Two Port Network - 2 | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE) PDF Download

Two Port network

h - PARAMETERS OR HYBRID PARAMETERS

• The hybrid parameters (h- parameters) would find wide usage in electronic circuit, especially in constructing models for transistors. 
• The par ameters of a transistor cannot be measured either by short-circuit admittance parameter measurement or open circuit impedance parameter measurement alone.

Because of the forward bias of the base-emitter junction, the device has a very low input resistance. For open-circuit impedance measurement of Z₁₂ and Z₂₂, it is very difficult to make the input open circuited. Z₁₁ and Z₁₂ can be measured by open circuit impedance measurements, since the collector-emitter junction is reversed biased.

• By making a short-circuit admittance parameter measurement, Y₁₂ and Y₂₂ can be measured by short-circuiting the input port, but Y₁₁ and Y₂₁ cannot be measured since the collector emitter junction is reverse biased. 
• By making a short-circuit admittance parameter measurement. Y₁₂ and Y₂₂ can be measured by short circuiting the input port, but Y₁₁ and Y₂₁ cannot be measured since the collector-emitter junction is reverse biased. 
• Some kind of parameter representation is required by which some parameters are measured by open circuiting the input ort, while the rest of the parameters can be measured by short-citcuiting the output port. This is the socalled hybrid parameter representation.
• This parameter representation is a hybrid of shortcircuit admittance and open-circuit impedance measurement. 
• One set of equations result when the voltage o the input port and the current of the output por are expressed in terms of the current of the input port and the voltage of the output port, in the form

V₁ = h₁₁I₁ + h₁₂V₂

I₂ = h₂₁I₁ +h₂₂V₂

where,h_i : input impedance with the output short-circuited.

h_r : reverse voltage gain with the input open-circuited.

h_f : Forward current gain with the output short-circuited.

h_o : Output admittance with the input open-circuited

Condition of Reciprocity

h₁₂ = - h ₂₁ 

Condition of Symmetry

h-parameter Equivalent Circuit

g-PARAMETERS OR INVERSE HYBRID PARAMETERS
I₁ = g₁₁ V₁ + g₁₂I₂

V₂ = g₂₁V₁ + g₂₂I₂

The g-parameters can be defined as,

..... (i.e. input admittance with output open circuited)

...(i.e. forward voltage gain with output open circuited)

......(i.e. forward voltage gain with output open circuited)

  ....(i.e. output impedance with input short circuited)

Condition of Reciprocity 

g₁₂ = - g₂₁

Condition of Symmetry

g-parameter Equivalent Circuit

INTER RELATIONS IN NETWORK PARAMETERS

Y - parameter in terms of Z-parameters

ABCD-parameters in terms of Z- parameters

 V₁ = Z₁₁I₁ + Z₁₂I₂
V₂ = Z₂₁I₁ + Z₂₂I₂

from equation (ii),

from equation (i) and (ii),

NOTE: ABCD parameters relate voltage and current in the primary to the voltage and current in the secondary

Z-parameters in terms of h-parameters

 V₁ = h₁₁I₁ + h₁₂V₂ ..................... (i)

I₂ = h₂₁I₁ + h₂₂V₂ .....................(ii)

from equation (ii)

Similarly 

• Find out Y parameters in terms of ABCD parameters. 
• h-parameters in terms of ABCD parameters.

Condition Under which passive Two-port Network is reciporcal and Symmeterical

INTERCONNECTION OF TWO-PORT NETWORKS

1. Series Connection

I_1a = I_1b = I₁  I_2a = I_2b = I₂

V₁ = V_1a + V_1b

so, in matrix form the Z-parameters of the seriesconnected combined network can be written as

2. Parallel Connection

3. Series Parallel Connection h-parameters

I_1a = I_1b = I₁

V₁ = V_1a + V_1b ........................(i)

I₂ = I_2a + I_2b ........................(ii)

 
Where

4. Parallel Series Connection

 




Where

5. Cascade Connection

 
I₁ = I_1a
V₁ = V_1a
I₂ = I_2b
I_1b = –I_2a
V_1b = V_2a

BARLETTS BISECTION THEOREM

• This theorem is applicable for symmetrical network. 
• A symmetrical network can be split into two half then the Z= parameters of the network are given by

Network is symmetrical as well as reciprocal Z_OC – Open circuited driving point impedances for half of section.
Z_SCH – Short circuited driving point impedances for half of section.

SYMMETRICAL LATTICE NETWORK

 


Z_b ~ Z_OCH
Z_a ~ Z_SCH

IMAGE IMPEDANCE

• If Z_i1 is the impedance seen at pot 1 with Z_i2 connected at port 2 and Z_i2 is impedance seenm at port 2 with Z_i1 connected at port 1. Then Z_i1 and Z_i2 are called image impedance at port 1 and 2 respectively.

• A pair of terminal through which a current may enter or leave a network is known as a port. Twoterminal devices or element (such as resistors, capacitors, and inductors) result in one-port networks.
• Here, we are mainly concerned with the twoport networks. A two-port network is an electrical network with two separate ports for input and output. Thus, a two-port network has two terminal pairs acting as access points. The current entering in the pair leaves the other terminal in the pair. Three-terminal devices such as transistors can be configured into two-port networks.
• Armed only with the knowledge that the circuit is linear, and the ability to measure the voltage and currents, we will shortly see that it is possible to characterize such a network with a set of parameters that allows us to predict how the network will interact with other networks.

Network parameters 

• Z– parameters
• Y= parameters
• ABCD parameters (Transmission parameter)
• A' B' C' D ' parameters (Inverse transmission parameter)
• h-parameters
• g-parameters

Z – PARAMETERS

.........the input driving- point impedance with the output port open-circuited

....The reverse transfer impedance with the input port open-circuited

....the forward transfer impedance with the output port open-circuited

.....the output driving point impedance with input port opencircuited

• The equivalent circuit representation of equation (i) is in figure below, where Z₁₂ I₂ and Z₂₁ I₁ are current controlled voltage sources (CCVS)
• Rewriting equation (i),

 V1 = (Z₁₁ –Z₁₂)I₁+ Z₁₂ (I₁ + I₂)
V₂ = (Z₂₁ – Z₁₂) I₁ + (Z₂₂ –Z₁₂)I₂
+ Z₁₂ (I₁ + I₂) ...........(ii)

• The equivalent circuit for equation (i) is given as,

• The equivalent circuit for equation (ii) is given as,

Condition of Reciprocity and Symmetry Reciprocal Network

• A network must be r eciproca l when r atio of response at port 2 to the excitation at port '1' is same as ratio of response at port 1 to the excitation at port 2

Z₁₁ = Z₂₂

..............(Because in Y₁₁ we have short circuit condition whereas in Z₁₁ we have open cuicuit condition) input driving point admittance

.....(reverse transfer admittance)

 ....(forward transfer admittance)

...(output driving point admittance)

Equivalent Circuit Y-parameter

• Rewriting equation of Y-parameters

For this equivalent II–network

For reciprocal network Y12 = Y21
For sysmmetrical network Y11  = Y22

 TRANSMISSION PARAMETERS (ABCD)

V1 = AV2 + B (–12)
I1 = CV2 + D (–I2)
The transmission parameters are

.....(i.e. the reverse voltage ratio with the receiving end open circuited)

....(i.e. the transfer admittance with the receiving end open-circuited)

...(i.e. the transfer impedance with the receiving end short-circuited )

....(i.e. the reverse  current ratio with the receiving end short circuited.)

Condition for Reciprocity

Condition for Symmetry

A = D
INVERSE TRANSMISSION PARAMETERS

The inverse transmission parameter can be defined as

.....(forward voltage ratio with sending end open circuited)

...( transfer a dmittance with sending end open circuited)

. .. (t ra ns fe r impeda nce wi th sending end short circuited)

....(forwar d current ratio with sending end short circuited)

Condition of Reciprocity

Condition for Symmetry

A'=D'

h–PARAMETERS OR HYBRID PARAMETERS

• The hybrid parameters (h- parameters) would find wide usage in electronic circuit, especially in constructing models for transistors.
• The parameters of a transistor cannot be measured either by short-circuit admittance parameter measurement or open circuit impedance parameter measurement alone. Because of the forward bias of the base-emitter junction, the device has a very low input resistance. For open-circuit impedance measurement of Z₁₂ and Z₂₂, it is very difficult to make the input open circuited. Z₁₁ and Z₁₂ can be measured by open circuit impedance measurements, since the collector-emitter junction is reversed biased.
• By making a short-circuit admittance parameter measurement, Y₁₂ and Y₂₂ can be measured by short-circuiting the input port, but Y₁₁ and Y₂₁ cannot be measured since the collector emitter junction is reverse biased.
• By making a short-circuit admittance parameter measurement. Y₁₂ and Y₂₂ can be measured by short circuiting the input port, but Y₁₁ and Y₂₁ cannot be measured since the collector-emitter junction is reverse biased.
• Some kind of parameter representation is required by which some parameters are measured by open circuiting the input ort, while the rest of the parameters can be measured by short-citcuiting the output port. This is the so-called hybrid parameter representation.
• This parameter representation is a hybrid of short-circuit admittance and open-circuit impedance measurement.
• One set of equations result when the voltage o the input port and the current of the output por are expressed in terms of the current of the input port and the voltage of the output port, in the form

where,
hi : input impedance with the output short-circuited.
hr : reverse voltage ga in with the input open-circuited.
hf : Forward cur rent gain with the output short-circuited.
ho : Output admittance with the input open-circuited.

Condition of Reciprocity
h₁₂ = - h₂₁

Condition of Symmetry

h-parameter Equivalent Circuit

g-PARAMETERS OR INVERSE HYBRID PARAMETERS

The g-parameters can be defined as,

...... (i.e. input admittance with output open circuited)

......(i.e. forward voltage gain with output open circuited)

......(i.e. forward voltage gain with output open circuited)

......(i.e. output impedance with input short circuited)

 Condition of Reciprocity
g12 = - g 21

Condition of Symmetry

g-parameter Equivalent Circuit

INTER RELATIONS IN NETWORK
 PARAMETERS
Y - parameter in terms of Z-parameters

ABCD-parameters in terms of Z- parameters

 V₁ = Z₁₁I₁ + Z₁₂I₂
V₂ = Z₂₁I₁ + Z₂₂I₂

from equation (ii),

from equation (i) and (ii),

NOTE: ABCD parameters relate voltage and current in the primary to the voltage and current in the secondary Z-parameters in terms of h-parameters

V₁ = h₁₁I₁ + h₁₂V₂ ..................... (i)
I₂ = h₂₁I₁ + h₂₂V₂ .....................(ii)

 from equation (ii),

 Similarly

• Find out Y parameters in terms of ABC D parameters.
• h-parameters in terms of ABCD parameters.

Condition Under which passive Two-port Network is reciporcal and Symmeterical 

INTERCONNECTION OF TWO-PORT NETWORKS
1 . Serie s Connection

From diagram,
I_1a = I_1b = I₁   I₂a = I₂b = I₂

V₁ = V₁a + V₁b

Also

so, in matrix form the Z-parameters of the seriesconnected combined network can be written as

2 . Parallel Connection

I₁ = I₁a + 1₁b
I, = I₂a + I₂b
V₁a = V₁b = V₁
V₂1 = V₂b = V₂

3 . Series Parallel Connection h-parameters

 




Where

4 . Parallel Series Connection

Where

5 . Cascade Connection

BARLETTS BISECTION THEOREM

• This t heor em is a pplicab le for symmetr ica l network.
• A symmetrical network can be split into two half then the Z= parameters of the network are given by

Network is symmetrical as well as reciprocal Z_OC – Open circuited driving point impedances for half of section.
Z_SCH – Short circuited driving point impedances for half of section.
 

SYMMETRICAL LATTICE NETWORK

Z_b » Z_OCH 
Z_a » Z_SCH

IMAGE IMPEDANCE

• If Z_i1 is the impedance seen at pot 1 with Z_i2 connected at port 2 and Z_i2 is impedance seenm at port 2 with Z_i1 connected at port 1. Then Z_i1 and Z_i2 are called image impedance at port 1 and 2 respectively.