Section 4 3_the_scattering_matrix_package

3/4/2009 4_3 The Scattering Matrix 1/3
Jim Stiles The Univ. of Kansas Dept. of EECS
4.3 – The Scattering Matrix
Reading Assignment: pp. 174-183
Admittance and Impedance matrices use the quantities I (z),
V (z), and Z (z) (or Y (z)).
Q: Is there an equivalent matrix for transmission line
activity expressed in terms of ( )V z+
, ( )V z−
, and ( )zΓ ?
A: Yes! Its called the scattering matrix.
HO: THE SCATTERING MATRIX
Q: Can we likewise determine something physical about our
device or network by simply looking at its scattering matrix?
A: HO: MATCHED, RECIPROCAL, LOSSLESS
EXAMPLE: A LOSSLESS, RECIPROCAL DEVICE
Q: Isn’t all this linear algebra a bit academic? I mean, it
can’t help us design components, can it?
A: It sure can! An analysis of the scattering matrix can tell
us if a certain device is even possible to construct, and if so,
what the form of the device must be.
HO: THE MATCHED, LOSSLESS, RECIPROCAL 3-PORT NETWORK

HO: THE MATCHED, LOSSLESS, RECIPROCAL 4-PORT NETWORK
Q: But how are scattering parameters useful? How do we
use them to solve or analyze real microwave circuit problems?
A: Study the examples provided below!
EXAMPLE: THE SCATTERING MATRIX
EXAMPLE: SCATTERING PARAMETERS
Q: OK, but how can we determine the scattering matrix of a
device?
A: We must carefully apply our transmission line theory!
EXAMPLE: DETERMINING THE SCATTERING MATRIX
Q: Determining the Scattering Matrix of a multi-port device
would seem to be particularly laborious. Is there any way to
simplify the process?
A: Many (if not most) of the useful devices made by us
humans exhibit a high degree of symmetry. This can greatly
simplify circuit analysis—if we know how to exploit it!
HO: CIRCUIT SYMMETRY
EXAMPLE: USING SYMMETRY TO DETERMINING A SCATTERING
MATRIX

Q: Is there any other way to use circuit symmetry to our
advantage?
A: Absolutely! One of the most powerful tools in circuit
analysis is Odd-Even Mode analysis.
HO: SYMMETRIC CIRCUIT ANALYSIS
HO: ODD-EVEN MODE ANALYSIS
EXAMPLE: ODD-EVEN MODE CIRCUIT ANALYSIS
Q: Aren’t you finished with this section yet?
A: Just one more very important thing.
HO: GENERALIZED SCATTERING PARAMETERS
EXAMPLE: THE SCATTERING MATRIX OF A CONNECTOR

02/23/07 The Scattering Matrix 723 1/13
The Scattering Matrix
At “low” frequencies, we can completely characterize a linear
device or network using an impedance matrix, which relates the
currents and voltages at each device terminal to the currents
and voltages at all other terminals.
But, at microwave frequencies, it
is difficult to measure total
currents and voltages!
* Instead, we can measure the magnitude and phase of
each of the two transmission line waves ( ) and ( )V z V z+ −
.
* In other words, we can determine the relationship
between the incident and reflected wave at each device
terminal to the incident and reflected waves at all other
terminals.
These relationships are completely represented by the
scattering matrix. It completely describes the behavior of a
linear, multi-port device at a given frequency ω , and a given line
impedance Z0.

Consider now the 4-port microwave device shown below:
Note that we have now characterized transmission line activity
in terms of incident and “reflected” waves. Note the negative
going “reflected” waves can be viewed as the waves exiting the
multi-port network or device.
Viewing transmission line activity this way, we can fully
characterize a multi-port device by its scattering parameters!
( )1 1V z+
( )4 4V z+
( )3 3V z+
( )2 2V z+
port 1
( )1 1V z−
( )4 4V z−
( )3 3V z−
( )2 2V z−
port 3
port
4
port
2
4-port
microwave
device
Z0 Z0
Z0
Z0
3 3Pz z=
2 2Pz z=
1 1Pz z=
4 4Pz z=

Say there exists an incident wave on port 1 (i.e., ( )1 1 0V z+
≠ ),
while the incident waves on all other ports are known to be zero
(i.e., ( ) ( ) ( )2 2 3 3 4 4 0V z V z V z+ + +
= = = ).
Say we measure/determine the
voltage of the wave flowing into
port 1, at the port 1 plane (i.e.,
determine ( )1 1 1PV z z+
= ).
Say we then measure/determine
the voltage of the wave flowing
out of port 2, at the port 2
plane (i.e., determine
( )2 2 2PV z z−
= ).
The complex ratio between 1 1 1 2 2 2( ) and ( )P PV z z V z z+ −
= = is know
as the scattering parameter S21:
( )
2
2 1
1
022 2 2 02
21
1 1 1 01 01
( )
( )
P
P P
P
j z
j z zP
j z
P
V eV z z V
S e
V z z V e V
β
β
β
+−− −
+ +
−+ + +
=
= = =
=
Likewise, the scattering parameters S31 and S41 are:
3 3 3 4 4 4
31 41
1 1 1 1 1 1
( )( )
and
( ) ( )
P P
P P
V z zV z z
S S
V z z V z z
−−
+ +
==
= =
= =
( )1 1V z+ port 1
Z0
1 1P
z z=
( )1 1 1 p
V z z
+
+
=
−
( )2 2V z−
port 2
Z0
2 2P
z z=
( )2 2 2p
V z z
−
+
=
−

We of course could also define, say, scattering parameter S34
as the ratio between the complex values 4 4 4( )PV z z+
= (the wave
into port 4) and 3 3 3( )PV z z−
= (the wave out of port 3), given
that the input to all other ports (1,2, and 3) are zero.
Thus, more generally, the ratio of the wave incident on port n to
the wave emerging from port m is:
( )
( )
(given that 0 for all )
( )
m m mP
mn k k
n n nP
V z z
S V z k n
V z z
−
+
+
=
= = ≠
=
Note that frequently the port positions are assigned a zero
value (e.g., 1 20, 0P Pz z= = ). This of course simplifies the
scattering parameter calculation:
0
0 0
0
00
( 0)
( 0)
j
mm m m
mn j
n n nn
V eV z V
S
V z VV e
β
β
+−− −
−+ ++
=
= = =
=
We will generally assume that the port
locations are defined as 0nPz = , and thus use
the above notation. But remember where this
expression came from!
Microwave
lobe

A: Terminate all other ports with a matched load!
( )1 1V z+
( )3 3 0V z+
=
( )3 3 0V z+
=
( )2 2 0V z+
=
( )1 1V z−
( )4 4V z−
( )3 3V z−
( )2 2V z−
4-port
microwave
device
Z0 Z0
Z0
Z0
4 0LΓ =
3 0LΓ =
2 0LΓ =
Q: But how do we ensure
that only one incident wave
is non-zero ?

Note that if the ports are terminated in a matched load (i.e.,
0LZ Z= ), then 0nLΓ = and therefore:
( ) 0n nV z+
=
In other words, terminating a port ensures
that there will be no signal incident on
that port!
A: Actually, both statements are correct! You must be careful
to understand the physical definitions of the plus and minus
directions—in other words, the propagation directions of waves
( )n nV z+
and ( )n nV z−
!
Q: Just between you and me, I think you’ve messed this up! In all
previous handouts you said that if 0LΓ = , the wave in the minus
direction would be zero:
( ) 0 if 0LV z−
= Γ =
but just now you said that the wave in the positive direction would
be zero:
( ) 0 if 0LV z+
= Γ =
Of course, there is no way that both statements can be correct!

( ) 0 if 0LV z−
= Γ =
For example, we originally analyzed this case:
In this original case, the wave incident on the load is ( )V z+
(plus direction), while the reflected wave is ( )V z−
(minus
direction).
Contrast this with the case we are now considering:
For this current case, the situation is reversed. The wave
incident on the load is now denoted as ( )n nV z−
(coming out of
port n), while the wave reflected off the load is now denoted as
( )n nV z+
(going into port n ).
As a result, ( ) 0n nV z+
= when 0nLΓ = !
LΓ
( )V z−
( )V z+
Z0
nLΓ
( )n nV z+
( )n nV z−
Z0
port n
N-port
Microwave
Network

Perhaps we could more generally state that for some load LΓ :
( ) ( )reflected incident
L L LV z z V z z= = Γ =
Now, back to our discussion of S-parameters. We found that if
0nPz = for all ports n, the scattering parameters could be
directly written in terms of wave amplitudes 0nV +
and 0mV −
.
( )0
0
(when 0 for all )m
mn k k
n
V
S V z k n
V
−
+
+
= = ≠
Which we can now equivalently state as:
0
0
(when all ports, except port , are terminated in )
m
mn
n
n
V
S
V
−
+
= matched loads
For each case, you must be able to
correctly identify the mathematical
statement describing the wave incident on,
and reflected from, some passive load.
Like most equations in engineering, the
variable names can change, but the physics
described by the mathematics will not!

One more important note—notice that for the ports terminated
in matched loads (i.e., those ports with no incident wave), the
voltage of the exiting wave is also the total voltage!
( ) 0 0
0
0
0
(for all terminated ports)
n n
m
m
j z j z
m m m m
j z
m
j z
m
V z V e V e
V e
V e
β β
β
β
− ++ −
+−
+−
= +
= +
=
Thus, the value of the exiting wave at each terminated port is
likewise the value of the total voltage at those ports:
( ) 0 0
0
0
0
0
(for all terminated ports)
m m m
m
m
V V V
V
V
+ −
−
−
= +
= +
=
And so, we can express some of the scattering parameters
equivalently as:
( )
0
0
(for port , , for )m
mn
n
V
S m i.e. m n
V +
= ≠terminated
You might find this result helpful if attempting to determine
scattering parameters where m n≠ (e.g., S21, S43, S13), as we can
often use traditional circuit theory to easily determine the
total port voltage ( )0mV .

However, we cannot use the expression above to determine the
scattering parameters when m n= (e.g., S11, S22, S33).
Think about this! The scattering parameters for these
cases are:
0
0
n
nn
n
V
S
V
−
+
=
Therefore, port n is a port where there actually is some
incident wave 0nV +
(port n is not terminated in a matched load!).
And thus, the total voltage is not simply the value of the exiting
wave, as both an incident wave and exiting wave exists at port n.
( )4 4 0V z+
=
( )3 3 0V z+
=
( )2 2 0V z+
=
( )1 1V z−
( )4 4V z−
( )3 3V z−
( )2 2V z−
4-port
microwave
device
Z0 Z0
Z0
Z0
4 0LΓ =
3 0LΓ =
2 0LΓ =
( )1 1 0V z+
≠
( ) ( ) ( )1 1 10 0 0V V V+ −
= + ( ) ( )3 30 0V V −
=

Typically, it is much more difficult to determine/measure the
scattering parameters of the form Snn , as opposed to
scattering parameters of the form Smn (where m n≠ ) where
there is only an exiting wave from port m !
We can use the scattering matrix to determine the
solution for a more general circuit—one where the ports
are not terminated in matched loads!
A: Since the device is linear, we can apply superposition.
The output at any port due to all the incident waves is
simply the coherent sum of the output at that port due
to each wave!
For example, the output wave at port 3 can be
determined by (assuming 0nPz = ):
03 33 03 32 02 31 0134 04V S V S V S V S V− + + + +
= + + +
More generally, the output at port m of an N-port device
is:
( )0 0
1
0
N
m mn n nP
n
V S V z− +
=
= =∑
Q: I’m not understanding the importance
scattering parameters. How are they
useful to us microwave engineers?

This expression can be written in matrix form as:
− +
=V VS
Where −
V is the vector:
01 02 03 0
T
NV ,V ,V , ,V− − − − −
⎡ ⎤= ⎣ ⎦V …
and +
V is the vector:
01 02 03 0
T
NV ,V ,V , ,V+ + + + +
⎡ ⎤= ⎣ ⎦V …
Therefore S is the scattering matrix:
11 1
1
n
m mn
S S
S S
⎡ ⎤
⎢ ⎥= ⎢ ⎥
⎢ ⎥⎣ ⎦
S
…
The scattering matrix is a N by N matrix that completely
characterizes a linear, N-port device. Effectively, the
scattering matrix describes a multi-port device the way that LΓ
describes a single-port device (e.g., a load)!

But beware! The values of the scattering matrix for a
particular device or network, just like LΓ , are
frequency dependent! Thus, it may be more
instructive to explicitly write:
( )
( ) ( )
( ) ( )
11 1
1
n
m mn
S S
S S
ω ω
ω
ω ω
⎡ ⎤
⎢ ⎥= ⎢ ⎥
⎢ ⎥⎣ ⎦
S
…
Also realize that—also just like ΓL—the scattering matrix
is dependent on both the device/network and the Z0
value of the transmission lines connected to it.
Thus, a device connected to transmission lines with
0 50Z = Ω will have a completely different scattering
matrix than that same device connected to transmission
lines with 0 100Z = Ω !!!

02/23/07 Matched reciprocal lossless 723 1/9
Matched, Lossless,
Reciprocal Devices
As we discussed earlier, a device can be lossless or reciprocal.
In addition, we can likewise classify it as being matched.
Let’s examine each of these three characteristics, and how
they relate to the scattering matrix.
Matched
A matched device is another way of saying that the input
impedance at each port is equal to Z0 when all other ports are
terminated in matched loads. As a result, the reflection
coefficient of each port is zero—no signal will be come out of a
port if a signal is incident on that port (but only that port!).
In other words, we want:
0 for allm mm mV S V m− +
= =
a result that occurs when:
= 0 for all if matchedmmS m

We find therefore that a matched device will exhibit a
scattering matrix where all diagonal elements are zero.
Therefore:
0 0.1 0.2
0.1 0 0.3
0.2 0.3 0
j
j
S
⎡ ⎤
⎢ ⎥
⎢ ⎥=
⎢ ⎥
⎢ ⎥⎣ ⎦
is an example of a scattering matrix for a matched, three port
device.
Lossless
For a lossless device, all of the power that delivered to each
device port must eventually find its way out!
In other words, power is not absorbed by the network—no
power to be converted to heat!
Recall the power incident on some port m is related to the
amplitude of the incident wave ( 0mV +
) as:
2
0
02
m
m
V
P
Z
+
+
=
While power of the wave exiting the port is:
2
0
02
m
m
V
P
Z
−
−
=

Thus, the power delivered to (absorbed by) that port is the
difference of the two:
2 2
00
0 02 2
mm
m m m
V V
P P P
Z Z
+ −
+ −
∆ = − = −
Thus, the total power incident on an N-port device is:
2
0
01 1
1
2
N N
m m
m m
P P V
Z
+ + +
= =
= =∑ ∑
Note that:
( )
2
0
1
N
H
m
m
V V V+ + +
=
=∑
where operator H indicates the conjugate transpose (i.e.,
Hermetian transpose) operation, so that ( )
H
V V+ +
is the inner
product (i.e., dot product, or scalar product) of complex vector
V+
with itself.
Thus, we can write the total power incident on the device as:
( )2
0
0 01
1
2 2
HN
m
m
P V
Z Z
V V+ +
+ +
=
= =∑
Similarly, we can express the total power of the waves exiting
our M-port network to be:
( )2
0
0 01
1
2 2
HN
m
m
P V
Z Z
V V− −
− −
=
= =∑

Now, recalling that the incident and exiting wave amplitudes are
related by the scattering matrix of the device:
− +
=V VS
Thus we find:
( ) ( )
0 02 2
H H H
P
Z Z
V V V VS S− − + +
−
= =
Now, the total power delivered to the network is:
1
M
m
P P P P+ −
=
∆ = ∆ = −∑
Or explicitly:
( ) ( )
( ) ( )
0 0
0
2 2
1
2
H H H
H H
P P P
Z Z
Z
V V V V
V V
S S
I S S
+ −
+ + + +
+ +
∆ = −
= −
= −
where I is the identity matrix.
Q: Is there actually some point to this long, rambling, complex
presentation?
A: Absolutely! If our M-port device is lossless then the total
power exiting the device must always be equal to the total
power incident on it.

If network is then .P Plossless, + −
=
Or stated another way, the total power delivered to the device
(i.e., the power absorbed by the device) must always be zero if
the device is lossless!
If network is then 0Plossless, ∆ =
Thus, we can conclude from our math that for a lossless device:
( ) ( )
0
1
0 for all
2
H H
P
Z
V V VI S S+ + +
∆ = − =
This is true only if:
0H H
I S S S S I− = ⇒ =
Thus, we can conclude that the scattering matrix of a lossless
device has the characteristic:
If a network is , then H
lossless S S I=
Q: Huh? What exactly is this supposed to tell us?
A: A matrix that satisfies H
S S I= is a special kind of
matrix known as a unitary matrix.

If a network is , then its scattering matrix is .lossless unitaryS
Q: How do I recognize a unitary matrix if I see one?
A: The columns of a unitary matrix form an orthonormal set!
12
22
33
4
13
23
33
43
14
22
11
21
31
41
3
2
3
44
S
S
S
S S
S
S
S
S
S
S
S
S
S
S
S
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥⎣ ⎦
=S
In other words, each column of the scattering matrix will have
a magnitude equal to one:
2
1
1 for all
N
m
mn nS
=
=∑
while the inner product (i.e., dot product) of dissimilar columns
must be zero.
1 2
1
1 2 0 for allnj j jni i i N Ni
N
n
jS S S SS S S S i j∗ ∗
=
∗ ∗
= + + + = ≠∑
In other words, dissimilar columns are orthogonal.
matrix
columns

Consider, for example, a lossless three-port device. Say a
signal is incident on port 1, and that all other ports are
terminated. The power incident on port 1 is therefore:
2
01
1
02
V
P
Z
+
+
=
while the power exiting the device at each port is:
2 2
0 1 01 2
1 1
0 02 2
m m
m m
V S V
P S P
Z Z
− −
− +
= = =
The total power exiting the device is therefore:
( )
1 2 3
2 2 2
11 21 311 1 1
2 2 2
11 21 31 1
P P P P
S P S P S P
S S S P
− − − −
+ + +
+
= + +
= + +
= + +
Since this device is lossless, then the incident power (only on
port 1) is equal to exiting power (i.e, 1P P− +
= ). This is true only
if:
2 2
1 3
2
1 21 1 1S S S+ + =
Of course, this will likewise be true if the incident wave is
placed on any of the other ports of this lossless device:
2 2 2
2 2 2
13 23 3
2 2
3
1 22 3 1
1S S S
S S S+ + =
+ + =

We can state in general then that:
3
2
1
1 for allmn
m
S n
=
=∑
In other words, the columns of the scattering matrix must have
unit magnitude (a requirement of all unitary matrices). It is
apparent that this must be true for energy to be conserved.
An example of a (unitary) scattering matrix for a lossless
device is:
Reciprocal
Recall reciprocity results when we build a passive (i.e.,
unpowered) device with simple materials.
For a reciprocal network, we find that the elements of the
scattering matrix are related as:
mn nmS S=
1 3
2 2
31
2 2
3 1
2 2
3 1
2 2
0 0
0 0
0 0
0 0
j
j
j
j
⎡ ⎤
⎢ ⎥
⎢ ⎥=
⎢ ⎥
⎢ ⎥
⎣ ⎦
S

For example, a reciprocal device will have 21 12S S= or
32 23S S= . We can write reciprocity in matrix form as:
if reciprocalTS S=
where T indicates (non-conjugate) transpose.
An example of a scattering matrix describing a reciprocal, but
lossy and non-matched device is:
0.10 0.20 0.050.40
0.40 0 0.100.20
0.20 0.10 0.30 0.120
0.05 0.12 00.10
j
jj
j j
j
−−⎡ ⎤
⎢ ⎥−
⎢ ⎥=
⎢ ⎥− − −
⎢ ⎥
−⎣ ⎦
S

2/23/2007 Example A Lossless Reciprocal Network 1/4
Example: A Lossless,
Reciprocal Network
A lossless, reciprocal 3-port device has S-parameters of
11 1 2S = , 31 1 2S = , and 33 0S = . It is likewise known that all
scattering parameters are real.
Find the remaining 6 scattering parameters.
A: Yes I have! Note I said the device was lossless and
reciprocal!
Start with what we currently know:
1
2 12 13
21 22 23
1
322
0
S S
S S S
S
⎡ ⎤
⎢ ⎥
= ⎢ ⎥
⎢ ⎥⎣ ⎦
S
Because the device is reciprocal, we then also know:
21 12S S= 1
13 31 2
S S= = 32 23S S=
Q: This problem is clearly
impossible—you have not provided
us with sufficient information!

And therefore:
1 1
2 21 2
21 22 32
1
322
0
S
S S S
S
⎡ ⎤
⎢ ⎥
= ⎢ ⎥
⎢ ⎥⎣ ⎦
S
Now, since the device is lossless, we know that:
( ) ( )
22 2
11 21 31
222
1 1
2 21 2
1 S S S
S
= + +
= + +
22 2
12 22 32
22 2
21 22 32
1 S S S
S S S
= + +
= + +
( ) ( )
2 2 2
13 23 33
222
1 1
2 32 2
1 S S S
S
= + +
= + +
and:
11 12 21 22 31 32
1 1
2 21 21 22 322
0 S S S S S S
S S S S
∗ ∗ ∗
∗ ∗ ∗
= + +
= + +
( ) ( )
11 13 21 23 31 33
1 1 1
2 21 322 2
0
0
S S S S S S
S S
∗ ∗ ∗
∗
= + +
= + +
( ) ( )
12 13 22 23 32 33
1
21 22 32 322
0
0
S S S S S S
S S S S
∗ ∗ ∗
∗
= + +
= + +
Columns have
unit magnitude.
Columns are
orthogonal.

These six expressions simplify to:
1
221S =
22 2
21 22 321 S S S= + +
1
32 2
S =
1 1
2 21 21 22 322
0 S S S S= + +
( )
1
21 322 2
0 S S= +
( )1
21 22 322
0 S S S= +
where we have used the fact that since the elements are all
real, then 21 21S S∗
= (etc.).
Q
A: Actually, we have six real equations and six real
unknowns, since scattering element has a magnitude and
phase. In this case we know the values are real, and thus
the phase is either 0 or 180 (i.e., 0
1j
e = or
1j
e π
= − ); however, we do not know which one!
From the first three equations, we can find the magnitudes:
Q: I count the expressions and find 6 equations yet
only a paltry 3 unknowns. Your typical buffoonery
appears to have led to an over-constrained condition
for which there is no solution!

1
221S = 1
222S = 1
32 2
S =
and from the last three equations we find the phase:
1
221S = 1
222S = 1
32 2
S = −
Thus, the scattering matrix for this lossless, reciprocal device
is:
1 1 1
2 2 2
1 1 1
2 2 2
1 1
2 2
0
−
−
⎡ ⎤
⎢ ⎥
= ⎢ ⎥
⎢ ⎥⎣ ⎦
S

3/4/2009 MLR 3 port network 1/2
A Matched, Lossless
Reciprocal 3-Port Network
Consider a 3-port device. Such a device would have a scattering
matrix :
11 12 13
21 22 23
31 32 33
S S S
S S S
S S S
⎡ ⎤
⎢ ⎥= ⎢ ⎥
⎢ ⎥⎣ ⎦
S
Assuming the device is passive and made of simple (isotropic)
materials, the device will be reciprocal, so that:
21 12 31 13 23 32S S S S S S= = =
Likewise, if it is matched, we know that:
11 22 33 0S S S= = =
As a result, a lossless, reciprocal device would have a scattering
matrix of the form:
21 31
21 32
31 32
0
0
0
S S
S S
S S
⎡ ⎤
⎢ ⎥= ⎢ ⎥
⎢ ⎥⎣ ⎦
S
Just 3 non-zero scattering parameters define the entire
matrix!

Likewise, if we wish for this network to be lossless, the
scattering matrix must be unitary, and therefore:
∗
∗
∗
+ = =
+ = =
+ = =
22
21 31 31 32
22
21 32 21 32
2 2
31 32 21 31
1 0
1 0
1 0
S S S S
S S S S
S S S S
Since each complex value S is represented by two real numbers
(i.e., real and imaginary parts), the equations above result in 9
real equations. The problem is, the 3 complex values S21, S31 and
S32 are represented by only 6 real unknowns.
We have over constrained our problem ! There are no solutions
to these equations !
As unlikely as it might seem, this means
that a matched, lossless, reciprocal 3-
port device of any kind is a physical
impossibility!
You can make a lossless reciprocal 3-
port device, or a matched reciprocal 3-
port device, or even a matched, lossless
(but non-reciprocal) 3-port network.
But try as you might, you cannot make a
lossless, matched, and reciprocal three
port component!

The Matched, Lossless,
Reciprocal 4-Port Network
The first solution is referred to as the symmetric solution:
0 0
0 0
0 0
0 0
j
j
j
j
α β
α β
β α
β α
⎡ ⎤
⎢ ⎥
⎢ ⎥=
⎢ ⎥
⎢ ⎥
⎣ ⎦
S
Note for this symmetric solution, every row and every column of
the scattering matrix has the same four values (i.e., α, jβ, and
two zeros)!
The second solution is referred to as the anti-symmetric
solution:
0 0
0 0
0 0
0 0
α β
α β
β α
β α
⎡ ⎤
⎢ ⎥−
⎢ ⎥=
⎢ ⎥
⎢ ⎥
−⎣ ⎦
S
Guess what! I have determined
that—unlike a 3-port device—a
matched, lossless, reciprocal 4-port
device is physically possible! In fact,
I’ve found two general solutions!

Note that for this anti-symmetric solution, two rows and two
columns have the same four values (i.e., α, β, and two zeros),
while the other two row and columns have (slightly) different
values (α, -β, and two zeros)
It is quite evident that each of these solutions are matched and
reciprocal. However, to ensure that the solutions are indeed
lossless, we must place an additional constraint on the values of
α, β. Recall that a necessary condition for a lossless device is:
2
1
1 for all
N
mn
m
S n
=
=∑
Applying this to the symmetric case, we find:
2 2
1α β+ =
Likewise, for the anti-symmetric case, we also get
2 2
1α β+ =
It is evident that if the scattering matrix is unitary (i.e.,
lossless), the values α and β cannot be independent, but must
related as:
2 2
1α β+ =

Generally speaking, we will find that α β≥ . Given the
constraint on these two values, we can thus conclude that:
10
2
β≤ ≤ and 1 1
2
α≤ ≤

2/23/2007 Example The Scattering Matrix 1/6
Example: The
Scattering Matrix
Say we have a 3-port network that is completely characterized
at some frequency ω by the scattering matrix:
0.0 0.2 0.5
0.5 0.0 0.2
0.5 0.5 0.0
⎡ ⎤
⎢ ⎥= ⎢ ⎥
⎢ ⎥⎣ ⎦
S
A matched load is attached to port 2, while a short circuit has
been placed at port 3:
1 (z)V +
3 (z)V +
2 (z)V +
port 1
1 (z)V −
3 (z)V −
2 (z)V −
port 3
port
2
3-port
microwave
device
Z0 Z0
Z0
3 0Pz =
2 0Pz =
1 0Pz =
0Z Z=
0Z =

Because of the matched load at port 2 (i.e., 0LΓ = ), we know
that:
022 2
2 2 02
( 0)
0
( 0)
VV z
V z V
++
− −
=
= =
=
and therefore:
02 0V +
=
NO!! Remember, the signal 2 ( )V z−
is incident on the matched
load, and 2 ( )V z+
is the reflected wave from the load (i.e., 2 ( )V z+
is incident on port 2). Therefore, 02 0V +
= is correct!
Likewise, because of the short circuit at port 3 ( 1LΓ = − ):
3 3 03
3 3 03
( 0)
1
( 0)
V z V
V z V
+ +
− −
=
= = −
=
and therefore:
03 03V V+ −
= −
You’ve made a terrible mistake!
Fortunately, I was here to
correct it for you—since 0LΓ = ,
the constant 02V −
(not 02V +
) is
equal to zero.

Problem:
a) Find the reflection coefficient at port 1, i.e.:
01
1
01
V
V
−
+
Γ
b) Find the transmission coefficient from port 1 to port 2, i.e.,
02
21
01
V
T
V
−
+
NO!!! The above statement is not correct!
Remember, 01 01 11V V S− +
= only if ports 2 and 3 are
terminated in matched loads! In this problem port 3
is terminated with a short circuit.
I am amused by the trivial
problems that you apparently
find so difficult. I know that:
01
1 11
01
0.0
V
S
V
−
+
Γ = = =
and
02
21 21
01
0.5
V
T S
V
−
+
= = =

Therefore:
01
1 11
01
V
S
V
−
+
Γ = ≠
and similarly:
02
21 21
01
V
T S
V
−
+
= ≠
To determine the values 21T and 1Γ , we must start with the
three equations provided by the scattering matrix:
01 02 03
02 01 03
03 01 02
0 2 0 5
0 5 0 2
0 5 0 5
V . V . V
V . V . V
V . V . V
− + +
− + +
− + +
= +
= +
= +
and the two equations provided by the attached loads:
02
03 03
0V
V V
+
+ −
=
= −

We can divide all of these equations by 01V +
, resulting in:
01 02 03
1
01 01 01
02 03
21
01 01
03 02
01 01
02
01
03 03
01 01
0 2 0 5
0 5 0 2
0 5 0 5
0
V V V
. .
V V V
V V
T . .
V V
V V
. .
V V
V
V
V V
V V
− + +
+ + +
− +
+ +
− +
+ +
+
+
+ −
+ +
= +
= = +
= +
=
= −
Γ =
Look what we have—5 equations and 5 unknowns! Inserting
equations 4 and 5 into equations 1 through 3, we get:
01 03
1
01 01
02 03
21
01 01
03
01
0 5
0 5 0 2
0 5
V V
.
V V
V V
T . .
V V
V
.
V
− +
+ +
− +
+ +
−
+
= −
= = −
=
Γ =

Solving, we find:
( )
( )
1
21
0 5 0 5 0 25
0 5 0 2 0 5 0 4
. . .
T . . . .
= − = −
= − =
Γ

2/23/2007 Example Scattering Parameters 1/4
Example: Scattering
Parameters
Consider a two-port device with a scattering matrix (at some
specific frequency 0ω ):
( )0
0 1 0 7
0 7 0 2
. j .
j . .
ω ω
⎡ ⎤
= = ⎢ ⎥−⎣ ⎦
S
and 0 50Z = Ω.
Say that the transmission line connected to port 2 of this
device is terminated in a matched load, and that the wave
incident on port 1 is:
( ) 1
1 1 2 j z
V z j e β−+
= −
where 1 2 0P Pz z= = .
Determine:
1. the port voltages ( )1 1 1PV z z= and ( )2 2 2PV z z= .
2. the port currents ( )1 1 1PI z z= and ( )2 2 2PI z z= .
3. the net power flowing into port 1

1. Since the incident wave on port 1 is:
( ) 1
1 1 2 j z
V z j e β−+
= −
we can conclude (since 1 0Pz = ):
( )
( )
1
1 1 1
0
2
2
2
Pj z
P
j
V z z j e
j e
j
β
β
−+
−
= = −
= −
= −
and since port 2 is matched (and only because its matched!),
we find:
( ) ( )
( )
1 1 1 11 1 1 1
0 1 2
0 2
P PV z z S V z z
. j
j .
− +
= = =
= −
= −
The voltage at port 1 is thus:
( ) ( ) ( )1 1 1 1 1 1 1 1 1
2
2 0 0 2
2 2
2 2
P P P
j
V z z V z z V z z
j . j .
j .
. e
π
+ −
−
= = = + =
= − −
= −
=
Likewise, since port 2 is matched:
( )2 2 2 0PV z z+
= =

And also:
( ) ( )
( )
2 2 2 21 1 1 1
0 7 2
1 4
P PV z z S V z z
j . j
.
− +
= = =
= −
=
Therefore:
( ) ( ) ( )2 2 2 2 2 2 2 2 2
0
0 1 4
1 4
1 4
P P P
j
V z z V z z V z z
.
.
. e
+ −
−
= = = + =
= +
=
=
2. The port currents can be easily determined from the
results of the previous section.
( ) ( ) ( )
( ) ( )
1 1 1 1 1 1 1 1 1
1 1 1 1 1 1
0 0
2
2 0 0 2
50 50
1 8
50
0 036
0 036
P P P
P P
j
I z z I z z I z z
V z z V z z
Z Z
. .
j j
.
j
j .
. e
π
+ −
+ −
−
= = = − =
= =
= −
= − +
= −
= −
=
and:

( ) ( ) ( )
( ) ( )
2 2 2 2 2 2 2 2 2
2 2 2 2 2 2
0 0
0 1 4
50 50
0 028
0 028
P P P
P P
j
I z z I z z I z z
V z z V z z
Z Z
.
.
. e π
+ −
+ −
+
= = = − =
= =
= −
= −
= −
=
3. The net power flowing into port 1 is:
( ) ( )
( )
1 1 1
2 2
01 01
0 0
2 2
2 2
2 0 2
2 50
0 0396
P P P
V V
Z Z
.
. Watts
+ −
+ −
∆ = −
= −
−
=
=

2/23/2007 Example Determining the Scattering Matrix 1/5
Example: Determining the
Scattering Matrix
Let’s determine the scattering matrix of this two-port device:
The first step is to terminate port 2 with a matched load, and
then determine the values:
( )1 1 1PV z z−
= and ( )2 2 2PV z z−
=
in terms of ( )1 1 1PV z z+
= .
0Z 0Z2Z0
1 0Pz =
z1
2 0Pz =
z2
0Z 2Z0
1 0Pz =
z1
2 0Pz =
Z0( )2 2
0V z
+
=
−
( )1 1
V z
+
−

Recall that since port 2 is matched, we know that:
( )2 2 2 0PV z z+
= =
And thus:
( ) ( ) ( )
( )
( )
2 2 2 2 2 2
2 2
2 2
0 0 0
0 0
0
V z V z V z
V z
V z
+ −
−
−
= = = + =
= + =
= =
In other words, we simply need to determine ( )2 2 0V z = in order
to find ( )2 2 0V z−
= !
However, determining ( )1 1 0V z−
= is a bit trickier. Recall that:
( ) ( ) ( )1 1 1 1 1 1V z V z V z+ −
= +
Therefore we find ( ) ( )1 1 1 10 0V z V z−
= ≠ = !
Now, we can simplify this circuit:
And we know from the telegraphers equations:
0Z
0
2
3
Z
1 0Pz =
z1
( )1 1
V z
+
−

( ) ( ) ( )
1 1
1 1 1 1 1 1
01 01
j z j z
V z V z V z
V e V e
+ −
− β + β+ −
= +
= +
Since the load 02 3Z is located at 1 0z = , we know that the
boundary condition leads to:
( ) ( )1 1
1 1 01
j z j z
LV z V e e− β + β+
= + Γ
where:
( )
( )
( )
( )
2
3 0 0
2
3 0 0
2
3
2
3
1
3
5
3
1
1
0 2
L
Z Z
Z Z
.
−
=
+
−
=
+
−
=
= −
Γ
Therefore:
( ) 1
1 1 01
j z
V z V e− β+ +
= and ( ) ( ) 1
1 1 01 0 2 j z
V z V . e+ β− +
= −
and thus:
( ) ( )0
1 1 01 010 j
V z V e V− β+ + +
= = =
( ) ( ) ( )0
1 1 01 010 0 2 0 2j
V z V . e . V+ β− + +
= = − = −
We can now determine 11S !
( )
( )
1 1 01
11
1 1 01
0 0 2
0 2
0
V z . V
S .
V z V
− +
+ +
= −
= = = −
=

Now its time to find ( )2 2 0V z−
= !
Again, since port 2 is terminated, the incident wave on port 2
must be zero, and thus the value of the exiting wave at port 2 is
equal to the total voltage at port 2:
( ) ( )2 2 2 20 0V z V z−
= = =
This total voltage is relatively easy to determine. Examining
the circuit, it is evident that ( ) ( )1 1 2 20 0V z V z= = = .
Therefore:
( ) ( )
( ) ( )
( )
( )
( )
2 2 1 1
0 0
01
01
01
0 0
0 2
1 0 2
0 8
j j
V z V z
V e . e
V .
V .
− β + β+
+
+
= = =
= −
= −
=
And thus the scattering parameter 21S is:
( )
( )
2 2 01
21
1 1 01
0 0 8
0 8
0
V z . V
S .
V z V
− +
+ +
=
= = =
=
0Z
2Z0
1 0Pz =
z1
2 0Pz =
Z0( )2 2
0V z
+
=
−
( )1 1
0V z
+
−
=

Now we just need to find 12S and 22S .
Q: Yikes! This has been an awful lot of work, and you mean that
we are only half-way done!?
A: Actually, we are nearly finished! Note that this circuit is
symmetric—there is really no difference between port 1 and
port 2. If we “flip” the circuit, it remains unchanged!
Thus, we can conclude due to this symmetry that:
11 22 0 2S S .= = −
and:
21 12 0 8S S .= =
Note this last equation is likewise a result of reciprocity.
Thus, the scattering matrix for this two port network is:
0 2 0 8
0 8 0 2
. .
. .
−⎡ ⎤
= ⎢ ⎥−⎣ ⎦
S
0Z 0Z2Z0
2 0Pz =
z2
1 0Pz =
z1

3/4/2009 Circuit Symmetry 1/14
Circuit Symmetry
One of the most powerful
concepts in for evaluating circuits
is that of symmetry. Normal
humans have a conceptual
understanding of symmetry, based
on an esthetic perception of
structures and figures.
On the other hand, mathematicians (as they
are wont to do) have defined symmetry in a
very precise and unambiguous way. Using a
branch of mathematics called Group
Theory, first developed by the young genius
Évariste Galois (1811-1832), symmetry is
defined by a set of operations (a group)
that leaves an object unchanged.
Initially, the symmetric “objects” under consideration by
Galois were polynomial functions, but group theory can
likewise be applied to evaluate the symmetry of structures.
For example, consider an ordinary
equilateral triangle; we find that it is a
highly symmetric object!
Évariste Galois

Q: Obviously this is true. We don’t need a mathematician to
tell us that!
A: Yes, but how symmetric is it? How does the symmetry of
an equilateral triangle compare to that of an isosceles
triangle, a rectangle, or a square?
To determine its level of symmetry, let’s first label each
corner as corner 1, corner 2, and corner 3.
First, we note that the triangle exhibits a plane of reflection
symmetry:
1
2
3
1
2
3

Thus, if we “reflect” the triangle across this plane we get:
Note that although corners 1 and 3 have changed places, the
triangle itself remains unchanged—that is, it has the same
shape, same size, and same orientation after reflecting across
the symmetric plane!
Mathematicians say that these two triangles are congruent.
Note that we can write this reflection operation as a
permutation (an exchange of position) of the corners, defined
as:
1 3
2 2
3 1
→
→
→
Q: But wait! Isn’t there is more than just one plane of
reflection symmetry?
A: Definitely! There are two more:
3
2
1

1 2
2 1
3 3
→
→
→
1 1
2 3
3 2
→
→
→
In addition, an equilateral triangle exhibits rotation
symmetry!
Rotating the triangle 120 clockwise also results in a
congruent triangle:
1 2
2 3
3 1
→
→
→
Likewise, rotating the triangle 120 counter-clockwise results
in a congruent triangle:
2
1
3
1
3
2
1
2
3
1
2
3
1
2
3
3
1
2

1 3
2 1
3 2
→
→
→
Additionally, there is one more operation that will result in a
congruent triangle—do nothing!
1 1
2 2
3 3
→
→
→
This seemingly trivial operation is known as the identity
operation, and is an element of every symmetry group.
These 6 operations form the dihedral symmetry group D3
which has order six (i.e., it consists of six operations). An
object that remains congruent when operated on by any and
all of these six operations is said to have D3 symmetry.
An equilateral triangle has D3 symmetry!
By applying a similar analysis to a isosceles triangle, rectangle,
and square, we find that:
1
2
3 2
3
1
1
2
3 1
2
3

An isosceles trapezoid has D1 symmetry, a
dihedral group of order 2.
A rectangle has D2 symmetry, a dihedral group
of order 4.
A square has D4 symmetry, a dihedral group of
order 8.
Thus, a square is the most symmetric object of the four we
have discussed; the isosceles trapezoid is the least.
Q: Well that’s all just fascinating—but just what the heck
does this have to do with microwave circuits!?!
A: Plenty! Useful circuits often display high levels of
symmetry.
For example consider these D1 symmetric multi-port circuits:
1 2
2 1
3 4
4 3
→
→
→
→
D1
D2
D4
50Ω
100Ω
200Ω200Ω
Port 1 Port 2
Port 4Port 3

1 3
2 4
3 1
4 2
→
→
→
→
Or this circuit with D2 symmetry:
which is congruent under these permutations:
1 3
2 4
3 1
4 2
→
→
→
→
1 2
2 1
3 4
4 3
→
→
→
→
1 4
2 3
3 2
4 1
→
→
→
→
50Ω
50 Ω
200Ω100Ω
Port 1 Port 2
Port 4Port 3
50Ω
50 Ω
200Ω200Ω
Port 1 Port 2
Port 4Port 3

Or this circuit with D4 symmetry:
which is congruent under these permutations:
1 3
2 4
3 1
4 2
→
→
→
→
1 2
2 1
3 4
4 3
→
→
→
→
1 4
2 3
3 2
4 1
→
→
→
→
1 4
2 2
3 3
4 1
→
→
→
→
1 1
2 3
3 2
4 4
→
→
→
→
The importance of this can be seen when considering the
scattering matrix, impedance matrix, or admittance matrix of
these networks.
For example, consider again this symmetric circuit:
50Ω
100Ω
200Ω200Ω
Port 1 Port 2
Port 4Port 3
50Ω
50 Ω
50 Ω50 Ω
Port 1 Port 2
Port 4Port 3

This four-port network has a single plane of reflection
symmetry (i.e., D1 symmetry), and thus is congruent under the
permutation:
1 2
2 1
3 4
4 3
→
→
→
→
So, since (for example) 1 2→ , we find that for this circuit:
11 22 11 22 11 22S S Z Z Y Y= = =
must be true!
Or, since 1 2→ and 3 4→ we find:
13 13 1324 24 24
31 31 3142 42 42
S S Z Z Y Y
S S Z Z Y Y
= = =
= = =
Continuing for all elements of the permutation, we find that
for this symmetric circuit, the scattering matrix must have
this form:
11 21 13 14
21 11 1314
31 3341 43
31 3341 43
S S S S
S S S S
S S S S
S S S S
⎡ ⎤
⎢ ⎥
⎢ ⎥=
⎢ ⎥
⎢ ⎥
⎣ ⎦
S

and the impedance and admittance matrices would likewise
have this same form.
Note there are just 8 independent elements in this matrix. If
we also consider reciprocity (a constraint independent of
symmetry) we find that 31 13
S S= and 41 14
S S= , and the matrix
reduces further to one with just 6 independent elements:
41
41
41 33
31
31
31
31
43
434
21
3
1
11
1 3
112
S
S
S S
S
S
S
S
S
S
S
S
S
SS S
⎡ ⎤
⎢ ⎥
⎢ ⎥=
⎢ ⎥
⎢ ⎥
⎣ ⎦
S
Or, for circuits with this D1 symmetry:
1 3
2 4
3 1
4 2
→
→
→
→
41
41
41
11
11
21
21
2
31
31
31
31 1
21 2
2
4 2
2
1
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S S
⎡ ⎤
⎢ ⎥
⎢ ⎥=
⎢ ⎥
⎢ ⎥
⎣ ⎦
S
Q: Interesting. But why do we care?
50Ω
50 Ω
200Ω100Ω
Port 1 Port 2
Port 4Port 3

A: This will greatly simplify the analysis of this symmetric
circuit, as we need to determine only six matrix elements!
For a circuit with D2 symmetry:
we find that the impedance (or scattering, or admittance)
matrix has the form:
41
41
41
11
11
11
21
21
21
41
31
31
3
21 1
1
131
Z
Z
Z
Z
Z Z
Z
Z
Z Z
ZZ
ZZ
Z
Z
⎡ ⎤
⎢ ⎥
⎢ ⎥=
⎢ ⎥
⎢ ⎥
⎣ ⎦
Z
Note here that there are just four independent values!
50Ω
50 Ω
200Ω200Ω
Port 1 Port 2
Port 4Port 3

For a circuit with D4 symmetry:
we find that the admittance (or scattering, or impedance)
matrix has the form:
41
41
41
11
11
11
21 21
21 21
21 21
21 24 1111
Y Y
Y Y
Y Y
Y Y
Y
Y
Y
Y
Y
Y
Y
Y
⎡ ⎤
⎢ ⎥
⎢ ⎥=
⎢ ⎥
⎢ ⎥
⎣ ⎦
Y
Note here that there are just three independent values!
One more interesting thing (yet another one!); recall that we
earlier found that a matched, lossless, reciprocal 4-port
device must have a scattering matrix with one of two forms:
50Ω
50 Ω
50 Ω50 Ω
Port 1 Port 2
Port 4Port 3

0 0
0 0
0 0
0 0
j
j
j
j
α β
α β
β α
β α
⎡ ⎤
⎢ ⎥
⎢ ⎥=
⎢ ⎥
⎢ ⎥
⎣ ⎦
S The “symmetric” solution
0 0
0 0
0 0
0 0
α β
α β
β α
β α
⎡ ⎤
⎢ ⎥−
⎢ ⎥=
⎢ ⎥
⎢ ⎥
−⎣ ⎦
S The “anti-symmetric” solution
Compare these to the matrix forms above. The “symmetric
solution” has the same form as the scattering matrix of a
circuit with D2 symmetry!
0
0
0
0
0
0
0
0
j
j
j
j
β
β
α
α
α
α
β
β
⎡ ⎤
⎢ ⎥
⎢ ⎥=
⎢ ⎥
⎢ ⎥
⎣ ⎦
S
Q: Does this mean that a matched, lossless, reciprocal four-
port device with the “symmetric” scattering matrix must
exhibit D2 symmetry?
A: That’s exactly what it means!

Not only can we determine from the form of the scattering
matrix whether a particular design is possible (e.g., a matched,
lossless, reciprocal 3-port device is impossible), we can also
determine the general structure of a possible solutions (e.g.
the circuit must have D2 symmetry).
Likewise, the “anti-symmetric” matched, lossless, reciprocal
four-port network must exhibit D1 symmetry!
0
0
0
0
00
0
0
α
α
α
β
β
β α
β
−
⎡ ⎤
⎢ ⎥
⎢ ⎥=
⎢
⎢ ⎥
⎣ ⎦−
⎥
S
We’ll see just what these symmetric, matched, lossless,
reciprocal four-port circuits actually are later in the course!

3/6/2009 Example Using Symmetry to Determine S 1/3
Example: Using Symmetry
to Determine a
Scattering Matrix
Say we wish to determine the scattering matrix of the simple
two-port device shown below:
We note that that attaching transmission lines of
characteristic impedance 0
Z to each port of our “circuit”
forms a continuous transmission line of characteristic
impedance 0
Z .
Thus, the voltage all along this transmission line thus has the
form:
( ) 0 0
j z j z
V z V e V eβ β− ++ −
= +
0 ,Z β
0z =z = −
port
1
port
2 0 ,Z β
0 ,Z β

We begin by defining the location of port 1 as 1Pz = − , and
the port location of port 2 as 2 0Pz = :
We can thus conclude:
( ) ( )
( ) ( )
( ) ( )
( ) ( )
1 0
1 0
2 0
2 0
0
0
j z
j z
j z
j z
V z V e z
V z V e z
V z V e z
V z V e z
β
β
β
β
−+ +
+− −
++ −
−− +
= ≤ −
= ≤ −
= ≥
= ≥
Say the transmission line on port 2 is terminated in a matched
load. We know that the –z wave must be zero ( )0
0V −
= , and so
the voltage along the transmission line becomes simply the +z
wave voltage:
( ) 0
j z
V z V e β−+
=
and so:
0z =z = −
( )V z
+
−
( )1V z+
( )1V z−
( )2V z−
( )2V z+
0
j z
V e β−+
0
j z
V e β+−

( ) ( ) ( )
( ) ( ) ( )
1 0 1
2 2 0
0
0 0
j z
j z
V z V e V z z
V z V z V e z
β
β
−+ + −
−+ − +
= = ≤ −
= = ≥
Now, because port 2 is terminated in a matched load, we can
determine the scattering parameters 11S and 21S :
( )
( )
( )
( ) ( )
22
1 1
11
1 1 000
0
0P
jβ
P VV
V z z V z
S
V z z V z V e++
− −
+ + − −+
==
= = −
= = = =
= = −
( )
( )
( )
( )
( )
( )
22
0
2 2 2 0
21
1 1 1 000
0 1
jβ
jβP
jβjβ
P VV
V z z V z V e
S e
V z z V z eV e++
−− − +
−
++ + − −+
==
= =
= = = = =
= = −
From the symmetry of the structure, we can conclude:
22 11 0S S= =
And from both reciprocity and symmetry:
12 21
j
S S e β−
= =
Thus:
0
0
j
j
e
e
β
β
−
−
⎡ ⎤
= ⎢ ⎥
⎢ ⎥⎣ ⎦
S

3/6/2009 Symmetric Circuit Analysis 1/10
Symmetric Circuit
Analysis
Consider the following D1 symmetric two-port device:
Q: Yikes! The plane of reflection symmetry slices through
two resistors. What can we do about that?
A: Resistors are easily split into two equal pieces: the 200Ω
resistor into two 100Ω resistors in series, and the 50Ω
resistor as two 100 Ω resistors in parallel.
I1
200Ω
I2
+
V2
-
+
100Ω 100Ω
50Ω
I1
100Ω
I2
+
V2
-
+
V1
-
100Ω 100Ω
100Ω
100Ω
100Ω

Recall that the symmetry of this 2-port device leads to
simplified network matrices:
21 21 21
21 2
11 11 11
1 1 1211 11 1
S Z Y
S
S Z Y
S Z Z Y Y
⎡ ⎤ ⎡ ⎤⎡ ⎤
= = =⎢ ⎥ ⎢ ⎥⎢ ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦
S Z Y
Q: Yes, but can circuit symmetry likewise simplify the
procedure of determining these elements? In other words,
can symmetry be used to simplify circuit analysis?
A: You bet!
First, consider the case where we attach sources to circuit in
a way that preserves the circuit symmetry:
Or,
I1
100Ω
I2
+
V2
-
+
V1
-
100Ω 100Ω
100Ω
100Ω
100Ω
+
-
+
-Vs
Vs

Or,
But remember! In order for symmetry to be preserved, the
source values on both sides (i.e, Is,Vs,Z0) must be identical!
Now, consider the voltages and currents within this circuit
under this symmetric configuration:
I1
100Ω
I2
+
V2
-
+
V1
-
100Ω 100Ω
100Ω
100Ω
100Ω
Is
Is
I1
100Ω
I2
+
V2
-
+
V1
-
100Ω 100Ω
100Ω
100Ω
100Ω
Vs
Vs
+
-
+
-
Z0 Z0

Since this circuit possesses bilateral (reflection) symmetry
(1 2 2 1,→ → ), symmetric currents and voltages must be equal:
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
a a
a a
b b
b b
c c
c c
d d
I I
V V
I I
V V
I I
V V
I I
V V
I I
=
=
=
=
=
=
=
=
=
Q: Wait! This can’t possibly be correct! Look at currents I1a
and I2a, as well as currents I1d and I2d. From KCL, this must be
true:
1 2 1 2a a d dI I I I= − = −
Yet you say that this must be true:
1 2 1 2a a d dI I I I= =
I1 I2
+
V2
-
+
V1
-
+
-
+
-Vs
Vs
+ V1a -
+ V1b -
+
V1c
-
+
V2c
-
- V2a +
- V2b +
I1a
I1b
I1c
I1d
I2a
I2d
I2c
I2b

There is an obvious contradiction here! There is no way that
both sets of equations can simultaneously be correct, is
there?
A: Actually there is! There is one solution that will satisfy
both sets of equations:
1 2 1 20 0a a d dI I I I= = = =
The currents are zero!
If you think about it, this makes perfect
sense! The result says that no current will
flow from one side of the symmetric
circuit into the other.
If current did flow across the symmetry
plane, then the circuit symmetry would be
destroyed—one side would effectively
become the “source side”, and the other
the “load side” (i.e., the source side
delivers current to the load side).
Thus, no current will flow across the reflection symmetry
plane of a symmetric circuit—the symmetry plane thus acts as
a open circuit!
The plane of symmetry thus becomes a virtual open!

Q: So what?
A: So what! This means that our circuit can be split apart
into two separate but identical circuits. Solve one half-
circuit, and you have solved the other!
I1 I2
+
V2
-
+
V1
-
+
-
+
-Vs
Vs
+ V1a -
+ V1b -
+
V1c
-
+
V2c
-
- V2a +
- V2b +
I1b
I1c I2c
I2b
Virtual Open
I=0
I1
+
V1
-
+
-Vs
+ V1a -
+ V1b -
+
V1c
-
I1b
I1c
I1a
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
0
2
2
200
0
200
200
0
s
a a
b b s
c c s
s
a a
b b s
c c s
d d
V V V
V V
V V V
V V V
I I V
I I
I I V
I I V
I I
= =
= =
= =
= =
= =
= =
= =
= =
= =

Now, consider another type of symmetry, where the sources
are equal but opposite (i.e., 180 degrees out of phase).
Or,
I1
100Ω
I2
+
V2
-
+
V1
-
100Ω 100Ω
100Ω
100Ω
100Ω
+
-
+
-Vs
-Vs
I1
100Ω
I2
+
V2
-
+
V1
-
100Ω 100Ω
100Ω
100Ω
100Ω
Is
-Is

Or,
This situation still preserves the symmetry of the circuit—
somewhat. The voltages and currents in the circuit will now
posses odd symmetry—they will be equal but opposite (180
degrees out of phase) at symmetric points across the
symmetry plane.
I1
100Ω
I2
+
V2
-
+
V1
-
100Ω 100Ω
100Ω
100Ω
100Ω
Vs
-Vs
+
-
+
-
Z0 Z0
I1 I2
+
V2
-
+
V1
-
+
-
+
-Vs
-Vs
+ V1a -
+ V1b -
+
V1c
-
+
V2c
-
- V2a +
- V2b +
I1a
I1b
I1c
I1d
I2a
I2d
I2c
I2b
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
a a
a a
b b
b b
c c
c c
d d
I I
V V
I I
V V
I I
V V
I I
V V
I I
= −
= −
= −
= −
= −
= −
= −
= −
= −

Perhaps it would be easier to redefine the circuit variables as:
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
a a
a a
b b
b b
c c
c c
d d
I I
V V
I I
V V
I I
V V
I I
V V
I I
=
=
=
=
=
=
=
=
=
Q: But wait! Again I see a problem. By KVL it is evident
that:
1 2c cV V= −
Yet you say that 1 2c c
V V= must be true!
A: Again, the solution to both equations is zero!
1 2 0c cV V= =
I1 I2
-
V2
+
+
V1
-
+
-
+
-Vs
-Vs
+ V1a -
+ V1b -
+
V1c
-
-
V2c
+
+ V2a -
+ V2b -
I1a
I1b
I1c
I1d
I2a
I2d
I2c
I2b

For the case of odd symmetry, the symmetric plane must be a
plane of constant potential (i.e., constant voltage)—just like a
short circuit!
Thus, for odd symmetry, the symmetric plane forms a virtual
short.
This greatly simplifies things, as we can again break the
circuit into two independent and (effectively) identical
circuits!
I1 I2
-
V2
+
+
V1
-
+
-
+
-Vs
-Vs
+ V1a -
+ V1b -
+
V1c
-
-
V2c
+
+ V2a -
+ V2b -
I1a
I1b
I1c
I1d
I2a
I2d
I2c
I2b
Virtual short
V=0
I1
+
V1
-
+
-Vs
+ V1a -
+ V1b -
+
V1c
-
I1a
I1b
I1c
I1d
1
1
1
1
1
1
1
1
1
50
100
100
0
0
100
s
s
a s
a s
sb
sb
c
c
sd
I V
V V
I V
V V
I V
V V
I
V
I V
=
=
=
=
=
=
=
=
=

3/6/2009 Odd Even Mode Analysis 1/10
Odd/Even Mode Analysis
Q: Although symmetric circuits appear to be plentiful in
microwave engineering, it seems unlikely that we would often
encounter symmetric sources . Do virtual shorts and opens
typically ever occur?
A: One word—superposition!
If the elements of our circuit are independent and linear, we
can apply superposition to analyze symmetric circuits when
non-symmetric sources are attached.
For example, say we wish to determine the admittance matrix
of this circuit. We would place a voltage source at port 1,
and a short circuit at port 2—a set of asymmetric sources if
there ever was one!
I1
100Ω
I2
+
V2
-
+
V1
-
100Ω 100Ω
100Ω
100Ω
100Ω
+
-
+
-Vs1=Vs Vs2=0

Here’s the really neat part. We find that the source on port
1 can be model as two equal voltage sources in series, whereas
the source at port 2 can be modeled as two equal but
opposite sources in series.
Therefore an equivalent circuit is:
+
-
+
-
2
sV
2
sV
I1
100Ω
I2
100Ω 100Ω
100Ω
100Ω
100Ω
+
-
2
sV
2
sV
−
+
-
Vs
+
-
0
+
-
+
-
2
sV
2
sV
−
+
-
+
-
+
-
2
sV
2
sV

Now, the above circuit (due to the sources) is obviously
asymmetric—no virtual ground, nor virtual short is present.
But, let’s say we turn off (i.e., set to V =0) the bottom source
on each side of the circuit:
Our symmetry has been restored! The symmetry plane is a
virtual open.
This circuit is referred to as its even mode, and analysis of it
is known as the even mode analysis. The solutions are known
as the even mode currents and voltages!
Evaluating the resulting even mode half circuit we find:
+
- 2
sV
I1
100Ω
I2
100Ω 100Ω
100Ω
100Ω
100Ω
2
sV +
-
1
e
I
+
-Vs/2
100Ω
100Ω
100Ω
1 2
1
2 200 400
e es sV V
I I= = =

Now, let’s turn the bottom sources back on—but turn off the
top two!
We now have a circuit with odd symmetry—the symmetry
plane is a virtual short!
This circuit is referred to as its odd mode, and analysis of it
is known as the odd mode analysis. The solutions are known
as the odd mode currents and voltages!
Evaluating the resulting odd mode half circuit we find:
+
- 2
sV
I1
100Ω
I2
100Ω 100Ω
100Ω
100Ω
100Ω
2
sV
−
+
-
1
o
I
+
-2
sV
100Ω
100Ω
100Ω
1 2
1
2 50 100
o os sV V
I I= = = −

Q: But what good is this “even mode” and “odd mode”
analysis? After all, the source on port 1 is Vs1 =Vs, and the
source on port 2 is Vs2 =0. What are the currents I1 and I2
for these sources?
A: Recall that these sources are the sum of the even and odd
mode sources:
1 2 0
2 2 2 2
s s s s
s s s
V V V V
V V V= = + = = −
and thus—since all the devices in the circuit are linear—we
know from superposition that the currents I1 and I2 are simply
the sum of the odd and even mode currents !
1 1 1 2 2 2
e o e o
I I I I I I= + = +
Thus, adding the odd and even mode analysis results together:
+
-
+
-
2
sV
2
sV
1 1 1
o e
I I I= +
100Ω
2 2 2
o e
I I I= +
100Ω 100Ω
100Ω
100Ω
100Ω
+
-
2
sV
2
sV
−
+
-

1 1 1 2 2 2
400 100 400 100
3
80 400
e o e o
s s s s
s s
I I I I I I
V V V V
V V
= + = +
= + = −
= = −
And then the admittance parameters for this two port
network is:
2
1
11
1 0
1 1
80 80
s
s
s sV
VI
Y
V V=
= = =
2
2
21
1 0
3 1 3
400 400
s
s
s sV
VI
Y
V V=
−
= = − =
And from the symmetry of the device we know:
22 11
1
80
Y Y= =
12 21
3
400
Y Y
−
= =
Thus, the full admittance matrix is:
31
80 400
3 1
80400
−
−
⎡ ⎤
= ⎢ ⎥
⎣ ⎦
Y
Q: What happens if both sources are non-zero? Can we use
symmetry then?

A: Absolutely! Consider the problem below, where neither
source is equal to zero:
In this case we can define an even mode and an odd mode
source as:
1 2 1 2
2 2
e os s s s
s s
V V V V
V V
+ −
= =
I1
100Ω
I2
+
V2
-
+
V1
-
100Ω 100Ω
100Ω
100Ω
100Ω
+
-
+
-Vs1
Vs2
2sV+
-
+
-
e
sV
o
sV−
+
-
Vs1
+
-
+
-
+
-
e
sV
o
sV
1
e o
s s sV V V= +
2
e o
s s sV V V= −

We then can analyze the even mode circuit:
And then the odd mode circuit:
And then combine these results in a linear superposition!
+
-
e
sV
I1
100Ω
I2
100Ω 100Ω
100Ω
100Ω
100Ω
e
sV +
-
+
-
o
sV
I1
100Ω
I2
100Ω 100Ω
100Ω
100Ω
100Ω
o
sV−
+
-

Q: What about current sources? Can I likewise consider
them to be a sum of an odd mode source and an even mode
source?
A: Yes, but be very careful! The current of two source will
add if they are placed in parallel—not in series! Therefore:
1 2 1 2
2 2
e os s s s
s s
I I I I
I I
+ −
= =
Is1 1
e o
s s sI I I= +e
sIo
sI
Is2 2
e o
s s sI I I= −e
sIo
sI−

One final word (I promise!) about circuit symmetry and
even/odd mode analysis: precisely the same concept exits in
electronic circuit design!
Specifically, the differential (odd) and common
(even) mode analysis of bilaterally symmetric
electronic circuits, such as differential amplifiers!
Hi! You might remember differential
and common mode analysis from such
classes as “EECS 412- Electronics
II”, or handouts such as
“Differential Mode Small-Signal
Analysis of BJT Differential Pairs”
BJT Differential Pair
Differential Mode
Common Mode

3/6/2009 Example Odd Even Mode Circuit Analysis 1/5
Example: Odd-Even Mode
Circuit Analysis
Carefully (very carefully) consider the symmetric circuit
below.
The two transmission lines each have a characteristic
impedance of .
Use odd-even mode analysis to determine the value of
voltage 1v .
-
+
λ
2
λ
+
v1
-
4.0 V
50Ω
50Ω 50Ω
50Ω
50Ω 50Ω
0 50Z = Ω
0 50Z = Ω

Solution
To simplify the circuit schematic, we first remove the bottom
(i.e., ground) conductor of each transmission line:
Note that the circuit has one plane of bilateral symmetry:
Thus, we can analyze the circuit using even/odd mode analysis
(Yeah!).
λ
2
λ
+
v1
-
50Ω
50Ω 50Ω
50Ω
50Ω 50Ω
+
-
λ
2
λ
+
v1
-
4.0 V
50Ω
50Ω 50Ω
50Ω
50Ω 50Ω
0 50Z = Ω
0 50Z = Ω

The even mode circuit is:
Whereas the odd mode circuit is:
We split the modes into half-circuits from which we can
determine voltages 1
e
v and 1
o
v :
+
-
λ
2
λ
+
1
e
v
-
2.0 V
50Ω
50Ω 50Ω
50Ω
50Ω 50Ω
+
-
2.0 V
I=0
+
-
λ
2
λ
+
1
o
v
-
2.0 V
50Ω
50Ω 50Ω
50Ω
50Ω 50Ω
+
-
V =0
-2.0 V

Recall that a 2= λ transmission
line terminated in an open
circuit has an input impedance
of in
Z = ∞ —an open circuit!
Likewise, a transmission line 4= λ
terminated in an open circuit has an
input impedance of 0in
Z = —a short
circuit!
Therefore, this half-circuit simplifies to:
And therefore the voltage 1
e
v is easily
determined via voltage division:
1
50
2 1.0
50 50
e
v V
⎛ ⎞
= =⎜ ⎟
+⎝ ⎠
Now, examine the right half-circuit of the odd mode:
Recall that a 2= λ transmission
line terminated in a short
circuit has an input impedance
of 0in
Z = — a short circuit!
Likewise, a transmission line 4= λ
terminated in an short circuit has an
input impedance of in
Z = ∞ — an open
circuit!
2.0 V
2
λ
4
λ
+
1
e
v
-
50Ω
50Ω
50Ω
+
-
+
1
e
v
-
50Ω
50Ω
50Ω
+
- 2.0 V
-2.0 V
2
λ
4
λ
+
1
o
v
-
50Ω
50Ω
50Ω
+
-

This half-circuit simplifies to
It is apparent from the circuit
that the voltage 1 0o
v = !
Thus, the superposition of the odd and even modes leads to
the result:
11 1 1 0.0 0 1.e o
v v Vv= + = + =
+
1
o
v
-
50Ω
50Ω
50Ω
+
- -2.0 V

3/6/2009 Generalized Scattering Parameters 1/8
Generalized Scattering
Parameters
Consider now this microwave network:
Q: Boring! We studied this before; this will lead to the
definition of scattering parameters, right?
A: Not exactly. For this network, the characteristic
impedance of each transmission line is different (i.e.,
01 02 03 04
Z Z Z Z≠ ≠ ≠ )!
( )1 1V z+
( )4 4V z+
( )3 3V z+
( )2 2V z+
port 1
( )1 1V z−
( )4 4V z−
( )3 3V z−
( )2 2V z−
port 3
port
4
port
2
4-port
microwave
device
Z01 Z03
Z02
Z04
3 3Pz z=
2 2Pz z=
1 1Pz z=
4 4Pz z=

Q: Yikes! You said scattering parameters are dependent on
transmission line characteristic impedance 0
Z . If these
values are different for each port, which 0
Z do we use?
A: For this general case, we must use generalized scattering
parameters! First, we define a slightly new form of complex
wave amplitudes:
0 0
0 0
n n
n n
n n
V V
a b
Z Z
+ −
= =
So for example:
01 03
1 3
01 03
V V
a b
Z Z
+ −
= =
The key things to note are:
A variable a (e.g., 1 2, ,a a ) denotes the complex
amplitude of an incident (i.e., plus) wave.
A variable b (e.g., 1 2, ,b b ) denotes the complex
amplitude of an exiting (i.e., minus) wave.
We now get to rewrite all our transmission line knowledge
in terms of these generalized complex amplitudes!
a
b

First, our two propagating wave amplitudes (i.e., plus and
minus) are compactly written as:
00 00n n nn nnV Va Z b Z+ −
= =
And so:
( )
( )
( )
0
0
2
n
n
n
n n
n
j z
n n
j z
n n
j z
n
n
n
n
a Z
b Z
V z e
V z e
z e
b
a
β
β
β
−+
+−
+
=
=
Γ =
Likewise, the total voltage, current, and impedance are:
( ) ( )
( )
( )
0
0
n n
n n
n n
n n
j z j z
n n n n n
j z j z
n n
n n
n
j z j z
n n
n j z j z
n n
V z Z a e b e
a e b e
I z
Z
a e b e
Z z
a e b e
β β
β β
β β
β β
− +
− +
− +
− +
= +
−
=
+
=
−
Assuming that our port planes are defined with 0nPz = , we can
determine the total voltage, current, and impedance at port n
as:

( ) ( ) ( )
( )
0
0
0 0
0
n n
n n n n n n n n n
n
n n
n n
n n
a b
V V z Z a b I I z
Z
a b
Z Z z
a b
−
= = + = =
+
= =
−
Likewise, the power associated with each wave is:
2 2
0 0
0
22
02 22 2
n n
n n
n
nn
n
V V
P P
Z Z
ba
+ −
+ −
= = = =
As such, the power delivered to port n (i.e., the power
absorbed by port n) is:
22
2
n n
n
n
n
P P P
a b+ − −
= − =
This result is also verified:
{ }
( )( ){ }
{ }
( ){ }
{ }{ }
22
22
22
1
Re
2
1
Re
2
1
Re
2
1
Re
2
1
Re Im
2
2
n n n
n n n n
n n n n n n n n
n n n n n n
n n
n n
n n
a
P V I
a b a b
a a b a a b b b
a b a b
b
a b
a j b a b
∗
∗ ∗
∗ ∗ ∗ ∗
∗∗ ∗
∗
=
= + −
= + − −
= +
+
=
−
− −
= −

Q: So what’s the big deal? This is yet another way to
express transmission line activity. Do we really need to know
this, or is this simply a strategy for making the next exam
even harder?
A: You may have noticed that this notation
( ,n na b ) provides descriptions that are a bit
“cleaner” and more symmetric between current
and voltage.
However, the main reason for this notation is for evaluating
the scattering parameters of a device with dissimilar
transmission line impedance (e.g., 01 02 03 04
Z Z Z Z≠ ≠ ≠ ).
For these cases we must use generalized scattering
parameters:
( )00
0 0
(when 0 for all )nm
mn k k
n m
ZV
S V z k n
V Z
−
+
+
= = ≠
1 1
1
1 1
a b
Z
a b
+
=
−

Note that if the transmission lines at each port are identical
( 0 0m nZ Z= ), the scattering parameter definition “reverts
back” to the original (i.e., 0 0mn m nS V V− +
= if 0 0m nZ Z= ). E.G.:
02
21 02
01
when 0
V
S V
V
−
+
+
= =
But, if the transmission lines at each port are dissimilar
( 0 0m nZ Z≠ ), our original scattering parameter definition is not
correct (i.e., 0 0mn m nS V V− +
≠ if 0 0m nZ Z≠ )! E.G.:
02
21 02
01
when 0
V
S V
V
−
+
+
≠ =
02
21 02
01
50
when 0
75
V
S V
V
−
+
+
= =
( )1 1V z+
( )2 2V z+
port 1
( )1 1V z−
( )2 2V z−port 2
2-port
microwave
device
Z01 =50 Ω Z02 = 50 Ω
( )1 1V z+
( )2 2V z+
port 1
( )1 1V z−
( )2 2V z−port 2
2-port
microwave
device
Z01 =50 Ω Z02 = 75 Ω

Note that the generalized scattering parameters can be more
compactly written in terms of our new wave amplitude
notation:
00
0 0
(when 0 for all )nm
mn
n nm
m
k
ZV
S k n
V Z
b
a
a
−
+
= = = ≠
Remember, this is the generalized form of scattering
parameter—it always provides the correct answer, regardless
of the values of 0mZ or 0nZ !
Q: But why can’t we define the scattering parameter as
0 0mn m nS V V− +
= , regardless of 0mZ or 0nZ ?? Who says we must
define it with those awful 0nZ values in there?
A: Good question! Recall that a lossless device is will always
have a unitary scattering matrix. As a result, the scattering
parameters of a lossless device will always satisfy, for
example:
2
1
1
M
mn
m
S
=
= ∑
This is true only if the scattering parameters are generalized!

The scattering parameters of a lossless device will form a
unitary matrix only if defined as mn m nS b a= . If we use
0 0mn m nS V V− +
= , the matrix will be unitary only if the connecting
transmission lines have the same characteristic impedance.
Q: Do we really care if the matrix of a lossless device is
unitary or not?
A: Absolutely we do! The:
lossless device ⇔ unitary scattering matrix
relationship is a very powerful one. It allows us to identify
lossless devices, and it allows us to determine if specific
lossless devices are even possible!

3/6/2009 The Scattering Matrix of a Connector 1/4
Example: The Scattering
Matrix of a Connector
First, let’s consider the scattering matrix of a perfect
connector—an electrically very small two-port device that
allows us to connect the ends of different transmission lines
together.
If the connector is ideal, then it will exhibit no series
inductance nor shunt capacitance, and thus from KVL and KCL:
( ) ( ) ( ) ( )1 1 2 2 1 1 2 20 0 0 0V z V z I z I z= = = = = − =
Terminating port 2 in a matched load, and then analyzing the
resulting circuit, we find that (not surprisingly!):
01 0V −
= and 02 01V V− +
=
0Z 0Z
Port
1
Port
2
( )1 1V z
+
−
( )2 2V z
+
−
( )1 1I z ( )2 2I z
1 0z = 2 0z =
1z 2z

From this we conclude that (since 02
0V +
= ):
01
11
01 01
0
0 0.
V
S
V V
−
+ +
= = = 02 01
21
01 01
1 0.
V V
S
V V
− +
+ +
= = =
This two-port device has D2 symmetry (a plane of bilateral
symmetry), meaning:
22 11 0 0.S S= = and 21 12 1 0.S S= =
The scattering matrix for such this ideal connector is
therefore:
0 1
1 0
⎡ ⎤
= ⎢ ⎥
⎣ ⎦
S
As a result, the perfect connector allows two transmission
lines of identical characteristic impedance to be connected
together into one “seamless” transmission line.
0Z 0Z

Now, however, consider the case where the transmission lines
connected together have dissimilar characteristic impedances
(i.e., 0 1Z Z≠ ):
Q: Won’t the scattering matrix of this ideal connector
remain the same? After all, the device itself has not
changed!
A: The impedance, admittance, and transmission matrix will
remained unchanged—these matrix quantities do not depend
on the characteristics of the transmission lines connected to
the device.
But remember, the scattering matrix depends on both the
device and the characteristic impedance of the transmission
lines attached to it.
After all, the incident and exiting waves are traveling on
these transmission lines!
The ideal connector in this case establishes a “seamless”
interface between two dissimilar transmission lines.
01Z 02Z
Port
1
Port
2

Remember, this is the same
structure that we evaluated
in an earlier handout!
In that analysis we found that—when 02 0V +
= :
01 02 01
01 02 01
V Z Z
V Z Z
−
+
−
=
+
and 02 02
01 02 01
2V Z
V Z Z
−
+
=
+
And so the (generalized) scattering parameters 11
S and 21
S
are:
0101 02 01
11
01 02 0101
ZV Z Z
S
V Z ZZ
−
+
−
= =
+
and 01 01 0202
21
01 02 0102
2Z Z ZV
S
V Z ZZ
−
+
= =
+
As a result we can conclude that the scattering matrix of the
ideal connector (when connecting dissimilar transmission lines)
is:
01 0202 01
02 01 01 02
01 02 01 02
01 02 01 02
2
2
Z ZZ Z
Z Z Z Z
Z Z Z Z
Z Z Z Z
⎡ ⎤−
⎢ ⎥
+ +⎢ ⎥=
⎢ ⎥
−⎢ ⎥
⎢ ⎥+ +⎣ ⎦
S
01Z 02Z

Section 4 3_the_scattering_matrix_package

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Section 4 3_the_scattering_matrix_package