Two-Port Networks
Linear circuit systems can be quantified by their terminal constraints Example: Thévenin Equivalent circuit:
Thévenin/Norton Circuits provide a “One-port” representation of a more complex circuit
Provides exactly the same terminal conditions for any arbitrary load circuit connected across the terminals

This can be generalized to circuits with an arbitrary number of terminals
Ex: 2-port network:

Ex: N-port network:

221 S. Gedney, University of Kentucky

Two-Port Parameterization
Two ports are parameterized based on their terminal currents and voltages , , and ,

The specific parameterization is dependent upon which quantities are the independent variables and which are the dependent variables.

EE 221 S. Gedney, University of Kentucky

Two Port Parameters
Parameter Type Independent Variables , Dependent Variables , Parameterization

Impedance (Z) Parameters Admittance (Y) Parameters Hybrid (h) parameters Inverse Hybrid (g) parameters Transmission (ABCD) parameters

 V1   Z1,1 V    Z  2   2,1

Z1,2   I1  Z 2,2   I 2   

,

,

 I1   Y1,1 Y1,2   V1   I   Y Y2,2   V2   2   2,1    V1   h1,1 I   h  2   2,1  I1   g1,1 V    g  2   2,1 h1,2   I1  h2,2   V2    g1,2   V1  g 2,2   I 2   

,

,

,

,

,

,

 V1   A B   V2   I    C D   I   2   1 

EE 221 S. Gedney, University of Kentucky

The Impedance (Z) Parameters
The impedance parameters are quantified by the Zmatrix which relates the port currents to the port voltages
 V1   Z1,1 V    Z  2   2,1
,

Z1,2   I1  Z 2,2   I 2   

The Z-parameters

have the units of

We can “measure” the Z-parameters of a linear twoport network in a very simple way:
Open circuit port 2 (This sets = 0).  V1   Z1,1 Z1,2   I1  V    Z  Z 2,2   0   2   2,1   Drive port 1 with a current source and measure port voltages V1  Z1,1 I1  Z1,1  V1 / I1

V2  Z 2,1 I1  Z 2,1  V2 / I1
EE 221 S. Gedney, University of Kentucky

Measuring the Z-parameters
Repeat by open circuiting port 1, drive port 2 with current, and measure the port voltages

 V1   Z1,1 V    Z  2   2,1

Z1,2   0  Z 2,2   I 2   

V1  Z1,2 I 2  Z1,2  V1 / I 2 V2  Z 2,2 I 2  Z 2,2  V2 / I 2
, ,

is the input impedance of port 2 with port 1 open circuited is the trans-impedance with port 1 open circuited

Similarly:
, ,

is the input impedance of port 1 with port 2 open circuited is the trans-impedance with port 2 open circuited

EE 221 S. Gedney, University of Kentucky

Simple Example
Find the Z-parameters of the two-port network:

1  1  5 j Answer: Z   1 1 5 j  
EE 221 S. Gedney, University of Kentucky

Another Example

4 9  Answer: Z     3 11
EE 221 S. Gedney, University of Kentucky

The Admittance (Y) Parameters
The admittance parameters are quantified by the Ymatrix which relates the port voltages to the port currents
 I1   Y1,1 Y1,2   V1   I   Y Y2,2   V2   2   2,1  
,

The Y-parameters

have the units of S (

)

We can “measure” the Y-parameters of a linear two-port network in a very simple way:
Short circuit port 2 (This sets  I1   Y1,1 Y1,2   V1   I   Y Y2,2   0   2   2,1   = 0).

Drive port 1 with a voltage source I1  Y1,1V1  Y1,1  I1 / V1

and measure port currents

I 2  Y2,1V1  Y2,1  I 2 / V1
EE 221 S. Gedney, University of Kentucky

Measuring the Y-parameters
Repeat by short circuiting port 1, drive port 2 with a voltage source, and measure the port currents

 I1   Y1,1 Y1,2   0   I   Y Y2,2   V2   2   2,1   I1  Y1,2V2  Y1,2  I1 / V2 I 2  Y2,2V2  Y2,2  I 2 / V2
, ,

is the input admittance of port 2 with port 1 short circuited is the trans-admittance with port 1 short circuited

Similarly
, ,

is the input admittance of port 1 with port 2 short circuited is the trans-admittance with port 2 short circuited

EE 221 S. Gedney, University of Kentucky

Simple Example
Find the Y-parameters of the two-port network:

1 1  25  5 j Answer: Y    1  25 
EE 221 S. Gedney, University of Kentucky

1 25 1 1  25 5

   j  

Another Example

EE 221 S. Gedney, University of Kentucky

 11  17 Answer: Y    3  17 

9  17   4 17  

Another Example

EE 221 S. Gedney, University of Kentucky

 11  17 Answer: Y    3  17 

9  17   4 17  

Relationship Between Z and Y Parameters
The Z and Y parameters provide a unique representation of the terminal relationships of the two-port networks
Any load or source connected to either end of the two-port can now be represented solely as a function of the Z or Yparameters.

Because each separately represents the two-port uniquely, there must be a relationship between the two parameters
What is this relationship?

 V1   Z1,1 V    Z  2   2,1

Z1,2   I1  Z 2,2   I 2   

 I1   Y1,1 Y1,2   V1   I   Y Y2,2   V2   2   2,1  

EE 221 S. Gedney, University of Kentucky

Relationship Between Z and Y Parameters
Note

 V1   Z1,1 V    Z  2   2,1
Therefore:

Z1,2   I1   I1   Z1,1        Z Z 2,2   I 2   I 2   2,1 Z1,2  Z 2,2  
1

Z1,2   V1  Z 2,2   V2    

1

Similarly:

 Y1,1 Y1,2   Z1,1 Y   Z  2,1 Y2,2   2,1  Z1,1 Z  2,1

Z1,2   Y1,1 Y1,2    Y Z 2,2   2,1 Y2,2   Consider the previous two examples:
Example 1 Example 2

1

1 1  25  5 j 1  1  5 j Z 1    1 1 5 j    1  25 
1

1 25 1 1  25 5

   j  

&

 11  17 4 9  1  Z  3 11    3  17 
1

9  17   4 17  

EE 221 S. Gedney, University of Kentucky

Lumped Circuit Model of the Z-parameters
The Z-parameters of an arbitrary circuit can be expressed as a lumped circuit model with a current controlled voltage source.
Z-parameters:

 V1   Z1,1 V    Z  2   2,1

Z1,2   I1  Z 2,2   I 2   

Two equations:

V1  Z1,1 I1  Z1,2 I 2

V2  Z 2,1 I1  Z 2,2 I 2

These are also the equations for the KVL loops of the twoloop network:

EE 221 S. Gedney, University of Kentucky

Mutual Inductor
Recall for the mutual inductor, we derived the equivalent circuit model:
ˆ I1 ˆ I2 ˆ I1 j L1 j L2

ˆ I2

ˆ V1

j L1

j L2

ˆ V2

ˆ V1

ˆ j MI 2

ˆ j MI1

ˆ V2

Therefore:  V1   Z1,1 V    Z  2   2,1

Z1,2   I1   j L1  Z 2,2   I 2   j M   

j M  I1  j L2  I 2   

This could easily be derived by computing the Z-matrix
EE 221 S. Gedney, University of Kentucky

Lumped Circuit Model of the Y-parameters
The Y-parameters can also be expressed as lumped circuit model
Y-parameters:

 I1   Y1,1 Y1,2   V1   I   Y Y2,2   V2   2   2,1  

Two equations I1  Y1,1V1  Y1,2V2

I 2  Y2,1V1  Y2,2V2

These are also the equations for the KCL equations of twonodes:

EE 221 S. Gedney, University of Kentucky

Hybrid Parameters
Another parameterization is the hybrid (H) parameters.
 V1   h1,1 I   h  2   2,1
, ,

h1,2   I1  h2,2   V2   

Hybrid, because the parameters have different units has the units of ohms (short ckt input impedance of port 1) has the units of Siemens (open ckt output admittance of port 2) and
,

and

,

are unitless ( , is the open ckt reverse voltage gain is the short-circuit forward current gain)
,

Equivalent circuit model:

EE 221 S. Gedney, University of Kentucky

Measuring the Hybrid Parameters
How would you measure the hybrid parameters?
 V1   h1,1 I   h  2   2,1 h1,2   I1  h2,2   V2   

EE 221 S. Gedney, University of Kentucky

Solution
 V1   h1,1 I   h  2   2,1
First column:

h1,2   I1  h2,2   V2   

V1  h1,1 I1 V 0  h1,1 
2

V1 I1 V 0
2

I 2  h2,1 I1 V 0  h2,1 
2

I2 I1

V2  0

Second column:

V1  h1,2V2 I 2  h2,2V2

I1  0

 h1,2 

V1 V2 I2 V2

I1  0

I1  0

 h2,2 

I1  0

EE 221 S. Gedney, University of Kentucky

Inverse Hybrid Parameters
Another parameterization is the inverse hybrid (G) parameters.
 I1   g1,1 V    g  2   2,1 g1,2   V1  g 2,2   I 2   

Hybrid, because the parameters have different units
, , ,

has the units of Siemens (open ckt input admittance) has the units of ohms (short circuit output impedance)
, ,

and gain and

are unitless ( , is the short ckt reverse current is the open-circuit forward voltage gain)

Equivalent circuit model: g1,1  I1 V1 V2 V1 g1,2 
I 2 0

I1 I2

V1  0

g 2,1 
EE 221 S. Gedney, University of Kentucky

g 2,2 
I 2 0

V2 I2

V1  0

Why Hybrid Parameters are used?
Hybrid parameters are often used as a two-port representation of transistor amplifiers
Represent physical characteristics of a small signal response of the amplifier. Either G or H-parameters can be used

EE 221 S. Gedney, University of Kentucky

Relationship Between Hybrid Parameters
 V1   h1,1 h1,2   I1  I   h h2,2   V2   2   2,1    V1   I1  or,    H    I2   V2   I1   g1,1 g1,2   V1  V    g g 2,2   I 2   2   2,1    I1   V1  or,    G    V2   I2 

What is the relationship between the two parameter sets?

EE 221 S. Gedney, University of Kentucky

Example
Given the circuit (a)
Compute vo:
Assume Vin = 1 cos(2t) V

EE 221 S. Gedney, University of Kentucky

Solution
First express the 2-port via the equivalent network:

Find below: v o  t   0.782cos( 2t  105.71 


180

)V

Nodal Analysis:
1) V1  Vin V1  0 V1  V2   0 2 2 j

MathCad Solution:

2)

Vo  V1 Vo  0 Vo  0  V      10 1   0 j 2 j 4 2  

EE 221 S. Gedney, University of Kentucky

Computing the Z-parameters from H-parmeters
Given:
 V1   h1,1 h1,2   I1  I   h h2,2   V2   2   2,1   V  I  or,  1   H  1   I2   V2 

Find the Z-parameters:
 V1   Z1,1 V    Z  2   2,1 Z1,2   I1  Z 2,2   I 2   

How did we solve for the Z-parameters?

EE 221 S. Gedney, University of Kentucky

Open Circuit Z-parameters
 V1   Z1,1 V    Z  2   2,1 Z1,2   I1   Z 2,2   0  

V2  

h2,1 h2,2

I1

EE 221 S. Gedney, University of Kentucky

Measure the Open Circuit Z-parameters
 V1   Z1,1 V    Z  2   2,1 Z1,2   I1   Z 2,2   0  

with port 2 open circuited, measure at port 2: h2,1 V2   I1 h2,2 Then, apply KVL at port 1: h2,1 h1,1h2,2  h1,2 h2,1 V1  h1,1 I1  h1,2V2  h1,1 I1  h1,2 I1  I1 h2,2 h2,2

 Z1,1 

V1 I1


I 2 0

h h2,2

Where

is the determinant of the H-matrix

EE 221 S. Gedney, University of Kentucky

Convert H-parameters to Z-parameters
 V1   Z1,1 V    Z  2   2,1
Open circuit port 1 (I1 = 0), drive port 2 with I2:
By KVL on port 2:

Z1,2   0   Z 2,2   I 2   

V2 

I2 V  Z 2,2  2 h2,2 I2


I1  0

1 h2,2

By KVL on port 1:

V1  h1,2V2  h1,2

I2 V  Z1,2  1 h2,2 I2


I1  0

h1,2 h2,2

EE 221 S. Gedney, University of Kentucky

Converting Between Two-Port Parameters

EE 221 S. Gedney, University of Kentucky

Which 2-Port Parameterization to Use?
Measurement
Determine the measurement which is best suited for extracting the two-port network parameters
Short circuit port 1 or 2
Measure port voltages or currents

Open circuit port 1 or 2
Measure port voltages or currents

Application
Determine the two-port network parameters that are best suited for the application
If different than the measured set, then simply convert from the measured parameter set to the application parameter set!

Some possible factors:
How combined with other parameter sets or with sources/loads Parameter set best suites the physical behavior of a device
EE 221 S. Gedney, University of Kentucky

Combining 2-ports in parallel/series
Two, 2-port networks are combined:

Are these in parallel or series? Which two-port parameters are best suites for finding a single parameter set for the combined 2ports?
EE 221 S. Gedney, University of Kentucky

Adding Two-Ports in Parallel a a  I1a   Y1,1 Y1,2   V1   a a  a  I 2   Y2,1 Y2,2   V2   b b  I1b   Y1,1 Y1,2   V1   b b  b  I 2   Y2,1 Y2,2   V2  

Apply KCL at port 1:

EE 221 S. Gedney, University of Kentucky

Adding Two-Ports in Parallel a a  I1a   Y1,1 Y1,2   V1   a a  a  I 2   Y2,1 Y2,2   V2   b b  I1b   Y1,1 Y1,2   V1   b b  b  I 2   Y2,1 Y2,2   V2  

Apply KCL at port 1: a a b b a b a b I1  I1a  I1b  Y1,1V1  Y1,2V2   Y1,1V1  Y1,2V2   Y1,1  Y1,1 V1  Y1,2  Y1,2 V2

a b a a b b a b a b I1  I 2  I 2  Y2,1V1  Y2,2V2   Y2,1V1  Y2,2V2   Y2,1  Y2,1 V1  Y2,2  Y2,2 V2

Similarly at port 2:

Conclude that parallel admittance parameters add: a b a b  I1   Y1,1  Y1,1 Y1,2  Y1,2   V1   I    Y a  Y b Y a  Y b  V   2   2,1 2,1 2,2 2,2   2 

EE 221 S. Gedney, University of Kentucky

Combining 2-ports in parallel/series
Next, two, 2-port networks are combined as:

Are these in parallel, series, or Neither? Which two-port parameters are best suites for finding a single parameter set for the combined 2ports?
EE 221 S. Gedney, University of Kentucky

Adding Two-Ports in Series a  V1a   Z1,1  a a  V2   Z 2,1 b  V1b   Z1,1  b b  V2   Z 2,1 a Z1,2   I1  a  Z 2,2   I 2    b Z1,2   I1  b  Z 2,2   I 2   

Apply KVL at port 1: a a b b a b a b V1  V1a  V1b   Z1,1 I1  Z1,2 I 2    Z1,1 I1  Z1,2 I 2    Z1,1  Z1,1  I1   Z1,2  Z1,2  I 2

a a b b a b a b V2  V2a  V2b   Z 2,1 I1  Z 2,2 I 2    Z 2,1 I1  Z 2,2 I 2    Z 2,1  Z 2,1  I1   Z 2,2  Z 2,2  I 2

Similarly at port 2:

Conclude that series impedance parameters add: a b  V1   Z1,1  Z1,1 V    Z a  Z b  2   2,1 2,1 a b Z1,2  Z1,2   I1  a b  Z 2,2  Z 2,2   I 2   

EE 221 S. Gedney, University of Kentucky

Cascading Two-port parameters
Two port parameters are often cascaded
Example: Port 1 of network Nb is directly connected to Port 2 of network Nb:

To accommodate this, another network parameter set is commonly used
The Transmission parameters (ABCD-Parameters)

 V1   A B   V2   I    C D  I   2   1 
Note that –I2a = +I1b
EE 221 S. Gedney, University of Kentucky

Measuring Transmission Parameters
How would one measure the transmission parameters?
 V1   A B   V2   I    C D  I   2   1 

EE 221 S. Gedney, University of Kentucky

Measuring Transmission Parameters
How would one measure the transmission parameters?
 V1   A B   V2   I    C D  I   2   1 
Measured as:

A

V1 V2

, C
I 2 0

I1 V2

S 
I 2 0

B

V1 I2

, D 
V2  0

I1 I2

V2  0

EE 221 S. Gedney, University of Kentucky

Cascading 2-port Networks
Two networks are cascaded:

How do the two networks combine?
 V1a   Aa  a  a  I1   C B a   V2a   a  Da   I2 

 V1b   Ab  b  b  I1   C

B b   V2b   b  Db    I 2 

EE 221 S. Gedney, University of Kentucky

Cascading 2-port Networks

Equating port voltages and currents:
 V2a   V1b   a b    I 2   I1 

Therefore
 V1a   Aa  a  a  I1   C B a   V2a   Aa  a a  a  D   I2   C B a   V1b   Aa  a a  b  D   I1   C B a  Ab  D a  C b B b   V2b   b  Db    I 2 

 V1   Aa Ab  B a C b I  a b C A  D aC b  1 

Aa B b  Aa D b   V2    C a Bb  D a Db    I 2 

EE 221 S. Gedney, University of Kentucky

Example
Find the transmission parameters of the Pi-network by cascading networks

Find the ABCD parameters of:

EE 221 S. Gedney, University of Kentucky

Example
Find the transmission parameters of the Pi-network by cascading networks

Find the ABCD parameters of:

A

V1 V2

 1, C 
I 2 0

I1 V2


I 2 0

1 S  5j

B

V1 I2

 0, D 
V2  0

I1 I2

1
V2  0

 1 V1       1  I1   5 j   EE 221 S. Gedney, University of Kentucky

0   V2  1   I2    

Center Element
Find the ABCD parameters of:

EE 221 S. Gedney, University of Kentucky

Series and Shunt Impedance
Find the ABCD parameters of:
A V1 V2  1, C 
I 2 0

I1 V2

 0 S 
I 2 0

B

V1 I2

 25 , D 
V2  0

I1 I2

1
V2  0

 V1   1 25   V2      I1   0 1    I 2    

Similarly:
 1 V1       1  I1   5 j  
EE 221 S. Gedney, University of Kentucky

0   V2  1   I2    

Cascading the three networks
Cascading of 3, two-port networks:

 1  V1   I  1  1 5j 

0   V2    1   I2  

 V1   1 25   V2   I    0 1  I   2   1 

 1  V1   I  1  1   5 j 

0   V2    1   I2  

How to combine?

EE 221 S. Gedney, University of Kentucky

Cascading the three networks

Cascading of 3, two-port networks:
 1  V1   I  1  1 5j  0  1   1 25   1 1  0 1    5 j    0 25   V2    V2   1  5 j 1   I2   1 1  5 j   I2       

Convert to a Z-matrix:
 1  5 j (1  5 j )(1  5 j )  25   1  5 j 1   Z1,1 Z1,2   1 1   Z Z 2,2   1 1 5 j 1 1 5 j 2,1         1  1 This is identical to what we derived before
EE 221 S. Gedney, University of Kentucky

Summary
Impedance Parameters
 V1   Z1,1 V    Z  2   2,1 Z1,2   I1  Z 2,2   I 2   

Admittance Parameters:
 I1   Y1,1 Y1,2   V1    I   Y Y2,2   V2   2   2,1  

Hybrid Parameters:
 V1   h1,1 I   h  2   2,1  I1   g1,1 V    g  2   2,1 h1,2   I1  h2,2   V2    g1,2   V1  g 2,2   I 2   
EE 221 S. Gedney, University of Kentucky

Inverse Hybrid Parameters

Summary
Transmission Parameters

 V1a   Aa  a  a  I1   C

B a   V2a   a  Da   I2 

 V1b   Ab  b  b  I1   C

B b   V2b   b  Db    I 2 

 V1a   Aa  a  a  I1   C

B a  Ab a  b D  C

B b   V2b  b b  I2  D 

EE 221 S. Gedney, University of Kentucky