Transmission-Line Differential Equations

Applying Kirchhoff's Voltage Law and Kirchhoff's Current Law to a line section of length Δx yields relationships for voltage and current variations. As Δx approaches zero, these relationships transform into linear, first-order homogeneous differential equations that describe how voltage and current change along the transmission line's length.

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

Characteristic impedance Zc reflects the intrinsic impedance of the transmission line and emerges from solving the differential equations. It appears in the current expression derived from the voltage equation and is fundamental to understanding how the line responds to applied signals and boundary conditions.

ABCD parameters, derived from hyperbolic functions cosh and sinh, provide a matrix representation relating voltage and current at any point along the line to receiving-end values. These transmission line constants are valid for any line length and offer exact solutions for transmission line behavior under various operating conditions.

Transmission lines are characterized by distributed resistance (R), inductance (L), capacitance (C), and conductance (G) per unit length. These combine into series impedance z=R+jωL and shunt admittance y=G+jωC, which determine how voltage and current propagate through the line and affect maximum power flow and line loadability.

Precise ABCD parameters derived from hyperbolic functions are valid for any line length but involve complex calculations. For practical hand calculations on short- and medium-length lines, approximations simplify analysis while maintaining reasonable accuracy for design and operation of power systems.

Solving the second-order homogeneous differential equation yields two integration constants. These constants are evaluated using boundary conditions at the receiving end, where voltage and current are measured. This process establishes the complete voltage and current expressions along the entire transmission line.