Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 81

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. The factor under the radical sign provides the propagation constant, i.e. the phase‐shift constant β. At the cut‐off frequency, ω = ω_c, β is zero. The cut‐off frequency is given by

(2.3.25)

where phase velocity of the voltage and current waves on the line at x = 0 is

(2.3.26)

The complex propagation constants can be rewritten as follows:

(2.3.27)

The propagation constants β₁ and β₂ are imaginary quantities for the signal below the cut‐off frequency ω < ω_c. Under such conditions, no wave propagates on the nonuniform line. The initial signal only gets attenuated. It is called the evanescent mode. The wave propagation takes place only for ω > ω_c. Therefore, a nonuniform transmission line behaves like a high‐pass filter (HPF). However, real parts of the complex propagation constants γ₁ and γ₂ are nonzero. For p > 0, the voltage wave gets attenuated while the current wave is increased in the positive direction of wave propagation. In the backward direction, the reflected voltage and current waves have opposite behavior. The attenuation in the signal is not due to any ohmic loss of a line. It is due to the continuous reflection of the wave as it progresses on the line. The opposite behavior of the voltage and current waves maintains the constant flow of power (P) at any location on a line:

(2.3.28)

where . Unlike a lossless uniform transmission line, the phase velocity of the voltage and current waves on a lossless nonuniform transmission line is dispersive as given below:

(2.3.29)

The phase velocity shows singularity at the cut‐off frequency. After the cut‐off frequency, i.e. for ω > > ω_c, it decreases, with an increase in frequency, to a value given by expression (2.3.26).