NETWORK ANALYSIS AND

SYNTHESIS

Chapter 3 Two port Network Analysis

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INTRODUCTION

 A pair of terminals through which a current may enter or leave a

network is known as a port.
 A port is an access to the network and consists of a pair of
terminals; the current entering one terminal leaves through the
other terminal so that the net current entering the port equals zero.
 For example, most circuits have two ports. We may apply an input
signal in one port and obtain an output signal from the other port.
 The parameters of a two-port network completely describes its
behavior in terms of the voltage and current at each port.
 Thus, knowing the parameters of a two port network permits us to
describe its operation when it is connected into a larger network.
Two-port networks are also important in modeling electronic devices
and system components.
 For example, in electronics, two-port networks are employed to
model transistors and Op-amps. Other examples of electrical
components modeled by two-ports are transformers and
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transmission lines.
A one port network is completely specified when the voltage-current
relationship at the terminals of the port given. A Typical one port or two
terminal network is shown in figure 1.1.

Fig. 1.2 represents a two-port network. A four terminal network is called

a two-port network when the current entering one terminal of a pair exits
the other terminal in the pair. For example, I1 enters terminal ‘a’ and exit
terminal ‘b’ of the input terminal pair ‘a-b’. Example for four-terminal or
two-port circuits are op amps, transistors, and transformers.

To characterize a two-port network requires that we relate the terminal

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quantities V1 ,V2 , I1 and I2 .The various terms that relate these voltages
and currents are called parameters.
TWO-PORT NETWORK PARAMETERS
 A general two –port network has two pair of voltage-current
relationship. The variables are

 Two of these are dependent variables; and the other two are
independent variables. The number of possible combinations
generated by the four variables, taken two at a time, is six.
 Thus there are six possible sets of equations describing a two-port
network.
 A two port network is a special case of multi-port network. Each
port consists of two terminals, one for entering the current and
the other for leaving.
 In order to describe the relationships between the port voltages
and port currents of a linear two port network, two linear equations
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are required among the four variables.
 In this chapter we will cover the six most used descriptions
 1.The z-parameter (or) open circuit impedance parameters

 Expressing two-port voltages in terms of two-port currents. i.e.,

(V1, V2)=f (I1, I2)

 Where [Z] is the open circuit impedance matrix of the two port
network and Zij are the open circuit impedance parameters. The
variables are dependent and are independent variables.
 In order to determine the Z parameters, open the output port and
applied some voltage in port1 and then, the input port is open
circuited and the output port is excited with the voltage as shown
in fig. i.e., The individual parameters are specified only when the
current in the one of the ports is zero.

Determination of Z parameters
The individual z parameters are defined by

Input driving point impedance with the output port

open circuited

Forward transfer impedance with the output port

open circuited

Reverse transfer impedance with the input port open

circuited

Output driving point impedance with the input port

open circuited

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 relates the current and voltage in the only; whereas gives the
voltage-current relationship for port. Such parameters are called
open circuit driving point impedances

The parameters relate the voltage in one port to the current in

the other port. These are known as open circuit transfer
impedances.

If Z12 = Z21 , the network is said to be reciprocal network.

if Z11 =Z22 then the network is called a symmetrical network.

A two-port is reciprocal if interchanging an ideal voltage source

at one port with an ideal ammeter at the other port gives the same
ammeter reading.

Impedance parameters is commonly used in the synthesis of

filters and also useful in the design and analysis of impedance7
matching networks and power distribution networks.
A two-port can be replaced by the following equivalent circuit.

In case the two-port is reciprocal, z12=z21, then it can also be represented

by the T-equivalent circuit.

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 Example 1. Given the following circuit. Determine the Z
parameters

To find Z11 and Z21 the

To find Z12 and Z22 , the input terminals are open
output terminals are open circuited. Also connect a voltage source V2 to the output
circuited. Also connect a terminals. This gives a circuit diagram as shown in Fig.
voltage source V1 to the input
terminals. This gives a circuit
diagram as shown in Fig

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2. THE Y PARAMETERS (OR) SHORT CIRCUIT ADMITTANCE
PARAMETERS
 Expressing two-port current in terms of two-port voltages. i.e.,

(I1, I2)=f (V1, V2)

 Where [Y] is the short circuit admittance matrix of the two port network and
Yij are the short circuit admittance parameters. The variables are dependent
and are independent variables.
 In order to determine the Y parameters, short the output port and applied
some voltage in port1 and then, the input port is short circuited and the
output port is excited with the voltage as shown in fig. In the following
diagrams the terminal currents can be expressed in terms of terminal
voltages as follows.

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Determination of Y parameters
The individual Y parameters are defined by

Input driving point admittance with the output port

short circuited

Forward transfer admittance with the output port

short circuited

Reverse transfer admittance with the input port

short circuited

Output driving point admittance with the output port

short circuited
If y12 = y21 , the network is said to be reciprocal network. Also, if y11 =
y22 then it is called a symmetrical network.
Notes:
1. and
3.The impedance and admittance parameters collectively known as
immittance parameters
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 A linear reciprocal two-port can be represented by the following
equivalent circuit

Similarly , a linear two-port can also be represented by the following

equivalent circuit with dependent sources .

Z in terms of y or y in terms of z

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 Find the Z and y parameters of the following circuit by definition of Z
parameters.

Step1: Let I2 = 0 and apply V1 Step2: Let I1 = 0 and apply V2

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Example Find the Z and Y – parameter of the circuit shown below.

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3. THE HYBRID (H) PARAMETERS
 The set of parameters that are extremely useful in describing
transistor circuits.
 In this case, voltage of the input port and the current of the output
port are expressed in terms of the current of the input port and the
voltage of the output port. Due to this reason, these parameters are
called as hybrid parameters, i.e.,
(V1, I2)=f (I1, V2)

The individual parameters are defined by the relationships

Input impedance with the output port short circuited

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Forward current gain with the output port short circuited

 Reverse voltage gain with the input port open circuited

Output admittance with the input port open circuited

Note that h11 and h21 are short circuit parameters, and h12 and h22
are open circuit type parameters
 is merely the reciprocal of .

An open circuit admittance parameter is related to as:

 Both the remaining h parameters are transfer functions.

 is short circuit current ratio, and is an open circuit voltage ratio.

Because of this mix of parameters, they are called hybrid

parameters.
If h12 = h21 , the network is said to be reciprocal network.
if h11 =h22 then the network is called a symmetrical network. 16
A linear two-port can be represented by the following equivalent circuit .

Relation ship between z,y and h

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 4.INVERSE HYBRID (OR g) PARAMETERS
If V 1 and I 2 are chosen as independent variables, the two-port network
equations may be written as

The g-parameters are defined as follows

If I 2 = 0 the output port is open circuit.

If V 1 = 0 the input port is short circuit.

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Find the h and g parameters of the two-port network

We obtain h11 and h21 by referring to the circuit in Fig. (a).

To get 22 h and h12 , refer to Fig. (b).

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 To get g11 and g21 ,

To get g22 and g12 ,

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5. THE ABCD PARAMETER OR TRANSMISSION LINE (T)
 Let us take the independent variables the voltage and current at port 1 and define the
following equation. (V1, I1)=f (V2, -I2)
 T – parameter also called transmission parameters because this parameter are useful
in the analysis of transmission lines because they express sending – end variables (V1
and I1) in terms of the receiving – end variables (V2 and -I2).

In explicit form the ABCD parameters can be as:

Reverse voltage ratio with the receiving end open

circuited

Reverse transfer admittance with the receiving end open

circuited

Reverse transfer impedance with the receiving end

short circuited
Reverse current ratio with the receiving end short
circuited
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If B = C , the network is said to be reciprocal network.
if A =D then the network is called a symmetrical network
Find the transmission parameters for the circuit shown in Fig.

To get A and C, refer to the circuit in Fig.(a).

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To get B and D, consider the circuit in Fig. (b).

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 6.Inverse Transmission Parameters
 The transmission parameter and the inverse transmission parameter are
duals of each other.
 If, instead of quantities V1 and I1, quantities V2 and I2 are expressed in
terms of V1 and I1, the resulting parameter (A′,B′,C′,D′) are called inverse
transmission parameter.
The inverse transmission parameters of the two port network in figure
having direction of voltages and current as shown, are given by

The inverse transmission parameters can be defined as

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Find the ABCD and inverse ABCD parameters for the
circuit shown below

To determine A and
C, consider the To determine B and D, consider the
circuit below. circuit below.

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 RELATIONSHIPS BETWEEN TWO PORT PARAMETERS

If all the two-port parameters for a network exist, it is possible to relate

one set of parameters to another, since these parameters interrelate the
variables V1 , I1, V2 and I2 .
Relation between the z parameters and y parameters:

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Relation between z and h parameters

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1. The y parameters of a network are:

Determine the z parameters for the

network.

2. Compute the z parameters of the circuit .

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3.5 INTERCONNECTION OF TWO PORT NETWORKS
 A large complex network may be divided in to sub networks for purposes of
analysis and design. The sub-networks are modeled as two-port networks
interconnected to form the original network.
 The two-port networks may therefore be regarded as building blocks that
can be interconnected to form a complex network.
 The interconnection can be in series, in parallel or in cascade. Although
interconnected network can be described by any of six parameter sets, a
certain set of parameters may have a definite advantage.
 When the networks are in series, their individual Z parameters adds up to
give the Z parameters of larger network.
 When the networks are in parallel their individual Y parameter add up to
give the Y parameter of larger network.
 When they are cascaded their individual transmission parameter can be
multiplied to get the transmission parameters of the larger network.
 The basic interconnections to be considered are: parallel, series and cascade.
 PARALLEL: Voltages are the same. Current of interconnection is the sum of
currents.
 SERIES: Currents are the same. Voltage of interconnection is the sum31of
voltages.
 CASCADE: Output of first subsystem acts as input for the second.
1. series connection of networks
When the circuit are laced in series, the common reference points of each
circuit are connected together.
For network

For network

From the diagram we notice that,

,
And that

Thus the Z parameters for overall network are

Or
Showing that Z parameters for the overall network are the sum of the Z
parameters for individual networks
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 2. PARALLEL NETWORKS
 Two port networks are in parallel when their
port voltages are equal and the port current
of the larger network are the sum of the
Individual port currents.

For network

For network

From the diagram we notice that,

and

So y parameters for the parallel connected

network can be

[ 𝑌 ] = [ 𝑌 𝑎 ] +[ 𝑌 𝑏 ]
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Showing that Y parameters for the overall network are the sum of the Y
parameters for individual networks
3. CASCADED NETWORKS
 Two networks are said to be cascaded when the output of one is the input of
the other.

For the two networks above:

From the fig

From the two equations we can rewrite

Thus the transmission parameter for the overall network are the product of
transmission parameters for the individual transmission parameters.
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Or
4. Series-parallel connection

I1 = I1a = I1b
V1 = V1a + V1b
I2 = I2a + I2b
V2 = V2a = V2b

5. Parallel-series connection
I1 = I1a + I1b ,
V1 = V1a = V1b
I2 = I2a = I2b
V2 = V2a + V2b

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Example Obtain the Z and Y parameters for the network in Fig
The network consists of two two-ports in series. It is better to
work with z parameters and then convert to y parameters.

Example :Obtain the y and h parameters for the network in Fig

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Example: For the parallel-series connection of the two two-ports in Fig.
find the g parameters.

We may obtain the g parameters from the given z parameters.

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 Network Functions for One Port and Two
Port Network

Driving Point Functions:

The impedance or admittance found at a given port is called a
driving point impedance(or admittance).
1.Driving Point Impedance:
Z11(s) = V1(s)/I1(s)
2. Driving Point Admittance:
Y11(s) = I1(s)/V1(s) = 1/Z11(s)

Transfer Function:
The transfer function relates the transform of a quantity at one
port to the transform of another quantity at another port. Thus
transfer functions which relate voltages and currents have
following possible forms:
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TRANSFER FUNCTION OF TERMINATED TWO-PORT NETWORKS

I. Two-port with out load and source impedances

Let us drive the expressions for open circuit voltage ratio by using Z and Y
parameters.

Consider the Z and Y parameter Consider the Z and Y parameter

equations for the two equations for the two
ports when port 2 is open ports when port 2 is short
Open circuit voltage ratio Short circuit current ratio

But in practice, we have to consider source and load impedances. So we go for

two-port with load and source impedances

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Example :For the network shown find the voltage ratio G21(s)=
V2(s) / V1(s) and Y11(s) = I(s) / V1(s) by considering no current in
the output terminals.

The network acts as a voltage divider .

G21(s)

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II. Two-port with load and source impedances
These transfer functions are functions of the two-port parameters z, h, or y
and the source and/or load impedance.
Example 1 : Drive transfer admittance of a two-ort network that is terminated
in resistor of R ohms as shown below.

The negative sign is because the

For the above N/W positive direction for I2 is directed
into the two-port instead of into the
load.

By eliminating V2 𝑰 𝟐 + 𝒚 𝟐𝟐 𝑰 𝟐 𝑹
⇒ 𝑽𝟏=
𝒚 𝟐𝟏

Note that is a transfer admittance of the two-ort network terminated

in a resistor R. And is a transfer admittance when port 2 is short circuited
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In order to solve for transfer functions of two-ports terminated at either port
by an impedance , let us consider the equivalent circuit below

Controlled
sources

Write a mesh equation for I2 The current ratio transfer function

for the terminated two-port
network is

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(or) Follow the previous example procedure.

 Suppose we required to find the transfer function for the two-ort
terminated at both ends as shown below.

Write the mesh equations:

Next solve for I2:

Since we arrive at the following situation

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Example Obtain the input impedance V1/I1 of a two port network terminated
with a load, using h parameters.
Solution:
We draw the circuit diagram of two port network using h parameters
including the load.

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Example : Use Z parameters as an illustration
two-port equation

input port constraint (2) Find I2 : From (A) and (C)

Output port constraint

Substitute (F) into (E)

(1) Find Zin = V1 / I1

From (D) and (B)

Substitute (E) into (A)

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Example : For the two-port shown in Fig

[h]=

We replace the two-port by its equivalent circuit as shown below.

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 Poles and Zeros of Network Function:
All the Poles and Zeros of Network Function have the form of a ratio of
two polynomials in s as,

The P(s) is the numerator polynomial in s having degree m while the

Q(s) is the denominator polynomial in s having degree n. Hence
network function can be expressed as,

Now the equation P(s) = 0 has m roots while the equation Q(s) = 0 has n
roots. Thus G(s) can be expressed in the factorized form as,

where z1, z2, … zm are the roots of the equation P(s) = 0 and P1, P2, …
Pn are the roots of the equation Q(s) = 0.

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The z1, z2, … zm,P1,P2 … Pn are the values of s and hence are complex
frequencies as s variable is a complex variable. 48
Poles:
• The values of ‘s’ i.e. complex frequencies, which make the network
function infinite when substituted in the denominator of a network
function are called poles of the network function.

• If such poles are real and non repeated, these are called simple poles.
If a particular pole has same value twice or more than that, it is called
repeated pole. A pair of poles with complex conjugate values is called a
pair of complex conjugate poles.

• The poles are the roots of the equation obtained by equating

denominator polynomial of a network function to zero. Such an
equation is called characteristic equation of a network.

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Zeros:
The values of ‘s’ i.e. complex frequencies, which make the network
function zero when substituted in the numerator of a network
function are called zeros of the network function.

• Similar to the poles, zeros also can be simple zeros, repeated

zeros or complex conjugate zeros.
• The zeros are the roots of the equation obtained by equating
numerator polynomial of a system function to zero.
• When the order m is greater than n then there are m-n poles
at infinity while if the order m is less than n then there are n-
m zeros at infinity.
• Hence if, for any rational network function, poles and zeros at
infinity and zero are taken into consideration in addition to
finite poles and zeros, the total number of zeros is equal to the
total number of poles.
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Pole zero diagram and stability
The network functions of all networks stable
• Must be rational functions with real coefficients.
• May not have poles in the right half s plane.
• May not have multiple poles on the jωaxis
 In the complex s –plane a pole is denoted by small cross(x)
and a zero by a small circle(o)

Example: obtain pole zero location for the function

The zeros of T(s) are: s = -2,s = -4 and poles are: s= 0,s= -1,s=-3

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Necessary conditions For Driving Point Function:

The restrictions on pole and zero locations in the Conditions For Driving
Point Function with common factors in P(s) and Q(s) cancelled are listed
below.
1. The coefficients in the polynomials P(s) and Q(s) of network function N(s)
= P(s)/Q(s) must be real and positive.
2. Complex poles or imaginary poles and zeros must occur in conjugate
pairs.
3. (a) The real parts of all poles and zeros must be zero, or negative. (b) If
the real part is zero, then the pole and zero must be simple.
4. The polynomials P(s) and Q(s) may not have any missing terms between
the highest and the lowest degrees, unless all even or all odd terms are
missing.
5. The degree of P(s) and Q(s) may differ by zero, or one only.
6. The lowest degree in P(s) and Q(s) may differ in degree by at the most
one.

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Example : check weather the following given function represent
deriving point function or not

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Necessary Conditions for Transfer Functions:
The restrictions on pole and zero location in transfer functions with
common factors in P(s) and Q(s) cancelled are listed below.
1. (a) The coefficients in the polynomials P(s) and Q(s) of N(s)=P(s)/Q(s)
must be real. (b) The coefficients in Q(s) must be positive, but some of the
coefficients in P(s) may be negative.
2. Complex or imaginary poles and zeros must occur in conjugate pairs.
3. The real part of poles must be negative, or zero. If the real part is zero,
then the pole must be simple.
4. The polynomial Q(s) may not have any missing terms between the
highest and the lowest degree, unless all even or all odd terms are
missing.
5. The polynomial P(s) may have missing terms between the lowest and
the highest degree.
6. The degree of P(s) may be as small as zero, independent of the degree
of Q(s).
7. (a) For the voltage transfer ratio and the current transfer ratio, the
maximum degree of P (s) must equal the degree of Q(s). (b) For transfer
impedance and transfer admittance, the maximum degree of P (s) must
equal the degree of Q(s) plus one.
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Example : check weather the following given function represent
transfer function or not

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