A new piece of equipment designed to make crystals from molten constituents is

Determine the current flowing through a 5 mF capacitor in response to a voltage v equal to: (a) ?20 V; (b) \(2e^{?5t}\ V\).

Determine the current through a 100 pF capacitor if its voltage as a function of time is given by Fig. 7.6.

Calculate the energy stored in a \(1000\ \mu F\) capacitor at \(t = 50\ \mu s\) if the voltage across it is \(1.5\ cos\ 10^5 t\) volts.

The current through a 200 mH inductor is shown in Fig. 7.13. Assume the passive sign convention, and find \(v_L\) at t equal to (a) 0; (b) 2 ms; (c) 6 ms.

The current waveform of Fig. 7.14a has equal rise and fall times of duration 0.1 s (100 ms). Calculate the maximum positive and negative voltages across the same inductor if the rise and fall times, respectively, are changed to (a) 1 ms, 1 ms; (b) \(12\ \mu s\), \(64\ \mu s\); (c) 1 s, 1 ns.

A 100 mH inductor has voltage \(v_L = 2e^{?3t}\ V\) across its terminals. Determine the resulting inductor current if \(i_L (?0.5) = 1\ A\).

Let L = 25 mH for the inductor of Fig. 7.10. (a) Find \(v_L\) at t = 12 ms if \(i_L = 10te^{?100t}\ A\). (b) Find \(i_L\) at t = 0.1 s if \(v_L = 6e^{?12t}\ V\) and \(i_L (0) = 10\ A\). If \(i_L = 8(1 ? e^{?40t})\ mA\), find (c) the power being delivered to the inductor at t = 50 ms and (d) the energy stored in the inductor at t = 40 ms.

Find \(C_{eq}\) for the network of Fig. 7.23.

If \(v_C(t) = 4\ cos\ 10^5 t\ V\) in the circuit in Fig. 7.26, find \(v_s(t)\).

Derive an expression for \(v_{out}\) in terms of \(v_s\) for the circuit shown in Fig. 7.29.

Making use of the passive sign convention, determine the current flowing through a 220 nF capacitor for \(t\ \geq\ 0\) if its voltage \(v_C(t)\) is given by (a) ?3.35 V; (b) \(16.2e^{?9t}\ V\); (c) 8 cos 0.01t mV; (d) 5 + 9 sin 0.08t V.

Sketch the current flowing through a 13 pF capacitor for \(t\ \geq\ 0\) as a result of the waveforms shown in Fig. 7.40. Assume the passive sign convention.

(a) If the voltage waveform depicted in Fig. 7.41 is applied across the terminals of a \(1\ \mu F\) electrolytic capacitor, graph the resulting current, assuming the passive sign convention. (b) Repeat part (a) if the capacitor is replaced with a 17.5 pF capacitor.

A capacitor is constructed from two copper plates, each measuring \(1\ mm\ \times\ 2.5\ mm\) and \(155\ \mu m\) thick. The two plates are placed such that they face each other and are separated by a \(1\ \mu m\) gap. Calculate the resulting capacitance if (a) the intervening dielectric has a permittivity of \(1.35 \varepsilon_{0}\); (b) the intervening dielectric has a permittivity of \(3.5 \varepsilon_{0}\); (c) the plate separation is increased by \(3.5\ \mu m\) and the gap is filled with air; (d) the plate area is doubled and the \(1\ \mu m\) gap is filled with air.

Two pieces of gadolinium, each measuring \(100\ \mu m\ \times\ 750\ \mu m\) and 604 nm thick, are used to construct a capacitor. The two plates are arranged such that they face each other and are separated by a 100 nm gap. Calculate the resulting capacitance if (a) the intervening dielectric has a permittivity of \(13.8 \varepsilon_{0}\); (b) the intervening dielectric has a permittivity of \(500 \varepsilon_{0}\); (c) the plate separation is increased by 100 nm and the gap is filled with air; (d) the plate area is quadrupled and the 100 nm gap is filled with air.

Design a 100 nF capacitor constructed from \(1\ \mu m\) thick gold foil, and which fits entirely within a volume equal to that of a standard AAA battery, if the only dielectric available has a permittivity of \(3.1\varepsilon_0\).

Design a capacitor whose capacitance can be varied mechanically with a simple vertical motion, between the values of 100 nF and 300 nF.

Design a capacitor whose capacitance can be varied mechanically over the range of 50 nF and 100 nF by rotating a knob \(90^{\circ}\).

A silicon pn junction diode is characterized by a junction capacitance defined as \(C_{j}=\frac{K_{s} \varepsilon_{0} A}{W}\) where \(K_s = 11.8\) for silicon, \(\varepsilon_0\) is the vacuum permittivity, A = the cross-sectional area of the junction, and W is known as the depletion width of the junction. Width W depends not only on how the diode is fabricated, but also on the voltage applied to its two terminals. It can be computed using \(W=\sqrt{\frac{2 K_{s} \varepsilon_{0}}{q N}\left(V_{\mathrm{bi}}-V_{A}\right)}\) Thus, diodes are frequently used in electronic circuits, since they can be thought of as voltage-controlled capacitors. Assuming parameter values of \(N = 10^{18}\ cm^{?3}\), \(V_{bi} = 0.57\ V\), and using \(q = 1.6\ \times\ 10^{?19}\ C\), calculate the capacitance of a diode with cross-sectional area \(A = 1\ \mu m\ \times\ 1\ \mu m\) at applied voltages of \(V_A = ?1,?5\), and ?10 volts.

The current flowing through a 33 mF capacitor is shown graphically in Fig. 7.43. (a) Assuming the passive sign convention, sketch the resulting voltage waveform across the device. (b) Compute the voltage at 300 ms, 600 ms, and 1.1 s.

Calculate the energy stored in a capacitor at time t = 1 s if (a) C = 1.4 F and \(v_C = 8\ V,\ t\ >\ 0\); (b) C = 23.5 pF and \(v_C = 0.8\ V,\ t\ >\ 0\); (c) C = 17 nF, \(v_C(1) = 12\ V\), \(v_C(0) = 2\ V\), and \(w_C(0) = 295\ nJ\).

A 137 pF capacitor is connected to a voltage source such that \(v_C(t) = 12e^{?2t}\ V,\ t\ \geq\ 0\) and \(v_C(t) = 12\ V,\ t\ <\ 0\). Calculate the energy stored in the capacitor at t equal to (a) 0; (b) 200 ms; (c) 500 ms; (d) 1 s.

For each circuit shown in Fig. 7.45, calculate the voltage labeled \(v_C\).

Design a 30 nH inductor using 29 AWG solid soft copper wire. Include a sketch of your design and label geometrical parameters as necessary for clarity. Assume the coil is filled with air only.

If the current flowing through a 75 mH inductor has the waveform shown in Fig. 7.46, (a) sketch the voltage which develops across the inductor terminals for \(t\ \geq\ 0\), assuming the passive sign convention; and (b) calculate the voltage at t = 1 s, 2.9 s, and 3.1 s.

The current through a 17 nH aluminum inductor is shown in Fig. 7.47. Sketch the resulting voltage waveform for \(t\ \geq\ 0\), assuming the passive sign convention.

Determine the voltage for \(t\ \geq\ 0\) which develops across the terminals of a 4.2 mH inductor, if the current (defined consistent with the passive sign convention) is (a) -10 mA; (b) 3 sin 6t A; (c) \(11 + 115 \sqrt{2}\ cos (100 \pi t - 9^{\circ})\ A\); (d) \(13e^{-t}\ nA\); (e) \(3 + te^{-14t}\ A\).

Determine the voltage for \(t\ \geq\ 0\) which develops across the terminals of an 8 pH inductor, if the current (defined consistent with the passive sign convention) is (a) 8 mA; (b) 800 mA; (c) 8 A; (d) \(4e^{?t}\ A\); (e) \(?3+ te^{?t}\ A\).

The current waveform shown in Fig. 7.14 has a rise time of 0.1 (100 ms) and a fall time of the same duration. If the current is applied to the “+” voltage reference terminal of a 200 nH inductor, sketch the expected voltage waveform if the rise and fall times are changed, respectively, to (a) 200 ms, 200 ms; (b) 10 ms, 50 ms; (c) 10 ns, 20 ns.

Determine the current flowing through a 6 mH inductor if the voltage (defined such that it is consistent with the passive sign convention) is given by (a) 5 V; (b) \(100\ sin\ 120 \pi t,\ t\ \geq\ 0\) and 0, t < 0.

The voltage across a 2 H inductor is given by \(v_L = 4.3t,\ 0\ \leq\ t\ \leq\ 50\ ms\). With the knowledge that \(i_L (?0.1) = 100\ \mu A\), calculate the current (assuming it is defined consistent with the passive sign convention) at t equal to (a) 0; (b) 1.5 ms; (c) 45 ms.

Calculate the energy stored in a 1 nH inductor if the current flowing through it is (a) 0 mA; (b) 1 mA; (c) 20 A; (d) 5 sin 6t mA, t > 0.

Determine the amount of energy stored in a 33 mH inductor at t = 1 ms as a result of a current \(i_L\) given by (a) 7 A; (b) \(3 ? 9e^{?10^3 t}\ mA\).

Making the assumption that the circuits in Fig. 7.50 have been connected for a very long time, determine the value for each current labeled \(i_x\).

Calculate the voltage labeled \(v_x\) in Fig. 7.51, assuming the circuit has been running a very long time, if (a) a \(10\ \Omega\) resistor is connected between terminals x and y; (b) a 1 H inductor is connected between terminals x and y; (c) a 1 F capacitor is connected between terminals x and y; (d) a 4 H inductor in parallel with a \(1\ \Omega\) resistor is connected between terminals x and y.

For the circuit shown in Fig. 7.52, (a) compute the Thévenin equivalent seen by the inductor; (b) determine the power being dissipated by both resistors; (c) calculate the energy stored in the inductor.

If each capacitor has a value of 1 F, determine the equivalent capacitance of the network shown in Fig. 7.53.

Determine an equivalent inductance for the network shown in Fig. 7.54 if each inductor has value L.

Using as many 1 nH inductors as you like, design two networks, each of which has an equivalent inductance of 1.25 nH.

Apply combinatorial techniques as appropriate to obtain a value for the equivalent inductance \(L_{eq}\) as labeled on the network of Fig. 7.57.

Reduce the circuit depicted in Fig. 7.58 to as few components as possible.

Refer to the network shown in Fig. 7.59 and find (a) \(R_{eq}\) if each element is a \(10\ \Omega\) resistor; (b) \(L_{eq}\) if each element is a 10 H inductor; and (c) \(C_{eq}\) if each element is a 10 F capacitor.

Determine the equivalent inductance seen looking into the terminals marked a and b of the network represented in Fig. 7.60.

Reduce the circuit represented in Fig. 7.61 to the smallest possible number of components.

Reduce the network of Fig. 7.62 to the smallest possible number of components if each inductor is 1 nH and each capacitor is 1 mF.

For the network of Fig. 7.63, \(L_1 = 1\ H,\ L_2 = L_3 = 2\ H,\ L_4 = L_5 = L_6 = 3\ H\). (a) Find the equivalent inductance. (b) Derive an expression for a general network of this type having N stages, assuming stage N is composed of N inductors, each having inductance N henrys.

Extend the concept of \(\Delta-Y\) transformations to simplify the network of Fig. 7.64 if each element is a 2 pF capacitor.

Extend the concept of \(\Delta-Y\) transformations to simplify the network of Fig. 7.64 if each element is a 1 nH inductor.

With regard to the circuit represented in Fig. 7.65, (a) write a complete set of nodal equations and (b) write a complete set of mesh equations.

(a) Write nodal equations for the circuit of Fig. 7.66. (b) Write mesh equations for the same circuit.

In the circuit shown in Fig. 7.67, let \(i_s = 60e^{?200t}\ mA\) with \(i_1(0) = 20\ mA\). (a) Find v(t) for all t. (b) Find \(i_1(t)\) for \(t\ \geq\ 0\). (c) Find \(i_2(t)\) for \(t\ \geq\ 0\).

Let \(v_s = 100e^{?80t}\ V\) and \(v_1(0) = 20\ V\) in the circuit of Fig. 7.68. (a) Find i(t) for all t. (b) Find \(v_1(t)\) for \(t\ \geq\ 0\). (c) Find \(v_2(t)\) for \(t\ \geq\ 0\).

If it is assumed that all the sources in the circuit of Fig. 7.69 have been connected and operating for a very long time, use the superposition principle to find \(v_C(t)\) and \(v_L (t)\).

For the circuit of Fig. 7.70, assume no energy is stored at t = 0, and write a complete set of nodal equations.

Interchange the location of \(R_1\) and \(C_f\) in the circuit of Fig. 7.27, and assume that \(R_i =\infty\), \(R_o = 0\), and \(A = \infty\) for the op amp. (a) Find \(v_{out}(t)\) as a function of \(v_s (t)\). (b) Obtain an equation relating \(v_o(t)\) and \(v_s (t)\) if A is not assumed to be infinite.

Derive an expression for \(v_{out}\) in terms of \(v_{s}\) for the amplifier circuit shown in Fig. 7.71.

In practice, circuits such as those depicted in Fig. 7.27 may not function correctly unless there is a conducting pathway between the output and input terminals of the op amp. (a) Analyze the modified integrating amplifier circuit shown in Fig. 7.72 to obtain an expression for \(v_{out}\) in terms of \(v_s\) , and (b) compare this expression to Eq. [17].

A new piece of equipment designed to make crystals from molten constituents is experiencing too many failures (cracked products). The production manager wants to monitor the cooling rate to see if this is related to the problem. The system has two output terminals available, where the voltage across them is linearly proportional to the crucible temperature such that 30 mV corresponds to \(30^{\circ}C\) and 1 V corresponds to \(1000^{\circ}C\). Design a circuit whose voltage output represents the cooling rate, calibrated such that \(1\ V = 1^{\circ}C/s\).

A confectionary company has decided to increase the production rate of its milk chocolate bars to compensate for a recent increase in the cost of raw materials. However, the wrapping unit cannot accept more than 1 bar per second, or it drops bars. A 200 mV peak-to-peak sinusoidal voltage signal is available from the bar-making system which feeds into the wrapping unit, such that its frequency matches the bar production frequency (i.e., 1 Hz = 1 bar/s). Design a circuit that provides a voltage output sufficient to power a 12 V audible alarm when the production rate exceeds the capacity of the wrapping unit.

One problem satellites face is exposure to high-energy particles, which can cause damage to sensitive electronics as well as solar arrays used to provide power. A new communications satellite is equipped with a high-energy proton detector measuring \(1\ cm\ \times\ 1\ cm\). It provides a current directly equal to the number of protons impinging the surface per second. Design a circuit whose output voltage provides a running total of the number of proton hits, calibrated such that 1 V = 1 million hits.

The output of a velocity sensor attached to a sensitive piece of mobile equipment is calibrated to provide a signal such that 10 mV corresponds to linear motion at 1 m/s. If the equipment is subjected to sudden shock, it can be damaged. Since \(force = mass\ \times\ acceleration\), monitoring of the rate of change of velocity can be used to determine if the equipment is transported improperly. (a) Design a circuit to provide a voltage proportional to the linear acceleration such that \(10\ mV = 1\ m/s^2\). (b) How many sensor-circuit combinations does this application require?

A floating sensor in a certain fuel tank is connected to a variable resistor (often called a potentiometer) such that a full tank (100 liters) corresponds to \(1\ \Omega\) and an empty tank corresponds to \(10\ \Omega\). (a) Design a circuit that provides an output voltage which indicates the amount of fuel remaining, so that 1 V = empty and 5 V = full. (b) Design a circuit to indicate the rate of fuel consumption by providing a voltage output calibrated to yield 1 V = 1 l/s.

(a) Draw the exact dual of the circuit depicted in Fig. 7.73. (b) Label the new (dual) variables. (c) Write nodal equations for both circuits.

(a) Draw the exact dual of the simple circuit shown in Fig. 7.74. (b) Label the new (dual) variables. (c) Write mesh equations for both circuits.

(a) Draw the exact dual of the simple circuit shown in Fig. 7.75. (b) Label the new (dual) variables. (c) Write mesh equations for both circuits.

(a) Draw the exact dual of the simple circuit shown in Fig. 7.76. (b) Label the new (dual) variables. (c) Write nodal and mesh equations for both circuits.

Draw the exact dual of the circuit shown in Fig. 7.77. Keep it neat!

Taking the bottom node in the circuit of Fig. 7.78 as the reference terminal, calculate (a) the current through the inductor and (b) the power dissipated by the \(7\ \Omega\) resistor. (c) Verify your answers with an appropriate PSpice simulation.

For the four-element circuit shown in Fig. 7.79, (a) calculate the power absorbed in each resistor; (b) determine the voltage across the capacitor; (c) compute the energy stored in the capacitor; and (d) verify your answers with an appropriate PSpice simulation. (Recall that calculations can be performed in Probe.)

(a) Compute \(i_L\) and \(v_x\) as indicated in the circuit of Fig. 7.80. (b) Determine the energy stored in the inductor and in the capacitor. (c) Verify your answers with an appropriate PSpice simulation.

For the circuit depicted in Fig. 7.81, the value of \(i_L (0) = 1\ mA\). (a) Compute the energy stored in the element at t = 0. (b) Perform a transient simulation of the circuit over the range of \(0\ \leq\ t\ \leq 500\ ns\). Determine the value of \(i_L\) at t = 0, 130 ns, 260 ns, and 500 ns. (c) What fraction of the initial energy remains in the inductor at t = 130 ns? At t = 500 ns?

Assume an initial voltage of 9 V across the \(10\ \mu F\) capacitor shown in Fig. 7.82 (i.e., v(0) = 9 V). (a) Compute the initial energy stored in the capacitor. (b) For t > 0, do you expect the energy to remain in the capacitor? Explain. (c) Perform a transient simulation of the circuit over the range of \(0\ \leq\ t\ \leq\ 2.5\ s\) and determine v(t) at t = 460 ms, 920 ms, and 2.3 s. (c) What fraction of the initial energy remains stored in the capacitor at t = 460 ms? At t = 2.3 s?

Referring to the circuit of Fig. 7.83, (a) calculate the energy stored in each energy storage element; (b) verify your answers with an appropriate PSpice simulation.

For the circuit of Fig. 7.28, (a) sketch \(v_{out}\) over the range of \(0\ \leq\ t\ \leq\ 5\ ms\) if \(R_f = 1\ k \Omega,\ C_1 = 100\ mF\), and \(v_s\) is a 1 kHz sinusoidal source having a peak voltage of 2 V. (b) Verify your answer with an appropriate transient simulation, plotting both \(v_s\) and \(v_{out}\) in Probe. (Hint: Between plotting traces, add a second y axis using Plot, Add Y Axis. This allows both traces to be seen clearly.)

(a) Sketch the output function \(v_{out}\) of the amplifier circuit in Fig. 7.29 over the range of \(0\ \leq\ t\ \leq\ 100\ ms\) if \(v_s\) is a 60 Hz sinusoidal source having a peak voltage of 400 mV, \(R_1\) is \(1\ k \Omega\), and \(L_f\) is 80 nH. (b) Verify your answer with an appropriate transient simulation, plotting both \(v_s\) and \(v_{out}\) in Probe. (Hint: Between plotting traces, add a second y axis using Plot, Add Y Axis. This allows both traces to be seen clearly.)

For the circuit of Fig. 7.71, (a) sketch \(v_{out}\) over the range of \(0\ \leq\ t\ \leq\ 2.5\ ms\) if \(R_f = 100\ k \Omega\), \(L_1 = 100\ mH\), and \(v_s\) is a 2 kHz sinusoidal source having a peak voltage of 5 V. (b) Verify your answer with an appropriate transient simulation, plotting both \(v_s\) and \(v_{out}\) in Probe. (Hint: Between plotting traces, add a second y axis using Plot, Add Y Axis. This allows both traces to be seen clearly.)

Consider the modified integrator depicted in Fig. 7.72. Take \(R_1 = 100\ \Omega\) , \(R_f = 10\ M \Omega\), and \(C_1 = 10\ mF\). The source \(v_s\) provides a 10 Hz sinusoidal voltage having a peak amplitude of 0.5 V. (a) Sketch \(v_{out}\) over the range of \(0\ \leq\ t\ \leq 500\ ms\). (b) Verify your answer with an appropriate transient simulation, plotting both \(v_s\) and \(v_{out}\) in Probe. (Hint: Between plotting traces, add a second y axis using Plot, Add Y Axis. This allows both traces to be seen clearly.)

Derive an expression for \(v_{out}\) in terms of \(v_s\) for the circuit shown in Fig. 7.29.

Write the single nodal equation for the circuit of Fig. 7.34a, and show, by direct substitution, that \(v = ?80e^{?10^6 t}\ mV\) is a solution. Knowing this, find (a) \(v_1\); (b) \(v_2\); and (c) i for the circuit of Fig. 7.34b.

Assuming the passive sign convention, sketch the voltage which develops across the terminals of a 2.5 F capacitor in response to the current waveforms shown in Fig. 7.42.

Calculate the power dissipated in the \(40\ \Omega\) resistor and the voltage labeled \(v_C\) in each of the circuits depicted in Fig. 7.44.

Calculate \(v_L\) and \(i_L\) for each of the circuits depicted in Fig. 7.48, if \(i_s = 1\ mA\) and \(v_s = 2.1\ V\).

Determine the inductor voltage which results from the current waveform shown in Fig. 7.49 (assuming the passive sign convention) at t equal to (a) ?1 s; (b) 0 s; (c) 1.5 s; (d) 2.5 s; (e) 4 s; ( f ) 5 s.

Compute the equivalent capacitance \(C_{eq}\) as labeled in Fig. 7.55.

Determine the equivalent capacitance \(C_{eq}\) of the network shown in Fig. 7.56.

For the integrating amplifier circuit of Fig. 7.27, \(R_1 = 100\ k \Omega\), \(C_f = 500\ \mu F\), and \(v_s = 20\ sin\ 540t\ mV\). Calculate \(v_{out}\) if (a) \(A = \infty\), \(R_i = \infty\), and \(R_o = 0\); (b) A = 5000, \(R_i = 1\ M \Omega\), and \(R_o = 3\ \Omega\).

Write the single nodal equation for the circuit of Fig. 7.34a, and show, by direct substitution, that \(v = ?80e^{?10^6 t}\ mV\) is a solution. Knowing this, find (a) \(v_1\); (b) \(v_2\); and (c) i for the circuit of Fig. 7.34b.

Assuming the passive sign convention, sketch the voltage which develops across the terminals of a 2.5 F capacitor in response to the current waveforms shown in Fig. 7.42.

Calculate the power dissipated in the \(40\ \Omega\) resistor and the voltage labeled \(v_C\) in each of the circuits depicted in Fig. 7.44.

Calculate \(v_L\) and \(i_L\) for each of the circuits depicted in Fig. 7.48, if \(i_s = 1\ mA\) and \(v_s = 2.1\ V\).

Determine the inductor voltage which results from the current waveform shown in Fig. 7.49 (assuming the passive sign convention) at t equal to (a) ?1 s; (b) 0 s; (c) 1.5 s; (d) 2.5 s; (e) 4 s; ( f ) 5 s.

Compute the equivalent capacitance \(C_{eq}\) as labeled in Fig. 7.55.

Determine the equivalent capacitance \(C_{eq}\) of the network shown in Fig. 7.56.

For the integrating amplifier circuit of Fig. 7.27, \(R_1 = 100\ k \Omega\), \(C_f = 500\ \mu F\), and \(v_s = 20\ sin\ 540t\ mV\). Calculate \(v_{out}\) if (a) \(A = \infty\), \(R_i = \infty\), and \(R_o = 0\); (b) A = 5000, \(R_i = 1\ M \Omega\), and \(R_o = 3\ \Omega\).