1. The University of Tennessee Electrical and Computer Engineering
An Introduction To
Two – Port Networks
The University of Tennessee
Electrical and Computer Engineering
Knoxville, TN
These notes were first prepared during the Spring Semester of
2002 in support of the ECE 202 course which will be changed
to ECE 300 in the Fall Semester of The material on two
Ports will be the same.
wlg

2. Two Port Networks Generalities: The Network
The standard configuration of a two port: I1 I2
The Network + +
Input
Port
Output
Port V1 V2 _ _
The two port network configuration shown with
this slide is practically used universally in all text
books. The same goes for the voltage and current
assumed polarity and direction at the input and
output ports. I think this is done because of how
we define parameters with respect the voltages
and currents at the input and out ports.
With respect to the network, it may be configured
with passive R, L, C, op-amps, transformers, dependent sources but not independent sources.
The network ?
The voltage and current convention ?
* notes

3. Two Port Networks Network Equations: V1 = z11I1 + z12I2
V2 = b11V1 - b12I1
I2 = b21V1 – b22I1
Impedance
Z parameters
I1 = y11V1 + y12V2
I2 = y21V1 + y22V2
V1 = h11I1 + h12V2
I2 = h21I1 + h22V2
Admittance
Y parameters
Hybrid
H parameters
The equations in light orange are the ones we will consider here. The other equations are also presented and consider in Nilsson & Riedel, 6th ed.
The parameters are defined in terms of open and short circuit conditions of the two ports.
This will be illustrated and some examples presented.
Transmission
A, B, C, D
parameters
V1 = AV2 - BI2
I1 = CV2 - DI2
I1 = g11V1 + g12I2
V2 = g21V1 + g22I2
* notes

4. Two Port Networks Z parameters:
z11 is the impedance seen looking into port 1
when port 2 is open.
z12 is a transfer impedance. It is the ratio of the
voltage at port 1 to the current at port 2 when
port 1 is open.
z21 is a transfer impedance. It is the ratio of the
voltage at port 2 to the current at port 1 when
port 2 is open.
Z11 and z22 are easy to remember. We look into
the input port and calculate the impedance with
the output terminals open, if the circuit is
contains only R, L. C elements.
We do the same at the output port to find z22.
However, if the network contains dependent
Sources, transformers, or op-amps, we must
find V1/I1 with I2 = 0. Similarly for z22
For z12 and z21 we still find ratios of port voltage
To port current with the opposite port open.
If you have only R, L, and C then z12 = z21.
z22 is the impedance seen looking into port 2
when port 1 is open.
* notes

5. Two Port Networks Y parameters:
y11 is the admittance seen looking into port 1
when port 2 is shorted.
y12 is a transfer admittance. It is the ratio of the
current at port 1 to the voltage at port 2 when
port 1 is shorted.
y21 is a transfer impedance. It is the ratio of the
current at port 2 to the voltage at port 1 when
port 2 is shorted.
Similar to input and output impedance, y11 and y22 are determined by looking into the input and output ports with the opposite ports short circuited.
When we look for y12 and y21 we adhere to the above equations with respect to shorting the
output terminals.
If the circuit is not passive, we need to be careful
and do exactly as the equations tell us to do.
y22 is the admittance seen looking into port 2
when port 1 is shorted.
* notes

6. Two Port Networks Z parameters: Example 1
Given the following circuit. Determine the Z parameters.
Find the Z parameters for the above network.

7. Two Port Networks Z parameters: = Example 1 (cont 1) For z11: For z22:
Therefore:  =

8. Two Port Networks Z parameters:
Example 1 (cont 2)
The Z parameter equations can be expressed in
matrix form as follows.

9. Two Port Networks Z parameters:
Example 2 (problem 18.7 Alexander & Sadiku)
You are given the following circuit. Find the Z parameters.

10. Two Port Networks Z parameters: but Z21 = -0.667  Substituting gives;
Example 2 (continue p2) ;
but
Other Answers
Z21 = 
Z12 = 
Z22 = 
Substituting gives; or

11. Two Port Networks Transmission parameters (A,B,C,D):
The defining equations are:
I2 = 0
V2 = 0
V2 = 0
I2 = 0

12. Two Port Networks Transmission parameters (A,B,C,D): Example
Given the network below with assumed voltage polarities and
Current directions compatible with the A,B,C,D parameters.
We can write the following equations.
V1 = (R1 + R2)I1 + R2I2
V2 = R2I R2I2
It is not always possible to write 2 equations in terms of the V’s and I’s
Of the parameter set.

13. Two Port Networks Transmission parameters (A,B,C,D): I2 = 0 V2 = 0
Example (cont.)
V1 = (R1 + R2)I1 + R2I2
V2 = R2I R2I2
From these equations we can directly evaluate the A,B,C,D parameters. = =
I2 = 0
V2 = 0
The answers to the above are underneath the gray boxes.
The A,B,C,D parameters are best used when we want to cascade two networks together such as follows = =
V2 = 0
I2 = 0
Later we will see how to interconnect two of these networks together for a final answer
Network 1
Network 2
* notes

14. Two Port Networks Hybrid Parameters:
The equations for the hybrid parameters are:
V2 = 0
I1 = 0
The H parameters are used almost solely in electronics. These parameters are used in the equivalent circuit
Of a transistor. As you will see, the H parameter equations directly set-up so as to describe the device in terms
Of the hij parameters which are given by the manufacturer. You will use this material in junior electronics.
V2 = 0
I1 = 0
* notes

15. Two Port Networks Hybrid Parameters:
The following is a popular model used to represent
a particular variety of transistors.
We can write the following equations:
In this example we consider a very popular model of
a transistor. We first assign some identifier, such
as K1, K2, K3, K4, to the model. Next we write the
equations at the input port and output port. In this
case they are extremely simple to write.
Then it turns out that we can use these equations to
directly identify what will be called the H parameters.
Note that one of the parameters is a voltage gain, one
is a current gain, one is an resistance and one is
a conductance. You will go into a fair amount of
detail on this model later in the curriculum
* notes

16. Two Port Networks Hybrid Parameters:
We want to evaluate the H parameters from the above set of equations. = K1 = K2
V2 = 0
I1 = 0 = K3 =
V2 = 0
I1 = 0

17. Two Port Networks Hybrid Parameters:
Another example with hybrid parameters.
Given the circuit below.
The equations for the circuit are:
V1 = (R1 + R2)I1 + R2I2
V2 = R2I R2I2
The H parameters are as follows. = = 1
V2=0
I1=0 =
- 1 =
V2=0
I1=0

18. Two Port Networks Modifying the two port network:
Earlier we found the z parameters of the following network.
Equations from two port networks are not considered
to be the end of the problem. We will practically
always add components to the input and output. When
we do this we must start with the two port parameter,
in a certain form, and modify the equations to
incorporate the changes at the two ports. For example,
we may place a voltage source in series with a resistor
at the input port and maybe a load resistor at the output port.
The example here illustrates how to handle this problem.
* notes

19. Two Port Networks Modifying the two port network: We now have:
We modify the network as shown be adding elements outside the two ports
We now have:
V1 = I1
V2 = - 4I2

20. Two Port Networks Modifying the two port network: V1 = 10 - 6I1
We take a look at the original equations and the equations describing
the new port conditions.
V1 = I1
V2 = - 4I2
So we have,
Remember that we are trying to solve for I1 and
I2 after changing the port conditions. This
involves doing some algebra but at a low level.
We recombine terms and arrange the original
equation in matrix form and we can easily
take the inverse to find the solution or else
we can use simultaneous equations with
our hand calculators.
10 – 6I1 = 20I1 + 8I2
-4I2 = 8I1 + 12I2
* notes

21. Two Port Networks Modifying the two port network:
Rearranging the equations gives, 26 8 10 8 16
0.4545

22. Two Port Networks Y Parameters and Beyond:
Given the following network.
Find the Y parameters for the network.
From the Y parameters find the z parameters

23. Two Port Networks so s + 0.5 I1 = y11V1 + y12V2 I2 = y21V1 + y22V2
Y Parameter Example
I1 = y11V1 + y12V2
I2 = y21V1 + y22V2
short
We use the above equations to
evaluate the parameters from the
network.
To find y11 so
s + 0.5 =

24. Two Port Networks
Y Parameter Example
We see
= 0.5 S

25. Two Port Networks To find y12 and y21 we reverse things and short V1
Y Parameter Example
To find y12 and y21 we reverse
things and short V1
short
We have
We have
= 0.5 S

26. Two Port Networks Y Y Parameter Example Summary:=
Now suppose you want the Z parameters for the same network.

27. Two Port Networks Going From Y to Z Parameters
For the Y parameters we have:
For the Z parameters we have:
From above;
Therefore
where

28. Two Port Parameter Conversions:

29. Two Port Parameter Conversions:
To go from one set of parameters to another, locate the set of parameters
you are in, move along the vertical until you are in the row that contains
the parameters you want to convert to – then compare element for element

30. Interconnection Of Two Port Networks
Three ways that two ports are interconnected: ya
* Parallel yb za
* Series zb
* Cascade Ta Tb

31. Interconnection Of Two Port Networks
Consider the following network: I1 I2 + + V1 V2 T1 T2 _ _
Referring to slide 13 we have;

32. Interconnection Of Two Port Networks
Multiply out the first row:
Set I2 = 0 ( as in the diagram)
Can be verified directly
by solving the circuit

34. Basic Laws of Circuits
End of Lesson
Two-Port Networks