Electric Circuits, 9th Edition P77 ppsx

736 More on Magnetically Coupled Coils and Ideal Transformers continuously, the use of a capacitor to simulate negative inductance is practically worthless. You can circumvent the problem of dealing with negative inductances by introducing an ideal transformer into the equivalent circuit. This doesn't completely solve the modeling problem, because ideal transformers can only be approximated. However, in some situations the approximation is good enough to warrant a discussion of using an ideal transformer in the T- and ^-equivalent circuits of magnetically coupled coils. An ideal transformer can be used in two different ways in either the T-equivalent or the -equivalent circuit. Figure C.12 shows the two arrange- ments for each type of equivalent circuit. Verifying any of the equivalent circuits in Fig. C.12 requires showing only that, for any circuit, the equations relating v x and v 2 to dijdt and di 2 /dt are identical to Eqs. C.l and C.2. Here, we validate the circuit shown in Fig. C.12(a); we leave it to you to verify the circuits in Figs. C.12(b), (c), and (d).To aid the discussion, we redrew the circuit shown in Fig. C.12(a) as Fig. C.13, adding the variables i {) and % From this circuit, v 1 = [L l M \ di } M d , a ) dt a dt (C.12) and v {) = hi M a di Q M d (C.13) a (a) L X L 2 - M : Ma ry~rv\— ••—n i:« rr UL X - M 2 ^L X L 2 -M 2 L-> - Ma 3 a 2 L\ - Ma (c) + th '\ + • »1 + »1 lia Ideal a L] - Ma L 2 - Ma Ma- th (b) a(L x L 2 - M 2 ) Ideal 1 :a Ideal M a 2 (L x L 2 -M 2 U a 2 (L x L 2 -M 2 ) U-Ma a 2 L x - Ma Vt (d) Figure C.12 • The four ways of using an ideal transformer in the T- and 7r-equivalent circuit for magnetically coupled coils. M ^2 _M L x - -Q a 2 a V\ M a h) r^i^^ir^t V(l V-7 Ideal (a) Figure C.13 A The circuit of Fig. C.12(a) with i 0 and v Q defined. C.2 The Need for Ideal Transformers in the Equivalent Circuits 737 The ideal transformer imposes constraints on v {) and / () : «o V2 a ' Substituting Eqs. C.14 and C.15 into Eqs. C.12 and C.13 gives dh . M d Vi = LTTT + —-J-U12) at a dt (C.14) (C.15) (C.16) and Vi = Lid_ a a 1 dt From Eqs. C.16 and C.17, , (ah) H —. 2 <** a dt (C.17) di\ di-} dt dt (C.18) and ^2 di\ dU M—- + L 2 -^ dt dt (C.19) Equations C.18 and C.19 are identical to Eqs. C.l and C.2; thus, insofar as terminal behavior is concerned, the circuit shown in Fig. C.13 is equivalent to the magnetically coupled coils shown inside the box in Fig. C.l. In showing that the circuit in Fig. C.13 is equivalent to the magneti- cally coupled coils in Fig. C.l, we placed no restrictions on the turns ratio a. Therefore, an infinite number of equivalent circuits are possible. Furthermore, we can always find a turns ratio to make all the inductances positive. Three values of a are of particular interest: and M_ hi L 2 (C.20) (C.21) (C22) The value of a given by Eq. C.20 eliminates the inductances L x — M/a and a 2 L x — aM from the T-equivalent circuits and the inductances (L X L 2 - M 2 )/(a 2 L { - aM) and a 2 (L { L 2 - M 2 )/{a 2 U - aM) from the 17-equivalent circuits. The value of a given by Eq. C.21 eliminates the inductances (L 2 /a 2 ) - (M/a) and L 2 — aM from the T-equivalent circuits and the inductances (L X L 2 - M 2 )J(L 2 - aM) and a 2 (LiL 2 — M 2 )j(L 2 — aM) from the 7r-equivalent circuits. Also note that when a = M/L h the circuits in Figs. C.l2(a) and (c) become identical, and when a = L 2 /M, the circuits in Figs. C.12(b) and (d) become identical. Figures C.14 and C.15 summarize these observations. _ynrvnr>_ "1 -U-i •I 1-O « Ideal (a) • 1:« Ideal (1 - k 2 )L 2 \k 2 L, v. (b) Figure C.14 • Two equivalent circuits when a = M/L h /.,(1 - k 2 ) i/c 2 L, • 1 : a • Ideal (a) HP-IJ *, 1 :a Ideal \U n (b) Figure C.15 • Two equivalent circuits when a = L 2 /M. • + NA \N 2 Figure C.16 A Experimental determination of the ratio MfL x . 738 More on Magnetically Coupled Coils and Ideal Transformers In deriving the expressions for the inductances there, we used the relationship M = kVL\L 2 . Expressing the inductances as functions of the self-inductances L\ and L 2 and the coefficient of coupling k allows the values of a given by Eqs. C.20 and C.21 not only to reduce the number of inductances needed in the equivalent circuit, but also to guarantee that all the inductances will be positive. We leave to you to investigate the conse- quences of choosing the value of a given by Eq. C.22. The values of a given by Eqs. C.20-C.22 can be determined experi- mentally. The ratio MjL x is obtained by driving the coil designated as hav- ing N\ turns by a sinusoidal voltage source. The source frequency is set high enough that coL\ 5$> R\, and the N 2 coil is left open. Figure C.16 shows this arrangement. With the N 2 coil open, V 2 = juiM\ { . (C.23) Now, as /a>L] » R h the current I\ is I. = 77 ( C - 24 ) ja)L [ Substituting Eq. C.24 into Eq. C23 yields (C.25) in which the ratio M/L l is the terminal voltage ratio corresponding to coil 2 being open; that is, I 2 = 0. We obtain the ratio L 2 /M by reversing the procedure; that is, coil 2 is energized and coil 1 is left open. Then Finally, we observe that the value of a given by Eq. C.22 is the geo- metric mean of these two voltage ratios; thus VxA-oVViA-o VL, M (C.27) For coils wound on nonmagnetic cores, the voltage ratio is not the same as the turns ratio, as it very nearly is for coils wound on ferromagnetic cores. Because the self-inductances vary as the square of the number of turns, Eq. C27 reveals that the turns ratio is approximately equal to the geometric mean of the two voltage ratios, or Appendix ~L ' D The Decibel Telephone engineers who were concerned with the power loss across the cascaded circuits used to transmit telephone signals introduced the deci- bel. Figure D.l defines the problem. There, p, is the power input to the system, p x is the power output of circuit A, p 2 is the power output of circuit B, and p (> is the power output of the system. The power gain of each circuit is the ratio of the power out to the power in. Thus Pi P\ B Pi -# •- C P<> Figure D.l • Three cascaded circuits. P\ P2 , Pa CTA = —, an = —, and err — —. Pi Pi Pi The overall power gain of the system is simply the product of the individ- ual gains, or Po Pi P^PlPo Pi Pi Pi = (T A <r B a c . The multiplication of power ratios is converted to addition by means of the logarithm; that is, log 10 — = logujo-A + log 1() o- B + log 1() cr c , Pi This log ratio of the powers was named the bel, in honor of Alexander Graham Bell. Thus we calculate the overall power gain, in bels, simply by summing the power gains, also in bels, of each segment of the transmission system. In practice, the bel is an inconveniently large quantity. One-tenth of a bel is a more useful measure of power gain; hence the decibel. The number of decibels equals 10 times the number of bels, so Po Number of decibels = 10 login — Pi When we use the decibel as a measure of power ratios, in some situa- tions the resistance seen looking into the circuit equals the resistance loading the circuit, as illustrated in Fig. D.2. When the input resistance equals the load resistance, we can convert the power ratio to either a voltage ratio or a current ratio: Po Pi v ou\ Via 'in '"in •* A 'out +- + { R, Figure D.2 • A circuit in which the input resistance equals the load resistance. or Po Pi if R 'in 739 740 The Decibel These equations show that the number of decibels becomes Number of decibels = 20 log] 0 'out = 20 log 'out 10 (D.l) TABLE D.l Some dB-Ratio Pairs dB 0 3 6 10 15 20 Ratio 1.00 1.41 2.00 3.16 5.62 10.00 dB 30 40 60 80 100 120 Ratio 31.62 100.00 10 3 10 4 10 5 10 6 The definition of the decibel used in Bode diagrams (see Appendix E) is borrowed from the results expressed by Eq. D.l, since these results apply to any transfer function involving a voltage ratio, a current ratio, a voltage-to-current ratio, or a current-to-voltage ratio. You should keep the original definition of the decibel firmly in mind because it is of fundamen- tal importance in many engineering applications. When you are working with transfer function amplitudes expressed in decibels, having a table that translates the decibel value to the actual value of the output/input ratio is helpful. Table D.l gives some useful pairs. The ratio corresponding to a negative decibel value is the reciprocal of the pos- itive ratio. For example, -3 dB corresponds to an output/input ratio of 1/1.41, or 0.707. Interestingly, —3 dB corresponds to the half-power fre- quencies of the filter circuits discussed in Chapters 14 and 15. The decibel is also used as a unit of power when it expresses the ratio of a known power to a reference power. Usually the reference power is 1 mW and the power unit is written dBm, which stands for "decibels rela- tive to one milliwatt." For example, a power of 20 mW corresponds to ±13 dBm. AC voltmeters commonly provide dBm readings that assume not only a 1 mW reference power but also a 600 ft reference resistance (a value commonly used in telephone systems). Since a power of 1 mW in 600 ft corresponds to 0.7746 V (mis), that voltage is read as 0 dBm on the meter. For analog meters, there usually is exactly a 10 dB difference between adjacent ranges. Although the scales may be marked 0.1, 0.3,1, 3,10, and so on, in fact 3.16 V on the 3 V scale lines up with 1 V on the 1 V scale. Some voltmeters provide a switch to choose a reference resistance (50, 135,600, or 900 ft) or to select dBm or dBV (decibels relative to one volt). Appendix Q Bode Diagrams As we have seen, the frequency response plot is a very important tool for analyzing a circuit's behavior. Up to this point, however, we have shown qualitative sketches of the frequency response without discussing how to create such diagrams. The most efficient method for generating and plot- ting the amplitude and phase data is to use a digital computer; we can rely on it to give us accurate numerical plots of \H(jm)\ and d{ja>) versus co. However, in some situations, preliminary sketches using Bode diagrams can help ensure the intelligent use of the computer. A Bode diagram, or plot, is a graphical technique that gives a feel for the frequency response of a circuit. These diagrams are named in recognition of the pioneering work done by H. W. Bode. 1 They are most useful for circuits in which the poles and zeros of H(s) are reasonably well separated. Like the qualitative frequency response plots seen thus far, a Bode diagram consists of two separate plots: One shows how the amplitude of H(jco) varies with frequency, and the other shows how the phase angle of H(j(o) varies with frequency. In Bode diagrams, the plots are made on semilog graph paper for greater accuracy in representing the wide range of frequency values. In both the amplitude and phase plots, the frequency is plotted on the horizontal log scale, and the amplitude and phase angle are plotted on the linear vertical scale. E,l Real, First-Order Poles and Zeros To simplify the development of Bode diagrams, we begin by considering only cases where all the poles and zeros of H(s) are real and first order. Later we will present cases with complex and repeated poles and zeros. For our purposes, having a specific expression for H(s) is helpful. Hence we base the discussion on K(s + zi) from which ui> \ ^O + *l) ]0)(j(0 + Pi) The first step in making Bode diagrams is to put the expression for H(jco) in a standard form, which we derive simply by dividing out the poles and zeros: Kzi(l +/w/zi) f s H(m) = ,. w< . , (E.3) p,(7a.)(l + /a>/A) 1 See II. W. Bode, Network Analysis and Feedback Design (New York: Van Nostrand, 1945). 742 Bode Diagrams Next we let K a represent the constant quantity Kz.[/p\, and at the same time we express H(jto) in polar form: M /90 |1 + J^/Pil /j3i K„|l + /w/zil Mil + WAI (E.4) From Eq. E.4, \H(ja>)\ = / ; ', (E.5) 0(«) = ?Ai - 90° - /3,. (E.6) By definition, the phase angles ip\ and /3j are 0, = tan ~ x wfz\\ (E.7) j3i = tan^w/Pi- ( E - 8 ) The Bode diagrams consist of plotting Eq. E.5 (amplitude) and Eq. E.6 (phase) as functions of o>. E.2 Straight-Line Amplitude Plots The amplitude plot involves the multiplication and division of factors associated with the poles and zeros of H(s). We reduce this multiplication and division to addition and subtraction by expressing the amplitude of H(j(o) in terms of a logarithmic value: the decibel (dB). 2 The amplitude of H(ja)) in decibels is /l tlB = 2()log 1() |//f>)|. (E.9) TABLE E.1 Actual Amplitudes and Their To § ive y° u a feel for the unit of decibels, Table E.l provides a translation Decibel Values between the actual value of several amplitudes and their values in deci- bels. Expressing Eq. E.5 in terms of decibels gives K 0 \l+jco/ Zl \ A dB = 20 log J() w|l + ja>/pi\ = 201og 1() /C + 20lQg t Jl + /»/*il - 20 log 10 <u - 20 log 10 |l -f- ja/pxl (E.10) See Appendix D for more information regarding the decibel. AiB 0 3 6 10 15 20 A 1.00 1.41 2.00 3.16 5.62 10.00 AiB 30 40 60 SO 100 120 A 31.62 100.00 10 3 10 4 10 5 10 6 The key to plotting Eq. E.10 is to plot each term in the equation sepa- rately and then combine the separate plots graphically. The individual fac- tors are easy to plot because they can be approximated in all cases by straight lines. The plot of 20 log 10 K (> is a horizontal straight line because K 0 is not a function of frequency. The value of this term is positive for K a > 1, zero for K 0 - 1, and negative for K 0 < 1. Two straight lines approximate the plot of 20 log 10 | 1 +• j(o/z\\. For small values of <w, the magnitude 11 + jafz\ | is approximately 1, and therefore 201og 1() |l + j(o/zi\^0 asw-^0. (E.ll) For large values of w, the magnitude |1 + jo)/z\\ is approximately o)/z\, and therefore 201og 1() |l + j(o/z.[\ ^20 \og u) ((o/Z]) asw—>oc. (E.12) On a log frequency scale, 20 \og m (a)/z\) is a straight line with a slope of 20 dB/decade (a decade is a 10-to-l change in frequency).This straight line intersects the 0 dB axis at w = z\. This value of o» is called the corner frequency.Thus, on the basis of Eqs. E.ll and E.12, two straight lines can approximate the amplitude plot of a first-order zero, as shown in Fig. E.l. The plot of — 201ogioa> is a straight line having a slope of -20 dB/decade that intersects the 0 dB axis at a» = l.Two straight lines approximate the plot of -20 log 10 |l 4- jco/p\\. Here the two straight lines 25 20 15 10 5 0 -5 z z ()I °8H.||T) \y 20 dB/dec - Uecade ade 1C Z) 1 2 3 4 5 6 7 8 910 o) (rad/s) 20 30 40 50 Figure E.l • A straight-line approximation of the amplitude plot of a first-order zero. intersect on the 0 dB axis at w = p\. For large values of w, the straight line 20 log 10 (o>/pi) has a slope of -20 dB/decade. Figure E.2 shows the straight-line approximation of the amplitude plot of a first-order pole. klB 5 0 -5 -10 -15 -20 p N Sj 201og 10 |j^j • -20 dB/decade ^\ iQPi 1 2 3 4 5 6 78910 o) (rad/s) 20 30 40 50 Figure E.2 • A straight-line approximation of the amplitude plot of a first-order pole. Figure E.3 shows a plot of Eq. E.10 for K 0 = VlO, Z\ = 0.1 rad/s, and pi = 5 rad/s. Each term in Eq. E.10 is labeled on Fig. E.3, so you can verify that the individual terms sum to create the resultant plot, labeled 201og 1() |//(/a>)|. Example E.l illustrates the construction of a straight-line amplitude plot for a transfer function characterized by first-order poles and zeros. A dB 50 40 30 20 10 0 -10 -20 \ \ \ \ S N yl \, > mi 0 log, 0 \ -1 N \ / / • \l )1 S > s III (/«01 Jgio^ • • <• • > s > / / \ s 20 s / C \ logio ^ V s \ s / *** * • K 0 V .'1111 -201o s * \ 01 I0\oi \ ogio \ ;iol^C ^ i+/| CO ) J \ 1 j, 0.05 0.1 0.5 1.0 5 10 50 100 500 a> (rad/s) Figure E.3 • A straight-line approximation of the amplitude plot for Eq. E.10. E.2 Straight-Line Amplitude Plots 745 Example E.l For the circuit in Fig. E.4: a) Compute the transfer function, H(s). b) Construct a straight-line approximation of the Bode amplitude plot. c) Calculate 20log 10 |.//(/cu)| at w = 50 rad/s and co = 1000 rad/s. d) Plot the values computed in (c) on the straight- line graph; and e) Suppose that v t {t) = 5cos (500* + 15°) V, and then use the Bode plot you constructed to pre- dict the amplitude of v a (t) in the steady state. 100 mH 10 .mF I 11 ill V, Figure E.4 • The circuit for Example E.l. c) We have //(/50) = o.ii(;50) (1 + /5)(1 + /0.5) Solution a) Transforming the circuit in Fig. E.4 into the s-domain and then using 5-domain voltage divi- sion gives = 0.9648/-15.25°, 20 log 10 |//(/50)| = 20 log 10 0.9648 H(s) = (R/L)s i • S* + (R/L)s + £ Substituting the numerical values from the cir- cuit, we get = -0.311 dB; //(/1000) = 0.11(/1000) (1 + /100)(1 + /10) H(s) = 110s 1105 s 2 + 110$ + 1000 (s + 10)(5 + 100) b) We begin by writing H(jto) in standard form: = 0.1094/-83.72°; 20 log 1() 0.1094 = -19.22 dB. H(ja>) = 0.11;a» [1 + /(a>/10)][l + /K100)]' The expression for the amplitude of H(J<&) in decibels is A dB = 201og 1() |//(/ w )| = 201og 10 0.11 + 201og 10 |H - 20 log 10 1+/ -20 log 10 Figure E.5 shows the straight-line plot. Each term contributing to the overall amplitude is identified. 40 30 20 10 0 Am -10 -20 -30 -40 -50 -60 1 5 10 50 100 5001000 co (rad/s) Figure E.5 • The straight-line amplitude plot for the transfer function of the circuit in Fig. E.4. •* ,*"' • ^20 lo T y y '" ^20'log ,.— SloO.ll imr -15 "'s& - | o \M \c Ml N —0.311 WrA NfFx . CO i 201og 10 |i +y- 1 | i i r i mi -1 2 .1 1 1 MINI 0 l0S,r 1 1 + / 1 II IIIII 10 log, k, ) *^i -^. __ v CO 10 1 »\H(jco ill -fn ;l^< T S * S N > i 1 N, T t 1"" 5) | -19.22)" V^ Yk 2s>. _.l.