Z0, the input impedance is response at the frequency at which unbalanced transformer). In this equal to nZ0 and the output the transmission-line is a quarter- case, if terminal 2 is grounded, it rep- impedance is equal to Z0/n. Since wavelength long. resents simply a delay line. In a par- Guanella adds voltages that have A coaxial cable transformer with ticular case, when terminals 2 and 3 equal delays through the transmis- the physical configuration and equiv- are grounded, the transformer per- sion lines, such a technique results in alent circuit representation shown in forms as a phase inverter. A series- the so called equal-delay transmis- Figures 2(a) and 2(b), respectively, parallel connection of a plurality of sion-line transformers. consists of the coaxial line arranged these coil pairs can produce a match The simplest transmission-line is inside the ferrite core or wound between unequal source and load a quarter-wave transmission line around the ferrite core. Due to its resistances. whose characteristic impedance is practical configuration, the coaxial Figure 1(b) shows a 4:1 chosen to give the correct impedance cable transformer takes a position impedance (2:1 voltage) transmis- transformation. However, this trans- between the lumped and distributed sion-line transformer where the two former provides a narrow-band per- systems. Therefore, at lower frequen- pairs of cylindrical transmission line formance valid only around frequencies its equivalent circuit represents coils are connected in series at the cies for which the transmission line is a conventional polarity reversing input and in parallel at the output. odd multiples of a quarter wave- low-frequency transformer shown in

For the characteristic impedance Z0 length. If a ferrite sleeve is added to Figure 2(c), while at higher frequency of each transmission line, this results the transmission line, common-mode it is a transmission line with the

22 High Frequency Electronics High Frequency Design COMBINERS & COUPLERS

Figure 3 · Low frequency models of a 1:1 coaxial cable transformer. Figure 4 · Schematic configurations of the Ruthroff 1:4 impedance transformer. characteristic impedance Z0 shown in Figure 2(d). The advantage of such a transformer is that the parasitic interturn capacitance determines its characteristic ⎡ ⎛ 2l⎞ ⎤ impedance, whereas in the conventional wire-wound L = 2l ⎢ln⎜ ⎟ − 1⎥ nH (2) m ⎣ ⎝ ⎠ ⎦ transformer with discrete windings this parasitic capaci- r tance has a negative effect on the transformer frequency response performance. where l is the length of the coaxial cable in cm and r is the

When RS = RL = Z0, the transmission line can be con- radius of the outer surface of the outer conductor in cm sidered a transformer with a 1:1 impedance transforma- [4]. tion. To avoid any resonant phenomena, especially for The use of high permeability core materials results in complex loads, which can contribute to the significant shorter transmission lines. If a toroid is used for the core, output power variations, as a general rule, the length l of the magnetizing inductance Lm is obtained by the transmission line is kept to no more than one-eighth λ A of wavelength min,or = πµ2 e Lm 4 n nH (3) Le λ l ≤ min (1) 8 where n is the number of turns, µ is the core permeability, Ae is the effective cross-sectional area of the core in λ 2 where min is the minimum wavelength in the transmis- cm , and Le is the average magnetic path length in cm sion line corresponding to the highest operating frequen- [11]. cy fmax. Considering the transformer equivalent circuit shown The low-frequency bandwidth limit of a coaxial cable in Fig. 2(a), the ratio between the power delivered to the 2 transformer is determined by the effect of the magnetiz- load PL and power available at the source Ps = Vs /8Rs ing inductance Lm of the outer surface of the outer con- when RS = RL can be obtained from ductor according to the equivalent low-frequency trans- 2 former model shown in Figure 3(a), where the transmis- P ()2ωL L = m (4) sion line is represented by the ideal 1:1 transformer [4]. P 2 + ()ω 2 S RLSm2 The resistance R0 represents the losses of the transmission line. An approximation to the magnetizing induc- which gives the minimum operating frequency fmin for a tance can be made by considering the outer surface of the given magnetizing inductance Lm, taking into account the coaxial cable to be the same as that of a straight wire (or maximum decrease of the output power by 3 dB, as linear conductor), which, at higher frequencies where the skin effect causes the current to be concentrated on the R f ≥ S (5) min π outer surface, would have the self-inductance of 4 Lm

24 High Frequency Electronics High Frequency Design COMBINERS & COUPLERS

Figure 5 · Schematic configurations of a 4:1 coaxial cable transformer.

A similar low-frequency model for Figure 6 · Schematic configura- a coaxial cable transformer using tions of a 9:1 coaxial cable trans- twisted or parallel wires is shown in former. Figure 3(b) [4]. Here, the model is symmetrical as both conductors are exposed to any magnetic material where RS is the source resistance and and therefore contribute identically RL is the load resistance. to the losses and low-frequency per- Figure 4(b) shows an impedance formance of the transformer. transformer acting as a phase invert- An approach using a transmission er, where the load resistance is line based on a single bifilar wound included between terminals 1 and 4 coil to realize a broadband 1:4 to become a 1:4 balun. This technique impedance transformation was intro- is called the bootstrap effect and duced by Ruthroff [12, 13]. In this doesn’t have the same high frequency case, by using a core material of suf- response as Guanella equal-delay ficiently high permeability, the num- approach because it adds a delayed ber of turns can be significantly voltage to a direct one [14]. The delay reduced. Figure 4(a) shows the circuit becomes excessive when the trans- schematic of an unbalanced-to-unbal- mission line reach a significant frac- anced 1:4 transmission line trans- tion of a wavelength. former where terminal 4 is connected Figure 5(a) shows the physical to the input terminal 1. As a result, implementation of the 4:1 impedance for V = V1 = V2, the output voltage is Ruthroff transformer using a coaxial twice the input voltage, and the cable arranged inside the ferrite core. transformer has a 1:2 voltage step-up At lower frequencies, such a trans- ratio. As the ratio of input voltage to former can be considered an ordinary input current is one-fourth the load 2:1 voltage autotransformer. To voltage to load current, the trans- improve the performance at higher former is fully matched for maximum frequencies, it is necessary to add an power transfer when RL = 4RS, and additional phase-compensating line the characteristic impedance of the of the same length shown in Fig. 5(b), transmission line Z0 is equal to the resulting in a Guanella ferrite-based geometric mean of the source and 4:1 impedance transformer. In this load impedances: case, a ferrite core is necessary only for the upper line because the outer = ZRR0 SL (6) conductor of the lower line is grounded at both ends and no current is flowing through it. A current I driven impedance coaxial cable transformer into the inner conductor of the upper shown in Figuer 6(b). The character- line produces a current I that flows in istic impedance of each transmission the outer conductor of the upper line, line is specified by the voltage resulting in a current 2I flowing into applied to the end of the line and the the load RL. Because the voltage 2V current flowing through the line and from the transformer input is divided is equal to Z0. in two equal parts between the coax- By using the transmission-line ial line and the load, such a trans- baluns with different integer-transformer provides impedance transfor- Figure 7 · Schematic configuration formation ratios in certain connec- mation from RS = 2Z0 into RL = Z0/2, of an equal-delay 2.25:1 unun. tion, it is possible to obtain the frac- where Z0 is the characteristic tional-ratio baluns and ununs [2, 19, impedance of each coaxial line. The 20]. Figure 7 shows a transformer bandwidth extension for the Ruthroff ments for the 3:1 voltage coaxial configuration for obtaining an transformers can also be achieved by cable transformers, which produce impedance ratio of 2.25:1, which con- using transmission lines with step- 9:1 impedance transformation. A cur- sists of a 1:1 Guanella balun on the function and exponential changes in rent I driven into the inner conductor top combined with a 1:4 Guanella their characteristic impedances [15, of the upper line in Figure 6(a) will balun where voltages on the left- 16]. To adopt this transmission line cause a current I to flow in the outer hand side are in series and on the transformer for microwave planar conductor of the upper line. This cur- right-hand side are in parallel [19]. applications, the coaxial line can be rent then produces a current I in the In this case, the left-hand side has replaced by a pair of stacked strip outer conductor of the lower line, the higher impedance. In a matched conductors or coupled microstrip resulting in a current 3I flowing into condition, this transformer should lines [17, 18]. the load RL. The lowest coaxial line have a high frequency response simi- Figure 6 shows similar arrange- can be removed, resulting in a 9:1 lar to a single transmission line. By

HFeLink 166 High Frequency Design COMBINERS & COUPLERS

Figure 9 · Coaxial cable combiner.

Figure 8 · Schematic configurations of a fractional Figure 10 · Two-cable hybrid combiner with grounded 1:2.25 impedance transformer. ballast resistor and load. grounding the corresponding terminals (shown by dashed a load impedance of 112.5 Ω. line), it becomes a broadband unun. Different ratios can By using the coaxial cable transformers, the output be obtained with other configurations. For example, using powers from two or more power sources can be combined. a 1:9 Guanella balun below the 1:1 unit results in a 1.78:1 Figure 9 shows an example of such a transformer, com- impedance ratio, whereas, with a 1:16 balun, the bining two in-phase signals when both signals are deliv- impedance ratio becomes 1.56:1. ered to the load RL and no signal will be dissipated in the On the other hand, the overall 1:1.5 voltage trans- ballast resistor R0 if their amplitudes are equal [12]. The former configuration can be achieved by using the cas- main advantage of this transformer is the zero longitudi- cade connection of a 1:3 voltage transformer to increase nal voltage along the line for equal input powers; as a the impedance by 9 times, and a 2:1 voltage transformer result, no losses occur in the ferrite core. When one input to decrease the impedance by 4 times, which block signal source (for example power amplifier) defaults or schematic is shown in Figure 8(a) [20]. The practical con- disconnects, the longitudinal voltage becomes equal to figuration using coaxial cables and ferrite cores is shown half a voltage of another input source. For this trans- in Figure 8(b). Here, the currents I/3 in the inner con- former, it is possible to combine two out-of-phase signals ductors of two lower lines cause an overall current 2I/3 in when the ballast resistor is considered the load, and the the outer conductor of the upper line, resulting in a cur- load resistor in turn is considered the ballast resistor. The rent 2I/3 flowing into the load RL. A load voltage 3V/2 is schematic of another hybrid coaxial cable transformer out of phase with a longitudinal voltage V/2 along the using as a combiner is shown in Figure 10. The advantage upper line, resulting in a voltage V at the transformer of this combiner is that both the load RL and the ballast input. The lowest line also can be eliminated with direct resistor R0 are grounded. These hybrid transformer-based connection of the points at both ends of its inner conduc- combiners can also be used for the power division when tor, as in the case of the 2:1 and 3:1 Ruthroff voltage the output power from a single source is divided and transformers shown in Figs. 5(a) and 6(b), respectively. If delivered into two independent loads. In this case, the the source impedance is 50 ohms, then the characteristic original load and the two signal sources should be impedance of all three transmission lines should be 75 switched. As it turns out, the term “hybrid” comes not ohms. In this case, the matched condition corresponds to from the fact that the transformer might be constructed

28 High Frequency Electronics High Frequency Design COMBINERS & COUPLERS

Figure 12 · Fully matched and isolated coaxial cable combiner.

RL of R0 = RL = Z0, where Z0 is the characteristic impedance of the each transmission line of the same length.

Baluns Baluns are very important elements in the design of mixers, push-pull amplifiers, or oscillators to link a symmetrical (balanced) circuit to an asymmetrical (unbalanced) circuit. Therefore, it makes sense to discuss their Figure 11 · Coaxial cable combiners with increased circuit configurations and performance in details sepa- isolation. rately.The main requirements to baluns are to provide an accurate 180-degree phase shift over required frequency bandwidth, with minimum loss and equal balanced of two different entities (for example, cable and resistor), impedances. In power amplifiers and oscillators, lack of but just because it is being driven by two signals as symmetry will degrade output power and efficiency. opposed to only one. Consequently, the hybrid trans- Besides, the symmetrical port must be well isolated from former represents a four-port device having two input ground to minimize an unwanted effect of parasitic capac- ports, one sum port and one difference port. The unique itances. characteristic of the hybrid transformer is its ability to A wire-wound transformer with a simplified equiva- isolate the two input signal sources. lent schematic, shown in Figure 13(a), provides an excel- Figure 11(a) shows a coaxial cable two-way combiner lent broadband balun covering in commercial applica- where the input signals having the same amplitudes and tions frequencies from low kHz to beyond 2 GHz. They are phases at ports 2 and 3 are matched at higher frequencies usually realized with a center-tapped winding that pro- when all lines are of the same lengths and RS = Z0 = RL/2 vides a short circuit to even-mode (common-mode) signals = R0/2 [2]. In this case, the isolation C23 between these while having no effect on the differential (odd-mode) sig- input ports can be calculated by nal. Wire-wound transformers are more expensive than the printed or lumped LC baluns, which are more suitable =+⎡ 2 θ ⎤ C2310log 10 ⎣ 4() 1 4 cot ⎦ dB (7) in practical mixer designs. However, unlike wire-wound transformers, the lumped LC baluns are narrow-band as where θ is the electrical length of each transmission line. containing the resonant elements. In order to improve the isolation, the symmetrical ballast Figure 13(b) shows the circuit schematic of a lattice- resistor R0 should be connected through two additional type LC balun that was proposed long ago for combining lines, as shown in Figure 11(b), where all transmission powers in push-pull amplifiers and their delivery to lines have the same electrical lengths. antenna [21]. It consists of two capacitors and two induc- Figure 12 shows a coaxial cable two-way combiner tors, which produce the ±90-degree phase shifts at the that is fully matched and isolated in pairs [2]. Such com- output ports. The values of identical inductances L and biners can be effectively used in high power broadcasting capacitances C can be obtained by VHF FM and VHF-UHF TV transmitters. In this case, for power amplifiers with the identical output impedances RR R and R when R = R = Z /2, it is necessary to L = out L (8) S1 S2 S1 S2 0 ω 0 choose the values of the ballast resistor R0 and the load

30 High Frequency Electronics High Frequency Design COMBINERS & COUPLERS

Figure 14 · Circuit arrangement with two cable transformers for push-pull operation.

their components. 1 In monolithic microwave applica- C = (9) ω 0 RRout L tions where the lumped inductances are usually replaced by transmission ω where 0 is the center bandwidth fre- lines, the designs with microstrip quency, Rout is the balanced output coupled lines, Lange couplers, or mul- resistance, and RL is the unbalanced tilayer coupled structures are very load resistance. When designing this popular. However, the electrical Figure 13 · Different circuit config- circuit, it is important to be confident length of the transmission lines at urations of 1:1 balun. that the operating frequency is well center bandwidth frequency is nor- below the self-resonant frequencies of mally set to a quarter-wavelength, which is too large for applications in wireless communication systems. Therefore, it is very attractive to use the lumped-distributed balun structures, which can significantly reduce the balun size and, at the same time, can satisfy the required electrical characteristics. Figure 13(c) shows such a compact balun with lumped- distributed structure consisting of the two coupled planar microstrip lines and two parallel capacitors, where the input transmission line is grounded at midpoint and the output transmission line is grounded at its one port [22]. Without these capacitors, it is necessary to leave a very small spacing between quarter-wave microstrip lines to achieve a 3-dB coupling between them. However, by optimizing the balun elements around the center bandwidth of 900 MHz, the planar structure of approximately one-sixteenth the size of the conventional quarter-wavelength structure was realized, with spacing S = 8 mils using an FR4 board with substrate thickness of 300 mils. Figure 14 shows the circuit Figure 15 · Schematic configurations of Marchand balun.

arrangement with two coaxial line T2 represents a 1:1 balun, in order to transformers combined to provide a provide maximum power delivery to push-pull operation of the power the load RL, the output equivalent amplifier by creating a balanced-to- resistance of each active device unbalanced impedance transforma- should be two times smaller. tion with higher spectral purity. For a simple 1:1 transmission-line Ideally, the out-of-phase RF signals balun realized with a twisted wire from both active devices will have pair or coaxial cable, the balanced pure half-sinusoidal waveforms, end is isolated from ground only at which contain (according to the the center bandwidth frequency. To Fourier series expansion) only funda- compensate for the short-circuited mental and even harmonic compo- line reactance over certain frequency nents. This implies a 180-degree shift bandwidth around center frequency, between fundamental components a series open-circuited transmission from both active devices and in-phase line was introduced by Marchand, condition for remaining even har- resulting in a compensated balun, monic components. In this case, the the simplified schematic of which is transformer T1 representing a phase shown in Figure 15(a) [23]. In this inverter is operated as a filter for case, at center bandwidth frequency even harmonics because currents when the electrical length of the com- flow through its inner and outer con- pensated line is a quarter-wave- ductors in opposite directions. For length, the load resistance RL is seen each fundamental flowing through unchanged. When this structure is its inner and outer conductors in the realized with coaxial cables, to elimi- same directions, it works as an RF nate unwanted current existing in choke, the impedance of which the outer conductor and correspond- depends on the core permeability. ing radiation, it is necessary to addi- Consequently, since the transformer tionally provide the certain coupling anced terminal is connected to the microstrip line located at the upper metallization level, whereas the balanced load is connected to the microstrip lines located at the lower metallization level. The transmission lines sections in different layers are not isolated from each other. It should be noted that, for a given set of the output balanced and load

unbalanced resistances Rout and RL, Figure 16 · Schematic configura- the characteristic impedances of the tions of coupled-line Marchand outer and inner microstrip lines Z01 balun. and Z02 are not unique, and they can be calculated from between the coaxial cables forming a 1 Z C = 02 (10) transmission line with two outer con- 2 Z01 ductors, as shown in Figure 15(b) [24]. Generally, the shunting reactance of this compensating line can RR ZZ=−4 out L (11) reduce the overall balun reactance 02 01 Z01 about center frequency or reverse its sign depending on the balanced load where C is the coupling factor [26]. resistance, characteristic impedance However, a different choice of Z01 and of the compensating line and cou- Z02 leads to a different frequency pling (characteristic impedance) bandwidth. For example, for Rout = 50 between the outer conductors of two ohms and RL = 100 ohms, it was lines. Hence, a compensating line can found that using the symmetrical create a complementary reactance to directional coupler with Z01 = Z02 = a balanced load and provide an 40.825 ohms results in a frequency ⏐ ⏐ improved match over broader fre- bandwidth of 48.4% with S11 < –10 quency range. At microwaves, wire- dB and amplitude imbalance within wound or coaxial cable transformers 0.91 dB, whereas the frequency band- are usually replaced by a pair of the width of 20.9% with amplitude quarterwave coupled transmission imbalance of less than 1.68 dB will be lines shown in Figure 15(c), thus realized for the nonsymmetrical case resulting in a compact planar struc- when Z01 = 38 ohms and Z02 = 20.42 ture. It should be noted that general- ohms. ly the characteristic impedances of The design of a three-line the coaxial or coupled transmission microstrip balun, which basic lines can be different to optimize the schematic is shown in Figure 16(b), is frequency-bandwidth response. based on the equivalence between a Multilayer configurations make six-port section of three coupled lines the Marchand balun even more com- and a six-port combination of two pact and can provide wide band- couplers [26]. The results of circuit widths due to the tight coupling analysis and optimization show that between coupled-line sections. the spacings between adjacent Modeling and synthesis results of a microstrip lines are so narrow that it two-layer monolithic Marchand is difficult to fabricate a single-layer balun configuration with two-coupled three-line balun. For a two-layer lines, the basic structure of which is three-line balun with two coupled shown in Figure 16(a), is discussed in outer lines on the top metallization [25]. In this configuration, the unbal- level, the spacing between these lines High Frequency Design COMBINERS & COUPLERS

Figure 17 · Schematic configurations of a planar Figure 18 · Broadband parallel-connected coaxial Marchand balun. cable 1:4 baluns. is significantly wider than in a single-layer case. planar Marchand baluns using microstrip lines and However, wider frequency range can be achieved using a lumped capacitors [29]. As an alternative, by employing two-layer three-line balun with two coupled outer lines at two additional inductors at each balanced output and the lower metallization level. For example, the measure- optimum coupling between the grounded strips shown in ments results for this balun show that, being fabricated Figure 17(c), a frequency bandwidth of 53% centered on the Duroid RT5880 substrate, it can provide a fre- around 6.2 GHz with size reduction of 64% over a con- quency range of 2.13 to 3.78 GHz with amplitude imbal- ventional coupled-line Marchand balun is achieved [30]. A ance within 2.12 dB and phase error of less than 4.51°. combined compensation technique uses a series capacitor To improve the performance of multilayer Marchand at the unbalanced input port to improve the matching balun based on microstrip-line technology over frequency bandwidth and inductors at the ground connections to range, a short transmission line can be included connect- minimize amplitude and phase imbalance [31]. ing the two couplers, as shown in Figure 17(a) [27]. This Figure 18 shows the broadband parallel-connected additional short microstrip line effectively compensates coaxial cable balun as an alternative to a series-connected for the amplitude and phase imbalance caused by the dif- Marchand balun [32]. It consists of an unbalanced input ference in even- and odd-mode phase velocities. Besides, coaxial cable connected to a dummy cable that maintains to minimize the balun size, the transmission lines of the symmetry. On the opposite side of the balun, the output coupler can be implemented in meander form that can inner and outer conductors are connected in parallel to give up to 90% reduction in size. As a result, the phase each other, while the input inner and outer conductors of and amplitude differences of the compensated balun were coaxial cables are cross-connected. The right-hand portion within 180 ±10° and 0 ±1 dB over the frequency range of of the balun forms a high impedance balanced load. By 5 to 30 GHz. The compensation can also be implemented means of the cross connection, the high impedance is by employing capacitors at each end of the coupled lines, reduced to a low impedance showing a 4:1 impedance as shown in Figure 4.19(b) [28]. In this case, the capacitor transformation ratio, for example, from a balanced load of will not affect the even-mode but effectively increases 200 ohms to a single-ended 50 ohms. The frequency band- odd-mode phase length, thus resulting in a minimum width of the balun is limited by the shunting effect at amplitude and phase imbalance over certain frequency lower frequencies and near half-wave resonance. These bandwidth. An exact synthesis technique that is widely parallel-connected baluns can provide approximately four used in filter design can be applied to develop and analyze times the operating frequency bandwidth of their series- new classes of miniaturized mixed lumped-distributed connected counterparts as covering in the experiment the

36 High Frequency Electronics High Frequency Design COMBINERS & COUPLERS frequency range from 160 to 4,000 1962. Design Procedure for Single-layer and 14. J. Sevick, “A Simplified Analysis of Two-Layer Three-Line Baluns,” IEEE MHz, a 25:1 bandwidth [33]. The par- the Broadband Transmission Line Trans. Microwave Theory Tech., vol. MTT- allel-connected balun may be realized Transformer,” High Frequency 46, Dec. 1998, 2514-2119. in a variety of configurations, some of Electronics, vol. 3, Feb. 2004, 48-53. 27. K. Nishikawa, I. Toyoda, and T. which are shown in Figure 18(b) and 15. R.T. Irish, “Method of Bandwidth Tokumitsu, “Compact and Broad-Band 18(c) [32, 33]. Extension for the Ruthroff Transformer,” Three-Dimensional MMIC Balun,” IEEE This article will be continued in Electronic Lett., vol. 15, Nov. 1979, 790- Trans. Microwave Theory Tech., vol. MTT- the next issue. 791. 47, Jan. 1999, 96-98. 16. S.C. Dutta Roy, “Optimum Design 28. C.Y. Ng, M. 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