For the op amp circuit in Fig. 8.108, find the differential equation for i(t)

The switch in Fig.8.4 was open for a long time but closed at \(t=0\). Determine: (a) \(i\left(0^{+}\right), v\left(0^{+}\right)\), (b) \(d i\left(0^{+}\right) / d t, d v\left(0^{+}\right) / d t\), (c) \(i(\infty), v(\infty)\). Equation Transcription: Text Transcription: t = 0 i(0^+), v(0^+) di(0^+)/dt, dv(0^+)/dt i(infinity), v(infinity)

For the circuit in Fig. 8.7, find: (a) \(i_{L}\left(0^{+}\right), v_{C}\left(0^{+}\right), v_{R}\left(0^{+}\right)\), (b) \(d i_{L}\left(0^{+}\right) / d t, d v_{C}\left(0^{+}\right) / d t, d v_{R}\left(0^{+}\right) / d t\), (c) \(i_{L}(\infty), v_{C}(\infty), v_{R}(\infty)\). Equation Transcription: Text Transcription: i_L(0^+), v_C(0^+), v_R(0^+) di_L(0^+)/dt, dv_C(0^+)/dt, dv_R(0^+)/dt i_L(infinity), v_C(infinity), v_R(infinity)

If \(R=10 \Omega, L=5 \mathrm{H}\), and \(C=2 \mathrm{mF}\) in Fig. 8.8, find \(\alpha, \omega_{0}, s_{1}\), and \(s_{2}\). What type of natural response will the circuit have? Equation Transcription: Text Transcription: R = 10 ohms, L = 5 H, C = 2 mF alpha, watt_0, s_1 s_2

The circuit in Fig. 8.12 has reached steady state at \(t = 0\). If the make before-break switch moves to position \(b\) at \(t = 0\), calculate \(i(t)\) for \(t > 0\). Equation Transcription: Text Transcription: t = 0 b t > 0

In Fig. 8.13, let \(R=2 \Omega, L=0.4 \mathrm{H}, C=25 \mathrm{mF}, v(0)=0, i(0)=50 \mathrm{~mA}\). Find \(v(t)\) for \(t>0\). Equation Transcription: Text Transcription: R = 2 ohms, L = 0.4 H, C = 25 mF, v(0) = 0, i(0) = 50 mA v(t) t > 0

Refer to the circuit in Fig. 8.17. Find \(v(t)\) for \(t > 0\). Equation Transcription: Text Transcription: v(t) t > 0

Having been in position \(a\) for a long time, the switch in Fig. 8.21 is moved to position \(b\) at \(t=0\). Find \(v(t)\) and \(v_{R}(t)\) for \(t>0\). Equation Transcription: Text Transcription: b t = 0 v(t) v_R (t) t > 0

Find \(i(t)\) and \(v(t)\) for \(t > 0\) in the circuit of Fig. 8.24. Equation Transcription: Text Transcription: i(t) v(t) t>0

Determine \(v\) and \(i\) for \(t > 0\) in the circuit of Fig. 8.28. (See comments Practice Problem 8.9 about current sources in Practice Prob. 7.5.) Equation Transcription: Text Transcription: v i t>0

For \(t>0\), obtain \(v_{o}(t)\) in the circuit of Fig. 8.32. ( Hint: First find \(v_{1}\) and \(v_{2}\).) Equation Transcription: Text Transcription: t>0 v_o(t) v_1 v_2

In the op amp circuit shown in Fig. 8.34, \(v_{s}=25 u(t) \mathrm{V}\), find \(v_{o}(t)\) for \(t>0 .\) Assume that \(R_{1}=R_{2}=10 \mathrm{k} \Omega, C_{1}=20 \mu \mathrm{F}\), and \(C_{2}=100 \mu \mathrm{F}\). Equation Transcription: Text Transcription: v_s= 25u(t) V v_o(t) t > 0 R_1= R_2= 10 k ohms, C_1= 20 mu F C_2= 100 mu F

Find \(i(t)\) using PSpice for \(0 < t < 4\ s\) if the pulse voltage in Fig. 8.35(a) is applied to the circuit in Fig. 8.38. Equation Transcription: Text Transcription: i(t) 0 < t < 4

Refer to the circuit in Fig. 8.21 (see Practice Prob. 8.7). Use PSpice to obtain \(v(t)\) for \(0 < t < 2\). Equation Transcription: Text Transcription: v(t) 0 < t < 2

Draw the dual circuit of the one in Fig. 8.46.

For the circuit in Fig. 8.50, obtain the dual circuit.

In Fig. 8.52, find the capacitor voltage \(v_{C}\) for \(t>0\). Equation Transcription: Text Transcription: t>0 v_C

Rework Example 8.17 if the output of the \(D/A\) converter is as shown in Fig. 8.56. Equation Transcription: Text Transcription: D/A

Find the voltage across the capacitor as a function of time for \(t > 0\) for the circuit in Fig. 8.72. Assume steady-state conditions exist at \(t = 0\)-. Equation Transcription: Text Transcription: t > 0 t = 0

Obtain \(v(t)\) for \(t > 0\) in the circuit of Fig. 8.73. Equation Transcription: Text Transcription: v(t) t > 0

The switch in the circuit of Fig. 8.74 has been closed for a long time but is opened at \(t = 0\). Determine \(i(t)\) for \(t > 0\). Equation Transcription: Text Transcription: t = 0 i(t) t > 0

Calculate \(v(t)\) for \(t > 0\) in the circuit of Fig. 8.75. Equation Transcription: Text Transcription: v(t) t > 0

Assuming \(R=2 \mathrm{k} \Omega\), design a parallel \(R L C\) circuit that has the characteristic equation \(s^{2}+100 s+10^{6}=0\). Equation Transcription: Text Transcription:

For the network in Fig. 8.76, what value of \(C\) is needed to make the response underdamped with unity neper frequency \((\alpha=1)\) ? Equation Transcription: Text Transcription: C (alpha= 1)

The switch in Fig. 8.77 moves from position \(A\) to position \(B\) at \(t = 0\) (please note that the switch must connect to point \(B\) before it breaks the connection at \(A\), a make-before-break switch). Determine \(i(t)\) for \(t > 0\). Equation Transcription: Text Transcription: A B t = 0 i(t) t > 0

Using Fig. 8.78, design a problem to help other students better understand source-free \(RLC\) circuits. Equation Transcription: Text Transcription: RLC

he step response of an \(R L C\) circuit is given by \(\frac{d^{2} i}{d t^{2}}+2 \frac{d i}{d t}+5 i=10\) Given that \(i(0)=2\) and \(d i(0) / d t=4\), solve for \(i(t)\). Equation Transcription: Text Transcription: RLC d^2i/dt^2+2di/dt+5i=10 i(0) = 2 di(0)/dt = 4 i(t)

A branch voltage in an \(RLC\) circuit is described by \(\frac{d^{2} v}{d t^{2}}+4 \frac{d v}{d t}+8 v=24\) If the initial conditions are \(v(0) = 0 = dv(0)/dt\), find \(v(t)\). Equation Transcription: Text Transcription: RLC d^2v/dt^2+4dv/dt+8v=24 v(0) = 0 = dv(0)/dt v(t)

A series \(R L C\) circuit is described by \(L \frac{d^{2} i}{d t^{2}}+R \frac{d i}{d t}+\frac{i}{C}=10\) Find the response when \(L=0.5 \mathrm{H}, R=4 \Omega\), and \(C=0.2 \mathrm{~F}\). Let \(i(0)=1, d i(0) / d t=0\). Equation Transcription: Text Transcription: RLC L d^2i/dt^2+R di/dt+i/C=10 L = 0.5 H, R = 4 ohms C = 0.2 F i(0) = 1, di(0)/dt = 0

Solve the following differential equations subject to the specified initial conditions (a) \(d^{2} v / d t^{2}+4 v=12, v(0)=0, d v(0) / d t=2\) (b) \(d^{2} i / d t^{2}+5 d i / d t+4 i=8, i(0)=-1\), \(d i(0) / d t=0\) (c) \(d^{2} v / d t^{2}+2 d v / d t+v=3, v(0)=5\), \(d v(0) / d t=1\) (d) \(d^{2} i / d t^{2}+2 d i / d t+5 i=10, i(0)=4\), \(d i(0) / d t=-2\) Equation Transcription: Text Transcription: d^2v/dt^2 + 4v = 12, v(0) = 0, dv(0)/dt = 2 d^2i/dt^2 + 5 di/dt + 4i = 8, i(0) =-1, di(0)/dt = 0 d^2v/dt^2+ 2 dv/dt + v = 3, v(0) = 5, dv(0)/dt = 1 d^2i/dt^2+ 2 di/dt + 5i = 10, i(0) = 4, di(0)/dt =-2

he step responses of a series \(R L C\) circuit are \(v_{C}=40-10 e^{-2000 t}-10 e^{-4000 t} \mathrm{~V}, t>0\) \(i_{L}(t)=3 e^{-2000 t}+6 e^{-4000 t} \mathrm{~mA}, t>0\) (a) Find \(C\). (b) Determine what type of damping is exhibited by the circuit. Equation Transcription: Text Transcription: v_C= 40 ? 10e^-2000t -10e^-4000t V, t > 0 i_L(t) = 3e^-2000t+ 6e^-4000t mA, t > 0

Consider the circuit in Fig. 8.79. Find \(v_{L}\left(0^{+}\right)\) and \(v_{C}\left(0^{+}\right)\). Equation Transcription: Text Transcription: v_L(0^+) v_C(0^+)

For the circuit in Fig. 8.80, find \(v(t)\) for \(t > 0\). Equation Transcription: Text Transcription: v(t) t > 0

Find \(v(t)\) for \(t > 0\) in the circuit of Fig. 8.81. Equation Transcription: Text Transcription: v(t) t > 0

Calculate \(i(t)\) for \(t > 0\) in the circuit of Fig. 8.82. Equation Transcription: Text Transcription: i(t) t > 0

Using Fig. 8.83, design a problem to help other students better understand the step response of series \(RLC\) circuits. Equation Transcription: Text Transcription: RLC

Obtain \(v(t)\) and \(i(t)\) for \(t > 0\) in the circuit of Fig. 8.84. Equation Transcription: Text Transcription: v(t) t > 0 i(t)

For the network in Fig. 8.85, solve for \(i(t)\) for \(t > 0\). Equation Transcription: Text Transcription: i(t) t > 0

Refer to the circuit in Fig. 8.86. Calculate \(i(t)\) for \(t > 0\). Equation Transcription: Text Transcription: i(t) t > 0

Determine \(v(t)\) for \(t > 0\) in the circuit of Fig. 8.87. Equation Transcription: Text Transcription: v(t) t > 0

The switch in the circuit of Fig. 8.88 is moved from position \(a\) to \(b\) at \(t = 0\). Assume that the voltage across the capacitor is equal to zero at \(t = 0\) and that the switch is a make before break switch. Determine \(i(t)\) for all \(t > 0\). Equation Transcription: Text Transcription: a b t=0 i(t) t > 0

For the network in Fig. 8.89, find \(i(t)\) for \(t > 0\). Equation Transcription: Text Transcription: i(t) t > 0

Given the network in Fig. 8.90, find \(v(t)\) for \(t > 0\). Equation Transcription: Text Transcription: v(t) t > 0

The switch in Fig.8.91 is opened at \(t=0\) after the circuit has reached steady state. Choose \(R\) and \(C\) such that \(\alpha=8 \mathrm{~Np} / \mathrm{s}\) and \(\omega_{d}=30 \mathrm{rad} / \mathrm{s}\). Equation Transcription: Text Transcription: t = 0 R C alpha= 8 Np/s watt_d= 30 rad/s

A series \(R L C\) circuit has the following parameters: \(R=1 \mathrm{k} \Omega, L=1 \mathrm{H}\), and \(C=10 \mathrm{nF}\). What type of damping does this circuit exhibit? Equation Transcription: Text Transcription: RLC R = 1 k ohms, L = 1 H C = 10 nF

In the circuit of Fig. 8.92, find \(v(t)\) and \(i(t)\) for \(t > 0\). Equation Transcription: Text Transcription: v(t) t > 0 i(t)

Using Fig. 8.93, design a problem to help other students better understand the step response of a parallel \(RLC\) circuit. Equation Transcription: Text Transcription: RLC

Find the output voltage \(v_{o}(t)\) in the circuit of Fig. $8.94$. Equation Transcription: Text Transcription: v_o(t)

Given the circuit in Fig. 8.95, find \(i(t)\) and \(v(t)\) for \(t > 0\). Equation Transcription: Text Transcription: v(t) t > 0 i(t)

Determine \(i(t)\) for \(t > 0\) in the circuit of Fig. 8.96. Equation Transcription: Text Transcription: i(t) t > 0

For the circuit in Fig. 8.97, find \(i(t)\) for \(t > 0\). Equation Transcription: Text Transcription: i(t) t > 0

Find \(v(t)\) for \(t > 0\) in the circuit of Fig. 8.98. Equation Transcription: Text Transcription: v(t) t > 0

The step response of a parallel \(R L C\) circuit is \(v=10+20 e^{-300 t}(\cos 400 t-2 \sin 400 t) \mathrm{V}, \quad t \geq 0\) when the inductor is \(25 \mathrm{mH}\). Find \(R\) and \(C\). Equation Transcription: Text Transcription: RLC v = 10 + 20e^-300t(cos 400t-2 sin 400t) V, t > or =0 25 mH R C

After being open for a day, the switch in the circuit of Fig. 8.99 is closed at \(t = 0\). Find the differential equation describing \(i(t), t > 0\). Equation Transcription: Text Transcription: t = 0 i(t), t > 0

Using Fig. 8.100, design a problem to help other students better understand general second-order circuits.

For the circuit in Fig. 8.101, find \(v(t)\) for \(t>0\). Assume that \(i\left(0^{+}\right)=2\) A. Equation Transcription: Text Transcription: v(t) t > 0 i(0^+) = 2 A

In the circuit of Fig. 8.102, find \(I(t)\) for \(t > 0\). Equation Transcription: Text Transcription: i(t) t > 0

Given the circuit shown in Fig. 8.103, determine the characteristic equation of the circuit and the values for \(i(t)\) and \(v(t)\) for all \(t > 0\). Equation Transcription: Text Transcription: v(t) t > 0 i(t)

In the circuit of Fig.8.104, the switch has been in position 1 for a long time but moved to position 2 at \(t=0\). Find: (a) \(v\left(0^{+}\right), d v\left(0^{+}\right) / d t\) (b) \(v(t)\) for \(t \geq 0\). Equation Transcription: Text Transcription: t = 0 v(0^+), dv(0^+)/dt v(t) t> or =0

The switch in Fig. 8.105 has been in position 1 for \(t < 0\). At \(t = 0\), it is moved from position 1 to the top of the capacitor at \(t = 0\). Please note that the switch is a make before break switch, it stays in contact with position 1 until it makes contact with the top of the capacitor and then breaks the contact at position 1. Given that the initial voltage across the capacitor is equal to zero, determine \(v(t)\). Equation Transcription: Text Transcription: t < 0 t = 0 v(t)

Obtain \(i_{1}\) and \(i_{2}\) for \(t>0\) in the circuit of Fig. 8.106. Equation Transcription: Text Transcription: i_1 i_2 t > 0

For the circuit in Prob. 8.5, find \(i\) and \(v\) for \(t > 0\). Equation Transcription: Text Transcription: v i t > 0

Find the response \(v_{R}(t)\) for \(t>0\) in the circuit of Fig. 8.107. Let \(R=8 \Omega, L=2 \mathrm{H}\), and \(C=125 \mathrm{mF}\). Equation Transcription: Text Transcription: v_R(t) t > 0 R = 8 ohms, L = 2 H C= 125 mF

For the op amp circuit in Fig. 8.108, find the differential equation for \(i(t)\). Equation Transcription: Text Transcription: i(t)

Using Fig. 8.109, design a problem to help other students better understand second-order op amp circuits.

Determine the differential equation for the op amp circuit in Fig. 8.110. If \(v_{1}\left(0^{+}\right)=2 \mathrm{~V}\) and \(v_{2}\left(0^{+}\right)=0 \mathrm{~V}\), find \(v_{o}\) for \(t>0\). Let \(R=100 \mathrm{k} \Omega\) and \(C=1 \mu \mathrm{F}\). Equation Transcription: Text Transcription: v_1(0^+) = 2 V v_2(0^+) = 0 V v_o t > 0 R = 100 k C = 1 mu F

Obtain the differential equations for \(v_{o}(t)\) in the op amp circuit of Fig. 8.111. Equation Transcription: Text Transcription: v_o(t)

In the op amp circuit of Fig. 8.112, determine \(v_{o}(t)\) for \(t>0\). Let \(v_{\mathrm{wn}}=u(t) \mathrm{V}, R_{1}=R_{2}=10 \mathrm{k} \Omega, C_{1}=C_{2}\) \(=100 \mu \mathrm{F}\). Equation Transcription: Text Transcription: v_o(t) t > 0 v_in= u(t) V, R_1= R_2= 10 k ohms, C_1= C_2= 100 mu F

For the step function \(v_{s}=u(t)\), use PSpice or MultiSim to find the response \(v(t)\) for \(0<t<6 \mathrm{~s}\) in the circuit of Fig. 8.113. Equation Transcription: Text Transcription: v_s= u(t) v(t) 0 < t < 6 s

Given the source-free circuit in Fig. 8.114, use PSpice or MultiSim to get \(i(t)\) for \(0 < t < 20 s\). Take \(v(0) = 30 V\) and \(i(0) = 2 A\). Equation Transcription: Text Transcription: i(t) 0 < t < 20 s v(0) = 30 V i(0) = 2 A

For the circuit in Fig. 8.115, use PSpice or MultiSim to obtain \(v(t)\) for \(0 < t < 4 s\). Assume that the capacitor voltage and inductor current at \(t = 0\) are both zero. Equation Transcription: Text Transcription: v(t) 0 < t < 4 s t = 0

Obtain \(v(t)\) for \(0 < t < 4 s\) in the circuit of Fig. 8.116 using PSpice or MultiSim. Equation Transcription: Text Transcription: v(t) 0 < t < 4 s

The switch in Fig. 8.117 has been in position 1 for a long time. At \(t = 0\), it is switched to position 2. Use PSpice or MultiSim to find \(i(t)\) for \(0 < t < 0.2 s\). Equation Transcription: Text Transcription: t = 0 i(t) 0 < t < 0.2 s

Design a problem, to be solved using PSpice or MultiSim, to help other students better understand source-free RLC circuits.

Draw the dual of the circuit shown in Fig. 8.118.

Obtain the dual of the circuit in Fig. 8.119.

Find the dual of the circuit in Fig. 8.120.

Draw the dual of the circuit in Fig. 8.121.

An automobile airbag igniter is modeled by the circuit in Fig. 8.122. Determine the time it takes the voltage across the igniter to reach its first peak after switching from \(A\) to \(B\). Let \(R=3 \Omega, C=1 / 30 \mathrm{~F}\), and \(L=60 \mathrm{mH}\). Equation Transcription: Text Transcription: A B R = 3 ohms, C = 1/30 F L = 60 mH

A load is modeled as a \(100-\mathrm{mH}\) inductor in parallel with a \(12-\Omega\) resistor. A capacitor is needed to be connected to the load so that the network is critically damped at \(60 \mathrm{~Hz}\). Calculate the size of the capacitor. Equation Transcription: Text Transcription: 100-mH 12-ohms 60 Hz

A mechanical system is modeled by a series \(R L C\) circuit. It is desired to produce an overdamped response with time constants \(0.1\) and \(0.5 \mathrm{~ms}\). If a series \(50-k \Omega\) resistor is used, find the values of \(L\) and \(C\). Equation Transcription: Text Transcription: RLC 0.5 ms 50-k ohms L C

An oscillogram can be adequately modeled by a second-order system in the form of a parallel \(R L C\) circuit. It is desired to give an underdamped voltage across a (200-k \Omega\) resistor. If the damped frequency is \(4 \mathrm{kHz}\) and the time constant of the envelope is \(0.25 \mathrm{~s}\), find the necessary values of \(L\) and \(C\). Equation Transcription: Text Transcription: RLC 200-ohms 4 kHz 0.25 s L C

The circuit in Fig.8.123 is the electrical analog of body functions used in medical schools to study convulsions. The analog is as follows: \(C_{1}=\) Volume of fluid in a drug \(C_{2}=\) Volume of blood stream in a specified region \(R_{1}=\) Resistance in the passage of the drug from the input to the blood stream \(R_{2}=\) Resistance of the excretion mechanism, such as kidney, etc. \(v_{0}=\) Initial concentration of the drug dosage \(v(t)=\) Percentage of the drug in the blood stream Find \(v(t)\) for \(t>0\) given that \(C_{1}=0.5 \mu \mathrm{F}, C_{2}=\) \(5 \mu \mathrm{F}, R_{1}=5 \mathrm{M} \Omega, R_{2}=2.5 \mathrm{M} \Omega\), and \(v_{0}=60 u(t) \mathrm{V}\). Equation Transcription: Text Transcription: C_1 C_2 R_1 R_2 v(t) v_o t > 0 C_1= 0.5 mu F, C_2= 5 mu F, R_1= 5 M ohms, R_2= 2.5 M ohms v_o= 60u(t) V

Figure 8.124 shows a typical tunnel-diode oscillator circuit. The diode is modeled as a nonlinear resistor with \(i_{D}=f\left(v_{D}\right)\), i.e., the diode current is a nonlinear function of the voltage across the diode. Derive the differential equation for the circuit in terms of \(v\) and \(i_{D}\). Equation Transcription: Text Transcription: i_D= f (v_D) v i_D