Week 13

Two-Port Networks I

PowerPoint® Slides
by Dr. Chow Li Sze

Email: lschow@um.edu.my
Last updated: 7 September 2017 1
Introduction
Terminal pairs represent the points where signals are either fed in
or extracted, they are referred to as the ports of the system.
Use of this building block is subject to
several restrictions:
Two-port building block.
1. No energy stored within the circuit.
2. No independent sources within the
circuit.
3. The current into the port must equal
the current out of the port. (i1 = i’1 ,
i2 = i’2)
4. All external connections must be
made to either the input port or the
output ports. (No connections
between a & c, a & d, b & c, b & d).
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Terminal Equations
 In two-port network, we want to relate the current and voltage at
one port to the current and voltage at the other port.
 The most general description of the two-port network is carried
out in the s domain.

s-domain two-port basic building block.

There are four terminal variables,

I1, V1, I2, V2.
Only two are independent. Once
we specify two variables, we can
find the other two unknowns.

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Terminal Equations
There are six different ways to combine the four variables:

V1  z11 I1  z12 I 2 V1  h11 I1  h12V2

Eq.1 Eq.5
V2  z 21 I1  z 22 I 2 I 2  h21 I1  h22V2
I1  y11V1  y12V2 I1  g11V1  g12 I 2 Eq.6
Eq.2
I 2  y21V1  y22V2 V2  g 21V1  g 22 I 2
V1  a11V2  a12 I 2 The coefficients of the current and/or
Eq.3
I1  a21V2  a22 I 2 voltage variables on the right hand side of
Eq.1-6 are called the parameters of the two-
V2  b11V1  b12 I1 port circuit.
Eq.4
I 2  b21V1  b22 I1 When using Eq.1, we refer to the z
parameters of the circuit. Similarly, we refer
to y, a, b, h, g parameters of the network.
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Two-Port Parameters
We can determine the parameters for any circuit by computation or
measurement. For example, to find the z parameters for a circuit:
V1 z11 is the impedance seen looking into port 1 when
z11  port 2 is open.
I1 I 2 0

V1 z12 is the transfer impedance. It is the ratio of the port

z12  1 voltage to the port 2 current when port 1 is open.
I2 I1  0

V2 z21 is the transfer impedance. It is the ratio of the port

z 21  2 voltage to the port 1 current when port 2 is open.
I1 I 2 0

V2 z22 is the impedance seen looking into port 2 when

z 22  port 1 is open.
I2 I1  0 5
Two-Port Parameters
 Impedance parameters may be either calculated or measured by:
1. first open port 2 and determining the ratios V1/I1 and V2/I1.
2. then open port 1 and determining the ratios V1/I2 and V2/I2.
 A port parameter is obtained by either opening or shorting a
port.
 A port parameter is an impedance, an admittance, or a
dimensionless ratio (of either two voltages or two currents).

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Example
Find the z parameters for the circuit below:

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Two-Port Parameters
I1 V1 V2
y11  S a11  b11 
V1 V V2 V1
2 0 I 2 0 I1  0

I1 V1 V2
y12  S a12    b12   
V2 V1  0
I2 V I1
2 0 V1  0

I2 I1 I2
y21  S a21  S b21  S
V1 V V2 V1
2 0 I 2 0 I1  0

I2 I1 I2
y22  S a22   b22  
V2 V1  0
I2 V2  0
I1 V1  0

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Two-Port Parameters
V1 I1
h11   g11  S
I1 V V1 I 2 0
2 0

V1 I1
h12  g12 
V2 I1  0
I2 V1  0

I2 V2
h21  g 21 
I1 V2  0
V1 I 2 0

I2 V2
h22  S g 22  
V2 I1  0
I2 V1  0

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Two-Port Parameters
 Immittance  a quantity that is either an impedance, z or
admittance, y.
 Transmission  a and b parameters, they describe the voltage
and current at one end of the two-port network in terms of the
voltage and current at the other end.
 Hybrid  h and g parameters, relate cross-variables, that is, an
input voltage and output current to an output voltage and input
current.

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Example
The following measurements pertain to a two-port circuit
operating in the sinusoidal steady state. With port 2 open, a voltage
equal to 150 cos 4000t V is applied to port 1. The current into port 1
is 25 cos (4000t – 45o)A, and the port 2 voltage is 100 cos (4000t +
15o)V. With port 2 short-circuited, a voltage equal to 30 cos 4000t V
is applied to port 1. The current into port 1 is 1.5 cos (4000t + 30o)A,
and the current into port 2 is 0.25 cos(4000t + 150o)A. Find the a
parameters that can describe the sinusoidal steady-state behavior
of the circuit.

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Two-Port Parameters
 Because the six sets of equations relate to the same variables, the
parameters associated with any pair of equations must be related
to the parameters of all the other pairs.
 If we know one set of parameters, we can derive all the other sets
from the known set.

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Two-Port Parameters
To find the z parameters as functions of the y parameters, first
solve Eq.2 for V1 and V2. Then, compare the coefficients of I1 and I2
in the resulting expressions to the coefficients of I1 and I2 in Eq.1.

V1  z11 I1  z12 I 2 I1 y12

Eq.1
V2  z 21 I1  z 22 I 2 I 2 y22 y y
From V1   22 I1  12 I 2 Eq.7
I1  y11V1  y12V2 y11 y12 y y
Eq.2
I 2  y21V1  y22V2 y21 y22

y11 I1
y21 I 2 y21 y11
V2   I1  I2 Eq.8
y y y
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Two-Port Parameters
Compare Eqs.7-8 with Eq.1 shows

y22
z11 
y
y12
z12  
y
y21
z 21  
y
y11
z 22 
y

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Two-Port Parameters
To find z parameters as functions of the a parameters, rearrange Eq.3 in
the form of Eq.1 and then compare coefficients.

V1  a11V2  a12 I 2 1 a22

From Eq.3 V2  I1  I2 Eq.9
I1  a21V2  a22 I 2 a21 a21
a11  a11a22 
V1  I1    a12  I 2 Eq.10
a21  a21 

a11 a
From Eq.10 z11  , z12 
a21 a21
1 a22
From Eq.9 z 21  , z 22 
a21 a21
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16
Example
Two sets of measurements are made on a two-port resistive
circuit. The first set is made with port 2 open, and the second
set is made with port 2 short-circuited. The results are as
follows:
Port 2 Open Port 2 Short-Circuited
V1 = 10 mV V1 = 24 mV
I1 = 10 A I1 = 20 A
V2 = -40 V I2 = 1 mA

Find the h parameters of the circuit.

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Reciprocal Two-Port Circuits
If a two-port circuit is reciprocal, the following relationships exist
among the port parameters:

z12  z 21
A two-port circuit is reciprocal if the
y12  y21 interchange of an ideal voltage source at
a11a22  a12 a21  a  1 one port with an ideal ammeter at the
b11b22  b12b21  b  1 other port produces the same ammeter
reading.
h12  h21
g12   g 21

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Reciprocal Two-Port Circuits
A reciprocal two-port circuit
When a voltage source of 15V is
applied to the port ad, it produces a
current of 1.75A in the ammeter at
port cd. The ammeter current can be
determined when we know the
voltage Vbd.

Vbd Vbd  15 Vbd

At node B:   0 Vbd  5V
60 30 20
5 15
At node C: I   1.75 A
20 10

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Reciprocal Two-Port Circuits
Voltage source and ammeter interchanged.

If the voltage source and ammeter

are interchanged, the ammeter will
still read 1.75A. It’s verified below:

Vbd Vbd Vbd  15 Vbd  7.5V

At node B:   0
60 30 20
7.5 15
At node A: I    1.75 A
30 10
A two-port circuit is also reciprocal if the interchange of an ideal current
source at one port with an ideal voltmeter at the other port produces the
same voltmeter reading. 20
Reciprocal Two-Port Circuits
 For a reciprocal two-port circuit, only three calculations or
measurements are needed to determine a set of parameters.
 A reciprocal two-port circuit is symmetric if its ports can be
interchanged without disturbing the values of the terminal currents
and voltages.
z11  z 22
A symmetric
These are additional y11  y22
reciprocal network
relationships among
a11  a22 only need two
the port parameters
b11  b22 calculations or
for symmetric
measurements to
reciprocal network: h h  h h  h  1
11 22 12 21 determine all the
g11 g 22  g12 g 21  g  1 two-port parameters.

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Reciprocal Two-Port Circuits
Examples of symmetric two-port circuits:
Symmetric tee Symmetric pi

Symmetric bridged tee Symmetric lattice

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Terminated Two-Port Circuit
In a typical application of a two-port
model, the circuit is driven at port 1 and
loaded at port 2.
Zg = internal impedance of the source
Vg = internal voltage of the source
ZL = load impedance

6 characteristics of the terminated two-port circuit define its terminal

behavior:
1. Input impedance, Zin = V1/I1, or admittance Yin = I1/V1
2. Output current I2.
3. Thevenin voltage & impedance (V Th, ZTh) with respect to port 2.
4. Current gain I2/I1.
5. Voltage gain V2/V1.
6. Voltage gain V2/Vg.
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Terminated Two-Port Circuit
Table below summarizes the expressions involving the y, a, b, h and g parameters,
for the terminated two-port equations:

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Terminated Two-Port Circuit

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Terminated Two-Port Circuit

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Terminated Two-Port Circuit
The derivation of any one of the desired expressions involves the
algebraic manipulation of the two-port equations along with the two
constraint equations imposed by the terminations.

If we use the z-parameter equations:

From Eq.1: V1  z11 I1  z12 I 2 Eq.1a

V2  z 21 I1  z 22 I 2 Eq.1b
From V1  Vg  I1Z g Eq.11
termination:
V2   I 2 Z L Eq.12
Looking into port 1: Zin = V1/I1
 z 21 I1
Substitute Eq.12 into Eq.1b: I2  Eq.13
Z L  z 22
z12 z 21
Then substitute into Eq.1a get Zin: Z in  z11 
z 22  Z L
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Terminated Two-Port Circuit
Vg  z12 I 2
Substitute Eq.11 into Eq.1a: I1  Eq.14
z11  Z g
 z 21Vg
Then substitute into Eq.13 get I2: I2 
( z11  Z g )( z 22  Z L )  z12 z 21

Thevenin voltage with respect to port 2 equals V2 when I2 = 0.

V1
From Eq.1a-1b: VTh  V2  z 21 I1  z 21
I 2 0
z11
z 21
Use Eq.14:  Vg
Z g  z11

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Terminated Two-Port Circuit
Thevenin/output impedance is the ratio V2/I2 when Vg is short-circuited (Vg = 0).

From Eq.11: V1   I1Z g

 z12 I 2
Substitute into Eq.1a: I1 
z11  Z g
V2 z12 z 21
Substitute into Eq.1b: Z Th   z 22 
I2 Vg  0
z11  Z g

I2  z 21
Current gain I2/I1 is obtained from Eq.13: 
I1 Z L  z 22

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Terminated Two-Port Circuit
To derive the voltage gain V2/V1, substitute Eq.12 into Eq.1b:

 V 
V2  z 21 I1  z 22  2  Eq.15
 ZL 
V z V
From Eq.1a: I1  1  12 2
z11 z11Z L

V2 z 21Z L
Substitute into Eq.15 to get the voltage gain: 
V1 z11Z L  z

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Terminated Two-Port Circuit
To derive the voltage ratio V2/Vg, combine Eq.1a, Eq.11, Eq.12 to find I1 as function
of V2 and Vg:
z12V2 Vg
I1  
Z L ( z11  Z g ) z11  Z g

Use Eq.16, Eq.12, Eq.1b to derive an expression for V2 and Vg, and get its ratio:

V2 z 21Z L

Vg ( z11  Z g )( z 22  Z L )  z12 z 21

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Example

The two-port circuit shown above is described in terms of its b

parameters: b11 = -20, b12 = -3000, b21 = -2mS, b22 = -0.2.
(a) Find the phasor voltage V2.
(b) Find the average power delivered to the 5k load.
(c) Find the average power delivered to the input port.
(d) Find the load impedance for maximum average power transfer.
(e) Find the maximum average power delivered to the load in (d).

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