Mwe Project Main

TABLE OF CONTENTS

Introduction

Two-Port S-Parameters

S-Parameter properties of 2-port networks

S-parameters

Types of S-parameters

The Scattering Matrix (S-parameter Matrix)

Features on S-parameters

Advantages/Disadvantages of S Parameters

Conclusion

Source code

Results

Bibliography

INTRODUCTION

Scattering parameters or S-parameters (the elements of a scattering matrix or S-

matrix) describe the electrical behavior of linear electrical networks when undergoing

various steady state. The parameters are useful for electrical engineering, electronics

engineering, and communication systems design, and especially for microwave

engineering.

The S-parameters are members of a family of similar parameters, S-parameters do

not use open or short circuit conditions to characterize a linear electrical network;

instead, matched loads are used. These terminations are much easier to use at high signal

frequencies than open-circuit and short-circuit terminations. Moreover, the quantities are

measured in terms of power. Many electrical properties of networks of components

(inductors, capacitors, resistors) may be expressed using S-parameters, such as

gain, return loss, voltage standing wave ratio (VSWR), reflection coefficient and

amplifier stability. The term 'scattering' is more common to optical engineering than RF

engineering, referring to the effect observed when a plane electromagnetic wave is

incident on an obstruction or passes across dissimilar dielectric media. In the

context of S-parameters, scattering refers to the way in which the traveling currents and

voltages in a transmission line are affected when they meet a discontinuity

caused by the insertion of a network into the transmission line. This is equivalent to the

wave meeting an impedance differing from the line's characteristic impedance.

Although applicable at any frequency, S-parameters are mostly used for

networks operating at radio frequency (RF) and microwave frequencies where signal

power and energy considerations are more easily quantified than currents and voltages.

S-parameters change with the measurement frequency, so frequency must be specified

for any S-parameter measurements stated, in addition to the characteristic impedance or

system impedance. S-parameters are readily represented in matrix form and obey the rules

of matrix algebra.

Two-Port S-Parameters:

The S-parameter matrix for the 2-port network is probably the most commonly used and

serves as the basic building block for generating the higher order matrices for larger

networks. In this case the relationship between the reflected, incident power waves and the

S-parameter matrix is given by:

Expanding the matrices into equations gives:

and

Each equation gives the relationship between the reflected and incident power waves at each

of the network ports, 1 and 2, in terms of the network's individual S-parameters, , ,

and . If one considers an incident power wave at port 1 ( ) there may result from it

waves exiting from either port 1 itself ( ) or port 2 ( ). However if, according to the

definition of S-parameters, port 2 is terminated in a load identical to the system impedance (

) then, by the maximum power transfer theorem, will be totally absorbed making

equal to zero. Therefore, defining the incident voltage waves as and

with the reflected waves being and ,

and

Similarly, if port 1 is terminated in the system impedance then becomes zero, giving

and

Each 2-port S-parameter has the following generic descriptions:

is the input port voltage reflection coefficient

is the reverse voltage gain

is the forward voltage gain

is the output port voltage reflection coefficient.

S-Parameter properties of 2-port networks:

An amplifier operating under linear (small signal) conditions is a good example of a

non-reciprocal network and a matched attenuator is an example of a reciprocal network. In

the following cases we will assume that the input and output connections are to ports 1 and 2

respectively which is the most common convention. The nominal system impedance,

frequency and any other factors which may influence the device, such as temperature, must

also be specified.

Complex linear gain:

The complex linear gain G is given by

.

That is simply the voltage gain as a linear ratio of the output voltage divided by the input

voltage, all values expressed as complex quantities.

Scalar linear gain:

The scalar linear gain (or linear gain magnitude) is given by

.

That is simply the scalar voltage gain as a linear ratio of the output voltage and the input

voltage. As this is a scalar quantity, the phase is not relevant in this case.

Scalar logarithmic gain:

The scalar logarithmic (decibel or dB) expression for gain (g) is

dB.

This is more commonly used than scalar linear gain and a positive quantity is normally

understood as simply a 'gain'... A negative quantity can be expressed as a 'negative gain' or

more usually as a 'loss' equivalent to its magnitude in dB. For example, a 10 m length of

cable may have a gain of - 1 dB at 100 MHz or a loss of 1 dB at 100 MHz

Insertion loss:

In case the two measurement ports use the same reference impedance, the insertion loss (IL)

is the magnitude of the transmission coefficient |S21| expressed in decibels. It is thus given by:

dB.

It is the extra loss produced by the introduction of the device under test (DUT) between the 2

reference planes of the measurement. Notice that the extra loss can be introduced by intrinsic

loss in the DUT and/or mismatch. In case of extra loss the insertion loss is defined to be

positive. The negative of insertion loss expressed in decibels is defined as insertion gain.

Input return loss:

Input return loss (RLin) can be thought of as a measure of how close the actual input

impedance of the network is to the nominal system impedance value. Input return loss

expressed in decibels is given by

dB.

Note that for passive two-port networks in which |S11| 1, it follows that return loss is a non-

negative quantity: RLin 0. Also note that somewhat confusingly, return loss is sometimes

used as the negative of the quantity defined above, but this usage is, strictly speaking,

incorrect based on the definition of loss.

Output return loss:

The output return loss (RLout) has a similar definition to the input return loss but applies to the

output port (port 2) instead of the input port. It is given by

dB.

Reverse gain and reverse isolation:

The scalar logarithmic (decibel or dB) expression for reverse gain ( ) is:

dB.

Often this will be expressed as reverse isolation ( ) in which case it becomes a positive

quantity equal to the magnitude of and the expression becomes:

dB.

Voltage reflection coefficient:

The voltage reflection coefficient at the input port ( ) or at the output port ( ) are

equivalent to and respectively, so

and .

As and are complex quantities, so are and .

Voltage reflection coefficients are complex quantities and may be graphically represented on

polar diagrams or Smith Charts.

Voltage standing wave ratio:

The voltage standing wave ratio (VSWR) at a port, represented by the lower case 's', is a

similar measure of port match to return loss but is a scalar linear quantity, the ratio of the

standing wave maximum voltage to the standing wave minimum voltage. It therefore relates

to the magnitude of the voltage reflection coefficient and hence to the magnitude of either

for the input port or for the output port.

At the input port, the VSWR ( ) is given by

At the output port, the VSWR ( ) is given by

This is correct for reflection coefficients with a magnitude no greater than unity, which is

usually the case. A reflection coefficient with a magnitude greater than unity, such as in a

tunnel diode amplifier, will result in a negative value for this expression. VSWR, however,

from its definition, is always positive. A more correct expression for port k of a multiport is;

S-parameters:

S-parameters are a useful method for representing a circuit as a black box.

The external behaviour of this black box can be predicted without any regard for the contents

of the black box.

This black box could contain anything :

a resistor,

a transmission line

or an integrated circuit.

A black box or network may have any number of ports.

S-parameters are measured by sending a single frequency signal into the network or

black box and detecting what waves exit from each port. Power, voltage and current can be

considered to be in the form of waves travelling in both directions.

For a wave incident on Port 1, some part of this signal reflects back out of that port and some

portion of the signal exits other ports.

S11 refers to the signal reflected at Port 1 for the signal incident at Port 1. Scattering

parameter S11 is the ratio of the two waves b1/a1.

S21 refers to the signal exiting at Port 2 for the signal incident at Port 1. Scattering parameter

S21 is the ratio of the two waves b2/a1.

Types of S-parameters:

When we are talking about networks that can be described with S-parameters, we are

usually talking about single-frequency networks. Receivers and mixers aren't referred to as

having S-parameters, although you can certainly measure the reflection coefficients at each

port and refer to these parameters as S-parameters. The trouble comes when you wish to

describe the frequency-conversion properties, this is not possible using S-parameters.

Small signal S-parameters are what we are talking about 99% of the time. By small signal,

we mean that the signals have only linear effects on the network, small enough so that gain

compression does not take place. For passive networks, small-signal is all you have to worry

about, because they act linearly at any power level.

Large signal S-parameters are more complicated. In this case, the S-matrix will vary with

input signal strength. Measuring and modeling large signal S-parameters will not be

described on this page (perhaps we will get into that someday)

Mixed-mode S-parameters refer to a special case of analyzing balanced circuits. We're not

going to get into that either!

Pulsed S-parameters are measured on power devices so that an accurate representation is

captured before the device heats up. This is a tricky measurement, and not something we're

gonna tackle yet.

The Scattering Matrix (S-parameter Matrix):

It is not practical to measure voltages and currents at the ports at microwave frequencies.

It is natural to deal with power in incident and reflected waves for microwave transmission

lines.

Active devices may not be stable with short or open terminations due to oscillation.

The Scattering matrix relates the voltage waves incident on the ports to those reflected from

the ports.

Most importantly, scattering matrix elements can be measured without open or short in the

load, just matching loads. There is no reflected wave regardless of the length of the

transmission lines used --- practical to implement.

Features on S-parameters:

The reflection coefficient looking into port n is not equal to Snn, unless all other ports are

connected to matched load.

The transmission coefficient from port m to port n is not equal to Snm, unless all other ports

are connected to matched load.

The S parameters are properties of the network itself, and are defined under the condition

that all ports are connected to matched loads. Changing the terminations or excitations of a

network does not change its S parameters, but may change the reflection and transmission

coefficients.

Advantages/Disadvantages of S Parameters:

Advantages:

Ease of measurement: It is much easier to measure power at high frequencies than

open/short current and voltage.

Disadvantages:

They are more difficult to understand and it is more difficult to interpret measurements.

CONLUSION:

In this project we have studied and implemented S- parametres of a two port R-C network

using matlab. We understood various properties of S- parameters.

Source code

clc;

clear all;

C=1e-9;

R=70;

freq=logspace(5,8);

omega=2.*pi.*freq;

Zo=50;

S_11=(R+j.*omega.*C.*Zo.*(R-Zo))./(R+2*Zo+2.*j.*omega.*C.*Zo.*(R+Zo));

S_21=(2*Zo) ./(R+2*Zo+2.*j.*omega.*C.*Zo.*(R+Zo));

S_22=(R-j.*omega.*C.*Zo.*(R+Zo))./(R+2*Zo+2.*j.*omega.*C.*Zo.*(R+Zo));

S_12=(2*Zo)./(R+2*Zo+2.*j.*omega.*C.*Zo.*(R+Zo));

subplot(2,1,1);

semilogx(freq, abs(S_11));

xlabel('Frequency (Hz)');

ylabel('Amplitude');

title('S_{11}') ;

subplot(2,1,2) ;

semilogx(freq, angle(S_11));

xlabel('Frequency (Hz)');

ylabel('Phase (rad)');

figure;

subplot(2,1,1);

semilogx(freq, abs(S_21));

xlabel('Frequency (Hz)');

ylabel('Amplitude');

title('S_{21}');

subplot(2,1,2);

semilogx(freq, angle(S_21));

xlabel('Frequency (Hz)');

ylabel('Phase (rad)');

figure;

subplot(2,1,1);

semilogx(freq, abs(S_22));

xlabel('Frequency (Hz)');

ylabel('Amplitude') ;

title('S_{22}') ;

subplot(2,1,2);

semilogx(freq, angle(S_22));

xlabel('Frequency (Hz)') ;

ylabel('Phase (rad)') ;

figure;

subplot(2,1,1);

semilogx(freq, abs(S_12));

xlabel('Frequency (Hz)');

ylabel('Amplitude') ;

title('S_{12}') ;

subplot(2,1,2);

semilogx(freq, angle(S_12)) ;

xlabel('Frequency (Hz)') ;

ylabel('Phase (rad)') ;

RESULTS

BIBLIOGRAPHY:

en.wikipedia.org/wiki/Scattering_parameters

www.microwaves101.com/.../438-s-parameters-microwave-encyclopedia...

www.antenna-theory.com/definitions/sparameters.php

en.wikipedia.org/wiki/Two-port_network

web.cecs.pdx.edu/~ece2xx/ECE222/Slides/TwoPorts.pdf

fourier.eng.hmc.edu/e84/lectures/ch2/node4.html