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,
V_{1}^{-} = S_{11} V _{1}^{+} + S _{12} V_{1}^{+}
V_{2}^{-} = S_{21} V_{1}^{+} + S _{22}V_{2}^{+}
At port 1, the total voltage is
V_{1} =V_{1}^{+} + 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
V_{2}|_{v2+=0} = V_{2}^{-}
Consequently, for this circuit the transmitted power is
P_{trans}|_{v2+=0} = |V_{2}^{-}|^{2} / 2z_{0}
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 | S_{11}|^{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 V_{2}^{+} = 0, then
This result states that the relative transmitted power to port 2 equals |S_{21}|^{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
S_{11} S_{11}^{*} + S_{21} S_{21}^{*} = 1 or |S_{11}|^{2} + | S_{21}|^{2} = 1
And
S_{12} S_{12}^{*} + S_{22} S_{22}^{*} = 1 or |S_{22}|^{2} + | S_{12}|^{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
S_{11} S_{12}^{*} + S_{21} S_{22}^{*} = 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 |S_{ij}|^{2} still relates to relative time average power flow.
For example, if Z_{0,1} ¹Z_{0,2} 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 |S_{ij}|^{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 S_{ij} 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 Z_{0,n} , we know that the total voltage is
V_{n} =V_{n}^{+} +V_{n}^{-}
while the current is
I_{n} = 1/ z_{0,n} [V_{n}^{+} =V_{n}^{-}]
Using it can be shown that
We will not be using a_{n}, b_{n} 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” a_{n} and b_{n}, which appear widely in the literature