two port network parameters

M i r M u h a m m a d L o d r o

L e c t u r e r D e p a r t m e n t o f E l e c t r i c a l E n g i n e e r i n g

S u k k u r I B A

2-Port Network Parameters

Contents

Introduction

Two Port Networks

Z Parameters

Y Parameters

S Parameters

Return Loss

Insertion Loss

Transmission (ABCD) Matrix

VNA Calibration

SOLT Structure

Microwave and Radar Engineering

Two Port Networks

Linear networks can be completely characterized by parameters measured at the network ports without knowing the content of the networks.

Networks can have any number of ports. Analysis of a 2-port network is sufficient to explain the theory and applies

to isolated signals (no crosstalk).

The ports can be characterized with many parameters (Z, Y, S, ABDC). Each has a specific advantage.

Each parameter set is related to 4 variables: 2 independent variables for excitation 2 dependent variables for response

2 Port

NetworkPo

rt 1

I1

+

-

V1

Po

rt 2

I2

+

-

V2


Z Parameters

Advantage: Z parameters are intuitive.

Relates all ports to an impedance & is easy to calculate.

Disadvantage: Requires open circuit voltage measurements, which are difficult to make.

Open circuit reflections inject noise into measurements.

Open circuit capacitance is non-trivial at high frequencies.

IZV

Impedance Matrix: Z Parameters for 2-port Network

2221212

2121111

IZIZV

IZIZV

2

1

2221

1211

2

1

I

I

ZZ

ZZ

V

V


Y Parameters

VYI

Admittance Matrix: Y Parameters for 2-port Networks

2221212

2121111

VYVYI

VYVYI

2

1

2221

1211

2

1

V

V

YY

YY

I

I

Advantage: Y parameters are also somewhat intuitive.

Disadvantage: Requires short circuit voltage measurements, which are difficult to make.

Short circuit reflections inject noise into measurements.

Short circuit inductance is non-trivial at high frequencies.


Example- Z-parameters

ZC

ZA

ZB

+

-

+

-

V1

V2

I1

I2

Po

rt 1

Po

rt 2

CA

CA

I

ZZ

ZZV

V

I

VZ

1

1

1

111

02

CC

I

ZI

ZI

I

VZ

2

2

2

112

01

CC

I

ZI

ZI

I

VZ

1

1

1

221

02

CB

CB

I

ZZ

ZZV

V

I

VZ

2

2

2

222

01

2221212

2121111

IZIZV

IZIZV


Frequency Domain: Vector Network Analyzer (VNA)

VNA offers a means to characterize circuit elements as a function of frequency.

VNA is a microwave based instrument that provides the ability to understand frequency dependent effects.

The input signal is a frequency swept sinusoid.

Characterizes the network by observing transmitted and reflected power waves.

Voltage and current are difficult to measure directly.

It is also difficult to implement open & short circuit loads at high frequency.

Matched load is a unique, repeatable termination, and is insensitive to length, making measurement easier.

Incident and reflected waves the key measures.

We characterize the device under test using S parameters.


S Parameters

We wish to characterize the network by observing transmitted and reflected power waves.

ai represents the square root of the power wave injected into port i.

bi represents the square root of the power wave injected into port j.

2 Port

Network

a1

+

-

V1

Po

rt 2

a2

+

-

V2

Po

rt 1

b1

b2

RVP

2

R

VPai

1

R

Vb

j

j


S Parameters

We can use a set of linear equations to describe the behavior of the network in terms of the injected and reflected power waves.

For the 2 port case:

2

1

2221

1211

2

1

a

a

SS

SS

b

b

2 Port

Network

a1

+

-

V1

Po

rt 2

a2

+

-

V2

Po

rt 1

b1

b2

2221212

2121111

aSaSb

aSaSb

iport at measuredpower

jport at measuredpower

i

j

ija

bS


Scattering Matrix – Return Loss

S11, the return loss, is a measure of the power returned to the source.

When there is no reflection from the load, or the line length is zero, S11 is

equal to the reflection coefficient.

50

50

0

00

1

1

0

1

0

1

1

111

02

Z

Z

V

V

V

V

Z

V

Z

V

a

bS

incident

reflected

a

0

0,0

jjai

iii

a

bS

In general:


Scattering Matrix – Return Loss

When there is a reflection from the load, S11 will be composed of multiple reflections due to standing waves.

Use input impedance to calculate S11 when the line is not perfectly terminated.

)0(1

)0(1)0(

z

zZzZZ oin

If the network is driven with a 50 source, S11 is calculated as follows:

RS = 50

Zin

S11 for a transmission line will exhibit periodic effects due to the standing waves.

In this case S11 will be maximum when Zin is real. An imaginary component implies a phase difference between Vinc and Vref. No phase difference means they are perfectly

aligned and will constructively add.

50

5011

in

inv

Z

ZS


Scattering Matrix – Insertion Loss

When power is injected into Port 1 and measured at Port 2, the power ratio reduces to a voltage ratio:

incident

dtransmitte

o

o

aV

V

V

V

Z

V

Z

V

a

bS

1

2

1

2

021

221

2 Port

Network

a1

+

-

V1

Po

rt 2

a2

+

-

V2

Po

rt 1

b1

b2

Z0

Z0

S21, the insertion loss, is a measure of the power transmitted from port 1 to port 2.


S Parameters

aSb

jkkaj

iij

a

bS

,0

jkk

jkk

Vj

j

i

i

aj

iij

Z

V

Z

V

a

bS

,0

,0

0

0

Sij = Gij is the reflection coefficient of the ith port if i=j with all other ports matched

Sij = Tij is the forward transmission coefficient of the ith port if I>j with all other ports matched

Sij = Tij is the reverse transmission coefficient of the ith port if I<j with all other ports matched


Comments on Losses

True losses come from physical energy losses.

Ohmic (i.e. skin effect)

Field dampening effects (loss tangent)

Radiation (EMI)

Insertion and return losses include other effects, such as impedance discontinuities and resonance, which are not true losses.

Loss free networks can still exhibit significant insertion and return losses due to impedance discontinuities.


Reflection Coefficients Reflection coefficient at the load:

0

0

ZZ

ZZ

L

LL

0

0

ZZ

ZZ

S

SS

L

L

L

Lin

S

SS

S

SSS

11

2

1211

22

211211

11

S

Sout

S

SSS

11

211222

1

Reflection coefficient at the source:

Input reflection coefficient:

Output reflection coefficient:

Assuming S12 = S21 and S11 = S22.


Transmission Line Z0 Measurements

Impedance vs. frequency

Recall

Zin vs f will be a function of delay () and ZL.

We can use Zin equations for open and short circuited lossy transmission.

lZZ openin tanh0,

lZZ shortin coth0,

lj

lj

ine

eZZ

2

2

01

1

openinshortin ZZZ ,,0

Using the equation for Zin, rin, and Z0, we can find the impedance.


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.



The transmission matrix describes the network in terms of both voltage and current waves (analagous to a Thévinin Equivalent).

The coefficients can be defined using superposition:

221

221

DICVI

BIAVV

2

2

1

1

I

V

DC

BA

I

V

02

1

2

IV

IC

2 Port

Network

I1

+

-

V1

Po

rt 2

I2

+

-

V2

Po

rt 1

02

1

2

VI

ID

02

1

2

VI

VB

02

1

2

IV

VA



Since the ABCD matrix represents the ports in terms of currents and voltages, it is well suited for cascading elements.

I1

+

-

V1

I2

V2

I1

I3

+

-

V3

The matrices can be mathematically cascaded by multiplication:

3

3

22

2

2

2

11

1

I

V

DC

BA

I

V

I

V

DC

BA

I

V

3

3

211

1

I

V

DC

BA

DC

BA

I

V

This is the best way to cascade elements in the frequency domain.

It is accurate, intuitive and simple to use.

2DC

BA

1DC

BA


Converting to and from the S-Matrix

The S-parameters can be measured with a VNA, and converted back and forth into ABCD, the Matrix

Allows conversion into a more intuitive matrix

Allows conversion to ABCD for cascading

ABCD matrix can be directly related to several useful circuit topologies


Advantages/Disadvantages of ABCD Matrix

Advantages: The ABCD matrix is intuitive: it describes all ports with voltages and

currents.

Allows easy cascading of networks.

Easy conversion to and from S-parameters.

Easy to relate to common circuit topologies.

Disadvantages: Difficult to directly measure: Must convert from measured

scattering matrix.


VNA Calibration

Proper calibration is critical!!!

There are two basic calibration methods

Short, Open, Load and Thru (SOLT)

Calibrated to known standard( Ex: 50)

Measurement plane at probe tip

Thru, Reflect, Line(TRL)

Calibrated to line Z0

Helps create matched port condition.


SOLT Calibration Structures

OPEN SHORT

LOAD THRU

Calibration Substrate

G

G

S

S

G

S

Signal

Ground

G

S

G

S



Thanks

two port network parameters

Documents

port i

2v2 z 21i1 z

example z

port case

port v1 v2 network

ith port

s11 v z

v1 v1 vincident z