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Microwave and Optical Technology Letters. Vol. 54, No. 8, August 2012
A Compact Broadband Nonsynchronous
Noncommensurate Impedance Transformer
Vitaliy Zhurbenko1, Kseniya Kim1, Kumar Narenda2

1
Electrical Engineering Department, Technical University of Denmark, Kgs. Lyngby, DK-2800 Denmark, phone: +45-45253820; fax: +45-4593-1634; e-mail: vz@elektro.dtu.dk.
2
Research & Development Center, Motorola Technology, Penang, 11800, Malaysia, e-mail: cnk020@motorola.com
Abstract— Nonsynchronous noncommensurate impedance transformers consist of a combination of high- and low-impedance transmission
lines. High-impedance lines have narrow tracks in strip and microstrip technology, which allows for great flexibility and miniaturization of the
layout in comparison to the traditional tapered line transformers. This flexibility of the broadband nonsynchronous noncommensurate
impedance transformers is experimentally demonstrated in this paper allowing the length reduction by almost 3 times.
Index Terms—impedance matching, wave transmission matrix.
I. INTRODUCTION
T
HE design of impedance matching circuits for ultra-wideband applications, such as Software Designed Radio (SDR) [1], is a
very challenging task, because the traditional designs based on distributed components lead to a large circuit size which is
highly undesirable in practice. The matching circuits based on lumped elements, however, are compact but suffer from low Qfactor at high frequencies.
Nonsynchronous noncommensurate altering transmission line impedance transformers (NN impedance transformers) are one
of the most promising types of transformers for a broadband matching. They provide a wide bandwidth, comparatively low
insertion loss, compact size, and exhibit a low sensitivity to fabrication errors.
NN impedance transformers consist of the sections of different lengths (noncommensurate) with the same characteristic
impedances as the impedances of the source and the load which should be matched. For the analysis, the NN impedance
transformer is assumed to be lossless, reciprocal and antimetric. These assumptions simplify the analysis, but still provide results
acceptable in practice, as it will be shown later. The electrical lengths of the sections are symmetrical regarding the transformer
centre and are related to each other linearly.
Although the algorithms for a design of 2-, 4-, and 6-section NN impedance transformers have been described in [2], [3], and
numerical data for a synthesis of 4-, 6-, and 8-section NN impedance transformers can be found in [4], [5] for limited
transformation ratios and frequency bandwidths, a synthesis of a 12-section NN impedance transformer is required in order to
ensure a desired reflection coefficient in a wider frequency range suitable for SDR applications [1]. According to the authors'
knowledge, the design data for a synthesis of a 12-section NN
impedance transformer is introduced here for the first time.
A schematic view of a 12-section NN impedance
transformer is shown in Fig.1. NN impedance transformer
consists of altering sections with the electrical lengths
Fig.1. Schematic view of a 12-section NN impedance transformer
and having the same characteristic impedances as the
impedance of the source
and the load .
Algorithm based on the wave transmission matrices for the analysis of the impedance transformer has been employed in order
to take into account the multiple reflections from the impedance transformer sections. According to this approach, the NN
impedance transformer is split into sections (two-port networks) formed by two transmission lines with characteristic
impedances
and . The resulting T-matrix of the overall NN impedance transformer is found by a consecutive T-matrices
This is pre-peer reviewed version of the following article:
Vitaliy Zhurbenko, Kseniya Kim, and Kumar Narenda, "A Compact Broadband Nonsynchro.nous Noncommensurate Impedance Transformer," Microwave
and Optical Technology Letters. Vol. 54, No. 8, August 2012, pp. 1832-1835,
which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/mop.26975/full
.
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Microwave and Optical Technology Letters. Vol. 54, No. 8, August 2012
multiplication of the impedance transformer sections.
The synthesis of the NN impedance transformer with Chebyshev characteristic has been performed solving a mini-max
problem for the magnitude of the total reflection coefficient
(1)
where
and
are the electrical lengths of the first section
the assumed matching bandwidth and
at the lowest frequency
and the highest frequency
of
(2)
where
is the scattering transfer coefficient of the overall NN impedance transformer. υ is the vector defining electrical
lengths of the remaining sections.
II. DESIGN SPECIFICATION AND LAYOUT DEVELOPMENT OF THE NN IMPEDANCE TRANSFORMER
In order to obtain the design data for the synthesis of the 12-section NN impedance transformers, (1) has been solved
numerically for the range of bandwidth ratio
and the transformation ratio
. Table I represents the design
data for the NN impedance transformer with a transformation ratio r=4.
TABLE I
DESIGN DATA FOR 12-SECTION NN IMPEDANCE TRANSFORMER WITH A TRANSFORMATION RATIO R=4
θtotal,
deg
-27.18
θ1=θ12,
deg
3.95
θ2=θ11,
deg
63.97
θ3=θ10,
deg
11.05
θ4=θ9,
deg
49.51
θ5=θ8,
deg
21.10
θ6=θ7,
deg
34.05
367.24
-20.93
5.49
58.39
12.74
46.20
21.82
33.02
355.31
6.0
-16.82
7.07
53.34
14.33
43.37
22.48
32.18
345.54
7.0
-14.00
8.56
48.99
15.75
40.94
23.04
31.45
337.47
8.0
-12.01
9.90
45.38
17.00
38.90
23.52
30.84
331.08
9.0
-10.55
11.10
42.44
18.08
37.22
23.93
30.33
326.18
10.0
-9.42
12.20
39.93
19.07
35.74
24.31
29.89
322.27
χ
|Γ|, dB
4.0
5.0
This table provides the parameters of the transformer in Fig. 1 for bandwidth ratios from 4 to 10. The second column also
gives the maximum achieved amplitude of the inband reflection coefficient.
The given range of bandwidth ratios has been chosen for practical reasons. χ > 10 would lead to poor matching. In most
practical cases, this would require implementation of a larger transformer with number of sections of more than twelve.
Choosing χ < 4 would lead to a very low level of reflection coefficient and in most practical cases it would be reasonable to use a
shorter transformer (with a number of sections of less than twelve).
Following the specification of the matching circuit for the SDR power amplifier in [1], an NN impedance transformer with a
transformation ratio r=4 (
and bandwidth ratio χ=5 (
has been
synthesized based on the design data from Table I. The calculated S-parameters of the impedance transformer are shown in
Fig.2.
0
-10
-6
-20
-12
-30
-18
-40
-24
-50
Magnitude of S21 (dB)
Magnitude of S11 (dB)
0
-30
0.0
f10.5
1.0
1.5
2.0
Frequency (GHz)
f2
2.5
3.0
Fig.2. A magnitude of S11 and S21 of the NN impedance transformer in Fig. 1.
The response of the NN impedance transformer exhibits six reflection coefficient minima. The same number of minima could
be achieved cascading six quarter-wave sections at expense of a longer matching circuit.
In order to further miniaturize the 12-section NN impedance transformer in Fig. 1, meandering of high-impedance lines has been
implemented. The meandered 12-section NN impedance transformer has been realized using RO4350 substrate with a
thickness
, relative dielectric constant
, dielectric loss tangent
, and conductor
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Microwave and Optical Technology Letters. Vol. 54, No. 8, August 2012
thickness
. It should also be noted that meandering of the traditional tapered line transformers would in many cases
require implementation of very thin substrates. The layout of the meandered microstrip 12-section NN impedance transformer is
shown in Fig.3.
50 Ohm
OUT
28.34 mm
θ=77°
12.5Ohm
IN
45.57 mm
θ=124°
Fig.3. Layout of the meandered microstrip 12-section NN
impedance transformer.
This final miniaturization step allowed to reduce the total electrical length from 355° (refer to Table I) to 124° (Fig. 3). The
physical dimensions of the meandered impedance transformer are given in Table II. The sections of the impedance transformer
in Fig.3 are numbered from the left (input port) to the right (output port).
TABLE II
THE DIMENSIONS OF THE MEANDERED MICROSTRIP NN IMPEDANCE TRANSFORMER DESIGNED USING RO4350
50 Ohm transmission line width
12.5 Ohm transmission line width
length of the section 1
length of the section 2
length of the section 3
length of the section 4
length of the section 5
length of the section 6
length of the section 7
length of the section 8
length of the section 9
length of the section 10
length of the section 11
length of the section 12
1.60 mm
10.18 mm
1.87 mm
19.55 mm
4.39 mm
15.42 mm
7.69 mm
11.30 mm
12.21 mm
7.50 mm
17.11 mm
4.37 mm
21.25 mm
1.86 mm
As one can see from the data in Table II, the microstrip realization of the transformer is not exactly symmetrical. This is due to
the fact that the propagation properties of the 50 Ohm and 12.5 Ohm transmission lines are slightly different and the same
electrical length leads to slightly different physical length of those two lines. It should be noted that
III. MEASUREMENT RESULTS
The photo of the fabricated NN impedance transformer is shown in Fig.4. It has been characterized using HP8720D Network
Analyzer. The scattering parameters of the impedance transformer could not be obtained directly, as the circuit has the input
impedance
and the output impedance
while the Network Analyzer utilizes standard 50 Ohm input
and output ports.
Therefore, the renormalization and de-embedding of the 50 Ohm
coaxial connector at the input port has been performed in order to obtain
the S-parameters of the impedance transformer. For de-embedding, the
coaxial connector and coaxial-to-microstrip line transition have been
modeled using a full-wave simulator. The obtained scattering parameters
of the SMA connector and the transition have been subtracted from the
measured data.
Fig.4. A photo of the meandered 12-section NN impedance
transformer manufactured on RO4350.
Simulated and measured S-parameters of the impedance transformer
are shown in Fig. 5.
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Microwave and Optical Technology Letters. Vol. 54, No. 8, August 2012
m1
0
Magnitude of S21 (dB)
Magnitude of S11 (dB)
0
-10
-20
-30
-40
-50
-1
-2
-3
-4
-5
-6
m1
freq=2.00GHz
dB(S(2,1))=-0.66
-7
-8
-9
-10
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.5
Frequency (GHz)
1.0
1.5
2.0
2.5
3.0
Frequency (GHz)
(a)
(b)
Measured and de-embedded
MoM simulations
Fig.5. S-parameters of the meandered NN impedance transformer in Fig. 4. (a) S11. (b) S21.
A method of moments (MoM) simulator from Agilent has been used in order to take into account the parasitic coupling
between the lines of the meandered transformer. It should be noted that the microstrip impedance transformer has been
implemented following the design data from Table I without any adjustments, which are usually required in microwave
component design due to discrepancies between schematics and full-wave simulations.
Measurements show that the magnitude of the reflection coefficient deteriorates at high frequencies. One minimum is not
visible compared to the MoM simulations in Fig. 5. This is most probably caused by the inaccuracy of the fixture model in the
de-embedding procedure and the fabrication errors. The measured maximum of inband reflection reaches the level of
approximately - 15 dB while the expected level is below - 20 dB (refer to Fig. 2). The magnitude of the transmission coefficient
is better than
up to 2 GHz. The measured characteristics are shifted in frequency. The shift of the frequency band is
most likely due to the discrepancy between the expected and actual permittivity of the substrate.
Compared to the compact two sections impedance transformer based on asymmetric coupled transmission lines which exhibits
the same transformation ratio, bandwidth, number of minima, and has a total electrical length of
[6], the meandered
12-section NN impedance transformer described here is more than 30 % shorter.
The low-pass behavior of the NN impedance transformer is also important in power amplifier matching applications because it
potentially allows to diminish the higher order harmonics of the amplifier.
IV. CONCLUSIONS
The fact that the length of NN impedance transformers decrease while increasing the transformation ratio makes them
attractive candidates for applications where a high transformation ratio is required, such as matching of power amplifiers. This,
however, leads to implementation of low impedance transmission lines, which in microstrip realization results in wide
conductors. Combination with high impedance narrow microstrip lines allows to simple miniaturization of the layout using
meandering which leads to effective use of the PCB area. The circuit is wideband and comparatively insensitive to fabrication
errors. It exhibits low pass characteristics which is useful in power amplifier applications.
The presented impedance transformer is designed for the SDR applications and allows 12.5 Ohm to 50 Ohm impedance
transformation with a reflection coefficient typically better than -15 dB in a fractional bandwidth of 133%. The occupied area is
approximately 0.35λ × 0.22λ.
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