Page 363 - Engineering Electromagnetics, 8th Edition

CHAPTER 10 Transmission Lines 345

D10.7. Standing wave measurements on a lossless 75- line show maxima
of 18 V and minima of 5 V. One minimum is located at a scale reading of 30 cm.
With the load replaced by a short circuit, two adjacent minima are found at scale
readings of 17 and 37 cm. Find: (a) s;(b) λ;(c) f ;(d) L ;(e) Z L .

Ans. 3.60; 0.400 m; 750 MHz; 0.704 −33.0; 77.9 + j104.7

D10.8. A normalized load, z L = 2− j1, is located at z = 0ona lossless 50-
line. Let the wavelength be 100 cm. (a)A short-circuited stub is to be located
at z =−d. What is the shortest suitable value for d?(b) What is the shortest
possible length of the stub? Find s:(c)on the main line for z < −d;(d)on the
main line for −d < z < 0; (e)on the stub.

Ans. 12.5 cm; 12.5 cm; 1.00; 2.62; ∞

10.14 TRANSIENT ANALYSIS
Throughout most of this chapter, we have considered the operation of transmission
linesundersteady-stateconditions,inwhichvoltageandcurrentweresinusoidalandat
a single frequency. In this section we move away from the simple time-harmonic case
and consider transmission line responses to voltage step functions and pulses, grouped
under the general heading of transients. These situations were briefly considered in
Section 10.2 with regard to switched voltages and currents. Line operation in transient
mode is important to study because it allows us to understand how lines can be
used to store and release energy (in pulse-forming applications, for example). Pulse
propagation is important in general since digital signals, composed of sequences of
pulses, are widely used.
We will confine our discussion to the propagation of transients in lines that are
lossless and have no dispersion, so that the basic behavior and analysis methods
may be learned. We must remember, however, that transient signals are necessarily
composed of numerous frequencies, as Fourier analysis will show. Consequently, the
question of dispersion in the line arises, since, as we have found, line propagation
constants and reflection coefficients at complex loads will be frequency-dependent.
So, in general, pulses are likely to broaden with propagation distance, and pulse
shapes may change when reflecting from a complex load. These issues will not be
considered in detail here, but they are readily addressed when the precise frequency
dependences of β and are known. In particular, β(ω) can be found by evaluating
the imaginary part of γ ,asgiven in Eq. (41), which would in general include the
frequency dependences of R, C, G, and L arising from various mechanisms. For
example, the skin effect (which affects both the conductor resistance and the internal
inductance) will result in frequency-dependent R and L. Once β(ω)is known, pulse
broadening can be evaluated using the methods to be presented in Chapter 12.
We begin our basic discussion of transients by considering a lossless transmission
line of length l terminated by a matched load, R L = Z 0 ,as shown in Figure 10.19a.