Traveling Waves: Lossless Lines

The power lines seen along streets can be modeled as a single-phase, two-wire, lossless transmission line.

Consider a line section characterized by series inductance and shunt capacitance, with directionality from the sending to the receiving end.

Using Kirchhoff's laws, the equations for voltage and current are written and divided by Delta x. As Delta x approaches zero, equations involving partial derivatives are derived since both position and time are variables.

Laplace transforms are applied assuming zero initial conditions and simplifying the derivatives to only one variable.

Differentiating these yields linear, second-order homogeneous differential equations with respective solutions. The velocity is a function of the inductance and capacitance values.

Taking inverse Laplace transforms and applying a time shift results in functions representing voltage and current waves. These expressions represent the forward and backward traveling waves.

To evaluate the constants, the solutions are substituted in the second-order equation.

The coefficients of the exponential functions on both sides are equated, yielding the forward and backward currents in terms of the forward and backward voltages, respectively, and the characteristic impedance.