Notes: Two Port Analysis | Network Theory (Electric Circuits) - Electrical Engineering (EE) PDF Download

In electronics, a two-port network (also known as a four-terminal network or quadripole) is an electrical circuit or device with two pairs of terminals for connecting to external circuits. A pair of terminals forms a port if the currents entering and exiting the terminals satisfy the port condition: the current entering one terminal must equal the current leaving the other terminal of the same port. These ports serve as the interfaces where the network connects to other networks, allowing signals to be applied or outputs to be taken. Typically, in a two-port network, port 1 is the input port and port 2 is the output port.

The first and second ports are called as port1 and port2 respectively.

One port network is a two terminal electrical network in which, current enters through one terminal and leaves through another terminal. Resistors, inductors and capacitors are the examples of one port network because each one has two terminals. One port network representation is shown in the following figure.
Here, the pair of terminals, 1 & 1’ represents a port. In this case, we are having only one port since it is a one port network.
Similarly, two port network is a pair of two terminal electrical network in which, current enters through one terminal and leaves through another terminal of each port. Two port network representation is shown in the following figure.
Here, one pair of terminals, 1 & 1’ represents one port, which is called as port1 and the other pair of terminals, 2 & 2’ represents another port, which is called as port2.
There are four variables V₁, V₂, I₁ and I₂ in a two port network as shown in the figure. Out of which, we can choose two variables as independent and another two variables as dependent. So, we will get six possible pairs of equations. These equations represent the dependent variables in terms of independent variables. The coefficients of independent variables are called as parameters. So, each pair of equations will give a set of four parameters.

Two Port Network Parameters

The parameters of a two port network are called as two port network parameters or simply, two port parameters. Following are the types of two port network parameters.

Now, let us discuss about these two port network parameters one by one.

Z parameters

We will get the following set of two equations by considering the variables V₁ & V₂ as dependent and I₁ & I₂ as independent. The coefficients of independent variables, I₁ and I₂ are called as Z parameters. 

The Z parameters are

Z parameters are called as impedance parameters because these are simply the ratios of voltages and currents. Units of Z parameters are Ohm (Ω).
We can calculate two Z parameters, Z₁₁ and Z₂₁, by doing open circuit of port2. Similarly, we can calculate the other two Z parameters, Z₁₂ and Z₂₂ by doing open circuit of port1. Hence, the Z parameters are also called as open-circuit impedance parameters.

Y parameters

We will get the following set of two equations by considering the variables I₁ & I₂ as dependent and V₁ & V₂ as independent. The coefficients of independent variables, V₁ and V₂ are called as Y parameters. 

The Y parameters are

Y parameters are called as admittance parameters because these are simply, the ratios of currents and voltages. Units of Y parameters are mho.
We can calculate two Y parameters, Y₁₁ and Y₂₁ by doing short circuit of port2. Similarly, we can calculate the other two Y parameters, Y₁₂ and Y₂₂ by doing short circuit of port1. Hence, the Y parameters are also called as short-circuit admittance parameters.

T parameter

We will get the following set of two equations by considering the variables V₁ & I₁ as dependent and V₂ & I₂ as independent. The coefficients of V₂ and -I₂ are called as T parameters.

The T parameters are 

T parameters are called as transmission parameters or ABCD parameters. The parameters, A and D do not have any units, since those are dimension less. The units of parameters, B and C are ohm and mho respectively.
We can calculate two parameters, A and C by doing open circuit of port2. Similarly, we can calculate the other two parameters, B and D by doing short circuit of port2.

T ’ parameters

We will get the following set of two equations by considering the variables V₂ & I₂ as dependent and V₁ & I₁ as independent. The coefficients of V₁ and -I₁ are called as T’ parameters. 

The T’ parameters are 

T’ parameters are called as inverse transmission parameters or A’B’C’D’ parameters. The parameters A’ and D’ do not have any units, since those are dimension less. The units of parameters, B’ and C’, are Ohm and Mho respectively.
We can calculate two parameters, A’ and C’, by doing an open circuit of port1. Similarly, we can calculate the other two parameters, B’ and D’, by doing a short circuit of port1.

h-parameters

We will get the following set of two equations by considering the variables V₁ & I₂ as dependent and I₁ & V₂ as independent. The coefficients of independent variables, I₁ and V₂, are called as h-parameters. 

The h-parameters are

h-parameters are called as hybrid parameters. The parameters, h₁₂ and h₂₁, do not have any units, since those are dimension-less. The units of parameters, h₁₁ and h₂₂, are Ohm and Mho respectively. 
We can calculate two parameters, h₁₁ and h₂₁ by doing short circuit of port2. Similarly, we can calculate the other two parameters, h₁₂ and h₂₂ by doing open circuit of port1.
The h-parameters or hybrid parameters are useful in transistor modelling circuits (networks).

g-parameters

We will get the following set of two equations by considering the variables I₁ & V₂ as dependent and V₁ & I₂ as independent. The coefficients of independent variables, V₁ and I₂ are called as g-parameters. 
The g-parameters are
g-parameters are called as inverse hybrid parameters. The parameters, g₁₂ and g₂₁ do not have any units, since those are dimension less. The units of parameters, g₁₁ and g₂₂ are mho and ohm respectively. 
We can calculate two parameters, g₁₁ and g₂₁ by doing open circuit of port2. Similarly, we can calculate the other two parameters, g₁₂ and g₂₂ by doing short circuit of port1. 

Sometimes, it is easy to find one set of parameters of a given electrical network easily. In those situations, we can convert these parameters into the required set of parameters instead of calculating these parameters directly with more difficulty.

Procedure of two port parameter conversions

Follow these steps, while converting one set of two port network parameters into the other set of two port network parameters.

• Step 1 − Write the equations of a two port network in terms of desired parameters.
• Step 2 − Write the equations of a two port network in terms of given parameters.
• Step 3 − Re-arrange the equations of Step2 in such a way that they should be similar to the equations of Step1.
• Step 4 − By equating the similar equations of Step1 and Step3, we will get the desired parameters in terms of given parameters. We can represent these parameters in matrix form.

Z parameters to Y parameters

Here, we have to represent Y parameters in terms of Z parameters. So, in this case Y parameters are the desired parameters and Z parameters are the given parameters.
Step 1 − We know that the following set of two equations, which represents a two port network in terms of Y parameters.
We can represent the above two equations in matrix form as

Step 2 − We know that the following set of two equations, which represents a two port network in terms of Z parameters. 

We can represent the above two equations in matrix form as 
Step 3 − We can modify it as
Step 4 − By equating Equation 1 and Equation 2, we will get 
Where, 
So, just by doing the inverse of Z parameters matrix, we will get Y parameters matrix. 

Z parameters to T parameters

Here, we have to represent T parameters in terms of Z parameters. So, in this case T parameters are the desired parameters and Z parameters are the given parameters.
Step 1 − We know that, the following set of two equations, which represents a two port network in terms of T parameters.

Step 2 − We know that the following set of two equations, which represents a two port network in terms of Z parameters.
Step 3 − We can modify the above equation as
Step 4 − The above equation is in the form of I₁=CV₂−DI₂ . Here,
Step 5 − Substitute I₁ value of Step 3 in V₁ equation of Step 2.

Step 6 − The above equation is in the form of V₁=AV₂−BI₂ . Here,

Step 7 − Therefore, the T parameters matrix is

Y parameters to Z parameters

Here, we have to represent Z parameters in terms of Y parameters. So, in this case Z parameters are the desired parameters and Y parameters are the given parameters.

Step 1 − We know that, the following matrix equation of two port network regarding Z parameters as
Step 2 − We know that, the following matrix equation of two port network regarding Y parameters as 
Step 3 − We can modify it as
Step 4 − By equating Equation 3 and Equation 4, we will get 
Where,
So, just by doing the inverse of Y parameters matrix, we will get the Z parameters matrix. 

Y parameters to T parameters

Here, we have to represent T parameters in terms of Y parameters. So, in this case, T parameters are the desired parameters and Y parameters are the given parameters.

Step 1 − We know that, the following set of two equations, which represents a two port network in terms of T parameters.

Step 2 − We know that the following set of two equations of two port network regarding Y parameters.

Step 3 − We can modify the above equation as
Step 4 − The above equation is in the form of V₁=AV₂−BI₂ . Here,
Step 5 − Substitute V₁ value of Step 3 in I₁ equation of Step 2.

Step 6 − The above equation is in the form of I₁=CV₂−DI₂ . Here,
Step 7 − Therefore, the T parameters matrix is

T parameters to h-parameters

Here, we have to represent h-parameters in terms of T parameters. So, in this case hparameters are the desired parameters and T parameters are the given parameters.
Step 1 − We know that, the following h-parameters of a two port network.
Step 2 − We know that the following set of two equations of two port network regarding T parameters.
Step 3 − Substitute V₂=0 in the above equations in order to find the two h-parameters, h₁₁ and h₂₁ .
Substitute, V₁ and I₁ values in h-parameter, h₁₁.
Substitute I₁ value in h-parameter h₂₁ .

Step 4 − Substitute I₁=0 in the second equation of step 2 in order to find the h-parameter h₂₂ .
Step 5 − Substitute in the first equation of step 2 in order to find the h-parameter, h₁₂ .
Step 6 − Therefore, the h-parameters matrix is 

h-parameters to Z parameters

Here, we have to represent Z parameters in terms of h-parameters. So, in this case Z parameters are the desired parameters and h-parameters are the given parameters.

Step 1 − We know that, the following set of two equations of two port network regarding Z parameters.
Step 2 − We know that, the following set of two equations of two-port network regarding h-parameters.
Step 3 − We can modify the above equation as 
The above equation is in the form of  

Step 4 − Substitute V₂ value in first equation of step 2.

The above equation is in the form of V₁=Z₁₁I₁+Z₁₂I₂ . Here,
Step 5 − Therefore, the Z parameters matrix is

In this way, we can convert one set of parameters into other set of parameters.

Application of Two Port Analysis 

Purpose and Usage:

• The two-port network model is utilized in mathematical circuit analysis to isolate specific parts of larger circuits.
• It treats a portion of the circuit as a "black box," defined by a matrix of numbers.

Advantages:

• Simplifies the calculation of the network's response to signals applied to the ports.
• Eliminates the need to solve for all internal voltages and currents within the network.
• Facilitates easy comparison of similar circuits or devices.

Applications:

• Commonly applied to devices like transistors, which are characterized by parameters (e.g., h-parameters) provided by manufacturers.
• Can be used for any linear circuit with four terminals, as long as it does not contain an independent source and satisfies the port conditions.

Examples of Analyzed Circuits:

• Filters
• Matching networks
• Transmission lines
• Transformers
• Small-signal transistor models (e.g., hybrid-pi model)

Historical Context:

• The analysis of passive two-port networks is based on reciprocity theorems first derived by Lorentz.

Mathematical Representation:

• Described by a 2x2 matrix of complex numbers.
• Various parameter models include z-parameters, y-parameters, h-parameters, g-parameters, and ABCD-parameters.
• Each model differs based on which variables (voltage or current) are considered independent.

Parameters and Variables:

• V₁: Voltage across port 1
• I₁: Current into port 1
• V₂: Voltage across port 2
• I₂: Current into port 2

Model Differences:

• Different models are used depending on which variables (voltage or current) are considered the independent variables.

Frequency Considerations:

• Current and voltage variables are most useful at low-to-moderate frequencies.
• At high frequencies (e.g., microwave frequencies), power and energy variables are more appropriate.
• The traditional current-voltage approach is replaced by scattering parameters at high frequencies.