Solved: The expressions for voltage, power, and energy derived in Example 6.5 involved

The current in a inductor is known to be a) Find the voltage across the inductor for (Assume the passive sign convention.) b) Find the power (in microwatts) at the terminals of the inductor when c) Is the inductor absorbing or delivering power at 5 ms? d) Find the energy (in microjoules) stored in the inductor at 5 ms. e) Find the maximum energy (in microjoules) stored in the inductor and the time (in milliseconds) when it occurs.

The triangular current pulse shown in Fig. P6.2 is applied to a 500 mH inductor. a) Write the expressions that describe in the four intervals 25 ms 50 ms, and b) Derive the expressions for the inductor voltage, power, and energy. Use the passive sign convention.

The current in a 50 mH inductor is known to be The voltage across the inductor (passive sign convention) is 3 V at a) Find the expression for the voltage across the inductor for b) Find the time, greater than zero, when the power at the terminals of the inductor is zero.

Assume in Problem 6.3 that the value of the voltage across the inductor at is instead of 3 V. a) Find the numerical expressions for i and for b) Specify the time intervals when the inductor is storing energy and the time intervals when the inductor is delivering energy. c) Show that the total energy extracted from the inductor is equal to the total energy stored.

The current in a 200 mH inductor is The voltage across the inductor (passive sign convention) is 4.25 V at Calculate the power at the terminals of the inductor at State whether the inductor is absorbing or delivering power.

Evaluate the integral for Example 6.2. Comment on the significance of the result.

The voltage at the terminals of the \(750 \ \mu \mathrm{H}\) inductor in Fig. P6.7(a) is shown in Fig. P6.7(b). The inductor current i is known to be zero for \(t \leq 0\). a) Derive the expressions for i for \(t \geq 0\). b) Sketch i versus t for \(0 \leq t \leq \infty\).

The current in the 50 mH inductor in Fig. P6.8 is known to be 100 mA for t < 0. The inductor voltage for \(t \geq 0\) is given by the expression \(v_{L}(t)=2 e^{-100 t} \mathrm{~V}, \quad 0^{+} \leq t \leq 100 \mathrm{~ms}\) \(v_{L}(t)=-2 e^{-100(t-0.1)} \mathrm{V}, \quad 100 \mathrm{~ms} \leq t<\infty\) Sketch \(v_{L}(t)\) and \(i_{L}(t)\) for \(0 \leq t<\infty\).

The current in and the voltage across a 10 H inductor are known to be zero for \(t \leq 0\). The voltage across the inductor is given by the graph in Fig. P6.9 for \(t \geq 0\). a) Derive the expression for the current as a function of time in the intervals \(0 \leq t \leq 25 \mathrm{~ms}\), \(25 \mathrm{~ms} \leq t \leq 75 \mathrm{~ms}, 75 \mathrm{~ms} \leq t \leq 125 \mathrm{~ms}\), \(125 \mathrm{~ms} \leq t \leq 150 \mathrm{~ms}\), and \(150 \mathrm{~ms} \leq t<\infty\). b) For t > 0, what is the current in the inductor when the voltage is zero? c) Sketch i versus t for \(0 \leq t<\infty\).

a) Find the inductor current in the circuit in Fig. P6.10 if and b) Sketch , i, p, and versus t. In making these sketches, use the format used in Fig. 6.8. Plot over one complete cycle of the voltage waveform. c) Describe the subintervals in the time interval between 0 and ms when power is being absorbed by the inductor. Repeat for the subintervals when power is being delivered by the inductor.

The current in a 25 mH inductor is known to be for -10 A for \(t \leq 0\) and ( -10 cos 400 t - 5 sin 400t )\(e^{-200 t}\) A for \(t \geq 0\). Assume the passive sign convention. a) At what instant of time is the voltage across the inductor maximum? b) What is the maximum voltage?

Initially there was no energy stored in the 5 H inductor in the circuit in Fig. P6.12 when it was placed across the terminals of the voltmeter. At t = 0 the inductor was switched instantaneously to position b where it remained for 1.6 s before returning instantaneously to position a. The d’Arsonval voltmeter has a full-scale reading of 20 V and a sensitivity of \(1000 \ \Omega / \mathrm{V}\). What will the reading of the voltmeter be at the instant the switch returns to position a if the inertia of the d’Arsonval movement is negligible?

The voltage across a \(5 \ \mu \mathrm{F}\) capacitor is known to be \(v_{c}=500 t e^{-2500 t} \mathrm{~V}\) for \(t \geq 0\). a) Find the current through the capacitor for t > 0. Assume the passive sign convention. b) Find the power at the terminals of the capacitor when \(t=100 \ \mu \mathrm{s}\). c) Is the capacitor absorbing or delivering power at \(t=100 \ \mu \mathrm{s}\)? d) Find the energy stored in the capacitor at \(t=100 \ \mu \mathrm{s}\). e) Find the maximum energy stored in the capacitors and the time when the maximum occurs.

The triangular voltage pulse shown in Fig. P6.14 is applied to a \(200 \ \mu \mathrm{F}\) capacitor. a) Write the expressions that describe v(t) in the five time intervals t < 0, \(0 \leq t \leq 2 \mathrm{~s}, 2 \mathrm{~s} \leq t \leq 6 \mathrm{~s}\), \(6 \mathrm{~s} \leq t \leq 8 \mathrm{~s}\), and t > 8 s. b) Derive the expressions for the capacitor current, power, and energy for the time intervals in part (a). Use the passive sign convention. c) Identify the time intervals between 0 and 8 s when power is being delivered by the capacitor. Repeat for the time intervals when power is being absorbed by the capacitor.

The voltage across the terminals of a \(5 \ \mu \mathrm{F}\) capacitor is \(v=\left\{\begin{array}{ll} 60 \mathrm{~V}, & t \leq 0 ; \\ \left(A_{1} e^{-1500 t}+A_{2} t e^{-1500 t}\right) \mathrm{V}, & t \geq 0 . \end{array}\right.\) The initial current in the capacitor is 100 mA. Assume the passive sign convention. a) What is the initial energy stored in the capacitor? b) Evaluate the coefficients \(A_{1}\) and \(A_{2}\). c) What is the expression for the capacitor current?

A \(100 \ \mu \mathrm{F}\) capacitor is subjected to a voltage pulse having a duration of 4 s. The pulse is described by the following equations: \(v_{c}(t)=\left\{\begin{array}{ll} 5 t^{3} \mathrm{~V}, & 0 \leq t \leq 2 \mathrm{~s}; \\ -5(t-4)^{3} \mathrm{~V}, & 2 \mathrm{~s} \leq t \leq 4 \mathrm{~s}; \\ 0 & \text { elsewhere .} \end{array}\right.\) Sketch the current pulse that exists in the capacitor during the 4 s interval.

The voltage at the terminals of the capacitor in Fig. 6.10 is known to be \(v=\left\{\begin{array}{ll} 60 \mathrm{~V}, & t \leq 0 ; \\ 30+5 e^{-500 t}(6 \cos 2000 t+\sin 2000 t) \mathrm{V}, & t \geq 0 . \end{array}\right.\) Assume \(C=120 \ \mu \mathrm{F}\). a) Find the current in the capacitor for t < 0. b) Find the current in the capacitor for t > 0. c) Is there an instantaneous change in the voltage across the capacitor at t = 0? d) Is there an instantaneous change in the current in the capacitor at t = 0? e) How much energy (in millijoules) is stored in the capacitor at \(t=\infty\)?

The expressions for voltage, power, and energy derived in Example 6.5 involved both integration and manipulation of algebraic expressions. As an engineer, you cannot accept such results on faith alone. That is, you should develop the habit of asking yourself, “Do these results make sense in terms of the known behavior of the circuit they purport to describe?” With these thoughts in mind, test the expressions of Example 6.5 by performing the following checks: a) Check the expressions to see whether the voltage is continuous in passing from one time interval to the next. b) Check the power expression in each interval by selecting a time within the interval and seeing whether it gives the same result as the corresponding product of v and i. For example, test at 10 and \(30 \ \mu \mathrm{s}\). c) Check the energy expression within each interval by selecting a time within the interval and seeing whether the energy equation gives the same result as \(\frac{1}{2} C v^{2}\). Use 10 and \(30 \ \mu \mathrm{s}\) as test points.

The initial voltage on the \(0.5 \ \mu \mathrm{F}\) capacitor shown in Fig. P6.19(a) is -20 V. The capacitor current has the waveform shown in Fig. P6.19(b). a) How much energy, in microjoules, is stored in the capacitor at \(t=500 \ \mu \mathrm{s}\)? b) Repeat (a) for \(t=\infty\).

The current shown in Fig. 6.20 is applied to a capacitor. The initial voltage on the capacitor is zero. a) Find the charge on the capacitor at b) Find the voltage on the capacitor at c) How much energy is stored in the capacitor by this current?

The rectangular-shaped current pulse shown in Fig. P6.21 is applied to a \(0.1 \ \mu \mathrm{F}\) capacitor. The initial voltage on the capacitor is a 15 V drop in the reference direction of the current. Derive the expression for the capacitor voltage for the time intervals in (a)–(d). a) \(0 \leq t \leq 10 \ \mu \mathrm{s}\); b) \(10 \ \mu \mathrm{s} \leq t \leq 20 \ \mu \mathrm{s}\); c) \(20 \ \mu \mathrm{s} \leq t \leq 40 \ \mu \mathrm{s}\) d) \(40 \ \mu \mathrm{s} \leq t<\infty\) e) Sketch v(t) over the interval \(-10 \ \mu \mathrm{s} \leq t \leq 50 \ \mu \mathrm{s}\).

Assume that the initial energy stored in the inductors of Figs. P6.22(a) and (b) is zero. Find the equivalent inductance with respect to the terminals a, b.

Use realistic inductor values from Appendix H to construct series and parallel combinations of inductors to yield the equivalent inductances specified below.Try to minimize the number of inductors used. Assume that no initial energy is stored in any of the inductors. a) 8 mH b) \(45 \ \mu \mathrm{H}\) c) \(180 \ \mu \mathrm{H}\)

The two parallel inductors in Fig. P6.24 are connected across the terminals of a black box at t = 0. The resulting voltage v for t > 0 is known to be \(64 e^{-4 t} \mathrm{~V}\). It is also known that \(i_{1}(0)=-10 \mathrm{~A}\) and \(i_{2}(0)=5 \mathrm{~A}\). a) Replace the original inductors with an equivalent inductor and find i(t) for \(t \geq 0\). b) Find \(i_{1}(t)\) for \(t \geq 0\). c) Find \(i_{2}(t)\) for \(t \geq 0\). d) How much energy is delivered to the black box in the time interval \(0 \leq t<\infty\)? e) How much energy was initially stored in the parallel inductors? f) How much energy is trapped in the ideal inductors? g) Show that your solutions for \(i_{1}\) and \(i_{2}\) agree with the answer obtained in (f).

The three inductors in the circuit in Fig. P6.25 are connected across the terminals of a black box at t = 0. The resulting voltage for t > 0 is known to be \(v_{o}=2000 e^{-100 t} \mathrm{~V}\). If \(i_{1}(0)=-6 \mathrm{~A}\) and \(i_{2}(0)=1 \mathrm{~A}\) find a) \(i_{o}(0)\); b) \(i_{o}(t), t \geq 0\); c) \(i_{1}(t), t \geq 0\); d) \(i_{2}(t), t \geq 0\); e) the initial energy stored in the three inductors; f) the total energy delivered to the black box; and g) the energy trapped in the ideal inductors.

For the circuit shown in Fig. P6.25, how many milliseconds after the switch is opened is the energy delivered to the black box 80% of the total energy delivered?

Find the equivalent capacitance with respect to the terminals a, b for the circuits shown in Fig. P6.27.

Use realistic capacitor values from Appendix H to construct series and parallel combinations of capacitors to yield the equivalent capacitances specified below. Try to minimize the number of capacitors used. Assume that no initial energy is stored in any of the capacitors. a) 480 pF b) 600 nF c) \(120 \ \mu \mathrm{F}\)

Derive the equivalent circuit for a series connection of ideal capacitors. Assume that each capacitor has its own initial voltage. Denote these initial voltages as \(v_{1}\left(t_{0}\right), v_{2}\left(t_{0}\right)\), and so on. (Hint: Sum the voltages across the string of capacitors, recognizing that the series connection forces the current in each capacitor to be the same.)

Derive the equivalent circuit for a parallel connection of ideal capacitors. Assume that the initial voltage across the paralleled capacitors is (Hint: Sum the currents into the string of capacitors, recognizing that the parallel connection forces the voltage across each capacitor to be the same.)

The two series-connected capacitors in Fig. P6.31 are connected to the terminals of a black box at t = 0. The resulting current i(t) for t > 0 is known to be \(800 e^{-25 t} \ \mu \mathrm{A}\). a) Replace the original capacitors with an equivalent capacitor and find \(v_{o}(t)\) for \(t \geq 0\). b) Find \(v_{1}(t)\) for \(t \geq 0\). c) Find \(v_{2}(t)\) for \(t \geq 0\). d) How much energy is delivered to the black box in the time interval \(10 \leq t<\infty\)? e) How much energy was initially stored in the series capacitors? f) How much energy is trapped in the ideal capacitors? g) Show that the solutions for \(v_{1}\) and \(v_{2}\) agree with the answer obtained in (f).

The four capacitors in the circuit in Fig. P6.32 are connected across the terminals of a black box at t = 0. The resulting current \(i_{b}\) for t > 0 is known to be \(i_{b}=-5 e^{-50 t} \mathrm{~mA}\). If \(v_{a}(0)=-20 \mathrm{~V}, \quad v_{c}(0)=-30 \mathrm{~V}\), and \(v_{d}(0)=250 \mathrm{~V}\), find the following for \(t \geq 0\); (a) \(v_{b}(t)\), (b) \(v_{a}(t)\), (c) \(v_{c}(t)\), (d) \(v_{d}(t)\), (e) \(i_{1}(t)\), and (f) \(i_{2}(t)\).

For the circuit in Fig. P6.32, calculate a) the initial energy stored in the capacitors; b) the final energy stored in the capacitors; c) the total energy delivered to the black box; d) the percentage of the initial energy stored that is delivered to the black box; and e) the time, in milliseconds, it takes to deliver 7.5 mJ to the black box.

At t = 0, a series-connected capacitor and inductor are placed across the terminals of a black box, as shown in Fig. P6.34. For t > 0, it is known that \(i_{o}=200 e^{-800 t}-40 e^{-200 t} \mathrm{~mA}\). If \(v_{\mathrm{c}}(0)=5 \mathrm{~V}\) find \(v_{o}\) for \(t \geq 0\).

The current in the circuit in Fig. P6.35 is known to be \(i_{o}=2 e^{-5000 t}(\cos 1000 t+5 \sin 1000 t) \mathrm{A}\) for \(t \geq 0^{+}\). Find \(v_{1}\left(0^{+}\right)\) and \(v_{2}\left(0^{+}\right)\).

a) Show that the differential equations derived in (a) of Example 6.6 can be rearranged as follows: \(4 \frac{d i_{1}}{d t}+25 i_{1}-8 \frac{d i_{2}}{d t}-20 i_{2}=5 i_{g}-8 \frac{d i_{g}}{d t}\); \(-8 \frac{d i_{1}}{d t}-20 i_{1}+16 \frac{d i_{2}}{d t}+80 i_{2}=16 \frac{d i_{g}}{d t}\). b) Show that the solutions for \(i_{1}\), and \(i_{2}\) given in (b) of Example 6.6 satisfy the differential equations given in part (a) of this problem.

Let \(v_{o}\) represent the voltage across the 16 H inductor in the circuit in Fig. 6.25.Assume \(v_{o}\) is positive at the dot. As in Example 6.6, \(i_{g}=16-16 e^{-5 t} \mathrm{~A}\). a) Can you find \(v_{o}\) without having to differentiate the expressions for the currents? Explain. b) Derive the expression for \(v_{o}\). c) Check your answer in (b) using the appropriate current derivatives and inductances.

Let \(v_{g}\) represent the voltage across the current source in the circuit in Fig. 6.25. The reference for \(v_{g}\) is positive at the upper terminal of the current source. a) Find \(v_{g}\) as a function of time when \(i_{g}=16-16 e^{-5 t} \mathrm{~A}\). b) What is the initial value of \(v_{g}\)? c) Find the expression for the power developed by the current source. d) How much power is the current source developing when t is infinite? e) Calculate the power dissipated in each resistor when t is infinite.

There is no energy stored in the circuit in Fig. P6.39 at the time the switch is opened. a) Derive the differential equation that governs the behavior of \(i_{2}\) if \(L_{1}=5 \mathrm{H}, \quad L_{2}=0.2 \mathrm{H}\), \(M=0.5 \mathrm{H}\), and \(R_{o}=10 \ \Omega\). b) Show that when \(i_{g}=e^{-10 t}-10 \mathrm{~A}, \mathrm{t} \geq 0\), the differential equation derived in (a) is satisfied when \(i_{2}=625 e^{-10 t}-250 e^{-50 t} \mathrm{~mA}, t \geq 0\). c) Find the expression for the voltage \(v_{1}\) across the current source. d) What is the initial value of \(v_{1}\)? Does this make sense in terms of known circuit behavior?

a) Show that the two coupled coils in Fig. P6.40 can be replaced by a single coil having an inductance of (Hint: Express as a function of ) b) Show that if the connections to the terminals of the coil labeled are reversed,

a) Show that the two magnetically coupled coils in Fig. P6.41 (see page 210) can be replaced by a single coil having an inductance of \(L_{\mathrm{ab}}=\frac{L_{1} L_{2}-M^{2}}{L_{1}+L_{2}-2 M}\). (Hint: Let \(i_{1}\) and \(i_{2}\) be clockwise mesh currents in the left and right “windows” of Fig. P6.41,respectively. Sum the voltages around the two meshes. In mesh 1 let \(v_{\mathrm{ab}}\) be the unspecified applied voltage. Solve for \(d i_{1} / d t\) as a function of \(v_{\mathrm{ab}}\).) b) Show that if the magnetic polarity of coil 2 is reversed, then \(L_{\mathrm{ab}}=\frac{L_{1} L_{2}-M^{2}}{L_{1}+L_{2}+2 M}\).

The polarity markings on two coils are to be determined experimentally. The experimental setup is shown in Fig. P6.42. Assume that the terminal connected to the positive terminal of the battery has been given a polarity mark as shown. When the switch is opened, the dc voltmeter kicks downscale. Where should the polarity mark be placed on the coil connected to the voltmeter?

The physical construction of four pairs of magnetically coupled coils is shown in Fig. P6.43. (See page 211.) Assume that the magnetic flux is confined to the core material in each structure. Show two possible locations for the dot markings on each pair of coils.

a) Starting with Eq. 6.59, show that the coefficient of coupling can also be expressed as \(k=\sqrt{\left(\frac{\phi_{21}}{\phi_{1}}\right)\left(\frac{\phi_{12}}{\phi_{2}}\right)}\). b) On the basis of the fractions \(\phi_{21} / \phi_{1}\) and \(\phi_{12} / \phi_{2}\), explain why k is less than 1.0.

Two magnetically coupled coils have self-inductances of 60 mH and 9.6 mH, respectively.The mutual inductance between the coils is 22.8 mH. a) What is the coefficient of coupling? b) For these two coils, what is the largest value that M can have? c) Assume that the physical structure of these coupled coils is such that \(\mathscr{P}_{1}=\mathscr{P}_{2}\). What is the turns ratio \(N_{1} / N_{2}\) if \(N_{1}\) is the number of turns on the 60 mH coil?

Two magnetically coupled coils are wound on a nonmagnetic core. The self-inductance of coil 1 is 288 mH, the mutual inductance is 90 mH, the coefficient of coupling is 0.75, and the physical structure of the coils is such that \(\mathscr{P}_{11}=\mathscr{P}_{22}\). a) Find \(L_{2}\) and the turns ratio \(N_{1} / N_{2}\). b) If \(N_{1}=1200\) what is the value of \(\mathscr{P}_{1}\) and \(\mathscr{P}_{2}\)?

The self-inductances of the coils in Fig. 6.30 are \(L_{1}=18 \mathrm{mH}\) and \(L_{2}=32 \mathrm{mH}\). If the coefficient of coupling is 0.85, calculate the energy stored in the system in millijoules when (a) \(i_{1}=6 \mathrm{~A}, i_{2}=9 \mathrm{~A}\); (b) \(i_{1}=-6 \mathrm{~A}, i_{2}=-9 \mathrm{~A}\); (c) \(i_{1}=-6 \mathrm{~A}, i_{2}=9 \mathrm{~A}\); and (d) \(i_{1}=6 \mathrm{~A}, i_{2}=-9 \mathrm{~A}\).

The coefficient of coupling in Problem 6.47 is increased to 1.0. a) If \(i_{1}\) equals 6 A, what value of \(i_{2}\) results in zero stored energy? b) Is there any physically realizable value of \(i_{2}\) that can make the stored energy negative?

The self-inductances of two magnetically coupled coils are 72 mH and 40.5 mH, respectively. The 72 mH coil has 250 turns, and the coefficient of coupling between the coils is \(2 / 3\). The coupling medium is nonmagnetic. When coil 1 is excited with coil 2 open, the flux linking only coil 1 is 0.2 as large as the flux linking coil 2. a) How many turns does coil 2 have? b) What is the value of \(\mathscr{P}_{2}\) in nanowebers per ampere? c) What is the value of \(\mathscr{P}_{11}\) in nanowebers per ampere? d) What is the ratio \(\left(\phi_{22} / \phi_{12}\right)\)?

The self-inductances of two magnetically coupled coils are and The coupling medium is nonmagnetic. If coil 1 has 300 turns and coil 2 has 500 turns, find and (in nanowebers per ampere) when the coefficient of coupling is 0.6.

Suppose a capacitive touch screen that uses the mutual-capacitance design, as shown in Fig. 6.33, is touched at the point x,y. Determine the mutual capacitance at that point, \(C_{\mathrm{mxy}}^{\prime}\), in terms of the mutual capacitance at the point without a touch, \(C_{\mathrm{mxy}}\), and the capacitance introduced by the touch, \(C_{t}\) .

a) Assume the parasitic capacitance in the self capacitance design, \(C_{p}=30\) pF, and the capacitance introduced by a touch is 15 pF (see Fig. 6.32[b]). What is the capacitance at the touch point with respect to ground for the x-grid and y-grid electrodes closest to the touch point? b) Assume the mutual capacitance in the mutual capacitance design, \(C_{\mathrm{mxy}}=30\) pF, and the capacitance introduced by a touch is 15 pF (see Fig. 6.33[b]). What is the mutual capacitance between the x- and y-grid electrodes closest to the touch point? c) Compare your results in parts (a) and (b) – does touching the screen increase or decrease the capacitance in these two different capacitive touch screen designs?

a) As shown in the Practical Perspective, the self capacitance design does not permit a true multitouch screen – if the screen is touched at two difference points, a total of four touch points are identified, the two actual touch points and two ghost points. If a self-capacitance touch screen is touched at the x, y coordinates (2.1, 4.3) and (3.2, 2.5), what are the four touch locations that will be identified? (Assume the touch coordinates are measured in inches from the upper left corner of the screen.) b) A self-capacitance touch screen can still function as a multi-touch screen for several common gestures. For example, suppose at time \(t_{1}\) the two touch points are those identified in part (a), and at time \(t_{2}\) four touch points associated with the x, y coordinates (1.8, 4.9) and (3.9, 1.8) are identified. Comparing the four points at \(t_{1}\) with the four points at \(t_{2}\), software can recognize a pinch gesture – should the screen be zoomed in or zoomed out? c) Repeat part (b),assuming that at time \(t_{2}\) four touch points associated with the x, y coordinates (2.8, 3.9) and (3.0, 2.8) are identified.