Using Fig. 7.131, design a problem to help other students better understand first-order

Refer to the circuit in Fig. 7.7. Let vC (0) = 60 V. Determine vC, vx, and io for t 0.

If the switch in Fig. 7.10 opens at t = 0, find v(t) for t 0 and wC (0).

Find i and vx in the circuit of Fig. 7.15. Let i(0) = 7 A.

For the circuit in Fig. 7.18, find i(t) for t > 0.

Determine i, io, and vo for all t in the circuit shown in Fig. 7.22. Assume that the switch was closed for a long time. It should be noted that opening a switch in series with an ideal current source creates an infinite voltage at the current source terminals. Clearly this is impossible. For the purposes of problem solving, we can place a shunt resistor in parallel with the source (which now makes it a voltage source in series with a resistor). In more practical circuits, devices that act like current sources are, for the most part, electronic circuits. These circuits will allow the source to act like an ideal current source over its operating range but voltage-limit it when the load resistor becomes too large (as in an open circuit).

Express the current pulse in Fig. 7.33 in terms of the unit step. Find its integral and sketch it.

Refer to Fig. 7.39. Express i(t) in terms of singularity functions.

If h(t) = { 0, 4, 3t 8, 0, t < 0 0 < t < 2 2 < t < 6 t > 6 express h(t) in terms of the singularity functions

Evaluate the following integrals: ( t3 + 5t2 + 10 ) ( t + 3 ) dt, 0 10 (t ) cos 3t dt

Find v(t) for t > 0 in the circuit of Fig. 7.44. Assume the switch has been open for a long time and is closed at t = 0. Calculate v(t) at t = 0.5

The switch in Fig. 7.47 is closed at t = 0. Find i(t) and v(t) for all time. Note that u(t) = 1 for t < 0 and 0 for t > 0. Also, u(t) = 1 u(t).

The switch in Fig. 7.52 has been closed for a long time. It opens at t = 0. Find i(t) for t > 0.

Switch S1 in Fig. 7.54 is closed at t = 0, and switch S2 is closed at t = 2s. Calculate i(t) for all t. Find i(1) and i(3).

For the op amp circuit in Fig. 7.56, find vo for t > 0 if v(0) = 4 V. Assume that Rf = 50 k, R1 = 10 k, and C = 10 F.

Find v(t) and vo(t) in the op amp circuit of Fig. 7.58.

Obtain the step response vo(t) for the circuit in Fig. 7.62. Let vi = 9u(t) V, R1 = 20 k, Rf = 40 k, R2 = R3 = 10 k, C = 2 F.

For the circuit in Fig. 7.66, use Pspice to find v(t) for t > 0.

The switch in Fig. 7.71 was open for a long time but closed at t = 0. If i(0) = 10A, find i(t) for t > 0 by hand and also by PSpice.

The RC circuit in Fig. 7.74 is designed to operate an alarm which acti vates when the current through it exceeds 90 A. If 0 R 6 k, find the range of the time delay that the variable resistor can create.

The flash unit of a camera has a 2-mF capacitor charged to 40 V. (a) How much charge is on the capacitor? (b) What is the energy stored in the capacitor? (c) If the flash fires in 0.8 ms, what is the average current through the flashtube? (d) How much power is delivered to the flashtube? After a picture has been taken, the capacitor needs to be recharged by a power unit that supplies a maximum of 5 mA. How much time does it take to charge the capacitor?

A relay has a resistance of 200 and an inductance of 500 mH. The relay contacts close when the current through the coil reaches 175 A. What time elapses between the application of 110 V to the coil and contact closure?

The spark coil of an automobile ignition system has a 20-mH induc tance and a 5- resistance. With a supply voltage of 12 V, calculate: the timeneeded for the coil to fully charge, the energy stored in the coil, and the voltage developed at the spark gap if the switch opens in 2 s.

Consider the circuit in Fig. 7.103. Given that vo(0) = 10 V, find vo and vx for t > 0.

Express the following signals in terms of singularity functions. (a) v(t) = { 0, 5, t < 0 t > 0 (b) i(t) = { 0, 10, 10, 0, t < 1 1 < t < 3 3 < t < 5 t > 5 (c) x(t) = { t 1, 1, 4 t, 0, 1 < t < 2 2 < t < 3 3 < t < 4 Otherwise (d) y(t) = { 2, 5, 0, t < 0 0 < t < 1 t > 1

Design a problem to help other students better understand singularity functions

Express the signals in Fig. 7.104 in terms of singularity functions.

Express v(t) in Fig. 7.105 in terms of step functions.

Sketch the waveform represented by i(t) = [r(t) r(t 1) u(t 2) r(t 2) + r(t 3) + u(t)(t 4)] A

Sketch the following functions: (a) x(t) = 10etu(t 1), (b) y(t) = 10 e (t1) u(t), (c) z(t) = cos 4t(t 1)

Evaluate the following integrals involving the impulse functions: (a) 4t2(t 1)dt (b) 4t2 cos 2 t(t 0.5)dt

Evaluate the following integrals: (a) e 4 t 2 (t 2)dt (b) [5(t) + et(t) + cos 2 t(t)]dt

Evaluate the following integrals: (a) 1 t u()d (b) 0 4 r(t 1)dt (c) 1 5 (t 6)2(t 2)dt

The voltage across a 10-mH inductor is 45(t 2)mV. Find the inductor current, assuming that the inductor is initially uncharged.

Evaluate the following derivatives: (a) d __ dt [u(t 1)u(t + 1)] (b) d __ dt [r(t 6)u(t 2)] (c) d __ dt [sin 4tu(t 3)]

Find the solution to the following differential equations: (a) dv ___ dt + 2v = 0, v(0) = 1 V (b) 2 di __ dt 3i = 0, i(0) = 2

Solve for v in the following differential equations, subject to the stated initial condition. (a) dvdt + v = u(t), v(0) = 0 (b) 2 dvdt v = 3u(t), v(0) = 6

A circuit is described by 4 dv ___ dt + v = 10 (a) What is the time constant of the circuit? (b) What is v(), the final value of v? (c) If v(0) = 2, find v(t) for t 0.

A circuit is described by di __ dt + 3i = 2u(t) Find i(t) for t > 0 given that i(0) = 0.

Calculate the capacitor voltage for t < 0 and t > 0 for each of the circuits in Fig. 7.106.

Find the capacitor voltage for t < 0 and t > 0 for each of the circuits in Fig. 7.107.

Using Fig. 7.108, design a problem to help other students better understand the step response of an RC circuit.

(a) If the switch in Fig. 7.109 has been open for a long time and is closed at t = 0, find vo(t). (b) Suppose that the switch has been closed for a long time and is opened at t = 0. Find vo(t).

Consider the circuit in Fig. 7.110. Find i(t) for t < 0 and t > 0.

The switch in Fig. 7.111 has been in position a for a long time. At t = 0, it moves to position b. Calculate i(t) for all t > 0.

Find vo in the circuit of Fig. 7.112 when vs = 30u(t)V. Assume that vo(0) = 5 V.

For the circuit in Fig. 7.113, is(t) = 5u(t). Find v(t).

Determine v(t) for t > 0 in the circuit of Fig. 7.114 if v(0) = 0.

Find v(t) and i(t) in the circuit of Fig. 7.115.

If the waveform in Fig. 7.116(a) is applied to the circuit of Fig. 7.116(b), find v(t). Assume v(0) = 0.

In the circuit of Fig. 7.117, find ix for t > 0. Let R1 = R2 = 1 k, R3 = 2 k, and C = 0.25 mF.

Rather than applying the shortcut technique used in Section 7.6, use KVL to obtain Eq. (7.60).

Using Fig. 7.118, design a problem to help other students better understand the step response of an RL circuit.

Determine the inductor current i(t) for both t < 0 and t > 0 for each of the circuits in Fig. 7.119.

Obtain the inductor current for both t < 0 and t > 0 in each of the circuits in Fig. 7.120.

Find v(t) for t < 0 and t > 0 in the circuit of Fig.7.121.

For the network shown in Fig. 7.122, find v(t) for t > 0.

Find i1(t) and i2(t) for t > 0 in the circuit of Fig.7.123

Rework Prob. 7.17 if i(0) = 10 A and v(t) = 20u(t) V.

Determine the step response vo(t) to is = 6u(t) A in the circuit of Fig. 7.124.

Find v(t) for t > 0 in the circuit of Fig. 7.125 if the initial current in the inductor is zero.

In the circuit in Fig. 7.126, is changes from 5 A to 10 A at t = 0; that is, is = 5u(t) + 10u(t). Find v and i.

For the circuit in Fig. 7.127, calculate i(t) if i(0) = 0.

Obtain \(v(t)\) and \(i(t)\) in the circuit of Fig. 7.128. Equation Transcription: Text Transcription: v(t) i(t)

Determine the value of \(i_{L}(t)\) and the total energy dissipated by the circuit from \(t=0 \mathrm{sec}\) to \(t=\infty \mathrm{sec}\). The value of \(i_{i n}(t)\) is equal to \([6-6 u(t)] \mathrm{A}\). Equation Transcription: Text Transcription: i_L(t) t = 0 t=infinity i_in(t) [6-6u(t)]A

If the input pulse in Fig. 7.130(a) is applied to the circuit in Fig. 7.130(b), determine the response \(i(t)\). Equation Transcription: Text Transcription: i(t)

Using Fig. 7.131, design a problem to help other students better understand first-order op amp circuits.

If \(v(0)=10 \mathrm{~V}\), find \(v_{o}(t)\) for \(t>0\) in the op amp circuit in Fig. 7.132. Let \(R=100 \mathrm{k} \Omega\) and \(C=20 \mu \mathrm{F}\). Equation Transcription: Text Transcription: v(0) = 10 V v_o(t) R = 100 k ohms C = 20 mu F

Obtain \(v_{o}\) for \(t>0\) in the circuit of Fig. 7.133. Equation Transcription: Text Transcription: v_o t > 0

For the op amp circuit in Fig. 7.134, find \(v_{o}(t)\) for \(t>0\). Equation Transcription: Text Transcription: v_o(t) t > 0

Determine vo for \(t>0\) when \(v_{s}= 20\ mV\) in the op amp circuit of Fig. 7.135. Equation Transcription: Text Transcription: t > 0 v_s= 20 mV

For the op amp circuit in Fig. 7.136, suppose \(v_{s}= 10u(t) V\). Find \(v(t)\) for \(t>0\). Equation Transcription: Text Transcription: v_s=10u(t) V t > 0 v(t)

Find \(i_{o}\) in the op amp circuit in Fig. 7.137. Assume that \(v(0)=-2 \mathrm{~V}, R=10 \mathrm{k} \Omega\), and \(C=10 \mu \mathrm{F}\). Equation Transcription: Text Transcription: i_o v(0)=-2 V, R = 10 k ohms C = 10 mu F

For the op amp circuit of Fig. 7.138, let \(R_{1}=10 \mathrm{k} . \Omega\), \(R_{f}=30 \mathrm{k\Omega}, C=20 \mu \mathrm{F}\), and \(v(0)=1 \mathrm{~V}\). Find \(v_{\mathrm{o}}\). Equation Transcription: Text Transcription: R_1= 10 k ohms R_f= 30 k ohms C = 20 mu F v(0) = 1 V v_o

Determine \(v_{i}(t)\) for \(t>0\) in the circuit of Fig. 7.139. Let \(i_{s}=10 \mathrm{u}(t) \mu \mathrm{A}\) and assume that the capacitor is initially uncharged. Equation Transcription: Text Transcription: v_o(t) t > 0 i_s= 10u(t) mu A

In the circuit of Fig- 7.140, find \(v_{o}\) and \(i_{o}\), given that \(v_{s}=10\left(1-e^{-1}\right] u(t) \mathrm{V}\). Equation Transcription: Text Transcription: v_o i_o v_s= 10[1-e^-t]u(t) V

Repeat Prob. 7.49 using PSpice or MultiSim.

The switch in Fig. 7.141 opens at \(t = 0\). Use PSpice or MultiSim to determine \(v(t)\) for \(t > 0\). Equation Transcription: Text Transcription: t = 0 v(t) t > 0 ________________

The switch in Fig. 7.142 moves from position \(a\) to \(b\) at \(t = 0\). Use PSpice or MultiSim to find \(i(t)\) for \(t > 0\). Equation Transcription: Text Transcription: a b t = 0 i(t) t > 0

In the circuit of Fig. 7.143, determine \(i_{o}(t)\). Equation Transcription: Text Transcription: i_o(t)

In the circuit of Fig. 7.144, find the value of \(i_{o}\) for all values of \(0 < t\). Equation Transcription: Text Transcription: i_o 0 < t

Repeat Prob. 7.65 using PSpice or MultiSim

In designing a signal-switching circuit, it was found that a \(100-\mu \mathrm{F}\) capacitor was needed for a time constant of \(3 \mathrm{~ms}\). What value resistor is necessary for the circuit? Equation Transcription: Text Transcription: 100-mu F 3 ms

An \(R C\) circuit consists of a series connection of a \(120-\mathrm{V}\) source, a switch, a 34-M \(\Omega\) resistor, and a \(15-\mu \mathrm{F}\) capacitor. The circuit is used in estimating the speed of a horse running a \(4-km\) racetrack. The switch closes when the horse begins and opens when the horse crosses the finish line. Assuming that the capacitor charges to \(85.6 \mathrm{~V}\), calculate the speed of the horse. Equation Transcription: Text Transcription: 120-V 34-M ohms 15-mu F 4-km 85.6 V

A capacitor with a value of \(10 \mathrm{mF}\) has a leakage resistance of \(2 \mathrm{M} \Omega\). How long does it take the voltage across the capacitor to decay to \(40 \%\) of the initial voltage to which the capacitor is charged? Assume that the capacitor is charged and then set aside by itself. Equation Transcription: Text Transcription: 10 mF 2 M ohms 40%

A simple relaxation oscillator circuit is shown in Fig. 7.145. The neon lamp fires when its voltage reaches \(75 \mathrm{~V}\) and turns off when its voltage drops to \(30 \mathrm{~V}\). Its resistance is \(120 \Omega\) when on and infinitely high when off. (a) For how long is the lamp on each time the capacitor discharges? (b) What is the time interval between light flashes? Equation Transcription: Text Transcription: 75 V 30 V 120 ohms

Figure 7.146 shows a circuit for setting the length of time voltage is applied to the electrodes of a welding machine. The time is taken as how long it takes the capacitor to charge from \(0\) to \(8\ V\). What is the time range covered by the variable resistor? Equation Transcription: Text Transcription: 8 V

A 120 -V dc generator energizes a motor whose coil has an inductance of \(50 \mathrm{H}\) and a resistance of \(100 \Omega\). A field discharge resistor of \(400 \Omega\) is connected in parallel with the motor to avoid damage to the motor, as shown in Fig. 7.147. The system is at steady state. Find the current through the discharge resistor \(100 \mathrm{~ms}\) after the breaker is tripped. Equation Transcription: Text Transcription: 120-V dc 50 H 100 ohms 400 ohms

The circuit in Fig. 7.148(a) can be designed as an approximate differentiator or an integrator, depending on whether the output is taken across the resistor or the capacitor, and also on the time constant \(\tau=R C\) of the circuit and the width \(T\) of the input pulse in Fig. 7.148(b). The circuit is a differentiator if \(\tau \ll T\), say \(\tau<0.1 T\), or an integrator if \(\tau \gg T\), say \(\tau>10 T\). (a) What is the minimum pulse width that will allow a differentiator output to appear across the capacitor? (b) If the output is to be an integrated form of the input, what is the maximum value the pulse width can assume? Equation Transcription: Text Transcription: tau = RC tau<<T tau>>T tau< 0.1T T tau> 10T

An RL circuit may be used as a differentiator if the output is taken across the inductor and \(\tau \ll T\) (say \(\tau<0.1 T\) ), where \(T\) is the width of the input pulse. If \(R\) is fixed at \(200 \mathrm{k} \Omega\), determine the maximum value of \(L\) required to differentiate a pulse with \(T=10 \mu \mathrm{s}\). Equation Transcription: Text Transcription: tau << T tau < 0.1T T L R 200 k ohms T = 10 mu s

An attenuator probe employed with oscilloscopes was designed to reduce the magnitude of the input voltage \(v_{i}\) by a factor of 10. As shown in Fig. 7.149, the oscilloscope has internal resistance \(R_{s}\) and capacitance \(C_{s}\), while the probe has an internal resistance \(R_{p}\). If \(R_{p}\) is fixed at \(6 \mathrm{M} \Omega\), find \(R_{s}\) and \(C_{s}\) for the circuit to have a time constant of \(15 \mu \mathrm{s}\). Equation Transcription: Text Transcription: v_i R_s C_s R_p 6 M ohms

The circuit in Fig. 7.150 is used by a biology student to study “frog kick.” She noticed that the frog kicked a little when the switch was closed but kicked violently for \(5\ s\) when the switch was opened. Model the frog as a resistor and calculate its resistance. Assume that it takes \(10\ mA\) for the frog to kick violently. Equation Transcription: Text Transcription: 5 s 10 mA

To move a spot of a cathode-ray tube across the screen requires a linear increase in the voltage across the deflection plates, as shown in Fig. 7.151. Given that the capacitance of the plates is \(4\ nF\), sketch the current flowing through the plates. Equation Transcription: Text Transcription: 4 nF