Determine the characteristic equation of each of the following differential equations

Determine the current \(i_R\) through the resistor of Fig. 8.3 at t = 1 ns if \(i_R(0) = 6\ A\).

Determine the inductor voltage v in the circuit of Fig. 8.6 for t > 0.

In a source-free series RL circuit, find the numerical value of the ratio: (a) \(i (2 \tau)/i(\tau)\); (b) \(i (0.5 \tau)/i (0)\); (c) \(t/ \tau\) if i (t)/i (0) = 0.2; (d) \(t/ \tau\) if i (0) ? i (t) = i (0) ln 2.

Noting carefully how the circuit changes once the switch in the circuit of Fig. 8.18 is thrown, determine v(t) at t = 0 and at \(t = 160\ \mu s\).

At t = 0.15 s in the circuit of Fig. 8.21, find the value of (a) \(i_L\); (b) \(i_1\); (c) \(i_2\).

Find values of \(v_C\) and \(v_o\) in the circuit of Fig. 8.23 at t equal to (a) \(0^?\); (b) \(0^+\); (c) 1.3 ms.

(a) Regarding the circuit of Fig. 8.25, determine the voltage \(v_C(t)\) for t > 0 if \(v_C(0^?) = 11\ V\). (b) Is the circuit “stable”?

Evaluate each of the following at \(t = 0.8\): (a) \(3u(t) ? 2u(?t) + 0.8u(1 ? t)\); (b) \([4u(t)]u(?t)\); (c) \(2u(t)\ sin\ \pi t\).

The voltage source 60 ? 40u(t) V is in series with a \(10\ \Omega\) resistor and a 50 mH inductor. Find the magnitudes of the inductor current and voltage at t equal to (a) \(0^?\); (b) \(0^+\); (c) \(\infty\); (d) 3 ms.

A voltage source, \(v_s = 20u(t)\ V\), is in series with a \(200\ \Omega\) resistor and a 4 H inductor. Find the magnitude of the inductor current at t equal to (a) \(0^?\); (b) \(0^+\); (c) 8 ms; (d) 15 ms.

The circuit shown in Fig. 8.41 has been in the form shown for a very long time. The switch opens at t = 0. Find \(i_R\) at t equal to (a) \(0^?\); (b) \(0^+\); (c) \(\infty\); (d) 1.5 ms.

For the circuit of Fig. 8.44, find \(v_C(t)\) at t equal to (a) \(0^?\); (b) \(0^+\); (c) \(\infty\); (d) 0.08 s.

Determine the capacitor voltage v in the circuit of Fig. 8.46 for t >0.

With regard to Fig. 8.50a, sketch \(i_L (t)\) in the range of \(0\ <\ t\ <\ 6\ s\) for (a) \(v_S(t) = 3u(t) ? 3u(t ?2) + 3u(t ? 4) ? 3u(t ? 6)+ · · ·\); (b) \(v_S(t) = 3u(t) ? 3u(t ? 2) + 3u(t ? 2.1) ? 3u(t ? 4.1)+· · ·\).

Setting \(R = 1\ k \Omega\) and L = 1 nH for the circuit represented in Fig. 8.1, and with the knowledge that \(i(0) = ?3\ mA\), (a) write an expression for i(t) valid for all \(t\ \geq\ 0\); (b) compute i(t) at t = 0, t = 1 ps, 2 ps, and 5 ps; and (c) calculate the energy stored in the inductor at t = 0, t = 1 ps, and t = 5 ps.

If \(i(0) = 1\ A\) and \(R = 100\ \Omega\) for the circuit of Fig. 8.1, (a) select L such that \(i(50 ms) = 368\ mA\); (b) compute the energy stored in the inductor at t = 0, 50 ms, 100 ms, and 150 ms.

Referring to the circuit shown in Fig. 8.1, select values for both elements such that L/R = 1 and (a) calculate \(v_R(t)\) at t = 0, 1, 2, 3, 4, and 5 s; (b) compute the power dissipated in the resistor at t = 0, 1 s, and 5 s. (c) At t = 5 s, what is the percentage of the initial energy still stored in the inductor?

The circuit depicted in Fig. 8.1 is constructed from components whose value is unknown. If a current i(0) of \(6\ \mu A\) initially flows through the inductor, and it is determined that \(i(1 ms) = 2.207\ \mu A\), calculate the ratio of R to L.

Determine the characteristic equation of each of the following differential equations: (a) \(5v + 14 \frac{dv}{dt} = 0\); (b) \(-9 \frac{di}{dt} - 18i = 0\); (c) \(\frac{di}{dt} + 18i + \frac{R}{B} i = 0\); (d) \(\frac{d^2 f}{dt^2} + 8 \frac{df}{dt} + 2f = 0\).

For the following characteristic equations, write corresponding differential equations and find all roots, whether real, imaginary, or complex: (a) 4s + 9 = 0; (b) 2s ? 4 = 0; (c) \(s^2 + 7s + 1 = 0\); (d) \(5s^2 + 8s + 18 = 0\).

With the assumption that the switch in the circuit of Fig. 8.54 has been closed a long, long, long time, calculate \(i_L(t)\) at (a) the instant just before the switch opens; (b) the instant just after the switch opens; (c) \(t = 15.8\ \mu s\); (d) \(t = 31.5\ \mu s\); (e) \(t = 78.8\ \mu s\).

The switch in Fig. 8.54 has been closed since Catfish Hunter last pitched for the New York Yankees. Calculate the voltage labeled v as well as the energy stored in the inductor at (a) the instant just prior to the switch being thrown open; (b) the instant just after the switch is opened; (c) \(t = 8\ \mu s\); (d) \(t = 80\ \mu s\).

Assuming the switch initially has been open for a really, really long time, (a) obtain an expression for \(i_W\) in the circuit of Fig. 8.56 which is valid for all \(t\ \geq\ 0\); (b) calculate \(i_W\) at t = 0 and t = 1.3 ns.

(a) Graph the function \(f(t) = 10e^{?2t}\) over the range of \(0\ \leq\ t\ \leq\ 2.5\ s\) using linear scales for both y and x axes. (b) Replot with a logarithmic scale for the y axis. [Hint: the function semilogy() can be helpful here.] (c) What are the units of the 2 in the argument of the exponential? (d) At what time does the function reach a value of 9? 8? 1?

The current i(t) flowing through a \(1\ \Omega\) resistor is given by \(i(t) = 5e^{?10t}\ mA,\ t\ \geq\ 0\). (a) Determine the values of t for which the resistor voltage magnitude is equal to 5 V, 2.5 V, 0.5 V, and 5 mV. (b) Graph the function over the range of \(0\ \leq\ t\ \leq\ 1\ s\) using linear scales for both axes. (c) Draw a tangent to your curve at t = 100 ms, and determine where the tangent intersects the time axis.

The thickness of a solar cell must be chosen carefully to ensure photons are properly absorbed; even metals can be partly transparent when rolled out into very thin foils. If the incident light flux (number of photons per unit area per unit time) at the solar cell surface (x = 0) is given by \(\Phi_{0}\), and the intensity of light a distance x inside the solar cell is given by \(\Phi(x)\), the behavior of \(\Phi (x)\) is described by the equation \(d \Phi/dx + \alpha \Phi = 0\). Here, \(\alpha\), known as the absorption coefficient, is a constant specific to a given semiconductor material. (a) What is the SI unit for \(\alpha\)? (b) Obtain an expression for \(\Phi(x)\) in terms of \(\Phi_{0}\), \(\alpha\), and x. (c) How thick should the solar cell be made in order to absorb at least 38% of the incident light? Express your answer in terms of \(\alpha\). (d) What happens to the light which enters the solar cell at x = 0 but is not absorbed?

For the circuit of Fig. 8.5, compute the time constant if the \(10\ \Omega\) resistor is replaced with (a) a short circuit; (b) a \(1\ \Omega\) resistor; (c) a series connection of two \(5\ \Omega\) resistors; (d) a \(100\ \Omega\) resistor. (e) Verify your answers with a suitable parameter sweep simulation. (Hint: the cursor tool might come in handy, and the answer does not depend on the initial current you choose for the inductor.)

The resistor in the circuit of Fig. 8.57 has been included to model the dielectric layer separating the plates of the 3.1 nF capacitor, and has a value of \(55\ M \Omega\). The capacitor is storing 200 mJ of energy just prior to t = 0. (a) Write an expression for v(t) valid for \(t\ \geq\ 0\). (b) Compute the energy remaining in the capacitor at t = 170 ms. (c) Graph v(t) over the range of \(0\ <\ t\ <\ 850\ ms\), and identify the value of v(t) when \(t = 2 \tau\).

The resistor in the circuit of Fig. 8.57 has a value of \(1\ \Omega\) and is connected to a 22 mF capacitor. The capacitor dielectric has infinite resistance, and the device is storing 891 mJ of energy just prior to t = 0. (a) Write an expression for v(t) valid for \(t\ \geq\ 0\). (b) Compute the energy remaining in the capacitor at t = 11 ms and 33 ms. (c) If it is determined that the capacitor dielectric is much leakier than expected, having a resistance as low as \(100\ k \Omega\), repeat parts (a) and (b).

Calculate the time constant of the circuit depicted in Fig. 8.57 if C = 10 mF and R is equal to (a) \(1\ \Omega\); (b) \(10\ \Omega\); (c) \(100\ \Omega\). (d) Verify your answers with an appropriate parameter sweep simulation. (Hint: the cursor tool might come in handy, and the time constant does not depend on the initial voltage across the capacitor.)

Design a capacitor-based circuit that will provide (a) a voltage of 9 V at some time t = 0, and a voltage of 1.2 V at a time 4 ms later; (b) a current of 1 mA at some time t = 0, and a reduced current of \(50\ \mu A\) at a time 100 ns later. (You can choose to design two separate circuits if desired, and do not need to show how the initial capacitor voltage is set.)

It is safe to assume that the switch drawn in the circuit of Fig. 8.58 has been closed such a long time that any transients which might have arisen from first connecting the voltage source have disappeared. (a) Determine the circuit time constant. (b) Calculate the voltage v(t) at \(t = \tau\), \(2 \tau\), and \(5 \tau\).

We can safely assume the switch in the circuit of Fig. 8.59 was closed a very long time prior to being thrown open at t = 0. (a) Determine the circuit time constant. (b) Obtain an expression for \(i_1(t)\) which is valid for t > 0. (c) Determine the power dissipated by the \(12\ \Omega\) resistor at t = 500 ms.

For the circuit represented schematically in Fig. 8.61, (a) calculate v(t) at t = 0, t = 984 s, and t = 1236 s; (b) determine the energy still stored in the capacitor at t = 100 s.

For the circuit depicted in Fig. 8.62, (a) compute the circuit time constant; (b) determine v in the instant just before the switch is closed; (c) obtain an expression for v(t) valid for t > 0; (d) calculate v(3 ms).

The switch drawn in Fig. 8.62 has been open a ponderously long time. (a) Determine the value of the current labeled i just prior to the switch being closed. (b) Obtain the value of i just after the switch is closed. (c) Compute the power dissipated in each resistor over the range of 0 < t < 15 ms. (d) Graph your answer to part (c).

(a) Obtain an expression for v(t), the voltage which appears across resistor \(R_3\) in the circuit of Fig. 8.63, which is valid for t > 0. (b) If \(R_1 = 2R_2 = 3R_3 = 4R_4 = 1.2\ k \Omega\), L = 1 mH, and \(i_L(0^?) = 3\ mA\), calculate v(t = 500 ns).

The switch shown in Fig. 8.65 has been closed for 6 years prior to being flipped open at t = 0. Determine \(i_L\), \(v_L\), and \(v_R\) at t equal to (a) \(0^?\); (b) \(0^+\); (c) \(1\ \mu s\); (d) \(10\ \mu s\).

Obtain expressions for both \(i_1(t)\) and \(i_L(t)\) as labeled in Fig. 8.66, which are valid for t > 0.

The voltage across the resistor in a simple source-free RL circuit is given by \(5e^{?90t}\ V\), t > 0. The inductor value is not known. (a) At what time will the inductor voltage be exactly one-half of its maximum value? (b) At what time will the inductor current reach 10% of its maximum value?

Referring to Fig. 8.67, calculate the currents \(i_1\) and \(i_2\) at t equal to (a) 1 ms; (b) 3 ms.

(a) Obtain an expression for \(v_x\) as labeled in the circuit of Fig. 8.68. (b) Evaluate \(v_x\) at t = 5 ms. (c) Verify your answer with an appropriate PSpice simulation. (Hint: employ the part named Sw_tClose.)

For the part Sw_tOpen, PSpice actually employs a sequence of simulations where the part is first replaced with a resistor having value \(1\ M \Omega\), and then replaced with a resistor having value \(10\ m \Omega\) corresponding to when the switch opens. (a) Evaluate the reliability of these default values by simulating the circuit of Fig. 8.55, and evaluating \(i_L\) at t = 1 ns. (b) Repeat part (a) with RCLOSED changed to \(1\ \Omega\). Did this change your answer? (c) Repeat part (a) with ROPEN changed to \(100\ k \Omega\) and RCLOSED reset to its default value. Did this change your answer? (Hint: double-click on the part to access its attributes.)

Select values for the resistors \(R_0\) and \(R_1\) in the circuit of Fig. 8.69 such that \(v_C(0.65) = 5.22\ V\) and \(v_C(2.21) = 1\ V\).

A quick measurement determines that the capacitor voltage \(v_C\) in the circuit of Fig. 8.70 is 2.5 V at \(t = 0^?\). (a) Determine \(v_C(0^+)\), \(i_1(0^+)\), and \(v(0^+)\). (b) Select a value of C so that the circuit time constant is equal to 14 s.

Determine \(v_C(t)\) and \(v_o(t)\) as labeled in the circuit represented by Fig. 8.71 for t equal to (a) \(0^?\); (b) \(0^+\); (c) 10 ms; (d) 12 ms.

For the circuit shown in Fig. 8.72, determine (a) \(v_C(0^?)\); (b) \(v_C(0^+)\); (c) the circuit time constant; (d) \(v_C(3\ ms)\).

The inductor in Fig. 8.74 is storing 54 nJ at \(t = 0^?\). Compute the energy remaining at t equal to (a) 0+; (b) 1 ms; (c) 5 ms.

Evaluate the following functions at t = ?2, 0, and +2: (a) f (t) = 3u(t); (b) g(t) = 5u(?t) + 3; (c) h(t) = 5u(t ? 3); (d) z(t) = 7u(1 ? t) + 4u(t + 3).

Evaluate the following functions at t = ?1, 0, and +3: (a) f(t) = tu(1 ? t); (b) g(t) = 8 + 2u(2 ? t); (c) h(t) = u(t + 1) ? u(t ? 1) + u(t + 2) ? u(t ? 4); (d) z(t) = 1 + u(3 ? t) + u(t ? 2).

Sketch the following functions over the range \(?3\ \leq\ t\ \leq\ 3\) (a) v(t) = 3- u(2 - t) - 2u(t) V; (b) i(t) = u(t) - u(t - 0.5) + u(t - 1) - u(t - 1.5) + u(t - 2) - u(t - 2.5) A; (c) q(t) = 8u(-t) C.

Use step functions to construct an equation that describes the waveform sketched in Fig. 8.75.

Employing step functions as appropriate, describe the voltage waveform graphed in Fig. 8.76.

For the circuit given in Fig. 8.78, (a) determine \(v_L(0^?)\), \(v_L(0^+)\), \(i_L(0^?)\), and \(i_L(0^+)\); (b) calculate \(i_L(150\ ns)\). (c) Verify your answer to part (b) with an appropriate PSpice simulation.

The circuit depicted in Fig. 8.79 contains two independent sources, one of which is only active for t > 0. (a) Obtain an expression for \(i_L(t)\) valid for all t; (b) calculate \(i_L(t)\) at \(t = 10\ \mu s\), \(20\ \mu s\), and \(50\ \mu s\).

The circuit shown in Fig. 8.80 is powered by a source which is inactive for t < 0. (a) Obtain an expression for i(t) valid for all t. (b) Graph your answer over the range of \(-1\ ms\ \leq\ t\ \leq\ 10\ ms\).

For the circuit shown in Fig. 8.81, (a) obtain an expression for i(t) valid for all time; (b) obtain an expression for \(v_{R}(t)\) valid for all time; and (c) graph both i(t) and \(v_{R}(t)\) over the range of \(-1\ s\ \leq\ t\ \leq\ 6\ s\).

For the two-source circuit of Fig. 8.82, note that one source is always on. (a) Obtain an expression for i(t) valid for all t; (b) determine at what time the energy stored in the inductor reaches 99% of its maximum value.

Obtain an expression for i(t) as labeled in the circuit diagram of Fig. 8.84, and determine the power being dissipated in the \(40\ \Omega\) resistor at t = 2.5 ms.

Obtain an expression for \(i_1\) as indicated in Fig. 8.85 that is valid for all values of t.

Plot the current i(t) in Fig. 8.86 if (a) \(R = 10\ \Omega\); (b) \(R = 1\ \Omega\). In which case does the inductor (temporarily) store the most energy? Explain.

(a) Obtain an expression for \(v_C\) in the circuit of Fig. 8.87 valid for all values of t. (b) Sketch \(v_C(t)\) over the range \(0\ \leq\ t\ \leq\ 4\ \mu s\).

Obtain an equation which describes the behavior of \(i_A\) as labeled in Fig. 8.88 over the range of \(-1\ ms\ \leq\ t\ \leq\ 5\ ms\)

The switch in the circuit of Fig. 8.89 has been open a really, really incredibly long time, before being closed without further fanfare at t = 0. (a) Evaluate the current labeled \(i_x\) at t = 70 ms. (b) Verify your answer with an appropriate PSpice simulation.

The “make-before-break” switch shown in Fig. 8.90 has been in position a since the first episode of “Jonny Quest” aired on television. It is moved to position b, finally, at time t = 0. (a) Obtain expressions for i(t) and \(v_C(t)\) valid for all values of t. (b) Determine the energy remaining in the capacitor at \(t = 33\ \mu s\).

The switch in the circuit of Fig. 8.91, often called a make-before-break switch (since during switching it briefly makes contact to both parts of the circuit to ensure a smooth electrical transition), moves to position b at t = 0 only after being in position a long enough to ensure all initial transients arising from turning on the sources have long since decayed. (a) Determine the power dissipated by the \(5\ \Omega\) resistor at \(t = 0^?\). (b) Determine the power dissipated in the \(3\ \Omega\) resistor at t = 2 ms.

The dependent source shown in Fig. 8.92 is unfortunately installed upside down during manufacturing, so that the terminal corresponding to the arrowhead is actually wired to the negative reference terminal of the voltage source. This is not detected by the quality assurance team so the unit ships out wired improperly. The capacitor is initially discharged. If the \(5\ \Omega\) resistor is only rated to 2 W, after what time t is the circuit likely to fail?

For the circuit represented in Fig. 8.93, (a) obtain an expression for v which is valid for all values of t; (b) sketch your result for \(0\ \leq\ t\ \leq\ 3\ s\).

Obtain an expression for the voltage \(v_x\) as labeled in the op amp circuit of Fig. 8.94.

Sketch the current \(i_L\) of the circuit in Fig. 8.50a if the 100 mH inductor is replaced by a 1 nH inductor, and is subjected to the waveform \(v_s(t)\) equal to (a) \(5u(t) ? 5u(t ? 10^{?9}) + 5u(t ? 2\ \times\ 10^{?9})\ V,\ 0\ \leq\ t\ \leq\ 4\ ns\); (b) \(9u(t) ? 5u(t ? 10^{?8}) + 5u(t ? 2\ \times\ 10^{?8})\ V,\ 0\ \leq\ t\ \leq\ 40\ ns\).

The 100 mH inductor in the circuit of Fig. 8.50a is replaced with a 1 H inductor. Sketch the inductor current \(i_L\) if the source \(v_s(t)\) is equal to (a) \(5u(t) ? 5u(t ? 0.01) + 5u(t ? 0.02)\ V,\ 0\ \leq\ t\ \leq\ 40\ ms\); (b) \(5u(t) ? 5u(t ? 10) + 5u(t ? 10.1)\ V,\ 0\ \leq\ t\ \leq\ 11\ s\).

Sketch the voltage \(v_C\) across the capacitor of Fig. 8.95 for at least 3 periods if \(R = 1\ \Omega\), C = 1 F, and \(v_s(t)\) is a pulsed waveform having (a) minimum of 0 V, maximum of 2 V, rise and fall times of 1 ms, pulse width of 10 s, and period of 10 s; (b) minimum of 0 V, maximum of 2 V, rise and fall times of 1 ms, pulse width of 10 ms, and period of 10 ms. (c) Verify your answers with appropriate PSpice simulations.

The circuit in Fig. 8.96 contains two switches that always move in perfect synchronization. However, when switch A opens, switch B closes, and vice versa. Switch A is initially open, while switch B is initially closed; they change positions every 40 ms. Using the bottom node as the reference node, determine the voltage across the capacitor at t equal to (a) \(0^?\); (b) \(0^+\); (c) \(40^?\ ms\); (d) \(40^+\ ms\); (e) 50 ms.

In the circuit of Fig. 8.96, when switch A opens, switch B closes, and vice versa. Switch A is initially open, while switch B is initially closed; they change positions every 400 ms. Determine the energy in the capacitor at t equal to (a) \(0^?\); (b) \(0^+\); (c) 200 ms; (d) \(400^?\ ms\); (e) \(400^+\ ms\); (f) 700 ms.

Refer to the circuit of Fig. 8.97, which contains a voltage-controlled dependent voltage source in addition to two resistors. (a) Compute the circuit time constant. (b) Obtain an expression for \(v_x\) valid for all t. (c) Plot the power dissipated in the resistor over the range of 6 time constants. (d) Repeat parts (a) to (c) if the dependent source is installed in the circuit upside down. (e) Are both circuit configurations “stable”? Explain.

In the circuit of Fig. 8.97, a 3 mF capacitor is accidentally installed instead of the inductor. Unfortunately, that’s not the end of the problems, as it’s later determined that the real capacitor is not really well modeled by an ideal capacitor, and the dielectric has a resistance of \(10\ k \Omega\) (which should be viewed as connected in parallel to the ideal capacitor). (a) Compute the circuit time constant with and without taking the dielectric resistance into account. By how much does the dielectric change your answer? (b) Calculate \(v_x\) at t = 200 ms. Does the dielectric resistance affect your answer significantly? Explain.

A voltage source, \(v_s = 20u(t)\ V\), is in series with a \(200\ \Omega\) resistor and a 4 H inductor. Find the magnitude of the inductor current at t equal to (a) \(0^?\); (b) \(0^+\); (c) 8 ms; (d) 15 ms.

The switch in the circuit of Fig. 8.55 has been closed a ridiculously long time before suddenly being thrown open at t = 0. (a) Obtain expressions for \(i_L\) and v in the circuit of Fig. 8.55 which are valid for all \(t\ \geq\ 0\). (b) Calculate \(i_L(t)\) and v(t) at the instant just prior to the switch opening, at the instant just after the switch opening, and at \(t = 470\ \mu s\).

Design a circuit which will produce a voltage of 1 V at some initial time, and a voltage of 368 mV at a time 5 seconds later. You may specify an initial inductor current without showing how it arises.

The switch above the 12 V source in the circuit of Fig. 8.60 has been closed since just after the wheel was invented. It is finally thrown open at t = 0. (a) Compute the circuit time constant. (b) Obtain an expression for v(t) valid for t > 0. (c) Calculate the energy stored in the capacitor 170 ms after the switch is opened.

For the circuit of Fig. 8.64, determine \(i_x\), \(i_L\), and \(v_L\) at t equal to (a) \(0^?\); (b) \(0^+\).

Design a complete circuit which provides a voltage \(v_{ab}\) across two terminals labeled a and b, respectively, such that \(v_{ab} = 5\ V\) at \(t = 0^?\), 2 V at t = 1 s, and less than 60 mV at t = 5. Verify the operation of your circuit using an appropriate PSpice simulation. (Hint: employ the part named Sw_tOpen or Sw_tClose as appropriate.)

The switch in Fig. 8.73 is moved from A to B at t = 0 after being at A for a long time. This places the two capacitors in series, thus allowing equal and opposite dc voltages to be trapped on the capacitors. (a) Determine \(v_1 (0^-)\), \(v_2 (0^-)\), and \(v_R (0^-)\). (b) Find \(v_1 (0^+)\), \(v_2 (0^+)\), and \(v_R (0^+)\). (c) Determine the time constant of \(v_R (t)\). (d) Find \(v_R (t)\), t > 0. (e) Find i(t). (f) Find \(v_1 (t)\) and \(v_2 (t)\) from i(t) and the initial values. (g) Show that the stored energy at \(t = \infty\) plus the total energy dissipated in the \(20\ k \Omega\) resistor is equal to the energy stored in the capacitors at t = 0.

With reference to the simple circuit depicted in Fig. 8.77, compute i(t) for (a) \(t = 0^?\); (b) \(t = 0^+\); (c) \(t = 1^?\); (d) \(t = 1^+\); (e) t = 2 ms.

(a) Obtain an expression for \(i_L\) as labeled in Fig. 8.83 which is valid for all values of t. (b) Sketch your result over the range \(-1\ ms\ \leq\ t\ \leq\ 3\ ms\).

The switch in the circuit of Fig. 8.89 has been closed an incredibly long time, before being thrown open at t = 0. (a) Evaluate the current labeled \(i_x\) at t = 70 ms. (b) Verify your answer with an appropriate PSpice simulation.

Referring to the circuit represented in Fig. 8.92, (a) obtain an equation which describes \(v_C\) valid for all values of t; (b) determine the energy remaining in the capacitor at \(t = 0^+\), \(t = 25\ \mu s\), and \(t = 150\ \mu s\).

Sketch the voltage \(v_C\) across the capacitor of Fig. 8.95 for at least 3 periods if \(R = 1\ \Omega\), C = 1 F, and \(v_s(t)\) is a pulsed waveform having (a) minimum of 0 V, maximum of 2 V, rise and fall times of 1 ms, pulse width of 10 s, and period of 10 ms; (b) minimum of 0 V, maximum of 2 V, rise and fall times of 1 ms, pulse width of 10 ms, and period of 10 s. (c) Verify your answers with appropriate PSpice simulations.

For the circuit of Fig. 8.98, assuming an ideal op amp, derive an expression for \(v_o(t)\) if \(v_s\) is equal to (a) 4u(t) V; (b) \(4e^{?130,000t} u(t)\ V\).

The switch in the circuit of Fig. 8.55 has been closed a ridiculously long time before suddenly being thrown open at t = 0. (a) Obtain expressions for \(i_L\) and v in the circuit of Fig. 8.55 which are valid for all \(t\ \geq\ 0\). (b) Calculate \(i_L(t)\) and v(t) at the instant just prior to the switch opening, at the instant just after the switch opening, and at \(t = 470\ \mu s\).

Design a circuit which will produce a voltage of 1 V at some initial time, and a voltage of 368 mV at a time 5 seconds later. You may specify an initial inductor current without showing how it arises.

The switch above the 12 V source in the circuit of Fig. 8.60 has been closed since just after the wheel was invented. It is finally thrown open at t = 0. (a) Compute the circuit time constant. (b) Obtain an expression for v(t) valid for t > 0. (c) Calculate the energy stored in the capacitor 170 ms after the switch is opened.

For the circuit of Fig. 8.64, determine \(i_x\), \(i_L\), and \(v_L\) at t equal to (a) \(0^?\); (b) \(0^+\).

Design a complete circuit which provides a voltage \(v_{ab}\) across two terminals labeled a and b, respectively, such that \(v_{ab} = 5\ V\) at \(t = 0^?\), 2 V at t = 1 s, and less than 60 mV at t = 5. Verify the operation of your circuit using an appropriate PSpice simulation. (Hint: employ the part named Sw_tOpen or Sw_tClose as appropriate.)

The switch in Fig. 8.73 is moved from A to B at t = 0 after being at A for a long time. This places the two capacitors in series, thus allowing equal and opposite dc voltages to be trapped on the capacitors. (a) Determine \(v_1 (0^-)\), \(v_2 (0^-)\), and \(v_R (0^-)\). (b) Find \(v_1 (0^+)\), \(v_2 (0^+)\), and \(v_R (0^+)\). (c) Determine the time constant of \(v_R (t)\). (d) Find \(v_R (t)\), t > 0. (e) Find i(t). (f) Find \(v_1 (t)\) and \(v_2 (t)\) from i(t) and the initial values. (g) Show that the stored energy at \(t = \infty\) plus the total energy dissipated in the \(20\ k \Omega\) resistor is equal to the energy stored in the capacitors at t = 0.

With reference to the simple circuit depicted in Fig. 8.77, compute i(t) for (a) \(t = 0^?\); (b) \(t = 0^+\); (c) \(t = 1^?\); (d) \(t = 1^+\); (e) t = 2 ms.

(a) Obtain an expression for \(i_L\) as labeled in Fig. 8.83 which is valid for all values of t. (b) Sketch your result over the range \(-1\ ms\ \leq\ t\ \leq\ 3\ ms\).

The switch in the circuit of Fig. 8.89 has been closed an incredibly long time, before being thrown open at t = 0. (a) Evaluate the current labeled \(i_x\) at t = 70 ms. (b) Verify your answer with an appropriate PSpice simulation.

Referring to the circuit represented in Fig. 8.92, (a) obtain an equation which describes \(v_C\) valid for all values of t; (b) determine the energy remaining in the capacitor at \(t = 0^+\), \(t = 25\ \mu s\), and \(t = 150\ \mu s\).

Sketch the voltage \(v_C\) across the capacitor of Fig. 8.95 for at least 3 periods if \(R = 1\ \Omega\), C = 1 F, and \(v_s(t)\) is a pulsed waveform having (a) minimum of 0 V, maximum of 2 V, rise and fall times of 1 ms, pulse width of 10 s, and period of 10 ms; (b) minimum of 0 V, maximum of 2 V, rise and fall times of 1 ms, pulse width of 10 ms, and period of 10 s. (c) Verify your answers with appropriate PSpice simulations.

For the circuit of Fig. 8.98, assuming an ideal op amp, derive an expression for \(v_o(t)\) if \(v_s\) is equal to (a) 4u(t) V; (b) \(4e^{?130,000t} u(t)\ V\).