6
IMPEDANCE TRANSFORMERS
In the preceding chapter, several techniques were considered to match a given load
impedance at a ®xed frequency. These techniques included transmission line stubs as
well as lumped elements. Note that lumped-element circuits may not be practical at
higher frequencies. Further, it may be necessary in certain cases to keep the
re¯ection coef®cient below a speci®ed value over a given frequency band. This
chapter presents transmission line impedance transformers that can meet such
requirements. The chapter begins with the single-section impedance transformer
that provides perfect matching at a single frequency. Matching bandwidth can be
increased at the cost of a higher re¯ection coef®cient. This concept is used to design
multisection transformers. The characteristic impedance of each section is controlled
to obtain the desired pass-band response.
Multisection binomial transformers exhibit almost ¯at re¯ection coef®cient about
the center frequency and increase gradually on either side. A wider bandwidth is
achieved with an increased number of quarter-wave sections. Chebyshev transfor-
mers can provide even wider bandwidth with the same number of sections but the
re¯ection coef®cient exhibits ripples in its pass-band. This chapter includes a
procedure to design these multisection transformers as well as transmission line
tapers. The chapter concludes with a brief discussion on the Bode-Fano constraints,
which provide an insight into the trade-off between the bandwidth and allowed
re¯ection coef®cient.
189
Radio-Frequency and Microwave Communication Circuits: Analysis and Design
Devendra K. Misra
Copyright # 2001 John Wiley & Sons, Inc.
ISBNs: 0-471-41253-8 (Hardback); 0-471-22435-9 (Electronic)
6.1 SINGLE SECTION QUARTER-WAVE TRANSFORMER
We considered a single-section quarter-wavelength transformer design problem
earlier in Example 3.5. This section presents a detailed analysis of such circuits.
Consider the load resistance R
Z
o
R
L
p
; Z
in
is equal to Z
o
and, hence,
there is no re¯ected wave beyond this point toward the generator. However, it
reappears at other frequencies when b` T 90
. The corresponding re¯ection
coef®cient G
in
can be determined as follows.
G
in
Z
in
À Z
o
Z
in
Z
o
R
L
À Z
o
R
L
Z
o
j2
Z
o
R
L
p
tanb`
r
in
exp jj
; r
in
R
L
À Z
o
fR
L
Z
o
9
=
;
1=2
6:1:2
Figure 6.1 A single-section quarter-wave transformer.
190
IMPEDANCE TRANSFORMERS
Variation in r
in
with frequency is illustrated in Figure 6.2. For b` near 90
, it can
be approximated as follows:
r
in
%
jR
L
À z
o
j
2
Z
o
R
L
p
tanb`
À Z
o
1 À r
2
M
p
; y
1
< p=2 6:1:4
In the case of a TEM wave propagating on the transmission line, b`
p
2
Â
f
f
o
,
where f
f
o
2 À
4
p
cos
À1
2r
M
Z
o
R
L
p
R
L
À Z
o
1 À r
2
M
p
3 Â 10
8
4 Â 900 Â 10
6
m 8:33 cm
Magnitude of the re¯ection coef®cient increases as b` changes from p=2 (i.e., the
signal frequency changes from 900 MHz). If the maximum allowed r is
r
M
0:05VSWR 1:1053, then fractional bandwidth is found to be
Df
f
o
2 À
4
p
cos
À1
2r
M
Z
o
R
L
p
R
L
À Z
o
Z
N
exp jb`G
N
expÀjb`
exp jb`ÀG
N
expÀjb`
6:2:1
where
G
N
R
L
À Z
N
R
L
Z
N
6:2:2
192
IMPEDANCE TRANSFORMERS
The re¯ection coef®cient seen by the (N-1)st section is
G
H
NÀ1
Z
jb`
G
N
e
Àjb`
Z
NÀ1
e
jb`
À G
N
e
Àjb`
or,
G
H
NÀ1
Z
N
À Z
NÀ1
e
jb`
G
N
Z
N
Z
G
NÀ1
e
Àj2b`
6:2:3
where
G
NÀ1
Z
N
À Z
NÀ1
Z
N
Z
NÀ1
6:2:4
If Z
N
is close to R
L
and Z
NÀ1
is close to Z
N
, then G
N
and G
NÀ1
N
e
Àj4b`
Figure 6.3 An N-section impedance transformer.
MULTI-SECTION QUARTER-WAVE TRANSFORMERS
193
Therefore, by induction, the re¯ection coef®cient seen by the feeding line is
G % G
0
G
1
e
Àj2b`
G
2
e
Àj4b`
ÁÁÁG
NÀ1
e
Àj2NÀ1b`
G
N
e
Àj2N b`
6:2:6
or,
G
P
N
2
ÁÁÁG
N
w
N
G
N
Q
N
n1
w À w
n
6:2:9
where
j À2b` 6:2:10
and,
w e
jj
6:2:11
Note that for b` 0 (i.e., l 3I), individual transformer sections in effect have
no electrical length and load R
L
appears to be directly connected to the main line.
Therefore,
G
P
N
n0
G
n
w
N1
À 1
w À 1
6:3:1
or,
G
G
N
e
jN1j
À 1
e
jj
À 1
e
jNj=2
sin
N 1
2
j
sin
j
2
Hence,
jGjrjr
N
j
N 1 sin
j
2
6:3:2
and, from (6.2.12),
P
N
n0
G
n
N 1r
sinfN 1b`g
N 1 sinb`
6:3:4
This can be viewed as an equation that describes magnitude r of the re¯ection
coef®cient as a function of frequency. As (6.3.4) indicates, a pattern of rb` repeats
periodically with an interval of p. It peaks at np, where n is an integer including
zero. Further, there are N À 1 minor lobes between two consecutive main peaks. The
number of zeros between the two main peaks of rb` is equal to the number of
quarter-wave sections, N.
Consider that there are three quarter-wave sections connected between a 100-ohm
load and a 50-ohm line. Its re¯ection coef®cient characteristics can be found from
(6.3.4), as illustrated in Figure 6.4. There are three zeros in it, one at b` p=2 and
the other two symmetrically located around this point. In other words, zeros occur at
b` p=4, p=2, and 3p=4. When the number of quarter-wave sections is increased
from 3 to 6, the rb` plot changes as illustrated in Figure 6.5.
UNIFORMLY DISTRIBUTED SECTION REFLECTION COEFFICIENTS
195
For a six-section transformer, Figure 6.5 shows ®ve minor lobes between two
main peaks of rb`. One of these minor lobes has its maximum value (peak) at
b` p=2. Six zeros of this plot are symmetrically located, b` np=7,
n 1; 2; ...; 6. Thus, characteristics of rb` can be summarized as follows:
Pattern of rb` repeats with an interval of p.
N
n0
w À w
n
6:3:5
where
w
n
e
j
2pn
N1
; n 1; 2; ...; N 6:3:6
(6.3.1) may be written as follows:
G
G
N
Y
N
n1
w À w
n
Y
N
n1
w À e
j
2pn
Y
N
n1
w À e
j
2pn
N1
6:3:8
Figure 6.6 Location of zeros on a unit circle on the complex w-plane for N 3.
UNIFORMLY DISTRIBUTED SECTION REFLECTION COEFFICIENTS
197
Since w e
jj
is constrained to lie on the unit circle, distance between w and w
n
is
given by
d
n
j e
jj
À e
j
Y
N
n1
d
n
j6:3:10
Thus, as j À2b` varies from 0 to 2 p, w e
jj
makes one complete traverse of
the unit circle, and distances d
1
; d
2
; ...; d
N
vary with j.Ifw coincides with the root
w
n
, then the distance d
n
is zero. Consequently, the product of the distances is zero.
Since the product of these distances is proportional to the re¯ection coef®cient,
rj
n
goes to zero. It attains a local maximum whenever w is approximately halfway
between successive roots.
Example 6.2: Design a four-section quarter-wavelength impedance transformer
with uniform distribution of re¯ection coef®cient to match a 100-O load to a 50-O
A 16Á Z
4
1400 A Z
4
87:5 O
Characteristic impedance of other sections can be determined from (6.2.8) as
follows:
1
15
Z
4
À Z
3
Z
4
Z
3
A 87:5 Z
3
15 Á 87:5 À 15 Á Z
3
A Z
3
76:5625 O
1
15
Z
3
The frequency range over which re¯ection coef®cient remains below 0.1 is
determined from (6.3.4) as follows:
0:1
1
3
Â
sin5 Á y
m
5 Á siny
m
A
sin5 Á y
m
siny
m
single section.
6.4 BINOMIAL TRANSFORMERS
As shown in Figures 6.4 and 6.5, there are peaks and nulls in the pass-band of a
multisection quarter-wavelength transformer with uniform section re¯ection coef®-
cients. This characteristic can be traced to equispaced roots on the unit circle. One
way to avoid this behavior is to place all the roots at a common point w equal to À1.
With this setting, distances d
n
are the same for all cases and
Q
N
n1
w À w
n
goes to
zero only once. It occurs for j equal to Àp, i.e., at b` p=2. Thus, r is zero only
for the frequency at which each section of the transformer is l=4 long. With
w
n
À1 for all n, equation (6.2.9) may be written as follows.
G
G
N
Q
N
n1
w À w
n
w 1
N !
n!N À n!
; n 0; 1; 2; ...; N : 6:4:2
and therefore, the section re¯ection coef®cients, normalized to G
N
, are binomially
distributed.
From equation (6.4.1),
G G
N
w 1
N
A Gb`G
N
e
Àj2b`
1
N
G
N
2
N
e
ÀjNb`
cosb`
N
or,
rb`r
N
and,
rb`
R
L
À Z
o
R
L
Z
o
Âjcosb`j
N
6:4:5
Re¯ection coef®cient characteristics of multisection binomial transformers versus
b` (in degrees) are illustrated in Figure 6.9. Re¯ection coef®cient scale is normal-
ized with the load re¯ection coef®cient r
L
. Unlike the preceding case of uniformly
distributed section re¯ection coef®cients, it shows a smooth characteristic without
lobes.
Example 6.3: Design a four-section quarter-wavelength binomial impedance trans-
former to match a 100-O load to a 50-O air-®lled coaxial line at 900 MHz.
3
G
4
A G
1
G
3
4 Â G
4
and,
G
2
G
4
4!
2!2!
6 A G
2
6 Â G
4
From (6.4.4),
r0r
N
2
N
R
L
À Z
1
48
0:020833
and from (6.2.8),
G
n
Z
n1
À Z
n
Z
n1
Z
n
A r
n
Z
n1
Z
n
Z
n1
À Z
n
Therefore,
Z
n
1 À r
48
 100 95:9184 O
Z
3
1 À
4
48
1
4
48
 95:9184 81:1617 O
Z
2
1 À
6
48
1
6
48
 81:1617 63:1258 O
and,
Z
1
1 À
4
48
1
1 À
1
48
 50 52:1277 O
BINOMIAL TRANSFORMERS
203
and,
Z
2
1
4
48
1 À
4
48
 50 61:6054 O
The frequency range over which the re¯ection coef®cient remains below 0.1 can
be determined from (6.4.5) as follows.
0:1
1
3
j cosW
M
j
4
A W
M
0:7376
Therefore,
raises the level of intervening lobes but reduces the main lobe. However, we need a
systematic method to determine the location of each zero for a maximum permis-
sible re¯ection coef®cient, r
M
, and the number of quarter-wave sections, N.An
optimal distribution of zeros around the unit circle will keep peaks of all pass-band
lobes at the same height of r
M
.
In order to have magnitudes of all minor lobes in the pass-band equal, section
re¯ection coef®cients are determined by the characteristics of Chebyshev functions,
Figure 6.11 Distribution of zeros for a uniform three-section impedance transformer (solid),
with two of those zeros moved to Æ120
(un®lled), or to Æ60
(hatched).
CHEBYSHEV TRANSFORMERS
205
named after a Russian mathematician who ®rst studied them. These functions satisfy
the following differential equation:
1 À x
2
d
2
T
m
x
dx
4
x8x
4
À 8x
2
1
:
:
T
m
x2xT
mÀ1
xÀT
mÀ2
x
Alternatively,
T
m
x
cosm cos
À1
x À 1 x 1
coshm cosh
À1
xx ! 1
À1
m
coshm cosh
À1
jxjx À1
and the distribution of zeros on the complex w-plane is selected according to that.
With x
o
properly selected, jT
m
xj precisely corresponds to rb`. This is done by
linking b` to x of Chebyshev polynomial as follows.
x x
o
cosb`6:5:4
Consider the design of a three-section equal-ripple impedance transformer. A
Chebyshev polynomial of order three is appropriate for this case. Figure 6.14
Figure 6.13 Chebyshev polynomials for m 1; 2; 3; 4.
CHEBYSHEV TRANSFORMERS
207
illustrates T
3
x together with (6.5.4). Note that Chebyshev variable x and angle j on
the complex w-plane are related through (6.5.4) because j is equal to À2b`.Asj
varies from 0 to À2p, x changes as illustrated in Figure 6.14 (b) and the
corresponding T
3
x in Figure 6.14 (a). Figure 6.15 shows jT
3
jj which can
represent the desired rj provided that
T
3
j 0
1
L
À Z
o
R
L
Z
o
Â
1
r
M
6:5:6
Location x
o
can now be determined from (6.5.2) as follows:
x
o
cosh
1
m
 cosh
À1
p
2
; n 1; 2; ...
where x
n
is the location of the nth zero.
Hence,
x
n
Æcos 2n À 1
p
2m
6:5:8
and corresponding j
n
can be evaluated from (6.5.4) as follows.
j
n
2 Â cos
À1
x
n
x
o
6:5:9
Zeros of rj; w
n
1
x
o
b` p À cos
À1
1
x
o
6:5:10
Example 6.4: Reconsider the matching of a 100-O to a 50-O line. Design an equal-
ripple four-section quarter-wavelength impedance transformer and compare its
bandwidth with those obtained earlier for r
M
0.1.
From (6.5.6),
T
4
x
o
100 À 50
100 50
e
Ày
2
. Hence,
e
y
e
Ày
2 Â 3:3333 6:6666
or,
e
2y
À 6:6666 Â e
y
1 0 A e
y
6:6666 Æ
6:6666
2
À 4
q
2
6:5131 and 0:1535
y ln6:51311:8738
Therefore,
1
4
cosh
; À139:71
; À220:29
; À292:39
Therefore,
w
1
0:381 À j0:925
w
2
À0:763 À j0:647
w
3
À0:763 j0:647
and,
w
4
0:381 j0:925
Now, from equation (6.2.9) we get
G G
4
1 0:764 Â w 0:837 Â w
2
0:764 Â w
3
w
4
Z
o
A 4:365 Â G
4
R
L
À Z
o
R
L
Z
o
G
4
1
4:365
Â
100 À 50
100 50
0:076
Hence,
G
o
G
4
0:076; G
1
G
Z
4
À Z
3
Z
4
Z
3
A Z
3
1 À G
3
1 G
3
Z
4
76:46 O
G
2
Z
3
À Z
2
Z
3
Z
2
A Z
Z
2
59:89 O
In order to minimize the accumulating error in Z
n
, it is advisable to calculate half
of the impedance values from the load side and the other half from the input side. In
other words, Z
1
and Z
2
should be determined as follows:
Z
1
1 G
o
1 À G
o
Z
o
58:2251 O
and
Z
2
1 G
1
1 À G
1
Power loss ratio
Incident power
Power delivered to the load
1
G.L. Matthaei, L. Young, and E.M.T. Jones, Microwave Filters, Impedance-Matching Networks, and
Coupling Structures, Dedham, MA: Artech House, 1980.
212
IMPEDANCE TRANSFORMERS
If P
in
represents the incident power and r
in
is the input re¯ection coef®cient, then
P
LR
P
in
1 À r
2
in
P
in
1
1 À r
2
in
A r
in
n
. For an equal
ripple characteristic in the pass-band, a Chebyshev polynomial can be used to
specify P
LR
as follows:
P
LR
1 k
2
T
2
N
sec y
M
cos y6:6:3
where k
2
is determined from the maximum value of P
LR
in the pass-band. y
M
represents the value of b` that corresponds to the maximum allowed re¯ection
coef®cient r
M
. Since T
2
N
sec y
M
À Z
o
2
4Z
L
Z
o
Â
sec
2
y
z
cos
2
y À 1
2
tan
4
y
z
6:6:5
Figure 6.16 A two-section impedance transformer.
EXACT FORMULATION OF MULTISECTION IMPEDANCE TRANSFORMERS
213
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