M07_RAO3333_1_SE_CHO7.QXD 4/9/08 2:37 PM Page 232 CHAPTER Transmission-Line 7 Analysis In the previous chapter, we introduced the transmission line and the transmission-line equations. The transmission-line equations enable us to discuss the wave propagation phenomena along an arrangement of two parallel conductors having uniform cross section in terms of circuit quantities instead of in terms of field quantities. This chapter is devoted to the analysis of lossless transmission-line systems first in frequency domain, that is, for sinusoidal steady state, and then in time domain, that is, for arbi- trary variation with time. In the frequency domain, we shall study the standing wave phenomenon by con- sidering the short-circuited line. From the frequency dependence of the input imped- ance of the short-circuited line, we shall learn that the condition for the quasistatic approximation for the input behavior of physical structures is that the physical length of the structure must be a small fraction of the wavelength. We shall study reflection and transmission at the junction between two lines in cascade and introduce the Smith® Chart, a useful graphical aid in the solution of transmission-line problems. In the time domain, we shall begin with a line terminated by a resistive load and learn the bounce diagram technique of studying the transient bouncing of waves back and forth on the line for a constant voltage source as well as for a pulse voltage source. We shall apply the bounce diagram technique for an initially charged line. Finally, we shall introduce the load-line technique of analysis of a line terminated by a nonlinear element and apply it for the analysis of interconnections between logic gates. A. FREQUENCY DOMAIN In Chapter 6, we introduced transmission lines, and learned that the voltage and cur- rent on a line are governed by the transmission-line equations 0V 0I =-l (7.1a) 0z 0t Smith® Chart is a registered trademark of Analog Instrument Co., P.O. Box 950, New Providence, NJ 07974, USA. 232 M07_RAO3333_1_SE_CHO7.QXD 4/9/08 2:37 PM Page 233 A. Frequency Domain 233 0I 0V =-gV - c (7.1b) 0z 0t For the sinusoidally time-varying case, the corresponding differential equations for the – - phasor voltage V and phasor current Iare given by – 0V - =-jvl I (7.2a) 0z - 0I – – – =-gV - jvcV =-(g + jvc)V (7.2b) 0z - – Combining (7.2a) and (7.2b) by eliminating I , we obtain the wave equation for V as 2 – - 0 V 0I – =-jvl = jvl(g + jvc)V 0z2 0z – 2 – = g V (7.3) where – g = jvl(g + jvc) (7.4) 2 is the propagation constant associated with the wave propagation on the line. The solu- – tion for V is given by – – g- – g- V(z) = Ae- z + Be z (7.5) – – where A and B are arbitrary constants to be determined from the boundary condi- - tions. The corresponding solution for I is then given by – - 1 0V 1 – – - g-z – – g-z I (z) =- =- (-gAe + gBe ) jvl 0z jvl g jvc + – - g-z – g-z = (Ae - Be ) jvl A 1 – - g-z – g-z = – (Ae - Be ) (7.6) Z0 where – jvl Z0 = (7.7) g + jvc A is known as the characteristic impedance of the transmission line. The solutions for the line voltage and line current given by (7.5) and (7.6), respec- tively, represent the superposition of (+) and (-) waves, that is, waves propagating in the positive z-andnegativez-directions, respectively. They are completely analogous to the solutions for the electric and magnetic fields in the medium between the conductors of the line. In fact, the propagation constant given by (7.4) is the same as the M07_RAO3333_1_SE_CHO7.QXD 4/9/08 2:37 PM Page 234 234 Chapter 7 Transmission-Line Analysis propagation constant jvm(s + jvP), as it should be. The characteristic impedance of the line is analogous to1 (but not equal to) the intrinsic impedance of the material medi- um between the conductors of the line. For a lossless line,that is,for a line consisting of a perfect dielectric medium between the conductors,g = 0 , and – g = a + jb = jvl # jvc = jv lc (7.8) 2 2 Thus, the attenuation constant a is equal to zero, which is to be expected, and the phase – - constant b is equal to v lc. We can then write the solutions for V and I as 1 – – -jbz – jbz V(z) = Ae + Be (7.9a) - 1 – b – b I(z) = (Ae-j z - Bej z) (7.9b) Z0 where l Z (7.10) 0 = c A is purely real and independent of frequency. Note also that v 1 1 vp = = = (7.11) b lc mP as it should be, and independent of frequency.2 2 Thus, provided that l and c are independent of frequency, which is the case if m and P are independent of frequency and the transmission line is uniform, that is, its di- mensions remain constant transverse to the direction of propagation of the waves, the lossless line is characterized by no dispersion, a phenomenon discussed in Section 8.3. We shall be concerned with such lines only in this book. 7.1 SHORT-CIRCUITED LINE AND FREQUENCY BEHAVIOR Let us now consider a lossless line short-circuited at the far end z = 0, as shown in Figure 7.1(a), in which the double-ruled lines represent the conductors of the transmis- sion line. Note that the line is characterized by Z0 and b, which is equivalent to specify- ing l,c , and v. In actuality, the arrangement may consist, for example, of a perfectly conducting rectangular sheet joining the two conductors of a parallel-plate line as in Figure 7.1(b) or a perfectly conducting ring-shaped sheet joining the two conductors of a coaxial cable as in Figure 7.1(c). We shall assume that the line is driven by a voltage generator of frequency v at the left end z =-l so that waves are set up on the line. The short circuit at z = 0 requires that the tangential electric field on the surface of the conductor comprising the short circuit be zero. Since the voltage between the conduc- tors of the line is proportional to this electric field, which is transverse to them, it fol- lows that the voltage across the short circuit has to be zero. Thus, we have – V(0) = 0 (7.12) M07_RAO3333_1_SE_CHO7.QXD 4/9/08 2:37 PM Page 235 7.1 Short-Circuited Line and Frequency Behavior 235 I(z) ϩ (b) V(z) z ϭ Ϫl z z ϭ 0 (c) (a) FIGURE 7.1 Transmission line short-circuited at the far end. – Applying the boundary condition given by (7.12) to the general solution for V given by (7.9a), we have – – -jb(0) – jb(0) V(0) = Ae + Be = 0 or – – B =-A (7.13) Thus, we find that the short circuit gives rise to a (- ) or reflected wave whose voltage is exactly the negative of the (+) or incident wave voltage, at the short circuit. Substi- tuting this result in (7.9a) and (7.9b), we get the particular solutions for the complex voltage and current on the short-circuited line to be – – -jbz – jbz – V (z) = Ae - Ae =-2jA sin bz (7.14a) – - 1 – b – b 2A I(z) = (Ae-j z + Aej z) = cos bz (7.14b) Z0 Z0 The real voltage and current are then given by – jvt -jp 2 ju jvt V (z, t) = Re[V(z)e ] = Re(2e Ae sin bz e ) > = 2A sin bz sin (vt + u) (7.15a) - v 2 u v I (z, t) = Re[I(z)ej t] = Re Aej cos bz ej t Z0 c d 2A = cos bz cos (vt + u) (7.15b) Z0 M07_RAO3333_1_SE_CHO7.QXD 4/9/08 2:37 PM Page 236 236 Chapter 7 Transmission-Line Analysis – u p where we have replaced A by Aej and -j by e-j 2. The instantaneous power flow down the line is given by > P (z, t) = V(z, t)I(z, t) 4A2 = sin bz cos bz sin (vt + u) cos (vt + u) Z0 A2 = sin 2bz sin 2(vt + u) (7.15c) Z0 These results for the voltage, current, and power flow on the short-circuited line given by (7.15a), (7.15b), and (7.15c), respectively, are illustrated in Figure 7.2, which shows the variation of each of these quantities with distance from the short circuit for several values of time. The numbers 1, 2, 3, ...,9 beside the curves in Figure 7.2 represent the order of the curves corresponding to values of (vt + u) equal to 0, p 4, p 2, ...,2p .It can be seen that the phenomenon is one in which the voltage, current, and power flow oscillate sinusoidally with time with different amplitudes at different> locations> on the line, unlike in the case of traveling waves, in which a given point on the waveform progresses in distance with time. These waves are therefore known as stand- ing waves.In particular,they represent complete standing waves,in view of the zero am- plitudes of the voltage, current, and power flow at certain locations on the line, as shown by Figure 7.2. The line voltage amplitude is zero for values of z given by sin bz = 0 or bz =-mp,,...,orm = 1, 2, 3 z =-ml 2,,...,thatm = 1, 2, 3 is,at multiples of l 2 from the short circuit. The line current amplitude is zero for values of z given by cos bz = 0 or bz =-(2m + 1)p 2,m> = 0, 1, 2, 3 , .