506 The Laplace Transform in Circuit Analysis Summary • We can represent each of the circuit elements as an 6-domain equivalent circuit by Laplace-transforming the voltage-current equation for each element: • Resistor: V = RI • Inductor: V - sLl - LI () • Capacitor: V = (l/sC)I + VJs In these equations, V = Z£{v}, I = ?£{i}, / 0 is the ini- tial current through the inductor, and V 0 is the initial voltage across the capacitor. (See pages 468-469.) • We can perform circuit analysis in the s domain by replacing each circuit element with its s-domain equiva- lent circuit. The resulting equivalent circuit is solved by writing algebraic equations using the circuit analysis techniques from resistive circuits. Table 13.1 summa- rizes the equivalent circuits for resistors, inductors, and capacitors in the s domain. (See page 470.) • Circuit analysis in the s domain is particularly advanta- geous for solving transient response problems in linear lumped parameter circuits when initial conditions are known. It is also useful for problems involving multiple simultaneous mesh-current or node-voltage equations, because it reduces problems to algebraic rather than differential equations. (See pages 476-478.) • The transfer function is the s-domain ratio of a circuit's output to its input. It is represented as where Y(s) is the Laplace transform of the output sig- nal, and X(s) is the Laplace transform of the input sig- nal. (See page 484.) • The partial fraction expansion of the product H(s)X(s) yields a term for each pole of H(s) and X(s). The H(s) terms correspond to the transient component of the total response; the X(s) terms correspond to the steady-state component. (See page 486.) • If a circuit is driven by a unit impulse, x(t) = 8(t), then the response of the circuit equals the inverse Laplace transform of the transfer function, y(t) = !£~ l {H(s)} = h(t). (See pages 488-489.) • A time-invariant circuit is one for which, if the input is delayed by a seconds, the response function is also delayed by a seconds. (See page 488.) • The output of a circuit, y(t), can be computed by con- volving the input, x(t), with the impulse response of the circuit, h(t): y{t) = h{t) * x{t) = / h{k)x{t - \)dk Jo = x{t) * h{t) = j x(\)h(t - A)d\. JO A graphical interpretation of the convolution integral often provides an easier computational method to gen- erate y(t). (See page 489.) • We can use the transfer function of a circuit to compute its steady-state response to a sinusoidal source. To do so, make the substitution s = j<o in H(s) and represent the resulting complex number as a magnitude and phase angle. If x(t) = A cos((ot + ¢), Hijco) = \H(jco)\e m "K then yjjt) = A\H(ja>)\ cos[e*t + ¢ + $(<*>)), (See page 496.) • Laplace transform analysis correctly predicts impulsive currents and voltages arising from switching and impul- sive sources. You must ensure that the ^-domain equiva- lent circuits are based on initial conditions at t = (T, that is, prior to the switching. (See page 498.) Problems 507 Problems Section 13.1 13.1 Derive the .s-domain equivalent circuit shown in Fig. 13.4 by expressing the inductor current i as a function of the terminal voltage v and then find- ing the Laplace transform of this time-domain integral equation. 13.2 Find the Thevenin equivalent of the circuit shown in Fig. 13.7. 13.3 Find the Norton equivalent of the circuit shown in Fig. 13.3. Section 13.2 13.4 A 2 kil resistor, a 312.5 mH inductor, and a 12.5 nF capacitor are in parallel. a) Express the s-domain impedance of this parallel combination as a rational function. b) Give the numerical values of the poles and zeros of the impedance. 13.5 A 1 kd resistor is in series with a 625 nF capaci- tor. This series combination is in parallel with a 100 mH inductor. a) Express the equivalent s-domain impedance of these parallel branches as a rational function. b) Determine the numerical values of the poles and zeros. 13.6 A 8 kO resistor, a 1 H inductor, and a 40 nF capaci- tor are in series. a) Express the s-domain impedance of this series combination as a rational function. b) Give the numerical value of the poles and zeros of the impedance. 13.7 Find the poles and zeros of the impedance seen looking into the terminals a,b of the circuit shown in Fig. PI3.7. Figure PI3.7 13.8 Find the poles and zeros of the impedance seen looking into the terminals a,b of the circuit shown in Fig. P13.8. Figure P13.8 h. 1F^ < IF;: IH 5 •in iia Section 13.3 13.9 The switch in the circuit shown in Fig. PI 3.9 has PSPICE been in position x for a long time. At t = 0, the * switch moves instantaneously to position y. a) Construct an .v-domain circuit for t > 0. b) Find V ot c) Find v 0 . Figure P13.9 13.10 The switch in the circuit in Fig. P13.10 has been in position a for a long time. At t - 0, it moves instan- taneously from a to b. a) Construct the s-domain circuit for t > 0. b) Find V a (s). c) Find v„(t) for t > 0. Figure P13.10 12511 50 V 137.5 V 10 mH 13.11 The switch in the circuit in Fig. P13.ll has been PSPICE closed for a long time before opening at t ~ 0, MULTISIM a) Construct the .y-domain equivalent circuit for t > 0. b) Find V a , 508 The Laplace Transform in Circuit Analysis c) Find v () for t > 0. Figure P13.ll 10 mF 8(2 2H PSPKE MULTISIM 13.12 The switch in the circuit in Fig. P13.12 has been in position a for a long time. At if = 0, the switch moves instantaneously to position b. a) Construct the ^-domain circuit for t > 0. b) Find V a . c) Find / L . d) Find v () for t > 0. e) Find i L for t > 0. Figure P13.12 6.25 JX¥ / = 0 20 ft son + 6.4 rnH v„ 13.13 The switch in the circuit in Fig. P13.13 has been PSPICE closed for a long time. At t = 0, the switch MULTISIM • j is opened. a) Find v a for t ^ 0. b) Find i a for t > 0. Figure P13.13 t = 0 PYn 125 nF 4kfl -AW- 20 V i 0.5 H- 13.14 The make-before-break switch in the circuit in PSPICE Fig. PI3.14 has been in position a for a long time. At MULTisiM ? = o, it moves instantaneously to position b. Find i 0 for t > 0. Figure P13.14 (3) 500 V 10 mF 13.15 Find V 0 and v a in the circuit shown in Fig. P13.15 if PSPICE the initial energy is zero and the switch is closed at MULTISIM f = 0 Figure P13.15 2.8 kll 200 mH t = 0 125 nF D„ 13.16 Repeat Problem 13.15 if the initial voltage on the PSPICE capacitor is 30 V positive at the upper terminal. MULTISIM 13.17 The switch in the circuit seen in Fig. PI3.17 has been PSPICE i n position a for a long time before moving instanta- MULTISIM neously to position b at t = 0. a) Construct the s-domain equivalent circuit for t > 0. b) Find V\ and v^. c) Find V 2 and v 2 . Figure P13.17 450 V 1.25 mH 16/UJF: 24^F 13.18 The switch in the circuit seen in Fig. P13.18 has been PSPICE i n position a for a long time. At t = 0, it moves MULTISIM instantaneously to position b. a) FindV fr b) Finder Problems 509 Figure P13.18 20 a 13.19 The switch in the circuit in Fig. P13.19 has been PSPICE closed for a long time before opening at t = 0. Find MULTISIM c + -^ n v n for t > 0. Figure P13.19 13.20 Find v () in the circuit shown in Fig. PI3.20 if PSP1CE L = 5u(t) mA. There is no energy stored in the cir- MULTISIM h ., n cuit at t = 0. Figure P13.20 13.21 There is no energy stored in the circuit in Fig. PI3.21 PSPICE at the time the switch is closed. MULTISIM a) Find v 0 for JsO. b) Does your solution make sense in terms of known circuit behavior? Explain. Figure P13.21 2H t = 0 + "A + 1H _ 4mF 35 V 0AvM)v o &V 13.22 There is no energy stored in the circuit in Fig. PI3.22 PSPICE MULTISIM at t = 0~. a) Use the mesh current method to find i a . b) Find the time domain expression for v 0 . c) Do your answers in (a) and (b) make sense in terms of known circuit behavior? Explain. Figure P13.22 10//(0 V in »1 H 1 F 13.23 a) Find the s-domain expression for V 0 in the circuit _™_ in Fig. PI3.23. MULTISIM b) Use the ^-domain expression derived in (a) to predict the initial- and final-values of v a . c) Find the time-domain expression for v () . Figure P13.23 |15w(0mA jlH 7n ^vw- + ;O.IF v„ 13.24 Find the time-domain expression for the current in PSPICE the inductor in Fig. P13.23. Assume the reference MULTISIM ¢^¢^0]} f or / £ j s down. 13.25 There is no energy stored in the capacitors in the PSPICE circuit in Fig. P13.25 at the time the switch is closed. MULTIsiM , , , n a) Construct the s-domain circuit tor t > 0. b) Find I h V h and V 2 . c) Find z'i, Vi, and i> 2 . d) Do your answers for i h V\, and v 2 make sense in terms of known circuit behavior? Explain. Figure PI3.25 50 kn 300 nF 20 V 100 nF 13.26 There is no energy stored in the circuit in Fig. PI3.26 PSPICE at the time the voltage source is turned on, and MULTISIM ^ = 75u{() y a) Find V 0 and I 0 . b) Find v 0 and i a . c) Do the solutions for v a and i a make sense in terms of known circuit behavior? Explain. 510 The Laplace Transform in Circuit Analysis Figure P13.26 4mF if ion 10 n :20 II 13.27 There is no energy stored in the circuit in Fig. PI 3.27 PSPI « at the time the current source is energized. MULTISIM a) Find / a and I b . b) Find / a and / b . c) Find V &i V bi and V c . d) Find t* a , v h , and v c . e) Assume a capacitor will break down whenever its terminal voltage is 1000 V. How long after the current source turns on will one of the capacitors break down? Figure P13.27 lOOmF + v« - ion 9«(0A(t 100 mF Z 100 mF ion PSPICE MULTISIM 13.28 There is no energy stored in the circuit in Fig. PI3.28 at t = 0". a) Find V a . b) Find v a . c) Does your solution for v 0 make sense in terms of known circuit behavior? Explain. Figure P13.28 30 n wv 40 mF 4H 50u(l)V| ion v„(J^$u(t)A 13.29 There is no energy stored in the circuit in Fig. PI 3.29 PSPICE at the time the sources are energized. MULTISIM a) Find I^s) and Ijis). b) Use the initial- and final-value theorems to check the initial- and final-values of i\{t) and /'2(f). c) Find i { (t) and i 2 (t) for t > 0. Figure P13.29 ion *-l\ 2.5 H 6«(/)A( f 200 mF :5n 75u(t) V 13,30 There is no energy stored in the circuit in Fig. P13.30 PSPICE at the time the current source turns on. Given that MULTISIM ig = 5QHW A . a) Find V„(s). b) Use the initial- and final-value theorems to find v o (0 + ) and y f) (oo). c) Determine if the results obtained in (b) agree with known circuit behavior. d) Find v 0 (t). Figure P13.30 13.31 The initial energy in the circuit in Fig. P13.31 is zero. PSPICE The ideal voltage source is 120«(7) V. MULTISIM a) Find I a (s). b) Use the initial- and final-value theorems to find i a (Q + ) and f 0 (oo). c) Do the values obtained in (b) agree with known circuit behavior? Explain. d) Find /„(0. Figure P13.31 20 <e> so n %, 20 H _4 CYV-V>_ + :4 mF 700 O 13.32 There is no energy stored in the circuit in Fig. P13.32 PSPICE at the time the voltage source is energized. MULTISIM a) Find V () , I () , and I L . b) Find v 0 , f ( „ and i L for t ^ 0. Figure P13.32 Figure P13.35 Problems 511 -25f 5Qfe~**u(0 A 5H r 13.33 Beginning with Eq. 13,65, show that the capacitor current in the circuit in Fig. 13.19 is positive for 0 < t < 200 (is and negative for t > 200 {is. Also show that at 200 (is, the current is zero and that this corresponds to when dv c /dt is zero. PSPICE MULTfSIM 13.34 The two switches in the circuit shown in Fig. P13.34 operate simultaneously. There is no energy stored in the circuit at the instant the switches close. Find /(f) for t & 0 + by first finding the s-domain Thevenin equivalent of the circuit to the left of the terminals a,b. Figure P13.34 40 V 2fjiF 13.35 The switch in the circuit shown in Fig. P13.35 has been open for a long time. The voltage of the sinusoidal source is v g = V m sin {cot + cj>). The switch closes at / = 0. Note that the angle cf) in the voltage expression determines the value of the voltage at the moment when the switch closes, that is, v g (0) = V m sin 4>- a) Use the Laplace transform method to find /" for t > 0. b) Using the expression derived in (a), write the expression for the current after the switch has been closed for a long time. c) Using the expression derived in (a), write the expression for the transient component of /'. d) Find the steady-state expression for i using the phasor method. Verify that your expression is equivalent to that obtained in (b). e) Specify the value of c/> so that the circuit passes directly into steady-state operation when the switch is closed. 13.36 The magnetically coupled coils in the circuit seen PSPICE m pjg_ pi 3.36 carry initial currents of 15 and 10 A, MULTISIM , as shown. a) Find the initial energy stored in the circuit. b) Find I { and / 2 . c) Find i] and i 2 . d) Find the total energy dissipated in the 120 and 270 H resistors. e) Repeat (a)-(d), with the dot on the 18 H induc- tor at the lower terminal. Figure P13.36 6H 120ft: / 8H T 15 A r 18 H ' hj i t 10 A :270 0 13.37 The switch in the circuit seen in Fig. PI3.37 has PSPICE been closed for a long time before opening at t = 0. 1 Use the Laplace transform method of analysis to find v„. Figure P13.37 X = 0 13.38 The make-before-break switch in the circuit seen in PSPICE pig. P13.38 has been in position a for a long time. At t = 0, it moves instantaneously to position b. Find L for t > 0. 512 The Laplace Transform in Circuit Analysis Figure P13.38 Figure P13.42 90 V 10 a 13.39 There is no energy stored in the circuit in Fig. PI3.39 PSPICE at the time the switch is closed. MULTISIM a) Find /,. b) Use the initial- and final-value theorems to find / t (0 + ) and/j(oo). c) Find /,. Figure P13.39 150 V 40 n 13.40 a) Find the current in the 40 XI resistor in the cir- pspicE cuit in Fig. PI3.39. The reference direction for the current is down through the resistor. b) Repeat part (a) if the dot on the 1.25 H coil is reversed. 13.41 In the circuit in Fig. P13.41, switch 1 closes at t = 0, PSPICE and the make-before-break switch moves instanta- MULTISIM neous iy f r om position a to position b. a) Construct the A-domain equivalent circuit for t > 0. b) Find/,. c) Use the initial- and final-value theorems to check the initial and final values of /,. d) Find /, for t > 0 + . Figure P13.41 120 a 10a 20 V 13.42 There is no energy stored in the circuit seen in PSPICE Fig. P13.42 at the time the two sources are energized. MULTISIM , ,, . . r . , T , a) Use the principle ot superposition to find V 0 . b) Find v 0 for t > 0. 10 a AAAr- 60K(/)V V 0 10H 12.5 mF f\ J1.5w(r)A |20a 13.43 Verify that the solution of Eqs. 13.91 and 13.92 for V 2 yields the same expression as that given by Eq. 13.90. 13.44 The op amp in the circuit shown in Fig. P13.44 is fR™. ideal. There is no energy stored in the circuit at the MULTISIM time it is energized. If v g = 16,000ta(/) V, find (a) V ( „ (b) v 0 , (c) how long it takes to saturate the operational amplifier, and (d) how small the rate of increase in v g must be to prevent saturation. Figure P13.44 12.5 nF 13.45 The op amp in the circuit seen in Fig. P13.45 is ideal. PSPICE There is no energy stored in the capacitors at the MULTISIM ^ m& ^g crrcm t j s energized. Determine (a) V () , (b) v m and (c) how long it takes to saturate the opera- tional amplifier. Figure P13.45 200 ka 200 ka •—wv + 250 nF -1(- 250 nF :100 Ml 13.46 PSPICE MULTISIM 0.5K(J) V ^- 500 nF Find v () (t) in the circuit shown in Fig. P13.46 if the ideal op amp operates within its linear range and v s = \6u(t) mV. Problems 513 Figure P13.46 13.47 The op amp in the circuit shown in Fig. PI3.47 is PSPICE ideal. There is no energy stored in the capacitors at MUITISIM jj 1£ j nstant th e c i rcu it is energized. a) Find v a if v gi = 40i/(f) V and V H2 = 16//(/) V. b) How many milliseconds after the two voltage sources are turned on does the op amp saturate? Sections 13.4-13.5 13.49 a) Find the numerical expression for the trans- fer function H(s) = V„/Vi for the circuit in Fig. PI3.49. b) Give the numerical value of each pole and zero of H{s). Figure P13.49 16 kO 100 kO 13.50 Find the numerical expression for the transfer func- tion (VJV,) of each circuit in Fig. P13.50 and give the numerical value of the poles and zeros of each transfer function. Figure P13.47 w«fion Figure P13.50 100 kO • VA— 40 nF 40 n F r-K- >\ 100 kO (a) 2kH v, 250 mPH v <> v < 2kfi 13.48 The op amps in the circuit shown in Fig. P13.48 are PSPICE ideal. There is no energy stored in the capacitors at MULTISIM t = () - Tf ^ = 16K ^ mVi how many millisecorids elapse before an op amp saturates? Figure P13.48 25 kf! »* (c) 40 kO (d) •— + •— - y WVv 1 10kO< ( i 1 t 250 nF^ » ( • * o + o (e) 13,51 a) Find the transfer function H(s) = VJVj for the circuit shown in Fig. PI3.51 (a). b) Find the transfer function H(s) = V 0 /V t for the circuit shown in Fig. PI 3.51(b). c) Create two different circuits that have the transfer function H(s) = V () /Vi = 1000/(5+1000). Use components selected from Appendix H and Figs.P13.51(a)and(b). 514 The Laplace Transform in Circuit Analysis Figure PI3.51 + •— R <>— !- —• (a) + (b) 13.54 The operational amplifier in the circuit in Fig. PI3.54 is ideal. a) Find the numerical expression for the transfer function H(s) = VJV S . b) Give the numerical value of each zero and pole of H(s). 13.52 a) Find the transfer function H(s) = VJV, for the circuit shown in Fig. PI3.52(a). b) Find the transfer function H(s) = V 0 /V t for the circuit shown in Fig. PI3.52(b). c) Create two different circuits that have the trans- fer function H(s) = VJV-, = s/(s + 10,000). Use components selected from Appendix H and Figs. P13.52(a) and (b). Figure P13.52 •—1(- Figure P13.54 13.53 + + (a) (b) a) Find the transfer function H(s) = V () /V, for the circuit shown in Fig. P13.53. Identify the poles and zeros for this transfer function. b) Find three components from Appendix H which when used in the circuit of Fig. P13.53 will result in a transfer function with two poles that are distinct real numbers. Calculate the values of the poles. c) Find three components from Appendix H which when used in the circuit of Fig. PI3.53 will result in a transfer function with two poles, both with the same value. Calculate the value of the poles. d) Find three components from Appendix H which when used in the circuit of Fig. P13.53 will result in a transfer function with two poles that are complex conjugate complex numbers. Calculate the values of the poles. C? = 25 nF + lkQ 200nF ^vw }|— ft c 13.55 The operational amplifier in the circuit in Fig. PI3.55 is ideal. a) Find the numerical expression for the transfer function//(5) = VJV r b) Give the numerical value of each zero and pole of H(s). Figure P13.55 400 pF Figure P13.53 ft L R 13.56 The operational amplifier in the circuit in Fig. PI3.56 is ideal. a) Derive the numerical expression of the trans- fer function H(s) = VJV g for the circuit in Fig. P13.56. b) Give the numerical value of each pole and zero of H(s). Problems 515 Figure P13.56 13.59 a) Find the transfer function I ( ,/I s as a function of PSPICE ^ f or the circuit seen in Fig. P13.59. MULnSIM b) Find the largest value of i± that will produce a bounded output signal for a bounded input signal. c) Find i t) for /x = -3,0,4,5, and 6 if L = 5u(t) A. Figure P13.59 8kO O 2kft |2H 13.57 There is no energy stored in the circuit in Fig. P13.57 PSPICE a t the time the switch is opened.The sinusoidal current MULTISIM source is generat j ng the signal 100 cos 10,000/ mA. The response signal is the current i ir a) Find the transfer function l 0 /l R . b) Find I a (s). c) Describe the nature of the transient component of 4(0 without solving for i n (t). d) Describe the nature of the steady-state compo- nent of i 0 (t) without solving for i 0 {t). e) Verify the observations made in (c) and (d) by finding i 0 (t). Figure P13.57 Section 13.6 13.60 a) Find h{t) * x{t) when h(t) and x(t) are the rec- tangular pulses shown in Fig. P13.60(a). b) Repeat (a) when x(t) changes to the rectangular pulse shown in Fig. P13.60(b). c) Repeat (a) when h(t) changes to the rectangular pulse shown in Fig. P13.60(c). Figure P13.60 HO 25 «.(t> K t= 0 100 nF x{t) 25 10 (a) 13.58 In the circuit of Fig. P13.58 i (> is the output signal and v g is the input signal. Find the poles and zeros of the transfer function, assuming there is no initial energy stored in the linear transformer or in the capacitor. Figure P13.58 x(t) 12.5 0 /7(0 25 10 20 0 (b) (c) 5H v P \ o 25 H 10 H 10 kO 62.5 nF 13.61 a) Given y{t) = h(t) * x(t), find y(t) when h(t) and x(t) are the rectangular pulses shown in Fig. PI3.61 (a). b) Repeat (a) when h{t) changes to the rectangular pulse shown in Fig. PI3.61(b). c) Repeat (a) when h(t) changes to the rectangular pulse shown in Fig. PI 3.61(c). d) Sketch y(t) versus t for (a)-(c) on a single graph. e) Do the sketches in (d) make sense? Explain.