Circuit Theory - Two Port Networks Complete Study Notes

 TWO PORT NETWORK 

• A pair of terminals through which a current may enter or leave a network is known as a port.
• Two-terminal devices or elements (such as resistors, capacitors, and inductors) result in one-port networks.
• The four-terminal or two-port circuits are used in op-amps, transistors, and transformers.
• Two Port Network circuit is shown below.

• The current entering one terminal leaves through the other terminal so that the net current entering the port equals zero.
• A two-port network consists of two pairs of terminals in which one pair of terminals is designated as input and other pair being output.
• Two port networks are useful in communications, control systems, power systems, and electronics.
• To characterise a two-port network requires that we relate the terminal quantities V₁, V₂, I₁, and I₂. The various terms that relate these voltages and currents are called parameters.
• When the voltage or current are compared at same port then the term defined the Driving Imminence Function,while on the other hand when the comparison is done at different port then the function will defined as Transfer Imminence Function.

 1. (Z-parameters) Open circuit Impedance Parameters

  Here, The voltages at input and output are expressed in the terms of input and output currents. The equations are given below.

V₁ = Z₁₁I₁ + Z₁₂I₂

V₂ = Z₂₁I₁ + Z₂₂I₂

where, Z₁₁, Z₁₂, Z₂₁ and Z₂₂ are called the Z-parameters. 

The Z-Parameter are found as follow:

 The z-parameters are also called as open-circuit impedance parameters because  they are obtained as the ratio of voltage and current when it is open-circuiting port 2 ( I₂ = 0) or port 1 ( I₁ = 0). 

• Z₁₁ is the driving Point impedance when output is open circuited.
• Z₁₂ is the reverse transfer impedance when input is open-circuited.
• Z₂₁ is the forward transfer impedance when output is open-circuit.
• Z₂₂ is the driving Point impedance when input is open-circuited.

  2. (Y-parameters) Short Circuit Admittance parameters 

  Y parameters are achieved by representing the currents at the two ports of the network in terms of voltages at two ports. Thus, voltages V₁ and V₂ are independent variables, while I₁ and I₂ are dependent variables.

   The equations are given below.

    I₁ = Y₁₁V₁ + Y₁₂V₂

    I₂ = Y₂₁V₁ + Y₂₂V₂

   Where, Y₁₁, Y₁₂, Y₂₁, Y₂₂ are called the Y-parameters.

  The y-parameters are also called short-circuit admittance parameters. These are obtained as a  ratio of current and voltage. The parameters are calculated by short-circuiting port 2 (V₂ = 0) or port 1 (V₁ = 0). 

  So the Y-Parameter can be found as follow:

• Y₁₁ Short-circuit driving point input admittance
• Y₁₂ Short-circuit reverse transfer admittance
• Y₂₁ Short-circuit forward transfer admittance
• Y₂₂ Short-circuit driving point output admittance

   3. h-parameters: Hybrid Parameters 

• These parameters are By expressing voltage at input port and the current at the output port, the h-parameters are obtained.
• Two-port network variables  are selected as independent of the input current (I₁ )and the output voltage(V₂).
• Here, the input voltage and the output current are the dependent variables of this model.  

   Equations for voltage at input port and current at the output port are given below.

   V₁ = h₁₁I₁ + h₁₂V₂

   I₂ = h₂₁I₁ + h₂₂V₂

    The h-Parameter can be found as follow:

• h₁₁ Short-circuit input impedance:
• h₂₁ Forward short-circuit current gain it is dimensionless .
• h₁₂ Reverse open-circuit voltage gain it is dimensionless :
• h₂₂ Open-circuit output admittance:

    The h-parameters are also called hybrid parameters since they consist of both open-circuit parameters (I_{1 )} and short-circuit parameters (V₂ )

    4. ( g-parameters ) Inverse hybrid parameters

• These parameters are obtained by expressing voltage at output port and the current at the input port.
• Off-diagonal g-parameters are dimensionless, while diagonal members have dimensions the reciprocal of one another. 

     I₁ = g₁₁V₁ + g₁₂I₂

     V₂ = g₂₁V₁ + g₂₂I₂

  Here g₁₂ and g₂₁ are dimensionless coefficients, g₂₂ is impedance and g₁₁ is admittance.

  5.(T-Parameters) or ABCD parameters Transmission Parameters

 These parameters are generally used in the analysis of power transmission in which the input port is considered as the sending side while the output port is considered as receiving side. These parameters are calculated by expressing voltage and current at the output port.

      ABCD parameters can be defined as following: 

• A is the reverse voltage ratio with open output.
• B is the reverse transfer impedance with shorted output.
• C is the  reverse transfer admittance with open output.
• D is the reverse current ratio with shorted output.

     V₁ = AV₂ + B(–I₂)

     I₁ = CV₂ + D(–I₂)

     The transmission parameters express the primary (sending end) variables V₁ and I₁ by dependence of the secondary variables i.e., V₂  (receiving end)  and '-I₂' . The negative sign of I₂ is used to mention that the current to enter the load at the receiving end.

• Symmetry Condition: if input impedance seen through both the port is same then both port are known as symmetric port.If port are symmetric then both port can be interchanged.
• Reciprocity Condition: If only single source is acting in the circuit then “By changing the position of response and excitation if ratio of response to excitation is constant then circuit is reciprocal”.

6.Conversion of Z-parameter in term of Y-parameter

• In the similar fashion we can obtain the other relationship

7.INTERCONNECTION OF TWO-PORT NETWORKS   

 The two-port networks can be connected in many ways such as series, parallel or cascade. the configuration listed below:

• Series Connection: When two 2-port network are connected in series configuration the z- parameter of each port will be directly added in the result of equivalent 2-port network.

This can be concluded  that if two-port networks with Z-parameters [Z]₁,[Z]₂,[Z]₃,[Z]_n , are connected in series, then the equivalent two port-parameters are given as

[Z]_eq = [Z]₁+ [Z]₂+ [Z]₃+ [Z]_n 

• Parallel-connected Two-port Network:   When two 2-port network are connected in Parallel configuration the Y- parameter of each port will be directly added in the result of equivalent 2-port network.

This can be concluded  that if two-port networks with Y-parameters [Y]₁,[Y]₂,[Y]₃,[Y]_n , are connected in Parallel, then the equivalent two port-parameters are given as

[Y]_eq = [Y]₁+ [Y]₂+ [Y]₃+ [Y]_n 

• Cascade Connection of Two-port Network:

When two 2-port network are connected in cascaded configuration then the T-parameter of equivalent two port network will be the Product of T-parameter of individual network.

networks have transmission parameters [A]₁,[A]₂,[A]₃,[A]_n, then the equivalent two-port parameter will have a transmission parameter given as 

[A]_eq = [A]_1*[A]_2*[A]_3*[A]_n

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