Lec.9 Generalized S-Parameters and Power Relations

RF & Microwave EngineeringBETE-Spring 2009

Department of Electrical EngineeringAir University

S-parameters and Time Average Power

Generalized S - Parameters

Lecture No. 9

RF & Microwave EngineeringBETE-Fall 2009

Basit Ali ZebDepartment of Electrical Engineering, AU

Properties of S-Matrices

• Recall, in case of reciprocal networks, it can be shown that:

– S-matrix is symmetric about the main diagonal i.e.,

• If a network is lossless, then S is a unitary

matrix.



Properties of S-Matrices

Expanding it:



Special Cases

CASE 1



Special Cases

CASE 2



Scattering Parameters &

Time Average Power Flow



Scattering Parameters

Recall the Base

Equations

Consider a generic two-port network connect to a TL

circuit as shown in the figure:



Time Average Power Flow

1. Referring to the previous shown 2-port network, the total

time average power flow at ports is comprised of two

terms: Power incident and power reflected.

2. We will calculate these time averaged power quantities

assuming both ports has TL of Z0 connected to them, and will relate them to the S-parameters of the network.

There is a simple and very important relationship between S

parameters and time average power flow between the ports.



At Port 1

The total voltage, incident power and reflected power at

port 1 are given as (see 2.37 in text book):

Further, since port 2 is

matched, the voltage there is given as:

Consequently, the transmitted power for this network is:



At Port 1

The ratio of incident and reflected time average power is:

By comparing the last

two equations, we get:

Power Reflection coefficient at Port 1



At Port 2

From the base equation no. 2 when V2+ = 0 we get S21 as:

The ratio of incident and transmitted time average power is:

By comparing the last

two equations, we get:

where



For Lossless Network

Recall that for a lossless network, the scattering

matrix must be unitary i.e., diagonal elements should be 1. As a direct result of this, for any

lossless 2-port, we get :

With port 2 matched, the first equation above can be under stood as conservation of power statement for the network.



Generalized S-parameters

We will consider the case if Z0,1 ≠ Z0,2,, with port 2 matched and calculate the incident, reflected and transmitted power

to generalize the S-parameters.



Generalized S-parameters

The incident, reflected and transmitted power for this 2-port network is given as:

While the S21 parameter is given as:



Useful Interpretation

In order to preserve the very useful interpretation of |Sij|2,

we need to redefine the S-parameter if port impedances

are not similar. For Example:

We redefine the S21 parameter as:

This redefinition leads to the so-called

normalized S-parameters.

Wave amplitude

towards port ‘n’ is:Wave amplitude

away from port ‘n’ is:



Generalized Parameters

These |Sij| are ‘generalized S-parameters’. They reduce to

regular S-parameters when all port impedances are equal.

If i ≠ j, we can substitute the

values of a and b in above

equation to recover:

If i = j, we can substitute the

values of a and b in above

equation to recover



Generalized Parameters an,bn

At the terminal planes for port n with characteristic impedance

Z0,n , we can find the normalized scattering parameters as:



Study

• Article 4.2 and 4.3 from the text book

• Next discussion on ABCD parameters

Lec.9 Generalized S-Parameters and Power Relations

Documents

Transcript of Lec.9 Generalized S-Parameters and Power Relations