User:Tanderson94/sandbox

Oftentimes, we are only interested in the terminal characteristics of the transmission line, which are the voltage and current at the sending and receiving ends. The transmission line itself is then modeled as a "black box" and a 2 by 2 transmission matrix is used to to model its behavior, as follows:

$$ \begin{bmatrix} V_S \\ I_S \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} V_R \\ I_R \end{bmatrix} $$

The line is assumed to be a reciprocal, symmetrical network, meaning that the receiving and sending labels can be switched with no consequence. The transmission matrix T also has the following properties:


 * det(T) = AD - BC = 1


 * A = D

The parameters A, B, C, and D differ depending on how the desired model handles the line's resistance (R), inductance (L), capacitance (C), and shunt (parallel) admittance Y. The four main models are the short line approximation, the medium line approximation, the long line approximation (with distributed parameters), and the lossless line. In all models described, a capital letter such as R refers to the total quantity summed over the line and a lowercase letter such as c refers to the per-unit-length quantity.

The lossless line approximation, which is the least accurate model, is often used on short lines when the inductance of the line is much greater than its resistance. For this approximation, the voltage and current are identical at the sending and receiving ends.

The short line approximation is normally used for lines less than 50 miles long. For a short line, only a series impedance Z is considered, while C and Y are ignored. The final result is that A = D = 1 per unit, B = Z ohms, and C = 0.

The medium line approximation is used for lines between 50 and 150 miles long. In this model, the series impedance and the shunt admittance are considered, with half of the shunt admittance being placed at each end of the line. This circuit is often referred to as a "nominal pi" circuit because of the shape that is taken on when admittance is placed on both sides of the circuit diagram. The analysis of the medium line brings one to the following result:



\begin{align} A &= D = 1 + \frac{XY}{2}\ \text{per unit} \\ B &= Z\Omega \\ C &= Y\left(1 + \frac{YZ}{4}\right)S \end{align} $$

The long line model is used when a higher degree of accuracy is needed or when the line under consideration is more than 150 miles long. Series resistance and shunt admittance are considered as distributed parameters, meaning each differential length of the line has a corresponding differential resistance and shunt admittance. The following result can be applied at any point along the transmission line, where gamma is defined as the propagation constant.



\begin{align} A &= D = \cosh(yx)\ \text{per unit} \\ B &= Z_c \sinh(yx)\Omega \\ C &= \frac{1}{Z_c} \sinh(yx)S \end{align} $$

To find the voltage and current at the end of the long line, x should be replaced with L (the line length) in all parameters of the transmission matrix.