02/03/05 The Scattering Matrix.

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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 V + (z ) and 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 .

Jim Stiles The Univ. of Kansas Dept. of EECS

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Consider the 4-port microwave device shown below:

V2− ( z 2 ) Z0 V2+ ( z 2 )

port
z 2 = z 2P
2
V1 + ( z1 ) port 1 port 3 V3− ( z 3 )
4-port
Z0 microwave Z0
device
V1 − ( z1 ) z1 = z1P z 3 = z 3P V3+ ( z 3 )
port
z 4 = z 4P
4

Z0
V4+ ( z 4 ) V4− ( z 4 )

Note in this example, there are four identical transmission lines

connected to the same “box”. Inside this box there may be a
very simple linear device/circuit, or it might contain a very large
and complex linear microwave system.

Æ Either way, the “box” can be fully characterized by its

scattering parameters!

Jim Stiles The Univ. of Kansas Dept. of EECS

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First, note that each transmission line has a specific location

that effectively defines the input to the device (i.e., z1P, z2P,
z3P, z4P). These often arbitrary positions are known as the port
locations, or port planes of the device.

Say there exists an incident wave on port 1 (i.e., V1 + ( z1 ) ≠ 0 ),

while the incident waves on all other ports are known to be zero
(i.e., V2 + ( z 2 ) = V3 + ( z 3 ) = V4+ ( z 4 ) = 0 ).

Say we measure/determine the voltage of the wave flowing into

port 1, at the port 1 plane (i.e., determine V1 + ( z1 = z1P ) ).
Say we then measure/determine the voltage of the wave flowing
out of port 2, at the port 2 plane (i.e., determine V2 − ( z 2 = z 2P ) ).

The complex ratio between V1 + (z1 = z1P ) and V2 − (z 2 = z 2P ) is know

as the scattering parameter S21:

V2− (z = z 2 ) V02− e + j β z P V02− + j β (z P +z P )

2

S21 = + = = e 2 1

V1 (z = z1 ) V01+ e − j β z P V01+1

Likewise, the scattering parameters S31 and S41 are:

V3− (z 3 = z 3P ) V4− (z 4 = z 4P )
S31 = + and S41 = +
V1 (z1 = z1P ) V1 (z1 = z1P )

Jim Stiles The Univ. of Kansas Dept. of EECS

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We of course could also define, say, scattering parameter S34

as the ratio between the complex values V4+ (z 4 = z 4P ) (the wave
into port 4) and V3 − (z 3 = z 3P ) (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:

Vm− (z m = z mP )
Smn = + (given that Vk + ( z k ) = 0 for all k ≠ n )
Vn (zn = znP )

Note that frequently the port positions are assigned a zero

value (e.g., z1P = 0, z 2P = 0 ). This of course simplifies the
scattering parameter calculation:

Vm− (z m = 0) V0−m e + j β 0 V0−m

Smn = + = =
Vn (zn = 0) V0n+ e − j β 0 V0n+

We will generally assume that the port

locations are defined as znP = 0 , and thus use
the above notation. But remember where this
expression came from!

Jim Stiles The Univ. of Kansas Dept. of EECS

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Q: But how do we ensure

that only one incident wave
is non-zero ?

A: Terminate all other ports with a matched load!

Γ2 L = 0

V2− ( z 2 ) Z0 V2+ ( z 2 ) = 0

V1 + ( z1 ) V3− ( z 3 )
4-port
Z0 microwave Z0 Γ3L = 0
device
V1 − ( z1 ) V3+ ( z 3 ) = 0

V3+ ( z 3 ) = 0 Z0 V4− ( z 4 )

Γ 4L = 0

Jim Stiles The Univ. of Kansas Dept. of EECS

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Note that if the ports are terminated in a matched load (i.e.,

Z L = Z 0 ), then ΓnL = 0 and therefore:

Vn+ ( zn ) = 0

In other words, terminating a port ensures

that there will be no signal incident on
that port!

Q: Just between you and me, I think you’ve messed this up! In all
previous handouts you said that if Γ L = 0 , the wave in the minus
direction would be zero:

V − (z ) = 0 if ΓL = 0

but just now you said that the wave in the positive direction would
be zero:
V + ( z ) = 0 if ΓL = 0

Of course, there is no way that both statements can be correct!

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
Vn+ ( zn ) and Vn− ( zn )!

Jim Stiles The Univ. of Kansas Dept. of EECS

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For example, we originally analyzed this case:

V + (z )

Z0 ΓL V − (z ) = 0 if ΓL = 0

V − (z )

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:

port n Vn − ( zn )
N-port
Microwave Z0 ΓnL
Network

Vn + ( zn )

For this current case, the situation is reversed. The wave

incident on the load is now denoted as Vn− ( zn ) (coming out of
port n), while the wave reflected off the load is now denoted as
Vn+ ( zn ) (going into port n ).

As a result, Vn+ ( zn ) = 0 when ΓnL = 0 !

Jim Stiles The Univ. of Kansas Dept. of EECS

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Perhaps we could more generally state that:

V reflected ( z = z L ) = ΓL V incident ( z = z L )

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!

Now, back to our discussion of S-parameters. We found that if

znP = 0 for all ports n, the scattering parameters could be
directly written in terms of wave amplitudes V0n+ and V0−m .

V0−m
Smn = + (given that Vk + ( z k ) = 0 for all k ≠ n )
V0n

Which we can now equivalently state as:

V0−m
Smn = + (given that all ports, except port n , are matched)
V0n

Jim Stiles The Univ. of Kansas Dept. of EECS

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One more important note—notice that for the matched ports

(i.e., those ports with no incident wave), the voltage of the
exiting wave is also the total voltage!

Vm ( z m ) =V0+m e − j β zn +V0−m e + j β zn
= 0 +V0−m e + j β zm
= V0−m e + j β zm (for all terminated ports)

Thus, the value of the exiting wave at each terminated port is

likewise the value of the total voltage at those ports:

Vm ( 0 ) = V0−m (for all terminated ports)

And so, we can express some of the scattering parameters

equivalently as:

Vm ( 0 )
Smn = (for matched port m , i.e., for m ≠ n )
V0n+

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 Vm ( 0 ) .

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:

Jim Stiles The Univ. of Kansas Dept. of EECS

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V0n−
Snn = +
V0n

Therefore, port n is a port where there actually is some

incident wave V0n+ (port n is not terminated in a matched load!).
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.
Γ2 L = 0
V1 ( 0 ) =V1 + ( 0 ) +V1 − ( 0 ) V3 ( 0 ) = V3− ( 0 )

V2− ( z 2 ) Z0 V2+ ( z 2 ) = 0

V1 + ( z1 ) ≠ 0 V3− ( z 3 )
4-port
Z0 microwave Z0 Γ3L = 0
device
V1 − ( z1 ) V3+ ( z 3 ) = 0

V4+ ( z 4 ) = 0 Z0 V4− ( z 4 )

Γ 4L = 0
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.
Jim Stiles The Univ. of Kansas Dept. of EECS
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Q: As impossible as it sounds, this

handout is even more boring and pointless
than any of your previous efforts. Why
are we studying this? After all, what is
the likelihood that a microwave network
will have only one incident wave—that all
of the ports will be matched?!

A: OK, say that our ports are not matched, such that we have
waves simultaneously incident on each of the four ports of our
device.

Since the device is linear, 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 znP = 0 ):

V03− = S34V04+ + S33V03+ + S32V02+ + S31V01+

More generally, the output at port m of an N-port device

is:

N
V0m = ∑ Smn V0n+
−
( znP = 0)
n =1

Jim Stiles The Univ. of Kansas Dept. of EECS

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This expression can be written in matrix form as:

V− = S V+
Where V − is the vector:

T
V − = ⎡⎣V01− ,V02− ,V03− , … ,V0−N ⎤⎦

and V + is the vector:

T
V + = ⎡⎣V01+ ,V02+ ,V03+ , … ,V0+N ⎤⎦

Therefore S is the scattering matrix:

⎡ S11 … S1n ⎤
S = ⎢⎢ ⎥
⎥
⎢⎣Sm 1 Smn ⎥⎦

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 ΓL
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 ΓL , are frequency dependent! Thus,
it may be more instructive to explicitly write:

⎡ S11 (ω ) … S1n (ω ) ⎤
S (ω ) = ⎢⎢ ⎥
⎥
⎢⎣Sm 1 (ω ) Smn (ω ) ⎥⎦

Jim Stiles The Univ. of Kansas Dept. of EECS