Anand K. Verma - Introduction To Modern Planar Transmission Lines

(2.3.2)

(2.3.3)

On substituting equations (2.3.1b)and (2.3.3)in equation (2.3.2):

(2.3.4)

This equation has both the voltage and current variables v(x, t) and i(x, t). However, most of the transmission lines are low‐loss lines. Thus, using R(x) → 0 G(x) → 0 in equations (2.3.1a)and (2.3.4), the following voltage wave equation is obtained for a lossless nonuniform transmission line:

(2.3.5)

Likewise, the current wave equation is obtained as,

(2.3.6)

If L(x) and C(x) are not a function of x, then equations (2.3.5)and (2.3.6)reduce to the familiar wave equations (2.1.24)and (2.1.25)on a uniform transmission line.

For a lossy nonuniform transmission line, it is not possible to get separate voltage and current wave equations in the time domain. However, separate voltage and current wave equations can be obtained in the frequency domain by using the phasor form of voltage and current. The transmission line equations in the phasor form are

(2.3.7)

where the line series impedance and shunt admittance p.u.l . are given by

(2.3.8)

The following wave equations for the nonuniform transmission line are obtained:

(2.3.9)

(2.3.10)

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

(2.3.11)

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):

(2.3.13)

In this expression Z 0(x) is the nominal characteristic impedance at any location x on the nonuniform transmission line, whereas characteristic impedance Z 0(0) is the nominal characteristic impedance at x = 0. For a uniform line, the phase velocity v p(x) is constant and . The wave equation (2.3.11)is reduced to the wave equation of a uniform transmission line. The solution (2.3.13)is also reduced to the standard solution, .

Equation (2.3.13)shows that for increasing characteristic impedance Z 0(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 Z Lat x = ℓ:

(2.3.14)

For a uniform transmission line Z 0(x = ℓ) = Z 0, and equation (2.3.14)is reduced to the nominal reflection coefficient,

(2.3.15)

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 Γ nom(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.

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 L 0and C 0are 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)