An 8 resistor, a 25 mH inductor, and a 62.5 pF capacitor

Derive the s-domain equivalent circuit shown in Fig. 13.4 by expressing the inductor current i as a function of the terminal voltage and then finding the Laplace transform of this time-domain integral equation.

Find the Thvenin equivalent of the circuit shown in Fig. 13.7.

Find the Norton equivalent of the circuit shown in Fig. 13.3.

A 400 resistor, a 12.5 mH inductor, and a 0.5 F capacitor 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.

An 400 resistor, a 25 mH inductor, and a 62.5 F 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.

An 8 resistor, a 25 mH inductor, and a 62.5 pF 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.

Find the poles and zeros of the impedance seen looking into the terminals a,b of the circuit shown in Fig. P13.7.

Find the poles and zeros of the impedance seen looking into the terminals a,b of the circuit shown in Fig. P13.8.

Find and in the circuit shown in Fig. P13.9 if the initial energy is zero and the switch is closed at t = 0.

Repeat Problem 13.9 if the initial voltage on the capacitor is 150 V positive at the upper terminal.

The switch in the circuit shown in Fig. P13.11 has been in position x for a long time. At the switch moves instantaneously to position y.a) Construct an s-domain circuit for t>0 b) Find Io. c)Find io

The switch in the circuit in Fig. P13.12 has been closed for a long time. At t=0 the switch is opened. a) Find for t 0 b) Find for v t 0.

The switch in the circuit in Fig. P13.13 has been in position a for a long time. At it moves instantaneously from a to b. a) Construct the s-domain circuit for b) Find Vo(s). c) Find for v t 0. o(t)

The switch in the circuit in Fig. P13.14 has been in position a for a long time. At the switch moves instantaneously to position b. a) Construct the s-domain circuit for t>0 b) Find Vo c) Find d) Find for e) Find for Figure P13.14

The switch in the circuit in Fig. P13.15 has been closed for a long time before opening at t = 0 a) Construct the s-domain equivalent circuit for t>0 b) Find I0 c) Find for i t 0.

The make-before-break switch in the circuit in Fig. P13.16 has been in position a for a long time.At t = 0 it moves instantaneously to position b. Find for t 0.

a) Find the s-domain expression for in the circuit in Fig. P13.17. b) Use the s-domain expression derived in (a) to predict the initial- and final-values of vo. c) Find the time-domain expression for vo Figure P13.17

Find the time-domain expression for the current in the capacitor in Fig. P13.17. Assume the reference direction for ic is down.

There is no energy stored in the circuit in Fig. P13.19 at t = 0- a) Use the mesh current method to find io. b) Find the time domain expression for vo c) Do your answers in (a) and (b) make sense in terms of known circuit behavior? Explain.

There is no energy stored in the circuit in Fig. P13.20 at the time the voltage source is turned on, and vg = 325u(t) V a) Find Vo and Io b) Find Vo and Io c) Do the solutions for Vo and Io make sense in terms of known circuit behavior? Explain.

There is no energy stored in the circuit in Fig. P13.21 at the time the sources are energized. a) Find I1 and I2 b) Use the initial- and final-value theorems to check the initial- and final-values of i1 (t) and i2(t)c) Find i1(t) and i2(t) for t 0

There is no energy stored in the circuit in Fig. P13.22 at t = 0- a) Find Vo b) Find vo c) Does your solution vo for make sense in terms of known circuit behavior? Explain.

Find vo in the circuit shown in Fig. P13.23 if ig = 20u(t) mA . There is no energy stored in the circuit at t = 0

The switch in the circuit in Fig. P13.24 has been closed for a long time before opening at t = 0 Find vo for t 0

There is no energy stored in the circuit in Fig. P13.25 at the time the switch is closed. a) Find vo for t 0 b) Does your solution make sense in terms of known circuit behavior? Explain.

The initial energy in the circuit in Fig. P13.26 is zero. The ideal voltage source is 600u(t) V. a) Find Vo(s).b) Use the initial- and final-value theorems to find vo(0+) and vo ().c) Do the values obtained in (b) agree with known circuit behavior? Explain. d) Find vo(t).

There is no energy stored in the circuit in Fig. P13.27 at the time the current source turns on. Given that ig = 100u(t) A: a) Find Io(s) b) Use the initial- and final-value theorems to find i (q). o(0+) and io i ( ) Determine if the results obtained in (b) agree with known circuit behavior. d) Find io(t).

The switch in the circuit seen in Fig. P13.28 has been in position a for a long time. At it moves instantaneously to position b. a) Find b) Find Figure P13.28 a b 5.625 H if 3  24 V 5  vo 0.1 F 20  25if t  0 vo. Vo.

The switch in the circuit seen in Fig. P13.29 has been in position a for a long time before moving instantaneously to position b at t = 0.

a) Construct the s-domain equivalent circuit for t 7 0 b) Find V1 and v1 c) Find V2 and v2.

There is no energy stored in the circuit in Fig. P13.31 at the time the current source is energized. a) Find and b) Find and c) Find and d) Find and 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?

The switch in the circuit shown in Fig. P13.32 has been open for a long time. The voltage of the sinusoidal source is The switch closes at Note that the angle in the voltage expression determines the value of the voltage at the moment when the switch closes, that is, a) Use the Laplace transform method to find i for 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 i. 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 so that the circuit passes directly into steady-state operation when the switch is closed. Figure P13.32

Beginning with Eq. 13.65, show that the capacitor current in the circuit in Fig. 13.19 is positive for 0 6 t 6 200 ms and negative for t 7 200 ms Also show that at the current is zero and that this corresponds to when dvC>dt is zero.

There is no energy stored in the circuit in Fig. P13.34 at the time the voltage source is energized. a) Find , , and b) Find , , and for i i L t t 0

The two switches in the circuit shown in Fig. P13.35 operate simultaneously. There is no energy stored in the circuit at the instant the switches close. Find for i(t) t 0+by first finding the s-domain Thvenin equivalent of the circuit to the left of the terminals a, b.

There is no energy stored in the circuit in Fig. P13.36 at the time the switch is closed. a) Find I1 b) Use the initial- and final-value theorems to find (q). 1(0+)and i1 c) Find i1 Figure P13.36

The magnetically coupled coils in the circuit seen in Fig. P13.38 carry initial currents of 300 and 200 A, as shown. a) Find the initial energy stored in the circuit. b) Find and I2 I .

The switch in the circuit seen in Fig. P13.40 has been closed for a long time before opening at Use the Laplace transform method of analysis to find Figure P13.40

The make-before-break switch in the circuit seen in Fig. P13.39 has been in position a for a long time.At t = 0 it moves instantaneously to position b. Find iofor t 0 Figure P13.39

The switch in the circuit seen in Fig. P13.40 has been closed for a long time before opening t = 0 at Use the Laplace transform method of analysis to find vo Figure P13.40

In the circuit in Fig. P13.41, switch 1 closes at and the make-before-break switch moves instantaneously from position a to position b. a) Construct the s-domain equivalent circuit for t>0 b) Find I1.c) Use the initial- and final-value theorems to check the initial and final values of d) Find i1 for t 0+.

Verify that the solution of Eqs. 13.91 and 13.92 for V2 yields the same expression as that given by Eq. 13.90.

There is no energy stored in the circuit seen in Fig. P13.43 at the time the two sources are energized. a) Use the principle of superposition to find Vo b) Find Vo for t >Figure P13.43

The op amp in the circuit shown in Fig. P13.44 is ideal. There is no energy stored in the circuit at the time it is energized. If vg = 5000tu(t) find (a)Vo (b)vo (c) how long it takes to saturate the operational amplifier, and (d) how small the rate of increase vg in must be to prevent saturation.

Find in the circuit shown in Fig. P13.45 if the ideal op amp operates within its linear range and vg = 16u(t) mV

The op amp in the circuit shown in Fig. P13.46 is ideal. There is no energy stored in the capacitors at the instant the circuit is energized. a) Find vo if vg1 40u(t) V and vg2 = 16u(t) V.b) How many milliseconds after the two voltage sources are turned on does the op amp saturate? Figure P13.46

a) Find the transfer function for the circuit shown in Fig. P13.48(a). b) Find the transfer function for the circuit shown in Fig. P13.48(b). c) Create two different circuits that have the transfer function Use components selected from Appendix H and Figs. P13.48(a) and (b)

a) Find the transfer function for the circuit shown in Fig. P13.48(a). b) Find the transfer function for the circuit shown in Fig. P13.48(b). c) Create two different circuits that have the transfer function Use components selected from Appendix H and Figs. P13.48(a) and (b)

a) Find the transfer function for the circuit shown in Fig. P13.49(a). b) Find the transfer function for the circuit shown in Fig. P13.49(b). c) Create two different circuits that have the transfer function Use components selected from Appendix H and Figs. P13.49(a) and (b)

a) Find the transfer function for the circuit shown in Fig. P13.50. Identify the poles and zeros for this transfer function. b) Find three components from Appendix H which when used in the circuit of Fig. P13.50 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. P13.50 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. P13.50 will result in a transfer function with two poles that are complex conjugate complex numbers. Calculate the values of the poles.

a) Find the numerical expression for the transfer function H(s) = Vo>Vi for the circuit in Fig. P13.51. b) Give the numerical value of each pole and zero of Figure H(s) P13.51

Find the numerical expression for the transfer function of each circuit in Fig. P13.52 and give the numerical value of the poles and zeros of each transfer function.

The operational amplifier in the circuit in Fig. P13.53 is ideal. a) Find the numerical expression for the transfer functionH(s) = Vo>Vg. b) Give the numerical value of each zero and pole of Figure P13.53 VCC VCC 125 k 400 pF 250 k 1.6 nF

The operational amplifier in the circuit in Fig. P13.54 is ideal. a) Find the numerical expression for the transfer function H(s) = Vo>V b) Give the numerical value of each zero and pole of H(s). Figure P13.54

The operational amplifier in the circuit in Fig. P13.55 is ideal. a) Derive the numerical expression of the transfer function for the circuit in Fig. P13.55. b) Give the numerical value of each pole and zero of H(s). Figure P13.55

There is no energy stored in the circuit in Fig. P13.56 at the time the switch is opened.The sinusoidal current source is generating the signal The response signal is the current a) Find the transfer function Io/Ig. b) Find Io(s). c) Describe the nature of the transient component of without solving for io i (t).

a) Find the transfer function as a function of for the circuit seen in Fig. P13.57. b) Find the largest value of that will produce a bounded output signal for a bounded input signal. c) Find i m = g = 5u(t) A for 3, 0, 4, 5, and 6 if Figure P13.57

In the circuit of Fig. P13.58 is the output signal and 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

A rectangular voltage pulse vi = [u(t) - u(t - 1)] V is applied to the circuit in Fig. P13.59. Use the convolution integral to find vo

a) Use the convolution integral to find the output voltage of the circuit in Fig. P13.52(a) if the input voltage is the rectangular pulse shown in Fig. P13.61. b) Sketch versus t for the time interval 0 t 100 msFigure P13.61

a) Use the convolution integral to find the output voltage of the circuit in Fig. P13.52(a) if the input voltage is the rectangular pulse shown in Fig. P13.61. b) Sketch versus t for the time interval Figure P

a) Repeat Problem 13.61, given that the resistor in the circuit in Fig. P13.52(a) is decreased to 200 . b) Does decreasing the resistor increase or decrease the memory of the circuit? c) Which circuit comes closer to transmitting a replica of the input voltage?

3 a) Given y(t) = h(t) * x(t) find when and are the rectangular pulses shown in Fig. P13.63(a). b) Repeat (a) when changes to the rectangular pulse shown in Fig. P13.63(b). c) Repeat (a) when changes to the rectangular pulse shown in Fig. P13.63(c). d) Sketch versus t for (a)(c) on a single graph. e) Do the sketches in (d) make sense? Explain.

a) Find when and are the rectangular pulses shown in Fig. P13.64(a). b) Repeat (a) when changes to the rectangular pulse shown in Fig. P13.64(b). c) Repeat (a) when changes to the rectangular pulse shown in Fig

The voltage impulse response of a circuit is shown in Fig. P13.65(a). The input signal to the circuit is the rectangular voltage pulse shown in Fig. P13.65(b). a) Derive the equations for the output voltage. Note the range of time for which each equation is applicable. b) Sketch for v-1 t 34 s.

Assume the voltage impulse response of a circuit can be modeled by the triangular waveform shown in Fig. P13.66.The voltage input signal to this circuit is the step function 10u(t) Va) Use the convolution integral to derive the expressions for the output voltage. b) Sketch the output voltage over the interval 0 to 15 s. c) Repeat parts (a) and (b) if the area under the voltage impulse response stays the same but the width of the impulse response narrows to 4 s. d) Which output waveform is closer to replicating the input waveform: (b) or (c)? Explain.

a) Assume the voltage impulse response of a circuit is h(t) = b 0, t 6 0; 10e-4t , t 0.Use the convolution integral to find the output voltage if the input signal is b) Repeat (a) if the voltage impulse response is c) Plot the output voltage versus time for (a) and (b) for 0 t 1 s

Use the convolution integral to find in the circuit seen in Fig. P13.68 vi = 75u(t) Vif Figure P13.68

a) Use the convolution integral to find in the circuit in Fig. P13.69(a) if is the pulse shown in Fig. P13.69(b). b) Use the convolution integral to find c) Show that your solutions for and are consistent by calculating and at and Figure P13.69 ig 100 k (a) vo io ig (mA) 50 50 (b) 0 200 100 t (ms) 0.2 mF 200 200 + ms. 10

The input voltage in the circuit seen in Fig. P13.70 is vi = 5[u(t) - u(t - 0.5)] V.a) Use the convolution integral to find b) Sketch for v 0 t 1 s Figure P13.70

a) Find the impulse response of the circuit shown in Fig. P13.71(a) if is the input signal and is the output signal. b) Given that vg has the waveform shown in Fig. P13.71(b), use the convolution integral to find vo c) Does vo. have the same waveform as vg Why or why not?

a) Find the impulse response of the circuit seen in Fig. P13.72 if vg is the input signal and vo is the output signal. b) Assume that the voltage source has the waveform shown in Fig. P13.71(b). Use the convolution integral to find vo c) Sketch 0 t 2 s for d) Does vo have the same waveform as vg? Why or why not?

The current source in the circuit shown in Fig. P13.73(a) is generating the waveform shown in Fig. P13.73(b). Use the convolution integral to find v t = 5 ms. o at Figure P13.73

The sinusoidal voltage pulse shown in Fig. P13.74(a) is applied to the circuit shown in Fig. P13.74(b). Use the convolution integral to find the value of vo at t = 75 ms.Figure P13.74

a) Show that if then Y(s) = H(s)X(s). b) Use the result given in (a) to find if F(s) = a s(s + a) 2

The operational amplifier in the circuit seen in Fig. P13.76 is ideal and is operating within its linear region. a) Calculate the transfer function Vo/Vg. b) If vg = cos 3000t V what is the steady-state expression for vo?

The op amp in the circuit seen in Fig. P13.77 is ideal. a) Find the transfer Vo/Vg.function b) Find vo if vg = 0.6u(t) V. c) Find the steady-state expression for vo if vg = 2 cos 10,000t V. Figure P13.77

The transfer function for a linear time-invariant circuit is H(s) = Vo Vg = 25(s + 8) s2 + 60s + 150 If what is the steady-state expression for ?

The transfer function for a linear time-invariant circuit is H(s) = Io Ig = 125(s + 400) s(s2 + 200s + 104) .If what is the steady-state expression for ?

When an input voltage of is applied to a circuit, the response is known to be vo = (50e-8000t - 20e-5000t)u(t) V.What will the steady-state response be if vg = 120 cos 6000 t V

Show that after V0Ce coulombs are transferred from to in the circuit shown in Fig. 13.47, the voltage across each capacitor is C1V0>(C1 + C2) (Hint: Use the conservation-of-charge principle.)

The voltage source in the circuit in Example 13.1 is changed to a unit impulse; that is,vg = d(t). a) How much energy does the impulsive voltage source store in the capacitor? b) How much energy does it store in the inductor? c) Use the transfer function to find d) Show that the response found in (c) is identical to the response generated by first charging the capacitor to 1000 V and then releasing the charge to the circuit, as shown in Fig. P13.82.

There is no energy stored in the circuit in Fig. P13.83 at the time the impulsive voltage is applied. a) Find v t 0. o(t) b) Does your solution make sense in terms of known circuit behavior? Explain.

The inductor in the circuit shown in Fig. P13.84 is carrying an initial current of at the instant the switch opens. Find (a) (b) (c) and (d) where is the total flux linkage in the circuit. Figure P13.84

a) Let in the circuit shown in Fig. P13.84, and use the solutions derived in Problem 13.84 to find and b) Let in the circuit shown in Fig. P13.84 and use the Laplace transform method to find v(t), (t i2

The parallel combination of and in the circuit shown in Fig. P13.86 represents the input circuit to a cathode-ray oscilloscope (CRO). The parallel combination of and is a circuit model of a compensating lead that is used to connect the CRO to the source. There is no energy stored in or at the time when the 10 V source is connected to the CRO via the compensating lead. The circuit values are and C 1 = 1.25 M a) Find b) Find c) Repeat (a) and (b) given is changed to 64 pF.

Show that if R1C1 = R2C2 in the circuit shown in Fig. P13.86, will be a scaled replica of the source voltage.

The switch in the circuit in Fig. P13.88 has been closed for a long time. The switch opens at Compute (a) ); 1(0-) (b) ); 1(0+ (c)ii ); 2(0- (d)i2(0+ (e)i (t); 1(t); (f) i v(t). 2 i (tand (g) v(t).Figure P13.87

There is no energy stored in the circuit in Fig. P13.89 at the time the impulse voltage is applied. a) Find i1 for t 0+ b) Find i2 for t 0+ c) Find v2 for t 0+ d) Do your solutions for i1,i2,v0 and make sense in terms of known circuit behavior? Explain.

The switch in the circuit in Fig. P13.90 has been in position a for a long time. At the switch moves to position b. Compute (a) (b) (c) (d) (e) (f) and (g) Figure P13.90 a b 20 k 100 V t  0 1.6 mF v3 2.0 mF v2 0.5 mF v1 i(t) v3(0+). v2(0+ v ); 1(0+ v i(t); ); 3(0-); v2(0- v ); 1(

There is no energy stored in the circuit in Fig. P13.91 at the time the impulsive current is applied. a) Find vo for t 0 b) Does your solution make sense in terms of known circuit behavior? Explain.

Assume the line-to-neutral voltage in the 60 Hz circuit of Fig. 13.59 is 120 l0 V (rms). Load is absorbing 1200 W; load is absorbing 1800 W; and load is absorbing 350 magnetizing VAR. The inductive reactance of the line (X1) is 1 k Assume Vg does not change after the switch opens. a) Calculate the initial value of and i (t). 2(tb) Find and using the s-domain circuit of Fig. 13.60. c) Test the steady-state component of using phasor domain analysis. d) Using a computer program of your choice, plot v vs. t for 0 t 20 ms.

3 Assume the switch in the circuit in Fig. 13.59 opens at the instant the sinusoidal steady-state voltage is zero and going positive, i.e.,vo = 12012 sin 120pt V. a) Find for v t 0 b) Using a computer program of your choice, plot vs.v 0 t 20 ms. t for c) Compare the disturbance in the voltage in part (a) with that obtained in part (c) of Problem 13.92.

The purpose of this problem is to show that the line-to-neutral voltage in the circuit in Fig. 13.59 can go directly into steady state if the load is disconnected from the circuit at precisely the right time. Let vo = Vm cos(120pt - u) V where Vm = 12012 Assume does not change after is disconnected. a) Find the value of (in degrees) so that goes directly into steady-state operation when the load is disconnected. b) For the value of u found in part t 0 (a), find for c) Using a computer program of your choice, plot on a single graph, for -10 ms t 10 ms before and after load is disconnected.