# S Parameters and Time Average Power

S Parameters and Time Average Power. Generalized S Parameters.
There are two remaining topics concerning S parameters we will cover in this lecture. The first is an important relationship between S parameters and relative time average power flow. The second topic is generalized scattering parameters, which are required if the port characteristic impedances are unequal.
S Parameters and Time Average Power
There is a simple and very important relationship between S parameters and relative time average power flow. To see this, consider a generic two-port connected to a TL circuit:

By definition,
V1- = S11 V 1+ + S 12 V1+
V2- = S21 V1+ + S 22V2+
At port 1, the total voltage is
V1 =V1+ + V 1-
and the total time average power at that port is comprised of the two terms (see 2.37):

Further, since port 2 is matched the total voltage there is
V2|v2+=0 = V2-
Consequently, for this circuit the transmitted power is
Ptrans|v2+=0 = |V2-|2 / 2z0
Using the results from (4), (5), and (7), we will consider ratios of these time average power quantities at each port and relate these ratios to the S parameters of the network.
. At Port 1. Using (4) and (5), the ratio of reflected and incident time average power is:

From (1) and noticing port 2 is matched so that

This result teaches us that the relative reflected time average power at port 1 equals | S11|2 when port 2 is matched.
. At Port 2. Using (7) and (4), the ratio of transmitted and incident time average power is:

However, from (2) and with V2+ = 0, then

This result states that the relative transmitted power to port 2 equals |S21|2 when port 2 is matched. Equations (9) and (11) provide an extremely useful physical interpretation of the S parameters as ratios of time average power. Note that this interpretation is valid regardless of the loss (or even gain) of the network. However, if the network is lossless we can use (9) and (11) to develop other very useful relationships. Recall that for a lossless network,

must be unitary. As a direct result of this
S11 S11* + S21 S21* = 1 or |S11|2 + | S21|2 = 1
And
S12 S12* + S22 S22* = 1 or |S22|2 + | S12|2 = 1
These equations are valid for all lossless two-ports. Furthermore, in the circuit above with port 2 matched, we can additionally interpret (12a) as a conservation of power statement for the network, based on (9) and (11). If port 2 is not matched, (12a) is still valid, of course, but it is no longer a conservation of power statement for the network. Tricky!
Lastly, since [S] is unitary, then
S11 S12* + S21 S22* = 0
which doesn’t appear to have a time average power interpretation. Can you devise a physical interpretation of ?
Generalized Scattering Parameters
If the characteristic impedances are different for some ports the network is connected, it becomes necessary to redefine the scattering parameters so that |Sij|2 still relates to relative time average power flow.
For example, if Z0,1 ¹Z0,2 in this circuit

then with port 2 matched the incident, reflected and transmitted time average power are, respectively,

And

Consequently,

which is a familiar result. However,

is not familiar. To preserve the very useful interpretation of |Sij|2 as a relativetime average power flow, we need to redefine the S parameters when the port impedances are not equal. For example, from

would preserve this interpretation. This redefinition leads to the so-called generalized S parameters. The “wave amplitude” towards port n is defined as

while the “wave amplitude” away from this port is defined as

As shown in text (pp. 181-182)
[b] = [s] . [a]
Where

These Sij are the generalized scattering parameters. They reduce to the “regular” S parameters when all port impedances are equal. If we substitute and note that we recover if i ¹ j and we recover if i = j . Consequently, we can interpret the generalized scattering parameters of terms of relative reflected time average power flows. Lastly, at the terminal plane for port n with characteristic impedance Z0,n , we know that the total voltage is
Vn =Vn+ +Vn-
while the current is
In = 1/ z0,n [Vn+ =Vn-]
Using it can be shown that

We will not be using an, bn or generalized scattering parameters very much in this course. This topic is mentioned primarily to reinforce the relationship of S parameters to relative time average power and to present the “wave amplitudes” an and bn, which appear widely in the literature