Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 80

Figure (2.13) shows a nonuniform transmission line. The line parameters (primary line constants) R(x), L(x), C(x), G(x) are distance‐dependent. It results in the distance‐dependent characteristic impedance, Z₀(x), and propagation constant, γ(x). Using equation (2.1.20), the voltage and current equations for a nonuniform transmission line are written as follows:

Figure 2.13 Nonuniform transmission line.

(2.3.1)

The following expressions are obtained on differentiating equation (2.3.1a) with x and equation (2.3.1b) with t:

(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 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 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 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 Z_L at x = ℓ:

(2.3.14)

For a uniform transmission line Z₀(x = ℓ) = Z₀, 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.