If Z(x) and Y(x) are not a function of x, the above wave equations reduce to wave equation (2.1.37a and b) for a uniform transmission line. For a lossless nonuniform line, the series impedance and shunt admittance per unit length are Z(x) = jωL(x), Y(x) = jωC(x). The voltage wave equation (2.3.9) could be written as
where position‐dependent nominal phase velocity of a nonuniform transmission line is given by
(2.3.12)
It is difficult to get a general solution for the above wave equations. However, under the case of no reflection on a line, and the line with a small fractional change in L(x) and C(x) over a wavelength, Lewis and Wells, and Wohler [B.17, J.11] have given the following solution of wave equation (2.3.11):
In this expression Z0(x) is the nominal characteristic impedance at any location x on the nonuniform transmission line, whereas characteristic impedance Z0(0) is the nominal characteristic impedance at x = 0. For a uniform line, the phase velocity vp(x) is constant and
Equation (2.3.13) shows that for increasing characteristic impedance Z0(x) along the line length, the voltage amplitude also increases as the square root of nominal characteristic impedance. Lewis and Wells [B.17] have also given an expression for the reflection coefficient of the nonuniform transmission line terminated in the load ZL at x = ℓ:
For a uniform transmission line Z0(x = ℓ) = Z0, and equation (2.3.14) is reduced to the nominal reflection coefficient,
At higher operating frequency ω, the reflection coefficient for any termination, given by equation (2.3.14), is also reduced to equation (2.3.15). However, reflection occurs at a lower frequency ω on a nonuniform transmission line, even if the nominal reflection coefficient Γnom(x = ℓ) zero, i.e. even if the line is matched at the load end. This behavior is different from that of a uniform transmission line.
2.3.2 Lossless Exponential Transmission Line
The general solution of the wave equation for a nonuniform transmission line is not available. However, the closed‐form solution is obtained for an exponential transmission line [J.11, J.13]. This case demonstrates the properties of a nonuniform line. The following exponential variation is assumed for the line inductance and capacitance of a nonuniform transmission line:
(2.3.16)
where L0 and C0 are primary line constants at x = 0 and p is a parameter controlling the propagation characteristics. The above choice of line inductance and capacitance maintains a constant phase velocity that is independent of the location along the line length. The characteristic impedance of a lossless exponential transmission line changes exponentially along the line length. Its propagation constant is also frequency‐dependent. Therefore, a lossless nonuniform line is dispersive. The nominal characteristic impedance at any location x on the line is
(2.3.17)
The parameter p, defined below, could be determined from the characteristic impedance at the input and output ends of the line:
(2.3.18)
If the impedances at both ends of a line are fixed, changing the line length, ℓ, can change the parameter p. The parameter p also determines the propagation characteristics of a nonuniform transmission line. The series impedance and shunt admittance p.u.l. of the exponential line can be written as follows:
(2.3.19)
In case of an exponential line, the voltage and current wave equations (2.3.9) and (2.3.10) reduce to
(2.3.20)
(2.3.21)
Let us assume the following exponential form of the solution for the above wave equations with separate propagation constants for the voltage and current waves:
(2.3.22)
The above differential equations provide the following characteristic equations:
(2.3.23)
On solving the above equations, the following expressions are obtained for the complex propagation constants:
(2.3.24)
In the case of a uniform transmission line (p = 0), the propagation constants for the voltage and current waves are identical. The parameter p determines the attenuation constant, i.e. α of a nonuniform line. It is positive for the condition Z0(x = ℓ) > Z0(x = 0). Thus, there is an attenuation factor even for a lossless nonuniform line.