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T-section dual-band impedance transformerfor frequency-dependent compleximpedance loads

M.A. Nikravan and Z. Atlasbaf 

A simple and successful design for matching complex impedance loads

with different values at two different frequencies is presented. The

explicit closed-form equations for the transformer parameters are

derived analytically. The validity of the proposed design is verified 

 by a numerical example of designing the input matching network of an amplifier.

 Introduction:   Impedance transformers are basic building blocks of 

many RF/microwave circuits. The growing trend towards dual-band  products has increased the demand for dual-band impedance transfor-

mers. A small dual-frequency transformer was introduced in   [1]   in

which a real impedance load is matched to a real impedance source

using two equal length series sections. An L-type impedance transfor-

mer is also reported for the same purpose in [2]. The problem of match-

ing frequency-dependent complex impedances was first discussed in [3]

where the impedances were assumed to be unequal in both frequencies.

A three-section impedance transformer is proposed in  [4]   to deal with

this problem. The proposed transformer matches a load with different 

complex impedances at two frequencies to a real impedance source.

Also, this case is studied in   [5]   in which four sections are used toachieve matching. In a more general problem, [6] discusses the situation

where both source and load impedances are complex and frequency-

dependent.

In this Letter, we propose a new approach to match a frequency-

dependent complex impedance load to a real impedance source at two

arbitrary frequencies by using T-section transmission lines. Analytical

closed-form equations are derived and a matching network is designed 

using these equations.

Z a ,q a 

  Z b ,    θ  b

Z c ,θ c Z L1 at f 1Z L2 at f 2

Z 0

Y left    Y right 

Y down 

open orshortedstub

commonnode

Fig. 1  Proposed T-section transformer 

Transformer structure and analysis:   The proposed T-section impe-

dance transformer is shown in   Fig. 1. Depending on the load and 

source impedances, an open or shorted stub can be used to maintain

the size of the transformer minimum.   Z  L1 ¼  R L1 + jX  L1   and   Z  L2 ¼

 R L2 + jX  L2   are the load impedances at the two design frequencies,

namely   f  1   and   f  2. Left, down and right sections of the transformer 

have characteristic impedances and lengths of   Z a,   La,   Z b,   Lb,   Z c   and 

 Lc, respectively.   Y right ,   Y left    and   Y down   are the input admittanceslooking from the common node towards right, left and down, respect-

ively. The key idea is to conjugate match these admittances at both fre-

quencies. It will be shown that design equations could be derived in

closed-form easily, if    Y right | f   1  =  Y right ∗| f   2,   Y left | f   1  =  Y left ∗| f   2   and 

Y down| f   1  =  Y down∗| f   2, which means conjugated relationship between

the two values. Let us start with   Z right   = 1/Y right   which is shown in

[4] to be conjugate at two frequencies if 

Z c  =

  R L1 R L2 +  X  L1 X  L2 +

 X  L1 +  X  L2

 R L2 −  R L1( R L1 X  L2 −  R L2 X  L1)

   (1)

 Lc   =

np + arctan  Z c( R L1 −  R L2)

 R L1 X  L2 −  R L2 X  L1

(m + 1)b c1

(2)

in which  b c1  =  u c/ Lc   is the propagation constant of the transmission

line at the first frequency  f  1. Second to first frequency ratio is denoted 

 by   m, i.e.   m =  f  2/ f  1.   n   is an arbitrary integer that should be chosen

minimum, while the transformer is easy to fabricate.

Y left  can be written as

Y left   =1

Z a×

Z a  +  jZ 0 tan(u a)

Z 0  +  jZ a tan(u a)(3)

in which   u a  =  u a1   = b a1 La   and   u a   = u a2   = b a2 La   at the first and 

second frequency bands, respectively. Since   Z 0   and   Z a   are both real,

Y left | f   1  =  Y left ∗| f   2   if 

tan(u a1) = − tan(u a2) (4)

and bearing in mind that  u a2/u a1  =  f  2/ f  1   = m, we conclude that 

u a1   =p 

1 + m  (5)

and hence

 La  =p 

(1 + m)b a1(6)

On the other hand, we have made   Y right | f   1  =  Y right ∗|f2   earlier, which

means the real parts of  Y right  at two frequencies are equal and the ima-

ginary parts are opposite in sign. Since the input impedance of an

open or shorted stub does not have any real part, in order to satisfy

the conjugate matching condition at the common node, the real part of 

Y left  should be equal to the real part of  Y right  at two frequencies, i.e.

G left   = G right    (7)

where G left  and  G right  are the real parts of  Y left  and  Y right , respectively.Using (3), (7) can be written as

Z aZ 0 +  Z 0Z a tan2(u a)

Z a(Z 20   + Z 

2a tan

2(u a)= G right    (8)

Solving for  Z a  results in

Z a   =

  Z 0(1 − Z 0G right  +  tan

2(u a))

G right  tan2(u a)

   (9)

The only remaining section to design is the open or shorted shunt stub.

Y down can be written as

Y down  =  + j tan(u b)/Z b,   open stub

− j cot (u b)/Z b,   shorted stub  (10)

The shunt stub should cancel out all imaginary impedances at the

common node, i.e.

Y down  = − j ( Bright  +  Bleft ) (11)

in which Bright  and  Bleft  are the imaginary parts of  Y right  and  Y left , respect-

ively. Obviously, in order to maintain the size of the stub minimum, i.e.

0 , u b , p /2; if ( Bright  + Bleft ) , 0, then Y down . 0 and we should usean open stub. On the other hand, if ( Bright  + Bleft ) . 0, then a shorted 

stub should be used. Therefore, regardless of what type the stub is, its

electrical and physical lengths can be calculated by

u b1   =p 

1 + m  (12)

 Lb  =p 

(1 + m)b b1

(13)

Derivation of these equations has been discussed in the design of the

left-hand section of the transformer. Using (10) and (11),   Z b   can be

written as

Z b  =

− tan(u b)

( Bright  +  Bleft ),   open stub

cot (u b)

( Bright  +  Bleft ),   shorted stub

(14)

If  Z b obtained by this formula is not feasible to fabricate, one can easily

double  u b and use the opposite kind of stub.

 Numerical example:   To verify the validity of the proposed transformer 

and design equations, we designed and studied the input matching

network of an amplifier. For this purpose, an ATF-33143 transistor was chosen, which has scattering parameters available up to 18 GHz

[7]. From calculations, this transistor is absolutely stable above about 

5 GHz. Thus, for simplicity, we designed our amplifier above 6 GHz

where conjugate impedance matching could be applied for maximum

ELECTRONICS LETTERS 28th April 2011 Vol. 47 No. 9 

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gain   [8]. The required source impedance for conjugate impedance

matching of the amplifier is calculated and plotted in  Fig. 2   in which

Z  s ¼ R + jX . Let us consider three design cases, namely A, B and C,

which have a first band at 6 GHz and second band at 8, 10, and 

12 GHz, respectively. T-section transformer parameters are tabulated 

in   Table 1. It should be mentioned that in case A, however, the

minimum length is obtained by a shorted stub, the required characteristic

impedance for a shorted stub is extremely low and we decided to double

the length of the stub and use an open stub instead.  Fig. 3  shows the

simulated return loss responses using AWR Microwave Office

 package. As can be seen, there is good matching at the designated fre-

quencies, which verifies the validity of the proposed solution.

5 6 7 8 9 10 11 12-400

-300

-200

-100

0

100

200

frequency, GHz

   i  m  p  e   d  a  n  c  e ,     Ω

  R

  X

Fig. 2  Required source impedance for conjugate matching of amplifier 

Table 1:  Transformer parameters for three design cases

Case f  1,

GHz f  2,

GHz   Z a, V   u a, deg   Z b, V   u b, deg   Z c, V   u c, degStubtype

A 6 8 68 77 10 154 43 196.6 Open

B 6 10 134 67.5 23 67.5 55 173 Short  

C 6 12 34 60 27 60 71 216 Open

6 8 10 12–50

–40

–30

–20

–10

0

frequency, GHz

   |   S   1   1   | ,   d   B

case A

case B

case C

Fig. 3  Simulated return loss of matching network 

Conclusion:   A T-section dual-band transformer is proposed to match a

frequency-dependent impedance load. Design equations are derived in

closed-form and three input matching networks are designed for three

amplifiers with different frequency bands. The simulated responses ver-

ified the validity of the design.

 Acknowledgment:   The authors thank the Iran Telecommunication

Research Center for supporting this work under contract T/18130/500.

# The Institution of Engineering and Technology 2011

2 December 2010doi: 10.1049/el.2010.7452

One or more of the Figures in this Letter are available in colour online.

M.A. Nikravan and Z. Atlasbaf ( Faculty of Electrical and Computer 

 Engineering, Tarbiat Modares University, Nasr Bridge, Tehran, Iran )

E-mail: atlasbaf@modares.ac.ir 

References

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2 Park, M.J., and Lee, B.: ‘Dual band design of single stub impedancematching networks with application to dual band stubbed T junctions’, Microw. Opt. Technol. Lett., 2010,  52, (6), pp. 1359–1362

3 Wu, Y., Liu, Y., and Li, S.: ‘A dual-frequency transformer for compleximpedances with two unequal sections’,  IEEE Microw. Wirel. Compon. Lett., 2009,  19, (2), pp. 77–79

4 Liu, X.,   et al .: ‘A three-section dual-band transformer for frequency-dependent complex load impedance’,   IEEE Microw. Wirel. Compon. Lett., 2009,  19, (10), pp. 611–613

5 Chuang, M.L.: ‘Dual-band impedance transformer using two-sectionshunt stubs’,   IEEE Trans. Microw. Theory Tech., 2010,   58, (5), pp. 1257– 1263

6 Wu, Y.,   et al .: ‘A generalized dual-frequency transformer for twoarbitrary complex frequency-dependent impedances’,   IEEE Microw.Wirel. Compon. Lett., 2009,  19, (12), pp. 792–794

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