Transmission-Line Differential Equations

Consider a circuit representing a line section.

The voltage and current are measured from the receiving end, with defined series impedance and shunt admittance per unit length.

Using Kirchhoff's laws and taking limits delta x approaches zero, two linear differential equations are derived. Differentiating the voltage and substituting it into the current equation yields a second-order homogeneous equation.

The solution involves two integration constants and a propagation constant. Substituting this into the voltage equation and rearranging the denominator yields the current expression with the characteristic impedance.

After evaluating the integration constants from the boundary conditions and identifying hyperbolic functions, the ABCD parameters of the distributed line are obtained. These provide the current and voltage at any point relative to the receiving-end values.

ABCD parameters are also evaluated at the sending end. The propagation constant, a complex quantity, is dimensionless when multiplied by length and aids in determining hyperbolic functions.

These parameters apply to any line length. Approximation can be used for hand calculations involving short- and medium-length lines.