What are T Parameters?
T parameters are defined as transmission line parameters or ABCD parameters. In a two-port network, port-1 is considered as sending end and port-2 is considered as receiving end. In the network diagram below, port-1 terminals represent the input (sending) port. Similarly, port-2 terminals represent the output (receiving) port.
For the above two-port network, equations of T-parameters are;
These parameters are used to make mathematical modeling of a transmission line. Parameter A and D are unitless. The unit of parameter B and C is ohm and mho, respectively.
To find the value of T-parameters, we need to open and short circuit the receiving end. When the receiving end is open-circuited, receiving end current IR is zero. Put this value in the equations and we get the value of A and C parameters.
When the receiving end is short-circuited, the voltage across the receiving terminals VR is zero. By putting this value in the equation, we can get the values of B and D parameters.
T Parameters Solved Example Problem
Consider an impedance is connected between the sending end and receiving end terminals as shown in the below figure. Find T-parameters of given network.
Here, the sending end current is the same as the receiving end current.
Now, we apply KVL to the network,
Compare equation-1 and 4;
Compare equation-2 and 3;
T Parameters of a Transmission Line
According to the length of line, transmission lines are classified as;
- Short transmission line
- Medium transmission line
- Long transmission line
Now, we find T-parameters for all types of transmission lines.
Short Transmission Line
The transmission line having a length of less than 80km and voltage level less than 20kV is considered a short transmission line. Due to the small length and lower voltage level, the capacitance of the line is neglected.
IR = Receiving end current
VR = Receiving end voltage
Z = Load Impedance
IS = Sending end current
VS = Sending end voltage
R = Line resistance
L = Line inductance
When current flows through the transmission line, IR drop occurs at line resistance and IXL drop occurs at inductive reactance.
From the above network, the sending end current is the same as receiving end current.
Now, compare these equations with the equations of the T-parameters (equation 1 & 2). And we get values of A, B, C, and D parameters for a short transmission line.
Medium Transmission Line
The transmission line having a length of 80km to 240km and voltage level is 20kV to 100kV is considered as a medium transmission line.
In the case of a medium transmission line, we cannot neglect the capacitance. We must consider the capacitance while modeling a medium transmission line.
According to the placement of capacitance, the medium transmission lines are classified into three methods;
- End Condenser Method
- Nominal T method
- Nominal π method
End Condenser Method
In this method, the capacitance of the line is assumed to be lumped at end of a transmission line. The graphical representation of End condenser method is shown below figure.
IC = Capacitor current = YVR
From the above figure,
By KVL, we can write;
Now, compare equations-5 and 6 with equations of T parameters;
Nominal T Method
In this method, the capacitance of the line is placed at the mid-point of the transmission line. The graphical representation of the Nominal T method is as shown below figure.
IC = Capacitor current = YVC
VC = Capacitor voltage
Now, compare equations-7 and 8 with equations of T parameter and we get,
Nominal π Method
In this method, the capacitance of the transmission line is divided into halves. One half is placed at sending end and the second half is placed at receiving end. Graphical representation of nominal π method is as shown below figure.
From the above figure, we can write;
Put the value of VS in this equation,
By comparing equations-9 and 10 with equations of T parameters, we get;
Long Transmission Line
The long transmission line is modeled as a distributed network. It cannot be assumed as a lumped network. The distributed model of a long transmission line is as shown below figure.
The length of a line is X km. To analyze the transmission line, we consider a small part (dx) of the line. And it is as shown below figure.
Zdx = series impedance
Ydx = shunt impedance
The voltage increases over the length increases. So, the rise of voltage is;
Similarly, current drawn by element is;
Differentiating above equations;
The general solution of above equation is;
Now, differentiate this equation with respect to X,
Now, we need to find constants K1 and K2;
For that assume;
Putting these values in above equations;
ZC = Characteristic Impedance
ɣ = Propagation Constant
Compare these equations with the equations of T-parameters;
Conversion of T parameters to other Parameters
We can find other parameters from the equations of T parameters. For that, we need to find a set of equations of other parameters in terms of T parameters.
Consider the generalized two-port network as shown below figure.
In this figure, the direction of receiving end current is changed. Therefore, we consider few changes in equations of T parameters.
Equations of T parameters is;
T parameter to Z parameters
The following set of equations represents Z parameters.
Now, we will find the equations of Z parameters in terms of T parameters.
Now compare equation-14 with equation-15
Compare equation-13 with equation-16;
T parameter to Y parameters
The set of equations of Y parameters is;
Put this value in equation-11;
Compare this equation with equation-17;
Compare this equation with equation-18;
T parameter to H parameters
The set of equations of H parameters is;
Compare this equation with equation-22;