Zhang K., Li D. Electromagnetic Theory for Microwaves and Optoelectronics
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3.6 Two-Port Networks 151
(3) The Scattering Matrix of Reciprocal, Lossless and Source-Free
Two-Port Networks.
The scattering matrix of reciprocal, lossless and source-free multi-port net-
works satisfies (3.106) and (3.109),
S
ij
= S
ji
, (S)
∗
(S) = (I).
For a two-port networks, the ab ove equations become
S
12
= S
21
, (3.122)
|S
11
|
2
+ |S
12
|
2
= 1, (3.123)
|S
12
|
2
+ |S
22
|
2
= 1, (3.124)
S
∗
11
S
12
+ S
∗
12
S
22
= 0, (3.125)
S
∗
12
S
11
+ S
∗
22
S
12
= 0. (3.126)
Suppose
S
11
= |S
11
|e
jφ
11
, S
22
= |S
22
|e
jφ
22
, S
12
= |S
12
|e
jφ
12
, S
21
= |S
21
|e
jφ
21
.
According to (3.122)–(3.126), we have
|S
12
| = |S
21
|, φ
12
= φ
21
, |S
11
| = |S
22
|, 2φ
12
= φ
11
+ φ
22
± π. (3.127)
(4) Symmetrical Two-Port Networks.
If the matched reflection coefficient of port 1 is equal to that of p ort 2 and
the network is reciprocal, i.e.,
S
11
= S
22
, S
12
= S
21
, (3.128)
the two-port network is known as a symmetrical network.
The other network parameters of a symmetrical network satisfy
z
11
= z
22
, z
12
= z
21
, y
11
= y
22
, y
12
= y
21
, (3.129)
a = d, T
12
= −T
21
. (3.130)
(5) A Reciprocal, Lossless, Source-Free Two-Port Network can Be-
come a Symmetrical Network by Means of Setting Appropriate
Reference Planes.
For an arbitrary reciprocal, lossless, source-free two-port network, according
to (3.122) and (3.127), S
12
= S
21
and |S
11
| = |S
22
|. Hence the difference
between S
11
and S
22
is only an angle. The angle of the reflection coefficient
changes linearly along the transmission line or waveguide. So we can choose
152 3. Transmission-Line and Network Theory for Electromagnetic Waves
Figure 3.15: A reciprocal, lossless, source-free two-port network can become
a symmetrical network.
appropriate reference planes on the input and output waveguides such that
S
11
= S
22
, and the network between these two reference planes becomes a
symmetrical network. See Fig. 3.15.
The network between reference planes 1 and 2 is a reciprocal, lossless,
source-free and symmetrical network (S). The scattering equation is
·
b
1
b
2
¸
=
·
S
11
S
12
S
21
S
22
¸·
a
1
a
2
¸
, (S) =
·
S
11
S
12
S
21
S
22
¸
,
where S
12
= S
21
and |S
11
| = |S
22
|. Suppose
S
11
= |S
11
|e
jφ
11
, S
22
= |S
22
|e
jφ
22
= |S
11
|e
jφ
22
.
If the reference plane 2 is moved to 3, the distance between planes 2 and
3 is l, and βl = θ, we have
b
2
= b
3
e
jθ
, a
2
= a
3
e
−jθ
.
Then the scattering equation becomes
·
b
1
b
3
e
jθ
¸
=
·
S
11
S
12
S
21
S
22
¸·
a
1
a
3
e
−jθ
¸
,
i.e.,
·
b
1
b
3
¸
=
·
S
11
S
12
e
−jθ
S
21
e
−jθ
S
22
e
−j2θ
¸·
a
1
a
3
¸
=
·
S
0
11
S
0
12
S
0
21
S
0
22
¸·
a
1
a
3
¸
.
The network between plane 1 and 3 is (S
0
):
(S
0
) =
·
S
0
11
S
0
12
S
0
21
S
0
22
¸
=
·
S
11
S
12
e
−jθ
S
21
e
−jθ
S
22
e
−j2θ
¸
The condition of a symmetrical network, S
0
11
= S
0
22
becomes
S
11
= S
22
e
−j2θ
, i.e., |S
11
|e
jφ
11
= |S
22
|e
j(φ
22
−2θ)
= |S
11
|e
j(φ
22
−2θ)
.
3.6 Two-Port Networks 153
Then we have
φ
11
= φ
22
−2θ ±2nπ, i.e., θ =
1
2
(φ
22
−φ
11
) ±nπ, or l =
φ
22
− φ
11
2β
±
nπ
β
.
(3.131)
Under this condition, the network between planes 1 and 3 becomes a sym-
metrical two-port network.
We come to the conclusion that any reciprocal, lossless, source-free two-
port network can become a symmetrical network by choosing appropriate
reference planes, even if the structure of the network is not symmetrical.
The features of various matrices for some special networks are shown in
Table 3.3.
Table 3.3 Features of matrices for source-free networks
(z) (y) (S) (a) (T )
Reciprocal (z)
T
= (z) (y)
T
= (y) (S)
T
= (S) |a| = 1 |T | = 1
z
ij
= z
ji
y
ij
= y
ji
S
ij
= S
ji
lossless (z)
†
= −(z) (y)
†
= −(y) (S)(S)
†
= (I)
Reciprocal |a| = 1 |T | = 1
lossless
(z)
∗
= −(z) (y)
∗
= −(y) (S)(S)
∗
= (I) a, d real T
11
= T
∗
22
b, c img. T
12
= T
∗
21
Symmetrical z
12
= z
21
y
12
= y
21
S
12
= S
21
|a| = 1 |T | = 1
z
11
= z
22
y
11
= y
22
S
11
= S
22
a = d T
12
= −T
21
where |a| and |T | are the determinants of matrices (a) and (T ), respectively.
3.6.3 The Working Parameters of Two-Port Networks
The insertion reflection, insertion attenuation, and insertion phase shift are
caused by connecting a two-port network in a transmission system. These
are the working parameters of the network.
(1) The Insertion Reflection Coefficient or Insertion VSWR.
The reflection co efficient of a port with another matched port is defined as
the insertion reflection coefficient of a two-port, denoted by Γ
1
and Γ
2
. They
are just the scattering parameters S
11
and S
22
, respectively, so that
Γ
1
= S
11
, Γ
2
= S
22
. (3.132)
154 3. Transmission-Line and Network Theory for Electromagnetic Waves
The insertion VSWR of port 1 and port 2, ρ
1
and ρ
2
, are given as
ρ
1
=
1 + |Γ
1
|
1 − |Γ
1
|
=
1 + |S
11
|
1 − |S
11
|
, ρ
2
=
1 + |Γ
2
|
1 − |Γ
2
|
=
1 + |S
22
|
1 − |S
22
|
. (3.133)
(2) The Insertion Attenuation, Absorption Attenuation, and Re-
flection Attenuation.
When a two-port is inserted in a matched transmission system, the output
power may be reduced b ecause of the insertion attenuation of the inserted
network. The insertion attenuation of a two-port is defined as the ratio of
the power of the inward wave at the input port to the power of the outward
wave at the output port, while the output port is matched, which is denoted
by L. According to the above definition, we may expresse L by
L =
a
1
a
∗
1
b
2
b
∗
2
¯
¯
¯
¯
2nd port matched
=
1
S
21
S
∗
21
=
1
|S
21
|
2
= |T
11
|
2
, (3.134)
or
L(dB) = −20 log |S
21
| = 20 log |T
11
|. (3.135)
The attenuation of a network consists of two parts, one is caused by the
lossy media in the network, and the p ower is absorbed by the media, which is
known as the absorption attenuation and denoted by L
A
; another is caused
by the reflection of the network, which is known as the reflection attenuation
and denoted by L
R
.
L = L
A
L
R
, L(dB) = L
A
(dB) + L
R
(dB). (3.136)
The absorption attenuation is the ratio of the power that entered the
network through the input port to the power out of the network through
the output port. The power that enters the network is the power of the
inward wave minus the power of the outward wave at the input port, so the
absorption attenuation is given by
L
A
=
a
1
a
∗
1
− b
1
b
∗
1
b
2
b
∗
2
=
1 − |S
11
|
2
|S
21
|
2
, (3.137)
or
L
A
(dB) = 10 log(1 − |S
11
|
2
) − 10 log |S
21
|
2
. (3.138)
The reflection attenuation is the ratio of the power of the inward wave at
the input port to the power that enters the network, so that
L
R
=
a
1
a
∗
1
a
1
a
∗
1
− b
1
b
∗
1
=
1
1 − |S
11
|
2
, or L
R
(dB) = −10 log(1 −|S
11
|
2
). (3.139)
For a lossless and source-free network, L
A
= 1 or L
A
= 0 dB, so that
|S
21
|
2
= 1 − |S
11
|
2
,
3.6 Two-Port Networks 155
and we have
L = L
R
=
1
1 − |S
11
|
2
=
1
1 − |Γ
1
|
2
=
(ρ + 1)
2
4ρ
. (3.140)
(3) The Insertion Phase Shift
The insertion phase shift of a two-port is defined as the phase shift between
the input wave and the output wave when the output port is matched, which
is denoted by φ. According to this definition, we have the angle of S
21
,
φ = arg S
21
= arg
1
T
11
. (3.141)
3.6.4 The Network Parameters of Some Basic Circuit
Elements
A two-port network may consists of basic elements. The network parameters
of some basic circuit elements are given as follows.
(1) The Series Impedance
From Fig. 3.16(a) and according to the definition of the admittance param-
eters given in Section 3.5.1, the normalized admittance matrix of a series
impedance z = Z/Z
C
as a two-port network is given by
(y) =
·
1/z −1/z
−1/z 1/z
¸
. (3.142)
All of the impedance parameters of a series impedance are infinite.
According to the definition of the transfer parameters given in Section
3.6, we have the normalized transfer matrix of a series impedance
(a) =
·
1 z
0 1
¸
. (3.143)
The S and T parameters of a series impedance can be derived from the y
or a parameters and are given in Table 3.4.
(2) The Parallel Admittance
From Fig. 3.16(b) and according to the definition of the impedance param-
eters given in Section 3.5.1, the normalized impedance matrix of a parallel
admittance y = Y /Y Z
C
as a two-port network is given by
(z) =
·
1/y 1/y
1/y 1/y
¸
. (3.144)
156 3. Transmission-Line and Network Theory for Electromagnetic Waves
Figure 3.16: (a) Series impedance and (b) parallel admittance as two-port
networks.
Figure 3.17: Ideal transformer as a two-port network.
All of the admittance parameters of a parallel admittance are infinite.
According to the definition of transfer parameters given in Section 3.6,
the normalized transfer matrix of a parallel admittance is
(a) =
·
1 0
y 1
¸
. (3.145)
The S and T parameters of a parallel admittance are given in Table 3.4.
(3) An Ideal Transformer
An ideal transformer is a transformer in which all of the exciting current, the
leakage impedance, the copper loss, and the iron loss are negligibly small.
An ideal transformer as a two-port network is shown in Fig. 3.17.
The relations between the voltages at the two ports and the currents at
the two ports are given by
U
2
= nU
1
, I
2
= −
I
1
n
,
where n denotes the transform ratio of the ideal transformer. Hence the
transfer matrix of an ideal transformer is given
(a) =
·
1/n 0
0 n
¸
. (3.146)
The scattering matrix of an ideal transformer is then given by
(S) =
·
(1 − n
2
)/(1 + n
2
) 2n/(1 + n
2
)
2n/(1 + n
2
) (n
2
− 1)/(n
2
+ 1)
¸
. (3.147)
3.6 Two-Port Networks 157
Figure 3.18: An arbitrary reciprocal, lossless and source-free two-p ort net-
work can be reduced to an ideal transformer.
The ideal transformer is an important element in microwave networks.
It can be proven that by appropriately choosing the two reference planes
on two waveguides connected at the input and output ports, or in other
words, by connecting two segments of transmission line at both sides of an
arbitrary two-port network, and by appropriately choosing the lengths of the
two lines, an arbitrary reciprocal, lossless and source-free two-port network
can be reduced to an ideal transformer. The lengths of the two lines l
1
and
l
2
must satisfy the following conditions:
φ
11
− 2βl
1
= mπ, φ
22
− 2βl
2
= (m ± 1)π, (3.148)
where m is an integer, φ
11
and φ
22
are the angles of the parameters S
0
11
and
S
0
22
of the original network. The ratio of the transformer n is equal to
√
ρ
for m even and 1/
√
ρ for m odd, refer to Fig. 3.18(a) and (b).
(4) The Connection of Two Transmission Lines or Waveguide
with Different Characteristic Impedance
Neglecting the reactance caused by the discontinuity in the connection of two
transmission lines, we have the relations for the voltages and currents in the
two lines:
U
1
= U
2
, I
1
= −I
2
.
See Fig 3.19(a).
Hence the transfer matrix is
(A) =
·
1 0
0 1
¸
. (3.149)
According to (3.117), the normalized transfer matrix becomes
(a) =
r
Z
C2
Z
C1
0
0
r
Z
C1
Z
C2
. (3.150)
158 3. Transmission-Line and Network Theory for Electromagnetic Waves
Figure 3.19: Connection of two transmission lines with different characteristic
impedance as a two-port network.
Figure 3.20: A segment of a transmission line as a two-port network.
Comparing the above equation with the normalized transfer matrix of an
ideal transformer (3.146), we see that the connection of two transmission lines
with different characteristic impedance is equivalent to an ideal transformer
connected between two transmission lines with a normalized characteristic
impedance of 1. The transform ratio of the equivalent ideal transformer is
given by
n =
r
Z
C1
Z
C2
, (3.151)
refer to Fig 3.19(b).
In practice, for a connector of two transmission lines or waveguides with
different cross-sections, the equivalent shunt capacitance across the connector
caused by the discontinuity must be considered.
(5) A Segment of Transmission Line
Referring to Fig. 3.20 and the definition of scattering parameters (3.91) and
(3.92), we have the scattering matrix of a segment of transmission line:
(S) =
·
0 e
−jβl
e
−jβl
0
¸
. (3.152)
According to the definition of A, B, C, and D (3.116), and by applying the
expressions of the distributions of voltages and currents on the shorted and
open transmission lines (3.49), (3.50), (3.52), and (3.53), we have the transfer
3.6 Two-Port Networks 159
matrix and the normalized transfer matrix of a segment of transmission line
(A) =
·
cos βl jZ
C
sin βl
(j/Z
C
) sin βl cos βl
¸
, (a) =
·
cos βl j sin βl
j sin βl cos βl
¸
.
(3.153)
The network matrices of basic elements are shown in Table 3.4.
(6) Cascade Connection of Two-Port Networks
For a two-port network consisting of several cascade-connected networks, the
transfer (or transmission) matrix of the network is equal to the product of
the transfer (or transmission) matrices of all the elementary networks. There
now follows two examples.
(1) For a segment of transmission line with characteristic impedance
Z
C
, connected between two transmission lines with different characteristic
impedances Z
C1
and Z
C2
, the normalized transfer matrix is equal to the
(a) matrix of a segment of transmission line (3.153) multiplied by the (a)
matrices of the connectors of the transmission lines (3.150) at both sides:
(a) =
r
Z
C
Z
C1
0
0
r
Z
C1
Z
C
·
cos βl j sin βl
j sin βl cos βl
¸
r
Z
C2
Z
C
0
0
r
Z
C
Z
C2
=
r
Z
CL
Z
Ci
cos βl j
Z
C
p
Z
Ci
Z
CL
sin βl
j
p
Z
Ci
Z
CL
Z
C
sin βl
r
Z
Ci
Z
CL
cos βl
. (3.154)
(2) For a connector of two transmission lines or waveguides with different
cross sections, the equivalent shunt capacitance across the connector caused
by the discontinuity can be calculated by means of the quasi-static approach.
The transfer matrix of the equivalent network of the discontinuity can then
be expressed as (3.145), and the transfer matrix of the connection of two
lines is given as (3.150). The transfer matrix of the connector is given by the
product of the above two matrices:
(a) =
·
1 0
y 1
¸
r
Z
C2
Z
C1
0
0
r
Z
C1
Z
C2
=
r
Z
C2
Z
C1
0
y
r
Z
C2
Z
C1
r
Z
C1
Z
C2
, (3.155)
(a) =
r
Z
C2
Z
C1
0
0
r
Z
C1
Z
C2
·
1 0
y 1
¸
=
r
Z
C2
Z
C1
0
y
r
Z
C1
Z
C2
r
Z
C1
Z
C2
. (3.156)
160 3. Transmission-Line and Network Theory for Electromagnetic Waves