a threesection dual band transformer

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IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 19, NO. 10, OCTOBER 2009 611

A Three-Section Dual-Band Transformer forFrequency-Dependent Complex Load Impedance

Xin Liu , Student Member, IEEE , Yuan’an Liu , Member, IEEE , Shulan Li, Fan Wu, andYongle Wu , Student Member, IEEE

Abstract—In this letter, we propose a practical three-section

dual-band transformer, which can terminate frequency-dependent

complex load impedance at two arbitrary bands simultaneously.

Analytical equations are derived to achieve the exact closed-form

solutions. Numerical examples are examined to verify the va-

lidity. This three-section transformer can be utilized to match

the complex load impedance with unequal values at two different

frequencies, such as microwave amplifiers based on transistors,

mixers, various kinds of antennas, and so forth.

Index Terms—Dual-band, impedance transformer, three-section.

I. INTRODUCTION

WITH the development of personal wireless communi-

cations, people show more and more interest in multi-

band devices. Achieving dual-band impedance matching is the

first step to realize these kinds of circuits. In 2002, Chow et al.

discovered a two-section transformer that can match at a fre-

quency and its first harmonic [1]. Then Monzon made a com-

prehensive analysis, derived its closed-form solutions [2], and

extended this construct to match at two arbitrary frequencies

[3]. Similar work has also been done [4] which made it equiv-alent to a two-section Chebyshev transformer. In [5], complex

conjugated loads were discussed at two frequencies. After that,

the complex impedance matching problem was solved [6] using

two unequal sections.

The transformers mentioned above are not feasible for a

number of matching problems for active devices whose load

impedances vary with frequencies, such as low noise amplifiers

(LNAs),poweramplifiers(PAs), mixers andmicrostripantennas.

Therefore, an improvement was proposed for the matching

problem of amplifiers [7], which has three sections of trans-

mission lines and two shorted stubs. However, this structure is

complicatedandnotconvenienttofabricatebecauseofgroundingstubs.

Manuscript received May 05, 2009; revised July 09, 2009. First publishedSeptember 04, 2009; current version published September 23, 2009. This work was supported by the National High Technology Research and DevelopmentProgram of China (863 Program, 2008AA01Z211), Sino-Swedish IMT-Ad-vanced Cooperation Project ( 2008DFA11780).

The authors are withthe School of Electronic Engineering, Beijing Universityof Posts and Telecommunications, Beijing 100876, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this letter are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/LMWC.2009.2029732

Fig. 1. Configuration of three-section transformer.

In this letter, we propose a three-section transformer for fre-

quency-dependent complex load impedance matching. The de-

sign equations are derived and analyzed. Numerical examples

are presented to verify its validity and exactness.

II. ANALYTICAL DERIVATION

The presented three-section transformer composes a tradi-

tional two-section transformer (section 1 and section 2 with

and as physical lengths, respectively) and an additional sec-tion 3 (with length of ) as illustrated in Fig. 1. The corre-

sponding input impedances and characteristic impedances are

defined as in Fig. 1.

At the two frequencies and (assuming ), the

complex load impedance has and , respec-

tively, which are not equal to each other in most cases. The sub-

scripts and denote the two frequencies and , respec-

tively. To eliminate the frequency-dependent character of the

load impedance, the section 3 is presented to transform the two

unequal values to . For this , we have the following two

choices.

1) Assign . Then the parameters of section

1 and section 2 can be obtained by the approach presentedin [6], in which the transcendental equations can only be

solved by optimization algorithms.

2) Assign , which means a conjugated

relationship between the two values. The following anal-

ysis indicates that the design equations can be solved in

closed-form as long as this conjugated condition can be

satisfied.

Therefore, the whole analysis is organized as the following

two parts. In Section II-A, section 3 transforms the terminated

load to conjugated and . In Section II-B, sec-

tion 1 and section 2 constitute the matching network between

and .

1531-1309/$26.00 © 2009 IEEE


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612 IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 19, NO. 10, OCTOBER 2009

A. Section 3 Transformer

Firstly we use section 3 tomatch between and , which

follows:

(1)

(2)

in which and represent the propagation constant for the

two bands, respectively. Two variables, namely and , are

desired to be solved from these two equations.

Equaling to , and separating it into real and

imagine parts, the equation can be rearranged as

(3)

(4)

1) If : Then (3) and (4) can be

rewritten as

(5)

(6)

Because

(7)

we know that , , simultaneously

and have the same sign. Thencannot be zero, resulting in

(8)

(9)

Therefore, if (7)–(9) can be satisfied simultaneously, the un-

known and can be solved. It should be demonstrated that

(8) is a strict limitation, because the load impedance may not

agree with (8) coincidently in most cases. This solution only

makes sense for the situations that the load is real, namely

, or frequency-independent that and .

If and , it can be calculated that, which means the section 3 is neglectable. In

this case, the presented transformer is exactly the same with the

two-section structure in [3]. The other case that and

rarely happens in practice.

2) If : (3) and (4) can be further

simplified utilizing the equation

(10)

Substituting (10) into (3) and (4), and assuming ,

, it can be deduced that

(11)

(12)

Hereby the section 3 can be designed according to the following

equations:

(13)

(14)

in which can be arbitrary integers but should be chosen care-

fully to make section 3 easy to fabricate in engineering.

B. Two-Section Transformer

Once section 3 is settled, the remaining work is to realize a

two-section transformer between and . As in Fig. 1, we

have the expressions that

(15)

(16)

Assume a real (mostly 50 or 75 in practice), and make

. Equaling to , the design equations

can be established as

(17)

(18)

Each expression is for two frequencies, which makes them tran-

scendental equations in four variables. Generally this kind of

equations can only be solved by numerical methods or optimiza-

tion algorithms. Thanks to the conjugated relationship of ,

just as the analysis in [3], the equations can be rewritten as

(19)

(20)

which result in multiple roots for and . Just as the analysis

in [3], to obtain a compact matching structure, we choose

(21)

Defining , accordingly, the equation for can be

deduced by eliminating in (15) and (16), as follows:

(22)

in which b, c, d, e are defined in Appendix A. This standard

fourth-order equation can be easily solved (see Appendix B),

hence is also determined. The unreasonable roots should beomitted artificially.


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LIU et al.: THREE-SECTION DUAL-BAND TRANSFORMER FOR FREQUENCY-DEPENDENT COMPLEX LOAD IMPEDANCE 613

Fig. 2. Simulated results of dB S11 versus frequency.

III. NUMERICAL EXAMPLES

In this section, a numerical example is given to verify the ap-

proach. A casually chosen example, transistor AT41511 is ex-amined atthe ISM bands of 2.4 GHz and 5.8 GHz. The input im-

pedances are at 2.4 GHz and

at 5.8 GHz [8], which need to be complex conju-

gated matched to 50 . The design procedure is:

1) Calculate using (13), then (14) gives the value of .

From this step it can be found that and

;

2) Given and , it is easy to obtain that is

at 2.4 GHz and

at 5.8 GHz, which agree with the request of conjugated

relationship;

3) From (21) we know that . Then

is obtained from (22) and .

This example is numerically simulated and the diagram of

S11 versus frequency is depicted in Fig. 2. It can be seen that

the three-section transformer matches well at the two required

frequencies, which proves the validity of the presented solution.

Moreover, the bandwidth is more than 300 MHz at each band

(12.5% for 2.4 GHz and 5.8% for 5.8 GHz), which is feasible

for most situations.

IV. CONCLUSION

A novel transformer has been proposed to match the

frequency-dependent complex impedance at two bands simul-taneously. In this letter, we derived the design equations for

three-section transformer, and obtained the closed-form solu-

tions. An example was used to testify the exactness and prove

the validity. The presented transformer solves the dual-band

match problem for frequency-dependent load and is practical

in designing the matching network for amplifiers, antennas and

other devices.

APPENDIX A

VARIABLE DEFINATION OF EQUATION (20)

(23)

(24)

(25)

(26)

APPENDIX B

SOLUTION OF

(27)

in which

(28)

(29)

(30)

(31)

(32)

REFERENCES

[1] Y. L. Chow and K. L. Wan, “A transformer of one-third wavelengthin two sections-for a frequency and its first harmonic,” IEEE Microw.Wireless Compon. Lett., vol. 12, no. 1, pp. 22–23, Jan. 2002.

[2] C. Monzon, “Analytical derivation of a two-section impedance trans-former for a frequency and it first harmonic,” IEEE Microw. WirelessCompon. Lett., vol. 12, no. 10, pp. 381–382, Oct. 2002.

[3] C. Monzon, “A small dual-frequency transformer in two sections,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 4, pp. 1157–1161,Apr. 2003.

[4] J. Sophocles and A. Orfanidis, “Two-section dual-band Chebyshevimpedance transformer,” IEEE Microw. Wireless Compon. Lett., vol.13, no. 9, pp. 382–384, Sep. 2003.

[5] P. Colantonio, F. Giannini,and L. Scucchia,“A new approachto designmatching networks with distributed elements,” in Proc. MIKON’04,May 2004, vol. 3, pp. 811–814.

[6] Y. Wu, Y. Liu, and S. Li, “A dual-frequency transformer for com-plex impedances with two unequal sections,” IEEE Microw. WirelessCompon. Lett., vol. 19, no. 9, pp. 77–79, Feb. 2009.

[7] P. Colantonio, F. Giannini,and L. Scucchia, “Matching network designcriteria for wideband high-frequency amplifiers,” Int. J. RF Microw.Computer-Aided Eng., vol. 15, no. 5, pp. 423–433, Sep. 2005.

[8] Avago Technologies, AT-41511, AT-41533 General Purpose, LowNoise NPN Silicon Bipolar Transistors Data Sheet [Online]. Available:www.avagotech.cn

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