Transmission-Line Differential Equations

Transmission lines are essential components of electrical power systems. They are characterized by the distributed nature of resistance (R), inductance (L), and capacitance (C) per unit length. To analyze these lines, differential equations are employed to model the variations in voltage and current along the line.

Line Section Model

A circuit representing a line section of length Δx helps in understanding the transmission line parameters. The voltage V(x) and current i(x) are measured from the receiving end. The line has a series impedance z=R+jωL per unit length and a shunt admittance y=G+jωC per unit length. Applying Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL) to this setup,  derive relationships for voltage and current variations along the line.

As Δx approaches zero, these relationships transform into linear, first-order, homogeneous differential equations

These differential equations, in turn, describe how voltage and current change over the length of the transmission line.

Solving these differential equations involves determining the propagation constant γ, which encapsulates the effects of both series impedance and shunt admittance. The general solutions for voltage and current along the line include two integration constants evaluated using boundary conditions at the receiving end.

The characteristic impedance Z_c is a key parameter, reflecting the intrinsic impedance of the transmission line. Using the solutions of the differential equations, the voltage and current expressed along the line in terms of hyperbolic functions, cosh (cosine hyperbolic), and sinh (sine hyperbolic), mathematical functions used to model the voltage and current distribution.

ABCD Parameters

The ABCD parameters, or transmission line constants, are derived from these hyperbolic functions. These parameters provide a matrix representation that relates the voltage and current at any point along the line to the values at the receiving end. The ABCD parameters are crucial for understanding how signals propagate through the transmission line.

Propagation Constant

The propagation constant γ is a complex quantity with real and imaginary parts representing attenuation and phase shift, respectively. When multiplied by the line length, this constant becomes dimensionless and is essential for evaluating the hyperbolic functions that describe voltage and current distribution.

The precise ABCD parameters are valid for any line length and offer an exact solution for transmission line behavior. For practical hand calculations involving short- and medium-length lines, approximations may be used. These parameters comprehensively analyze transmission lines under various operating conditions, ensuring efficient and reliable power transmission.