Design of Multi-Band Transmission Line Transformer Using Particle Swarm
Majid Khodier*, and Nihad Dib
EE Dept., Jordan Univ. of Science and Technology
P. O. Box 3030, Irbid, Jordan
Abstract: The design of N-section matching transformer operating at N arbitrary
frequencies using the particle swarm optimization (PSO) method is demonstrated.
Although analytical methods based on standard transmission line theory can be used in
such designs, however, the analysis becomes cumbersome if N exceeds three, and
numerical methods should be used to solve the resulting nonlinear equations. The design
using the PSO, however, is much easier, and gives the same results as the analytical
methods. Different examples are presented and compared with published literature.
With the advent of multi-band operation in wireless communication systems, it becomes
essential to have matching transformers that operate at several frequencies. Recently,
several papers have been published in which different techniques were proposed to
design dual-frequency matching transformers [1-6]. In , a dual-frequency four-section
alternating impedance transmission line transformer (TLT) was proposed and used to
design a dual-band Wilkinson power divider. In [3, 4], a dual-band two-section TLT was
studied and simple design equations for the impedances and lengths of the two sections
were given in . In , an extension of this dual-band TLT to match complex
impedances was presented and applied to wideband high-frequency amplifiers. In , a
new synthesis method was proposed to design equal-ripple dual-band impedance
transformer. Very recently, a three-section TLT, extended from the two-section TLT
concept, was investigated and applied for a tri-band application in . After somewhat
lengthy procedure, design expressions for the 3-section TLT for three arbitrary operating
frequencies were given in . However, two nonlinear equations had to be solved
simultaneously via an optimization process.
In this paper, instead of going through lengthy analytical derivations, which could end up
with the need for the use of optimization process too, we propose the use of the particle
swarm optimization (PSO) technique to design multi-band multi-section TLT. First, a
general expression for the reflection coefficient of N-section TLT is derived. Then, the
PSO technique is used to find the characteristic impedances and lengths of these sections
such that perfect match is achieved at N arbitrary frequencies. The PSO has been recently
used in many applications in electromagnetics, antennas, and microwave circuits. The
details of the PSO can be found elsewhere in the literature [8, 9].
Results for the dual-frequency two-section and tri-frequency three-section TLT are given
which agree exactly with those given in the literature. Results for quad-frequency four-
section TLT are also presented.
1-4244-0123-2/06/$20.00 ©2006 IEEE 3305
2. Design Method
The design is concerned with finding the characteristic impedances and lengths of N-
section TLT that matches a transmission line of characteristic impedance Z0 to a load ZL
at N arbitrary frequencies (see Figure 1). The impedance transformer ratio is defined as k
= ZL/Z0. The particle swarm optimization (PSO) method is used to find the parameters of
the matching transformer (i.e., the characteristic impedance and length of each section).
The most important part of the PSO method is the evaluation of the fitness function. For
the current problem, the following fitness function is used:
| )(| min Fitness
parameter. The input reflection coefficient can be easily calculated using standard
transmission line theory. It is obvious that for N frequency points, the PSO will search for
2N unknowns: N characteristic impedances, and N lengths. Also, to make the final design
compact, we restricted the length of each section to be less than λ/4 at the lowest design
frequency. At the beginning of the PSO algorithm, the impedance values and lengths are
randomly initialized within the intervals [Z0 , ZL], and [0, λ/4], respectively.
is the input reflection coefficient at frequency fn, and an is a weighting
The results from the PSO algorithm are verified by comparing them with published
results using analytical methods. The first example illustrates the results for a two-section
transformer that matches Z0 = 200 Ω to ZL = 50 Ω at two arbitrary frequencies. The
results are summarized in Table I. Our results are identical to the results given in [Table I,
Ref. 3] using analytical methods. It should be mentioned here that in Table I, and all the
tables that follow, impedance values are normalized to Z0, and lengths are normalized to
the wavelength at the lowest design frequency.
The second example illustrates the design of a three-section transformer operating at
three arbitrary frequencies. The results in Table II are obtained for Z0 = 50 Ω and k = 0.6,
1.5 and 2, operating at f = 0.9, 1.8, and 2.4 GHz. Our results are very close to the results
given in [Table I, Ref. 7], which are obtained using a lengthy analytical method.
The third example illustrates the design of a four-section alternating-impedance
transformer operating at two arbitrary frequencies . The impedance values of this
transformer are fixed at [Z1, Z2, Z3, Z4] = [ZL, Z0, ZL, Z0], while the lengths are chosen as
[l1, l2, l3, l4] = [l1, l2, l2, l1]. Therefore, we have only two unknowns: l1, and l2. Table III
Figure 1: An N-section, N-band matching transformer.
shows the results for k = 1/2. The results are in excellent agreement with those given in
[Table I, Ref. 2].
The fourth and last example demonstrates the design of a four-section transformer
operating at four arbitrary frequencies with various values of k. The design parameters
are listed in Table IV, and the variation of the input reflection coefficient with frequency
is shown in Figure 2. It is clear that perfect match is obtained at the four design
frequencies. This quad-frequency transformer will be implemented using microstrip
transmission lines, for which full-wave and experimental results will be presented in the
Table I: Design parameters for a two-section transformer with Z0 = 200 Ω and ZL = 50
f1 (GHz) f2 (GHz)
15 15 0.705
13 17 0.6952
12 18 0.6797
10 20 0.6248
Table II: Design parameters for f1 = 0.9 GHz, f2 = 1.8 GHz, and f3 = 2.4 GHz and
different values of k.
k Z1 Z2 Z3
0.6 0.8935 0.7748 0.6717
1.5 1.0934 1.2247 1.3718
2.0 1.1655 1.4142 1.7160
Table III: Design parameters for the alternating-impedance four-section transformer
with k = 1/2, and f1 = 1 GHz.
1.2 1.4 1.6
Table IV: Design parameters for the four-section transformer with f1 = 1.0 GHz, f2 = 2.0
GHz, f3 = 2.6 GHz, f4 = 3.4 GHz, and different values of k.
0.5 0.8941 0.7516 0.64580.5538
1.5 1.0641 1.1719 1.2823 1.4105
2.0 1.1280 1.3232 1.5423 1.8239
10 0.1103 2.5295 4.0093 6.8306
l2 l3 l4
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Figure 2: Input reflection coefficient as a function of frequency for different values of k
for a quad-band 4-section TLT.