EETOMAGNETI OMPATIBIITY HANDBOOK 1 hapter 9: Transent Behavor n the Tme Doman 9.1 Desgn a crcut usng reasonable values for the components that s capable of provdng a tme delay of 100 ms to a dgtal sgnal. State all assumptons. 9.2 Startng wth the ntegral equaton for an ntegrator provded n ths chapter t 1 v t t dt v ( ) = ( ) ( 0) c 0 determne the expresson for the output voltage f αt ( ) = 5 ( ) t e u t Then, plot both the nput and output versus tme on the same set of axes for each of the gven cases: 1 5 1 α =, α =, α = 5 9.3 Startng wth the dfferental equaton for an dfferentator provded n ths chapter dv v dv = 0 dt dt determne the expresson for the output voltage f αt ( ) = 4.5( 1 ) ( ) v t e u t Then, plot both the nput and output versus tme on the same set of axes for each of the gven cases: 1 5 1 α =, α =, α = 5 9.4S epeat Problem 9.3 for the followng ramp sgnal: ( ) = αtu( t) v t 9.5. Verfy that the results gven for ase 1 for the general crcut analyss gven n ths chapter are correct by reanalyzng the ase 1 crcut. Do not merely use opyrght 2002 by Kenneth. Kaser, Verson 08/12/05
2 EETOMAGNETI OMPATIBIITY HANDBOOK the expresson provded for the general crcut but repeat the analyss begnnng wth t A Be τ. 9.6 epeat Problem 9.5 for ase 2. 9.7 epeat Problem 9.5 for ase 3. 9.8 epeat Problem 9.5 for ase 4. 9.9 Determne the expresson for the voltage across the capactor, v, and the current through the load resstor,, for the pre-emphass crcut gven n Fgure 1. P s v v = u( t) Fgure 1 9.10S For the general crcut example gven n ths chapter, repeat the entre analyss but replace and s wth ts Norton equvalent; that s, replace the voltage source and correspondng resstance wth an ndependent current source of value V o / s n parallel wth a resstance s. Do not used source transformatons but repeat the analyss usng ths current source. A current source s sometmes approxmated by placng a hgh resstance n seres wth a hgh-voltage source. Why does ths approxmate a current source? 9.11 epeat the analyss for the statc charge buldup dscusson n ths chapter but assume that a resstor, p, s temporarly placed n shunt wth the capactor at t = t1 ( t2 t1 ) 2. It s removed from the crcut at t = t2 as shown n Fgure 2. For the numercal analyss, let p =, p = 10, and p = /10 for all three specfc values of gven. t = t 2 t = t 2 I s s t = t 1 t = t x p t x t2 = t 1 t 2 1 opyrght 2002 by Kenneth. Kaser, Verson 08/12/05
EETOMAGNETI OMPATIBIITY HANDBOOK 3 Fgure 2 9.12 Determne for t > 0, v o (t), (t), and the energy dsspated n the 1 kω resstor from to 2τ and from t = 2τ to 10τ for the crcut gven n Fgure 3. 1.2 kω 5 V 1 kω v o 0.1F Fgure 3 9.13 For t > 0, determne v o (t), (t), and the maxmum energy that can be dsspated n both of the 2.2 kω resstors for the crcut gven n Fgure 4. 1 kω 5 V 2.2 kω 2.2 kω vo 0.1F Fgure 4 9.14 Determne for t > 0 the current (t), the voltage across each of the capactors, and the energy released from both capactors from to two tme constants for the crcut gven n Fgure 5. opyrght 2002 by Kenneth. Kaser, Verson 08/12/05
4 EETOMAGNETI OMPATIBIITY HANDBOOK 10 Ω 0.1Ω 1 Ω 2.2F 9 V 3 V 1F Fgure 5 9.15 For the crcut gven n Fgure 6, determne the voltage across each of the capactors for t 0. Hnt: frst, determne the current through the net capactance and the 220 Ω resstor. Second, use ths current expresson n the ntegral defnton for the capactor voltage for each capactor: t 1 vc t t dt v ( ) = ( ) ( 0) 0 c 1 Ω 220 Ω 1.5 V 1F 2.2F 680 Ω Fgure 6 9.16 For the crcut gven n Fgure 7, determne (t) for t > 0. 10 Ω 3 V 0.1F 10 kω 100 ma Fgure 7 9.17 For the crcut gven n Fgure 8, determne the tme requred for the voltage v c (t) to rse to 6 V. opyrght 2002 by Kenneth. Kaser, Verson 08/12/05
EETOMAGNETI OMPATIBIITY HANDBOOK 5 0.1 Ω 12 V 1F v 0.1F 10 kω 0.1 F 0.22 F c Fgure 8 9.18 For the crcut gven n Fgure 9, determne (t) for t > 0. 1 kω 1.2 kω 5 V 3 2.2 kω 0.1F Fgure 9 9.19 For the gven gate n Fgure 10, determne the voltage across the load resstor,, for t > 0 f a) v1 ( t) = 5 u( t), v2 ( t) = 5u ( t 4τ ) b) v1 ( t) = 5u ( t 4 τ ), v2 ( t) = 5u ( t) c) v1 ( t) = 5 u( t), v2 ( t) = 5u ( t 4τ ) d) v ( t) = 5u ( t 4 τ ), v ( t) = 5u ( t) 1 2 where τ s the tme constant for the system. What s ths tme constant n terms of,, and? opyrght 2002 by Kenneth. Kaser, Verson 08/12/05
6 EETOMAGNETI OMPATIBIITY HANDBOOK v 1 v 2 v Fgure 10 9.20S an the state of the nternal gates for a dgtal memory element change more than once durng a clock cycle? Explan. 9.21 Startng wth the general exponental soluton, determne drectly the followng expresson for the voltage across the resstor of a seres crcut to a step voltage nput by determnng the ntal and fnal voltage across the resstor: t τ v ( t) = ( t) = Vs1 e u ( t) whereτ = 9.22 Startng wth the general exponental soluton, determne drectly the followng expresson for the voltage across the nductor of a seres crcut to a step voltage nput by determnng the ntal and fnal voltage across the nductor drectly: t τ ( ) = f > 0 v t V e t 9.23 For a step current source (a current source that turns on at ) of strength I n parallel wth a parallel crcut, verfy the followng expressons for the voltage across the and and the respectve currents through the and : s t t t v t I e u t t I e u t t Ie u t ( ) = 1 ( ), ( ) = 1 ( ), ( ) = ( ) learly state all assumptons. 9.24 For a step current source (a current source that turns on at ) of strength I n parallel wth a parallel crcut, verfy that the followng expressons for the voltage across the and and the respectve currents through the and : t t t v t I u t t Ie u t t I e u t ( ) = e ( ), ( ) = ( ), ( ) = 1 ( ) opyrght 2002 by Kenneth. Kaser, Verson 08/12/05
EETOMAGNETI OMPATIBIITY HANDBOOK 7 learly state all assumptons. 9.25 For a current sgnal source n parallel wth a parallel crcut, repeat the analytcal analyss gven n the ntegrator and dfferentator dscussons provded n the chapter. 9.26 For a current sgnal source n parallel wth a parallel crcut, repeat the analytcal analyss gven n the ntegrator and dfferentator dscussons provded n the chapter. 9.27 Determne (t), v (t), and the energy stored n the nductor from to 5τ for the crcut gven n Fgure 11. v 1.2 kω 1 kω 5 V 0.1 mh Fgure 11 9.28 Determne (t) and the energy dsspated by the 1 kω resstor between 0.1 and 1 ns for the two-supply crcut gven n Fgure 12. Determne the energy delvered by the current supply over ths same tme range. 0.1H 3 V 1 kω 10 kω 100 ma Fgure 12 9.29 For the crcut gven n Fgure 13, determne the voltage across the nductor and the voltage across the 680 Ω resstor for t > 0 (where τ corresponds to the tme constant of the crcut before the second swtchng). Determne the net energy stored by the nductor between and 3τ. opyrght 2002 by Kenneth. Kaser, Verson 08/12/05
8 EETOMAGNETI OMPATIBIITY HANDBOOK 1 Ω 10 mh t = 3τ 1.5 V 470 Ω 220 Ω 680 Ω Fgure 13 9.30 For the two-nductor crcut gven n Fgure 14, determne v (t) and (t). After three tme constants determne the energy dsspated by the 10 Ω resstor. After fve tme constants determne the energy dsspated by the 10 Ω resstor. 10 Ω 1 Ω 9 V 1 Ω 3 V t = t 1 2.2 mh v 1 mh Fgure 14 9.31 The swtch has been closed a long tme before openng at for the crcut shown n Fgure 15. Fnd the voltage v x (t) for t > 0. The method for determnng the equvalent resstance for an crcut wth a dependent supply s dentcal to that for an crcut. 100 Ω 9 V 2 Ω 4v x v x 2H Fgure 15 opyrght 2002 by Kenneth. Kaser, Verson 08/12/05
EETOMAGNETI OMPATIBIITY HANDBOOK 9 9.32 For the crcut gven n Fgure 15, assume that the swtch has been open for a long tme before closng at. Fnd the voltage v x (t) for t > 0. 9.33 For the crcut gven n Fgure 16, determne all labeled currents and voltages mmedately after the swtch s closed and after a very long tme. Assume that the swtch has been open for a very long tme before closng. Is ths crcut a seres or parallel crcut? p v s V p s v v Fgure 16 9.34 For the crcut gven n Fgure 17, determne all labeled currents and voltages mmedately after the swtch s opened and after a very long tme. Assume that the swtch has been closed for a very long tme before openng. Is ths crcut a seres or parallel crcut? I s p v Fgure 17 9.35 For the crcut gven n Fgure 18, determne all labeled currents and voltages mmedately after both swtches are closed and after a very long tme. Assume that the swtches have been open for a very long tme before closng. Is ths crcut a seres or parallel crcut? opyrght 2002 by Kenneth. Kaser, Verson 08/12/05
10 EETOMAGNETI OMPATIBIITY HANDBOOK I s p v s Fgure 18 9.36 Determne and sketch (t) and v 100 (t) for the crcut gven n Fgure 19 after the swtch s closed. Is ths system oscllatory? Then, determne and sketch the energy dsspated n the resstor and energy stored n the nductor and capactor after the swtch s closed. Approxmately, how long does t take the transents to de off. v 100 100 Ω 1F 10 mh Fgure 19 9.37 For the crcut gven n Fgure 20, f I s = 10 ma, = 10 Ω, = 2.2 µh, and = 0.01 µf, determne and sketch (t), (t), (t), and v(t) after the swtch s thrown. Is ths system oscllatory? Determne and sketch the energy dsspated by the resstor and energy stored by nductor and capactor for t > 0. Approxmately, how long does t take the transents to de off. I s v Fgure 20 9.38 Determne the expresson for (t), v x (t), and v (t) for the crcut gven n Fgure 21 after the swtch s thrown open. Assume that = 9 V, s = 10 Ω, = 10 mh, and = 0.001 µf. Select a value for x so that the system s oscllatory opyrght 2002 by Kenneth. Kaser, Verson 08/12/05
EETOMAGNETI OMPATIBIITY HANDBOOK 11 wth a rngng frequency of 30 khz. Then, select another value for x so that the system s overdamped wth a neglgble response after 10 ms. s v v x x Fgure 21 9.39E Assumng that the swtch has been open for a long tme before closng at, determne the expressons for v (t) and x (t) for the crcut gven n Fgure 22. Use superposton to determne the ntal values. Assume that = 6 V, I s = 100 ma, s = 10 Ω, x = 220 Ω, p = 10 kω, = 1 mh, and = 0.22 µf. Estmate the tme requred for the transents to decay to a neglgble level. Is ths a seres or parallel crcut? s x v x p I s Fgure 22 9.40E Assumng that the swtch has been closed for a long tme before openng at for the crcut gven n Fgure 23, determne the expressons for v (t) and x (t). Use superposton to determne the ntal values. Assume that I x = 10 ma, I s = 20 ma, x = 22 kω, y = 10 kω, = 100 mh, and = 2.2 µf. Estmate the tme requred for the transents to decay to a neglgble level. What s smallest possble rngng frequency f x s allowed to vary? Is ths a seres or parallel crcut? y x I x v x I s Fgure 23 9.41 Assumng that the swtch has been open for a long tme before closng at t = t x for the crcut gven n Fgure 24, determne the expressons for v (t) and x (t). opyrght 2002 by Kenneth. Kaser, Verson 08/12/05
12 EETOMAGNETI OMPATIBIITY HANDBOOK Assume that = 15 V, x = 2.2 kω, y = 4.7 kω, z = 6.8 kω, = 10 mh, and = 3.3 µf. Estmate the tme requred for the transents to decay to a neglgble level. Is ths a seres or parallel crcut? z x t = t x v x y Fgure 24 9.42 Usng real component values, desgn a crcut capable of oscllatng for 1 mnute wth only a 1% reducton n ampltude at a frequency of 10 khz. The ampltude of the ntal oscllaton should be 10 mv. epeat the desgn for a 10 MHz rngng frequency. Whch specfcaton s easer to meet and why? 9.43E Derve the equatons for the dampng coeffcent and resonant frequency gven n the chapter for the network consstng of two seres- crcuts that are n parallel. heck ether the dampng coeffcent or resonant frequency aganst another known result by allowng certan terms to approach zero or nfnty. 9.44E Derve the equatons for the dampng coeffcent and the resonant frequency gven n the chapter for the network consstng of two seres- crcuts that are n parallel heck ether the dampng coeffcent or resonant frequency aganst another known result by allowng certan terms to approach zero or nfnty. 9.45E Derve the equatons for the dampng coeffcent and resonant frequency gven n the chapter for the network consstng of a parallel crcut n seres wth a parallel crcut. heck ether the dampng coeffcent or resonant frequency aganst another known result by allowng certan terms to approach zero or nfnty. 9.46E Derve the equatons for the dampng coeffcent and resonant frequency gven n ths chapter for the network consstng of a parallel crcut n seres wth another parallel crcut. heck ether the dampng coeffcent or resonant frequency aganst another known result by allowng certan terms to approach zero or nfnty. 9.47E. Derve the equatons for the dampng coeffcent and resonant frequency gven n the chapter for the network consstng of a parallel crcut n seres wth another parallel crcut. heck ether the dampng coeffcent or resonant frequency aganst another known result by allowng certan terms to approach zero or nfnty. 9.48E Determne the expresson for the voltage across 1 for the crcut gven n Fgure 25, assumng that the swtch has been opened for a long tme before closng at t opyrght 2002 by Kenneth. Kaser, Verson 08/12/05
EETOMAGNETI OMPATIBIITY HANDBOOK 13 = 0. learly show the polarty of ths voltage. et = 6 V, = 22 Ω, 1 = 2 Ω, 2 = 4 Ω, 1 = 2 mh, and 2 = 10 mh. 1 2 1 2 Fgure 25 9.49E Determne the expresson for the current through 1 for the crcut gven n Fgure 26, assumng that the swtch has been opened for a long tme before closng at. learly show the drecton of ths current. et = 9 V, = 2 Ω, 1 = 220 Ω, 2 = 2 kω, 1 = 0.22 µf, and 2 = 1.0 µf. 1 1 2 2 Fgure 26 9.50E Determne the expresson for the voltage across for the crcut gven n Fgure 27, assumng that the swtch has been opened for a long tme before closng at t = 0. learly show the polarty of ths voltage. et = 15 V, = 22 Ω, 1 = 1 kω, 2 = 22 kω, = 2.2 µf, and = 100 mh. opyrght 2002 by Kenneth. Kaser, Verson 08/12/05
14 EETOMAGNETI OMPATIBIITY HANDBOOK 1 2 Fgure 27 9.51E Determne whether the gven crcut n Fgure 28 oscllates. If t does oscllate, determne ts frequency and the number of whole nteger cycles that occur before the oscllatory sgnal falls below approxmately 25% of ts maxmum ampltude. For all of the swtchng ntervals, estmate the tme requred for the respectve transents (voltage for capactors and current for nductors) to decay wthn 5% of ther fnal value. Fnally, estmate the current through each nductor and voltage across each capactor at t = t x, and determne ther exact values at t =. et = 12 V, = 22 Ω, 1 = 1 kω, 2 = 22 kω, 1 = 2.2 µf, 2 = 0.22 µ, and t x = 2 ms. 1 t = t x 1 2 2 Fgure 28 9.52E epeat Problem 9.51 for the crcut gven n Fgure 29. et = 6 V, = 220 Ω, 1 = 10 Ω, 2 = 10 kω, 3 = 10 Ω, = 22 µf, = 100 mh, and t x = 10 ms. opyrght 2002 by Kenneth. Kaser, Verson 08/12/05
EETOMAGNETI OMPATIBIITY HANDBOOK 15 1 2 t = t x 3 Fgure 29 9.53E epeat Problem 9.51 for crcut gven n Fgure 30. et = 12 V, = 1 Ω, 1 = 10 kω, 2 = 22 kω, = 0.022 µf, = 10 mh, and t x = 200 µs. 1 t = t x 2 Fgure 30 9.54E epeat Problem 9.51 for crcut gven n Fgure 31. et I s = 10 ma, = 220 Ω, 1 = 10 Ω, 2 = 10 kω, 3 = 10 kω, 1 = 220 mh, 2 = 10 mh, and t x = 2 ms. Hnt: source transformatons. 1 2 t = t x I s 3 1 2 Fgure 31 9.55 Verfy that for the functon opyrght 2002 by Kenneth. Kaser, Verson 08/12/05
16 EETOMAGNETI OMPATIBIITY HANDBOOK the maxmum percent overshoot s ω ω = o αt 1 ( ) sn ω tan d x t A e dt F ω d α A o % overshoot 100 e ω = F πα 2 2 α opyrght 2002 by Kenneth. Kaser, Verson 08/12/05