velocity is also independent of frequency. A lossy line is dispersive. However, it also becomes dispersionless under the Heaviside's condition – (2.1.48). A transmission line, such as a microstrip in the inhomogeneous medium, can have dispersion even without losses.
2.1.6 Wave Equation with Source
In the above discussion, the development of the voltage and current wave equations has ignored the voltage or current source. However, a voltage or current source is always required to launch the voltage and current waves on a line. Therefore, it is appropriate to develop the transmission line equation with a source [B.8]. The consideration of a voltage/current source is important to solve the electromagnetic field problems of the layered medium planar lines, discussed in chapters 14 and 16.
Shunt Current Source
Figure (2.7) shows the lumped element model of a transmission line section of length Δx with a shunt current source
The loop and node equations are written below to develop the Kelvin–Heaviside transmission line equations with a current source:
Figure 2.7 Equivalent lumped circuit of a transmission line with a shunt current source.
The Loop Equation
The Node Equation
For Δx → 0, the above equations are reduced to
(2.1.52)
The above equations are rewritten below in term of the characteristic impedance (Z0) and propagation constant (γ) of a transmission line:
(2.1.53)
On solving the above equations for the voltage, the following inhomogeneous voltage wave equation, with a current source, is obtained:
Away from the location of the current source, i.e. for x ≠ x0, equation (2.1.54) reduces to the homogeneous equation (2.1.37a). The wave equation for the current wave, with a shunt current source, could also be rewritten.
2.1.7 Solution of Voltage and Current‐Wave Equation
The voltage and current wave equations in the phasor form are given in equation (2.1.37). The solution of a wave equation is written either in terms of the hyperbolic functions or in terms of the exponential functions. The first form is suitable for a line terminated in an arbitrary load. A section of the line transforms the load impedance into the input impedance at any location on the line. The impedance transformation takes place due to the standing wave formation. The hyperbolic form of the solution also provides the voltage and current distributions along the line. The exponential form of the solution demonstrates the traveling waves on a line, both in the forward and in the backward directions. A combination of the forward‐moving and the backward‐moving waves produces the standing wave on a transmission line.
The Hyperbolic Form of a Solution
Figure (2.8a) shows a section of the transmission line having a length ℓ. It is fed by a voltage source,
At any section on the line, its characteristic impedance Z0 relates the line voltage
