analytical design of dual-band impedance transformer with additional transmission zero

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Published in IET Microwaves, Antennas & PropagationReceived on 2nd November 2013Revised on 18th May 2014Accepted on 22nd May 2014doi: 10.1049/iet-map.2014.0181

120The Institution of Engineering and Technology 2014

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ISSN 1751-8725

Analytical design of dual-band impedance transformerwith additional transmission zeroMing-Lin Chuang

Department of Communication Engineering, National Penghu University of Science and Technology, No. 300,

Liu-Ho Road, Magung City, Penghu, Taiwan

E-mail: [email protected]

Abstract: This study presents a three-stage dual-band impedance transformer, capable of providing good match between twocircuit modules and generating an additional and selectable transmission zero. The input impedances of the output circuitmodule are unequally complex at two uncorrelated frequencies. Among the three stages of the proposed transformer, the firststage transforms the complex impedance of the load to become equal resistance at the two frequencies. The second stagecancels out the susceptance produced by the first stage at the two frequencies concurrently and generates the desiredtransmission zero. The final stage transforms the equal and real impedance to the input circuit module. This study providesanalytical formulation of each stage to obtain the required circuit parameters instead of numerical solution or optimisation.The proposed impedance transformer has up to four configurations, subsequently decreasing the possibility of using anextremely narrow or wide transmission line. Simulations and measurement demonstrate the reliability of the proposedstructure and design formula. The simulation results agree measured results well, and comply with the specifications.

1 Introduction

The emergence of various wireless communication standardshas increased the importance of multi-band radiofrequency(RF)/microwave circuits. One of the critical circuits ismulti-band impedance transformers or multi-band matchingnetworks. The impedance transformer is a widely usedcircuit in RF/microwave systems. In addition to providing agood match between two circuits, an impedance transformeralso plays an important role in filter, amplifier and oscillatordesigns.For the load with real impedance, the three-section

dual-band impedance transformer is first designed tooperate at the fundamental frequency and the first harmonicfrequency [1]. Monzon [2] proposed a two-sectionstepped-impedance transmission line to match a load withequal and real impedances at the two uncorrelatedfrequencies. By using two dual-band impedancetransformers, Fan et al. [3] constructed a wide-bandimpedance transformer. Cogollos et al. [4] combined thesimilar concept and low-pass filter theory to obtain adual-band impedance transformer with a wide matchingbandwidth. Sophocles and Orfanidis [5] and Castaldi [6]developed dual-band impedance transformers withChebyshev response. Tsai et al. [7] proposed a four-sectiondual-band matching network to avoid using an extremelynarrow line. Chuang [8] presented a modified two-sectiontransformer to match a load with unequal impedance.For the load with purely imaginary impedance, Giofre et al.

[9] proposed a concept of impedance buffer, making itpossible to develop a multi-band matching network.

As to the load with complex impedance, Giannini andScucchia [10] developed a dual-stub transformer to match acomplex load at the fundamental frequency and firstharmonic frequency. Wu et al. [11] modified thetwo-section transformer [2] to match a load with equalcomplex impedance. Liu et al. [12] extended to athree-section dual-band transformer to match a load withunequal complex impedance. Wu et al. [13] proposed afour-section dual-band transformer to match source andload with unequal complex impedances. By using thetransmission line and two shunt stubs, Colantonio et al.[14] transformed unequal complex impedances into equalreal impedances. Chuang [15] extended the single-bandshunt stub structure to a dual two-section structure to matcha load with unequal complex impedance. The requiredcircuit parameters are obtained by numerically solving thederived object function of single parameter. This impedancetransformer is applied to design a dual-band poweramplifier [16]. Rawat and Ghannouchi [17] developedT-type and pi-type quarter-wavelength impedance invertersto build a dual-band impedance transformer. By adopting adual-band filter, Fu et al. [18] constructed a dual-bandimpedance transformer. He et al. [19] using a traditionalsingle-band shunt stub transformer to dual-band application.Nikravan and Atlasbaf [20] constructed a T-structuredual-band transformer. In the above two studies, all circuitparameters of the structures are uniquely determined,implying the inability to tune the transformer if the requiredtransmission line width is too wide or too narrow.Although the above studies have successfully achieved a

sufficient match at the desired frequencies, the out-of-band

T Microw. Antennas Propag., 2014, Vol. 8, Iss. 13, pp. 1120–1126doi: 10.1049/iet-map.2014.0181

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response has not yet been considered. For amplifier design,one of the major applications of impedance transformer, thetransformer plays a role of controlling the matchingcondition and amplifier gain. Considering the performanceof the whole transceiver, Wang et al. [21] and Chang andLin [22] connected a transistor amplifier and a notch filterto suppress the harmonic responses. Anand et al. [23] andBen Yishay et al. [24] designed an image-reject amplifierby adding a notch filter to suppress image signal. Theabove studies require an additional notch filters to design asingle-band amplifier that can suppress unwanted signals. Ifthe impedance transformer has the ability to control theamplifier gain as well as generate a controllabletransmission zero, the notch filter is no longer needed.This study proposes a dual-band impedance transformer
that not only connects two circuit modules and providesgood match at the two operating frequencies, but alsoinherently generates an additional transmission zero tosuppress unwanted signals out of the operating band. Thisadditional transmission zero is not available in the previousstudies [13–20]. The considered load has unequal compleximpedance at two uncorrelated frequencies. The proposedimpedance transformer consists of three stages. Each stageis determined individually, and all the circuit parameters areobtained using the derived analytical formulation ratherthan numerical solution or optimisation techniques.Measurements are compared with simulations to validatethe proposed structure and design theory.

2 Transformer structure

Fig. 1 shows the structure of the proposed dual-bandimpedance transformer. The transformer connects twocircuit modules. The input circuit module has an outputimpedance of Z0, that is, source impedance of thetransformer at the two desired frequencies, whereas theoutput circuit module has an input impedance of unequalcomplex value, that is, the load impedance of thetransformer at the two desired frequencies. The proposedtransformer consists of three stages with each one describedas follows.The first stage transforms the two unequal complex

impedances to an input admittance, Yin,I, which has anequal conductance at the two desired frequencies.

Gin,I = Re Yin,I∣∣f1

{ }= Re Yin,I

∣∣f2

{ }(1)

where Yin,I| f1 and Yin,I| f2 denote the input admittances at

Fig. 1 Structure of the proposed three-stage dual-band impedancetransformer

IET Microw. Antennas Propag., 2014, Vol. 8, Iss. 13, pp. 1120–1126doi: 10.1049/iet-map.2014.0181

frequencies f1 and f2, respectively. These two desiredfrequencies are usually uncorrelated with each other. Thesusceptance of Yin, I is then cancelled out by stage II. Twoconfigurations are available to realise the shunt susceptance−jB as discussed in the next section. Stage II also providesan additional transmission zero between the two circuitmodules. Finally, stage III transforms the resulting inputimpedance Zin,II = 1/Gin,I into port 1 which has a real andequal impedance Z0 at the two desired frequencies. Thecharacteristic impedances Zb and Zc, and the electricallengths θb and θc can be determined by Monzon’s study[2]. In this work, all electrical lengths are defined at the firstoperating frequency, f1.

3 Determination of stage I

As shown in Fig. 1, the input impedances of port 2 are ZL1 =RL1 + jXL1 and ZL2 = RL2 + jXL2 at the two frequencies,respectively. In most cases, ZL1≠ ZL2. Substituting theabove impedances to (1) results in a set of complicatedsimultaneous equations that is not easily solved. However,two special choices satisfying (1) leads to a set of relativelycompact simultaneous equations, which can be solved moreeasily. In addition, related stage II is also easier to obtain.The two choices are discussed below.

3.1 A-1 Yin,I|f1 = Y*in,I|f2

The first choice is one in which the input admittances, Yin, I,conjugate each other at the two frequencies. Similar to theprocedure described in [12], the following equation is derived

Za − XL1 tan ua + jRL1 tan uaRL1 + jXL1 + jZa tan ua

= Za − XL2 tanRfua − jRL2 tanRfuaRL2 − jX2 − jZa tanRfua

(2)

Separating the above equation into real part and imaginarypart leads to the following two simultaneous equations

RL2XL2 − RL1XL1

( )tan ua + tanRfua( )

+ Za RL1 − RL1

( )1− tan ua tanRfua( ) = 0 (3a)

Z2a − RL1RL2 − XL1XL2

( )tan ua + tanRfua( )

+ Za XL1 + XL2

( )1− tan ua tanRfua( ) = 0 (3b)

where Rf is the frequency ratio and is defined as Rf≡ f2/f1.The electrical length θa can be expressed as a function of Za

by rearranging (3a)

ua =1

Rf + 1tan−1 Za RL1 − RL2

( )RL1XL2 − RL2XL1

+ np

[ ](4)

Substituting (4) into (3b) yields

Za =RL1RL2 + XL1XL2 +

XL1 + XL2

RL1 − RL2RL1XL2 − RL2XL1

( )√

(5)

Therefore stage I can be designed using (4) and (5).

1121& The Institution of Engineering and Technology 2014

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3.2 A-2 Yin,I|f1 = Yin,I|f2
The second choice to satisfy (1) sets the input admittances,Yin,I, to achieve an equal complex value at the two desiredfrequencies. The required equation is

Za − XL1 tan ua + jRL1 tan uaRL1 + jXL1 + jZa tan ua

= Za − XL2 tanRfua + jRL2 tanRfuaRL2 + jX2 + jZa tanRfua

(6)

Separating the above equation into real and imaginary partsallows us to derive two simultaneous equations as follows

RL2XL1 + RL1XL2

( )tan ua − tanRfua( )

+ Za RL1 − RL2

( )1+ tan ua tanRfua( ) = 0 (7a)

Z2a − RL1RL2 + XL1XL2

( )tan ua − tanRfua( )

+ Za Xa − Xb

( )1+ tan ua tanRfua( ) = 0 (7b)

Rearranging (7a) to express θa as a function of Za leads to

ua =1

Rf − 1tan−1 Za(RL1 − RL2)

RL1XL2 + RL2XL1+ np

[ ], n [ Z (8)

Substituting (8) into (7b) leads to

Za =RL1RL2 − XL1XL2 +

XL1 − XL2

RL1 − RL2RL1XL2 + RL2XL1

( )√

(9)

Stage I can thus be designed by (8) and (9).

4 Determination of stage II – A-1 condition

The proposed impedance transformer not only transforms thetwo unequal complex impedances into the same realimpedance, Z0, at the two desired frequencies concurrently,but also generates an additional and selectable transmissionzero. Therefore stage II is implemented by connecting shuntstubs to generate a virtual short to ground at the main path.Fig. 2 shows the entire transformer, where two shunt stubs

Fig. 2 Three-stage dual-band impedance transformer with dual shunt s


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with uniform impedances are connected on both sides ofthe main path.To cancel out the susceptance produced by stage I, these

two stubs must satisfy the simultaneous equations accordingto the choice of stage I. In addition, to generate anadditional transmission zero, regardless of the cases of stageI, the electrical length, θS1, of the upper open stub is chosento be quarter-wavelength at the desired frequency of thetransmission zero. According to the choice of stage I, thedetermination of stage II is discussed below separately. Thissection discusses the A-1 condition.Corresponding to condition A-1, assume that the resulting

input admittance of stage I is

Yin,I1 = Gin,I + jBin,I at f1 (10a)

Yin,I2 = Gin,I − jBin,I at f2 (10b)

Lengths and impedances of the two shunt stubs must satisfythe following simultaneous equations

1

ZS1tan uS1 +

1

ZS2tan uS2 = −Bin,I at f1 (11a)

1

ZS1tanRfuS1 +

1

ZS2tanRfuS2 = Bin,I at f2 (11b)

uS1 =p

2at fZ (11c)

where fz denotes the desired frequency of the transmissionzero.These three equations have four design parameters,

allowing us the freedom to choose arbitrary θS2 and therequired characteristic impedances of the open stubs arethus solved as

ZS1 = − 1

Bin,I

a

tan uS2 + tanRfuS2(12a)

ZS2 =1

Bin,I

a

tan uS1 + tanRfuS1(12b)

where

a = tan uS1 tanRfuS2 − tanRfuS1 tan uS2 (13)

tubs


Fig. 3 Simulated results with A-1 condition and open-ended stub 2for ZL1 = 30 + j15 Ω at 2.4 GHz and ZL2 = 40− j25 Ω at 5.8 GHz

Transmission is chosen at fz = 8.2 GHz. The solid line denotes |S11| for ZL1and the dashed line denotes |S11| for ZL2. The curves marked with circleand triangle symbols denote |S21| for ZL1 and ZL2, respectively

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If the lower stub is chosen short-ended instead of open-ended,as depicted in the dashed circle in Fig. 2, the requiredsimultaneous equations are rewritten as
1

ZS1tan uS1 −

1

ZS2cot uS2 = −Bin,I at f1 (14a)

1

ZS1tanRfuS1 −

1

ZS2cotRf uS2 = Bin,I at f2 (14b)

The characteristic impedances of the both stubs are thensolved as

ZS1 =w1 cot uS2 − w2 tan uS1

w2Bin,I(15a)

ZS2 =w1 cot uS2 − w2 tan uS1

w1Bin,I(15b)

where

w1 = tan uS1 + tanRfuS1 (16a)

w2 = cot uS2 + cotRfuS2 (16b)

This section presents the numerical results for A-1 conditionsusing Ansoft Designer with the ideal transmission line model.Performance of the proposed transformer is verified bychoosing the input impedance of port 2 arbitrarily anduncorrelated, which are ZL1 = 30 + j15 at 2.4 GHz and ZL2 =40− j25 at 5.8 GHz. Table 1 lists the detailed parameters ofthe designed impedance transformers with A-1 condition,where θS1 = 26.34° at 2.4 GHz is chosen to generate atransmission zero at 8.2 GHz.Figs. 3 and 4 display the frequency responses with

open-ended stub 2 and short-ended stub 2, respectively.Both Figs. 3 and 4 shows the agreement on thespecifications, match at 2.4 and 5.8 GHz, and atransmission zero at 8.2 GHz. Because the completeimpedance of port 2 varied with frequency is unknown, thesimulation is carried using two constant impedances ZL1and ZL2 individually for the whole band. Therefore thefrequency responses are only valid near 2.4 GHz for ZL1and near 5.8 GHz for ZL2, separately. However, |S21| nearthe transmission zero is valid because the virtual groundalways holds, if the shunt open stub satisfies (11c),regardless of load impedance. In Fig. 4, an extratransmission zero generated at 4.9 GHz where the length ofopen-ended stub 2 is about quarter-wavelength. Fig. 6reveals a similar phenomenon. An extra transmission zero is

Table 1 Designed parameters with A-1 condition for ZL1 = 30 +j15 Ω at 2.4 GHz, ZL2 = 40− j25 Ω at 5.8 GHz and the transmissionzero at 8.2 GHz

Open-endedstub 2

Short-endedstub 2

θa(°) θb(°) θc(°) θS1(°) θS2(°) θS1(°) θS2(°)

5.58 52.68 52.68 26.34 44.00 26.34 85.00

Za(Ω) Zb(Ω) Zc(Ω) ZS1(Ω) ZS2(Ω) ZS1(Ω) ZS2(Ω)

46.64 45.21 46.82 111.86 108.83 29.68 25.87


generated at 5.4 GHz, in which the length of short-endedstub 2 is about half-wavelength.

5 Determination of stage II – A-2 condition

Corresponding to section A-2, if the resulting inputadmittance of stage I is

Yin,I1 = Gin,I + jBin,I at f1 (17a)

Yin,I2 = Gin,I + jBin,I at f2 (17b)

The two open stubs must satisfy the same equations as (11),except for that (11b) is replaced by

1

ZS1tanRfus1 +

1

ZS2tanRfuS2 = −Bin,I at f2 (18)

Again, we can choose arbitrary θS2 and the requiredcharacteristic impedances of both open stubs are then

Fig. 4 Simulated results with A-1 condition and short-ended stub 2for ZL1 = 30 + j15 Ω at 2.4 GHz and ZL2 = 40− j25 Ω at 5.8 GHz



Fig. 5 Simulated results with A-2 condition and open-ended stub 2for ZL1 = 30 + j15 Ω at 2.4 GHz and ZL2 = 40− j25 Ω at 5.8 GHz


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solved as
ZS1 =g1 tan uS2 − g2 tan uS1

g2Bin,I(19a)

ZS2 =g2 tan uS1 − g1 tan uS2

g1Bin,I(19b)

where

g1 = tan uS1 − tanRfuS1 (20a)

g2 = tan uS2 − tanRfuS2 (20b)

Similar to the discussion in Section 4, the lower stub can beshort-ended. The corresponding equations are the same as(14), except for that (14b) is replaced by

1

ZS1tanRfuS1 −

1

ZS2cotRf uS2 = −Bin,I at f2 (21)

The required characteristic impedances of the both stubs aresolved as

ZS1 =f1 cot uS2 − f2 tan uS1

f2Bin,I(22a)

ZS2 =f1 cot uS2 − f2 tan uS1

f1Bin,I(22b)

where

f1 = tan uS1 − tanRfuS1 (23a)

f2 = cot uS2 − cotRfuS2 (23b)

The impedance transformer with A-2 condition to transform thesame load impedance as in Section 5 is also designed andverified. The other specifications are the same. Table 2 liststhe detailed parameters and Figs. 5 and 6 show the relatedfrequency responses. A good match is achieved at the twospecified frequencies for the target impedances, and thetransmission zeros occur at 8.2 GHz for both impedances asdesired. Two extra transmission zeros are generated at 1.89and 5.68 GHz in Fig. 5 because the length of open-endedstub 2 is odd multiple of quarter-wavelength at thesefrequencies. The redundant transmission zero at 5.68 GHzcompresses the bandwidth of matching at 5.8 GHz. Fig. 6also reveals an extra transmission zero at 4.91 GHz, wherethe short-ended stub 2 has a length of half-wavelength andgenerates virtual ground at the main path.

Table 2 Designed parameters with A-2 condition for ZL1 = 30 +j15 Ω at 2.4 GHz, ZL2 = 40− j25 Ω at 5.8 GHz and the transmissionzero at 8.2 GHz

Open-endedstub 2

Short-endedstub 2

θa(°) θb(°) θc(°) θS1(°) θS2(°) θS1(°) θS2(°)

46.64 52.68 52.68 26.34 114.00 26.34 88.00

Za(Ω) Zb(Ω) Zc(Ω) ZS1(Ω) ZS2(Ω) ZS1(Ω) ZS2(Ω)

50.94 67.12 60.66 22.94 122.49 141.84 141.85


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In this example, the configuration with open-ended stub 2has narrow bandwidth at 5.8 GHz. Fortunately, other threeconfigurations can still be applied to avoid such narrowbandwidth. The other possible limitation is that the requiredtransmission lines may have high impedance for someparticular load. If designers simply want to design adual-band impedance transformer without any controllabletransmission zero, they can choose arbitrary length of thestub 1 and avoid the above limitations. If designers want tointroduce this selectable transmission zero, they can chooseone of the four proposed configurations. That reduces thepossibility of using high impedance transmission line orobtaining very narrow bandwidth.

6 Experimental results

This section presents the experimental results to validate thedesign procedure. The fabricated transformers are etched ona 1.8 mm thick FR4 substrate with a dielectric constant of4.35 and a loss tangent of 0.016. Performances of thefabricated transformers are measured using R&S ZVB8network analyser.To emulate the frequency dependent and uncorrelated input

impedance of port 2, two identical transformers are

Fig. 6 Simulated results with A-2 condition and short-ended stub 2for ZL1 = 30 + j15 Ω at 2.4 GHz and ZL2 = 40− j25 Ω at 5.8 GHz



Table 3 Physical parameters (unit: mm) of the fabricatedtransformer with A-1 condition and open-ended stub 2 for ZL1 =56− j17.68 Ω at 0.9 GHz, ZL2 = 33− j3.68 Ω at 2.2 GHz and thetransmission zero at 3.5 GHz

La Lb Lc LS1 LS29.9 28.6 27.1 11.4 66.9Wa Wb Wc WS1 WS23.3 3.0 3.2 1.0 1.1

Fig. 8 Simulated results and measured results of the fabricateddual-band transformer for ZL1 = 56− j17.68 Ω at 0.9 GHz andZL2 = 33− j3.28 Ω at 2.2 GHz

Solid line denotes measured results and the dashed line denotes simulatedresults

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manufactured and connected to different loads made by aseries-connected lumped resistor and capacitor. Moreover,because the parasitic effects because of via-holes andsoldering are serious at a higher frequency, the fabricatedtransformer is designed to operate at lower frequencies, thatis, 0.9 and 2.2 GHz. The frequency of transmission zero ischosen as 3.5 GHz. The input impedance of port 2 at 0.9GHz is emulated by a 56 Ω SMD resistor series connectedto a 10 pF SMD capacitor. Meanwhile, the input impedanceof port 2 at 2.2 GHz is emulated by a 33 Ω SMD resistorseries connected to a 22 pF SMD capacitor. Thecorresponding impedances at the two specified frequenciesare 56− j17.68 and 33− j3.28 Ω, respectively.The transformer with A-1 condition and open-ended stub 2

is manufactured and measured because it has shortertransmission lines and stubs, implying a smaller circuit size.Table 3 lists the physical parameters of the designedtransformers. In this table, Wa, Wb, Wc, WS1 and WS2

correspond to line widths for Za, Zb, Zc, ZS1 and ZS2,whereas La, Lb, Lc, LS1 and LS2 correspond to line lengthsfor θa, θb, θc, θS1 and θS2, respectively. Fig. 7 displays thephotographs of the fabricated transformer compared withguided wavelength of a 50 Ω microstrip at 0.9 GHz. Thesize of the transformer is ∼0.38× 0.15l2g. Since inputimpedance of port 2 is emulated by a lumped resistor andcapacitor, only reflection is measured and displayed. Thetransmission response is not measured because there is noinstrument having port impedance of unequal complexvalue at two uncorrelated frequencies. However, thequarter-wavelength shunt open stub assures the generationof transmission zero. Fig. 8 compared the measuredreflection with the numerical responses. Since the loadimpedances at the two frequencies are unequal, Fig. 8 onlydisplays reflection for Z1 = 56− j17.68 Ω between 0.4 and1.4 GHz and reflection for Z2 = 33− j3.28 Ω between 1.8

Fig. 7 Photograph of the fabricated dual-band transformer

56 Ω resistor series connected with a 10 pF capacitor are at the end of thetransformer to emulate ZL1 = 56− j17.68 Ω. Meanwhile, a 33 Ω resistorseries connected with a 22 pF capacitor are at the end of the other identicaltransformer to emulate ZL1 = 33− j3.68 Ω


and 2.8 GHz. Both numerical results and measured resultsshow an acceptable match. A slightly large deviationbetween numerical results and measured results at higherfrequency may arise from serious parasitic effects whichchange the value of the load impedance.

7 Conclusions

This work presents a three-stage dual-band impedancetransformer to match an output circuit module, which hasunequal complex impedances at two uncorrelatedfrequencies. Importantly, besides providing a goodmatching between two circuit modules, this dual-bandimpedance transformer generates an additional andselectable transmission zero to suppress unwanted signals orimprove the selectivity of the whole RF circuit. In addition,the required circuit parameters of each stage are obtained byanalytical formulations instead of time-consumingnumerical solutions or optimisation techniques. Up to fourpossible circuit configurations are available leading to ahigher probability of not using an extremely narrow orextremely wide transmission line. The simulated andmeasured results validate the proposed structures and designprocedure.

8 Acknowledgment

This work was supported by the National Science Council,Taiwan, under Contract NSC-100-2221-E-346-009.

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analytical design of dual-band impedance transformer with additional transmission zero

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