lect13 - Peterson Transmission Lines for Electrical and...

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Unformatted text preview: Peterson: Transmission Lines for Electrical and Computer Engineers 2/99 Lecture 13 Frequency-domain Analysis of Transmission Lines Objectives: Introduce the treatment of sinusoidal signals on transmission lines, through the use of phasors. Define phase constant, frequency, and related frequency-domain parameters. Convert the transmission line equations into their phasor form, and present the general solution. Finally, derivexthe input impedance of a line with a general load. Previous lectures have focused on transmission lines excited with general time-dependent functions. In many practical applications involving transmission lines, the signals are sinusoids or modulated sinusoids. The study of a linear system response for sinusoidal excitation is known as frequency-domain analysis, as opposed to the general time-domain analysis considered previously. Applications involving modulated sinusoids include cable TV distribution lines, transmission lines feeding broadcast antennas for radio or television, and other elements of “wireless” communication systems. In addition, measurements are often made using sinusoidal signals, since the quantity being measured can be made more sensitive (and therefore measured more accurately) if the measurement set-up incorporates resonant circuits tuned to that particular frequency. Finally, losses are frequency- dependent, and are usually easier to characterize from a frequency-domain perspective. In our previous discussions, we considered waves having the general form f(t :t Z/VP). Beginning in this lecture, we confine our attention to sinusoidal waves such as V*(z,t) = V+ cos[u)(t — z/vp)] ' (13.1) Equation, (13.1) represents a voltage wave propagating in the +2 direction with velocity VP. The parameter (c in (13.1) is the radian frequency, typically in units of radians/second. The radian frequency is related to the frequency f, with units of Hertz, by u) = 23rf ‘ (13.2) Equation (13.1) can be rewritten as V+(Z,t) = V+ cos(wt — fiz) (13.3) where p = m/vp = m sqrt(LC) (13.4) is the phase constant of the transmission line at radian frequency m, with units of radians/meter. Equivalently, the phase constant is given by B = Zarf/vp = 23:0» (13.5) Peterson: Transmission Lines for Electrical and Computer Engineers 2/99 where 7» is the line wavelength at this frequency, typically in meters. It is apparent that the argument of (13.1) exhibits the functional form of a one-dimensional wave, and therefore the previous aspects of transmission line theory can be applied. However, there are additional simplifications to the theory that can be realized when all the signals in a problem are sinusoidal at the same frequency. In a linear electrical system, all the signals arising from a sinusoidal excitation are also sinusoidal with the same frequency as the excitation, but generally having different amplitudes and phases. Under these circumstances, it is convenient to use a shorthand notation that deals directly with the amplitudes and phases of the signals, and suppresses the sinusoidal time dependence. This notation is called phasor notation. [We mention here that phasors are directly related to Fourier transform quantities; since we do not assume the reader has a background in Fourier analysis, we do not pursue this connection] A phasor is a complex—valued quantity, whose magnitude and phase represent the amplitude and phase angle of the associated sinusoid. Phasors representing waves on transmission lines are also a function of position on the line (2). The manipulation of phasors requires a familiarity with complex exponential functions. Most of our readers should already be familiar with the Taylor series for e", given by the expansion e":1+x+x2/2!+...+x“/n!+... (13.6) If we consider the exponential function of an imaginary argument, say jB , we obtain e‘°=1+j6—62/2!—j63/3!+ 94/4! + (13.7) By separating the real and imaginary parts, the series can be rewritten as L eie={l—62/2!+64/4!+...}+j{0 —63/3!+05/5!+ ...} (13.8) However, these expansions are readily recognized as the Taylor series for cosine and sine, respectively. Therefore, we obtain Euler’s identity. ei" = cos(6) +j sin(6) (13.9) Euler’s identity suggests a graphical interpretation (Figure 13.1). Given an angle 6, the function ei8 is a point on the unit circle in the complex plane corresponding to the angle 6 measured from the positive real axis. The real part of 6‘9 is the projection onto the real axis (cosB), while the imaginary part of ei9 is the projection onto the imaginary axis (sinB). The magnitude of the complex exponential function ei9 is always one. Thus, one way of representing a general sinusoid with amplitude A and phase angle 9 is to express it as FLXM \3.‘ Peterson: Transmission Lines for Electrical and Computer Engineers 2/99 A cos 6 = Re{A cw} (13.10) and work with the complex function {A e59}. The symbol “ e” means to take the real part of the complex quantity within the brackets, or equivalently to project it onto the real axis in the complex plane. The phasor can be denoted using an arrow in the complex plane pointing from the origin to the point {A 6°}. This arrow indicated the magnitude (A) and the phase angle (6) of the sinusoid. [Insert Figure 13.1 here] A voltage waveform such as (13.1) that is a function of position can be expressed as a phasor V(z), where the phasor is defined so that v+ cos(wt— [32) = Re{V(z) em} (13.11) For this specific signal, the phasor is given by V(z) = V+ €531 (13.12) Using Euler’s identity, we observe that (since the coefficient V+ is presumed to be real- valued in this situation) Re{V(z) eimt} = Re{V” ei("’“flz’} = V+ cos((nt — 52) (13.13) Note that the phasor does not include the factor e3“. ' By convention, all the phasor quantities we work with will be assumed to have the identical frequency and therefore will exhibit the same eimt time dependence, which will be suppressed. For visualization purposes, we can conceptually think of the phasor in Figure 13.1 rotating around the origin with radian frequency w as time progresses. To summarize, the real-valued sinusoidal function v(z,t) = V+ cos(wt — 62) has been replaced by a complex—valued phasor V(z) = V+ e’j‘”. We have traded the time dependence of v(z,t) for the complex-valued nature of V(z). To minimize confusion, we Will usually denote functions of time with lower—case letters, and phasors with upper—case letters. As an example of constructing phasors, consider the time function u(z,t) = A sin(u)t — 62 + 113/6) (13.14) To convert this to a phasor, we rewrite the function as a cosine, to obtain u(z,t) = A cos((nt — $2 + n/6 — at/Z) (13.15) Peterson: Transmission Lines for Electrical and Computer Engineers 2/99 u(z,t) = Re{A em" ““"’3)} (13.16) and identify the phasor as U(z) = A 6“" e"""3 (13.17) As an example illustrating the construction of the time function from a phasor, consider the phasor given by W(z) = —j B 6““ (13.18) Where B is real-valued. This function is equivalent to W(z) = B either?"2 ' (13.19) Therefore, the time function is w(z,t) = Re{B elm" 5" m’} ( 13.20) or w(z,t) = B cos(u)t + Bz - 3/2) = B sin(u)t + 52) (13.21) This function represents a wave propagating in the —z direction. We now focus more specifically on transmission line problems. For a general line, the transmission line equations derived in Lecture 1 are 3v- ~ ._ a' 53‘ R’- L37: (13.22) 34’. E Manew- c.“ (13.23) To recast these equations into phasor form, we replace the time functions v(z,t) and i(z,t) with phasors V(z) and 1(2), and replace all time derivatives by multiplications with ju). The phasor equations take the form 1! a —(m3u.\ 12(2) A: (13.24) £5 . -(& firm) We) (13.25) A: Peterson: Transmission Lines for Electrical and Computer Engineers 2/99 For the moment, we consider only the lossless case, and set R and G equal to zero. The, lossless equations are $1 _._ -5uL 32(2) as (13.26) AI . It is a relatively straightforward process to show that these equations have the general solution V(z) = v; e'sz + V; cm“ (13.28) I(z) = (V0+ 0) e‘jfiz — (VO‘IZO) 6““31 (13.29) where the phase constant p = w sqrt(LC) and the characteristic impedance Z0 = sqrt(UC) have been defined previously. We introduce two coefficients, V0+ and Vo', which represent the complex-valued coefficients associated with the waves traveling in the +2 and —z directions. [The reader should substitute V(z) and 1(2) back into equations (13.26) and (13.27) to confirm that they are the solution!] [Insert Figure 13.2 here] Consider a transmission line driven from the left by a sinusoidal generator and loaded at the right end with an impedance ZL (Figure 13.2). The generator end of the line is located at z=0 while the load impedance is located at z=L. The line has characteristic impedance Z0 and phase constant 3. It is assumed that the generator has been turned on for a long time, so that the initial transients have died out and the remaining signals are entirely sinusoidal at a single frequency. The voltage and current waves are given by equations (13.28) and (13.29). However, at the load VL = V(z=L) and IL = I(z=L) are related by IL = VL / ZL (13.30) Imposing this constraint yields V(L) = VL = v; e-j'3L + V; eff“L (13.31) 1(2) = vL / zL = (v0+ 0) e'i"L — (V0720) eat? (13.32) These equations can be solved to find V0+ and V0" in terms of VL. We determine that +__ ‘ §flL - % V0 — as (H’f‘: v._ (13.33) Peterson: Transmission Lines for Electriail and Computer Engineers 2/99 Vo-= 13" aft—3h __ (13.34) After substituting these coefficients back into (13.28) and (13.29), we construct the impedance at a location z as , ' (H) _~ (1,9 We) Y§[(le%)edl3 +(l»§—‘i)em I 3““ “‘- = » (13.35) I“) )3: £2 ' we) . -' (L-e) a2. [("eJedF "(“§;)€,3F ] which can be rewritten using Euler’s identity as EL cos/ewe.) + 4%.: sin/5&1) } 20» = Z. 2° MPG,” +3, 2‘- sync-e) (13.36) and finally simplified to EL + ,3 z. to... fan—2) zca= 2 2° +3 a Jew “Ti—(L's) (13.37) If we specialize this result to the input end of the line, we obtain the input impedance in the form V“ Z VZL+5 “Ea '6pr } Zia: E— -— 0 204‘3 JEL- tam/5L (13.38) Equation (13.38) is a useful result that we will frequently refer back to. Among other things, it specifies the form of an equivalent circuit for the line, as depicted in Figure 13.3. [Insert Figure/l3 .3 here] MwQPMAMSJRMWM... Mam“.-- . , .WUf ( %,A+.\W .=_W A (305(40’1001-12 tWWe \WW um... 2. .. when... ed....£...m.c a Fad A MA "96%). So‘ujlalq I tWW (wi-\ 12-) \r(2-t\= Ke{ «£39,130 __}= 2.:{43 6.0 an; ,4 3 ¢._(.§_—nt) _;..;. (4.36; we A , f f ' “:54 gm-..) A V ...
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