With respect to Fig. 13.36, assume L1 = 400 mH, L2 = 230 mH, and M = 10 mH. Determine

Assuming M = 10 H, coil \(L_2\) is open-circuited, and \(i_1 = -2e^{-5t}\ A\), find the voltage \(v_2\) for (a) Fig. 13.2a; (b) Fig. 13.2b.

For the circuit of Fig. 13.9, write appropriate mesh equations for the left mesh and the right mesh if \(v_s = 20e^{-1000t}\ V\).

Let \(i_s = 2\ cos\ 10t\ A\) in the circuit of Fig. 13.14, and find the total energy stored in the passive network at t = 0 if k = 0.6 and terminals x and y are (a) left open-circuited; (b) short-circuited.

Element values for a certain linear transformer are \(R_1 = 3\ \Omega\), \(R_2 = 6\ \Omega\), \(L_1 = 2\ mH\), \(L_2 = 10\ mH\), and \(M = 4\ mH\). If \(\omega = 5000\ rad/s\), find \(\mathbf{Z}_{\mathrm {in}}\) for \(\mathbf{Z}_{L}\) equal to (a) \(10\ \Omega\); (b) \(j20\ \Omega\); (c) \(10 + j20\ \Omega\); (d) \(?j20\ \Omega\).

If the networks in Fig. 13.23 are equivalent, specify values (in mH) for \(L_A\), \(L_B\), and \(L_C\).

Repeat Example 13.7 using voltages to compute the dissipated power.

Let \(N_1 = 1000\) turns and \(N_2 = 5000\) turns in the ideal transformer shown in Fig. 13.34. If \(\mathbf{Z}_L=500-j400\ \Omega\), find the average power delivered to \(\mathbf{Z}_L\) for (a) \(\mathbf{I}_2=1.4\ \angle20^{\circ}\mathrm{\ A\ rms}\); (b) \(\mathbf{V}_2=900\ \angle 40^{\circ}\mathrm{\ V\ rms}\); (c) \(\mathbf{V}_1=80\ \angle 100^{\circ}\mathrm{\ V\ rms}\); (d) \(\mathbf{I}_1=6\ \angle 45^{\circ}\mathrm{\ A\ rms}\); (e) \(\mathbf{V}_s=200\ \angle 0^{\circ}\mathrm{\ V\ rms}\).

Consider the two inductances depicted in Fig. 13.35. Set \(L_1 = 10\ mH\), \(L_2 = 5\ mH\), and \(M = 1\ mH\). Determine the steady-state expression for (a) \(v_1\) if \(i_1 = 0\) and \(i_2 = 5\ cos\ 8t\ A\); (b) \(v_2\) if \(i_1 = 3\ sin\ 100t\ A\) and \(i_2 = 0\); (c) \(v_2\) if \(i_1 = 5\ cos\ (8t – 40^{\circ})\ A\) and \(i_2 = 4\ sin\ 8t\ A\).

With respect to Fig. 13.36, assume \(L_1 = 400\ mH\), \(L_2 = 230\ mH\), and \(M = 10\ mH\). Determine the steady-state expression for (a) \(v_1\) if \(i_1 = 0\) and \(i_2 = 2\ cos\ 40t\ A\); (b) \(v_2\) if \(i_1 = 5\ cos\ (40t + 15^{\circ})\ A\) and \(i_2 = 0\). (c) Repeat parts (a) and (b) if M is increased to 300 mH.

In Fig. 13.37, set \(L_1 = 1\ \mu H\), \(L_2 = 2\ \mu H\), and \(M = 150\ nH\). Obtain a steady state expression for (a) \(v_1\) if \(i_2 = –cos\ 70t\ mA\) and \(i_1 = 0\); (b) \(v_2\) if \(i_1 = 55\ cos\ (5t - 30^{\circ})\ A\); (c) \(v_2\) if \(i_1 = 6\ sin\ 5t\ A\) and \(i_2 = 3\ sin\ 5t\).

For the configuration of Fig. 13.38, \(L_1 = 0.5L_2 = 1\ mH\) and \(M = 0.85 \sqrt{L_1 L_2}\). Calculate \(v_2 (t)\) if (a) \(i_2 = 0\) and \(i_1 = 5e^{-t}\ mA\); (b) \(i_2 = 0\) and \(i_1 = 5\ cos\ 10t\ mA\); (c) \(i_2 = 5\ cos\ 70t\ mA\) and \(i_1 = 0.5i_2\).

The physical construction of three pairs of coupled coils is shown in Fig. 13.39. Show the two different possible locations for the two dots on each pair of coils.

In the circuit of Fig. 13.40, \(i_1 = 5\ sin\ (100t - 80^{\circ})\ mA\), \(L_1 = 1\ H\), and \(L_2 = 2\ H\) If \(v_2 = 250\ sin\ (100t - 80^{\circ})\ mV\), calculate M.

In the circuit represented in Fig. 13.40, determine \(i_1\) if \(v_2(t) = 4\ cos\ 5t\ V\), \(L_1 = 1\ mH\), \(L_2 = 4\ mH\), and \(M = 1.5\ mH\).

Calculate \(v_1\) and \(v_2\) if \(i_1 = 3\ cos\ (2000t + 13^{\circ})\ mA\) and \(i_2 = 5\ sin\ 400t\ mA\), \(L_1 = 1\ mH\). \(L_2 = 3\ mH\), and \(M = 200\ nH\), for the coupled inductances shown in (a) Fig. 13.35; (b) Fig. 13.36.

For the circuit of Fig. 13.41, calculate \(\mathbf{I}_{1}\), \(\mathbf{I}_{2}\), \(\mathbf{V}_{2} / \mathbf{V}_{1}\), and \(\mathbf{I}_{2} / \mathbf{I}_{1}\).

For the circuit of Fig. 13.42, plot the magnitude of \(\mathbf{V}_{2} / \mathbf{V}_{1}\) as a function of frequency \(\omega\), over the range \(0\ \leq\ \omega\ \leq\ 2\ rad/s\).

For the circuit of Fig. 13.43, (a) draw the phasor representation; (b) write a complete set of mesh equations; (c) calculate \(i_2(t)\) if \(v_1(t) = 8\ sin\ 720t\ V\).

In the circuit of Fig. 13.43, M is reduced by an order of magnitude. Calculate \(i_3\) if \(v_1 = 10\ cos\ (800t - 20^{\circ})\ V\).

In the circuit shown in Fig. 13.44, find the average power absorbed by (a) the source; (b) each of the two resistors; (c) each of the two inductances; (d) the mutual inductance.

Consider the circuit of Fig. 13.46. The two sources are \(i_{s1} = 2\ cos\ t\ mA\) and \(i_{s2} = 1.5\ sin\ t\ mA\). If \(M_1 = 2\ H\), \(M_2 = 0\ H\), and \(M_3 = 10\ H\), calculate \(v_{AG}(t)\).

For the circuit of Fig. 13.46, \(M_1 = 1\ H\), \(M_2 = 1.5\ H\), and \(M_3 = 2\ H\). If \(i_{s1} = 8\ cos\ 2t\ A\) and \(i_{s2} = 7\ sin\ 2t\ A\), calculate (a) \(\mathbf{V}_{A B}\); (b) \(\mathbf{V}_{A G}\); (c) \(\mathbf{V}_{C G}\).

Determine an expression for \(i_C(t)\) valid for t > 0 in the circuit of Fig. 13.48, if \(v_s(t) = 10t^2 u(t) / (t^2 + 0.01)\ V\).

For the coupled inductor network of Fig. 13.49a, set \(L_1 = 20\ mH\), \(L_2 = 30\ mH\), \(M = 10\ mH\), and obtain equations for \(v_A\) and \(v_B\) if (a) \(i_1 = 0\) and \(i_2 = 5\ sin\ 10t\); (b) \(i_1 = 5\ cos\ 20t\) and \(i_2 = 2\ cos\ (20t - 100^{\circ})\ mA\). (c) Express \(\mathbf{V}_{1}\) and \(\mathbf{V}_{2}\) as functions of \(\mathbf{I}_{A}\) and \(\mathbf{I}_{B}\) for the network shown in Fig. 13.49b.

Find \(\mathbf{V}_1\ (j\omega)\) and \(\mathbf{V}_2\ (j\omega)\) in terms of \(\mathbf{I}_1\ (j\omega)\) and \(\mathbf{I}_2\ (j\omega)\) for each circuit of Fig. 13.51.

(a) Find \(\mathbf{Z}_{\mathrm{in }}\ (j\omega)\) for the network of Fig. 13.52. (b) Plot \(\mathbf{Z}_{\mathrm{in}}\) over the frequency range of \(0\ \leq\ \omega\ \leq\ 1000\ rad/s\). (c) Find \(\mathbf{Z}_{\mathrm{in }}\ (j\omega)\) for \(\omega = 50\ rad/s\).

For the coupled coils of Fig. 13.53, \(L_1 = L_2 = 10\ H\), and M is equal to its maximum possible value. (a) Compute the coupling coefficient k. (b) Calculate the energy stored in the magnetic field linking the two coils at t = 200 ms if \(i_1 = 10\ cos\ 4t\ mA\) and \(i_2 = 2\ cos\ 4t\ mA\).

With regard to the coupled inductors shown in Fig. 13.53, \(L_1 = 10\ mH\), \(L_2 = 5\ mH\), and k = 0.75. (a) Compute M. (b) If \(i_1 = 100\ sin\ 40t\ mA\), and \(i_2 = 0\), compute the energy stored in each coil and in the magnetic field coupling the two inductors at t = 2 ms. (c) Repeat part (b) if \(i_2\) is set to \(75\ cos\ 40t\ mA\).

For the circuit of Fig. 13.54, \(L_1 = 2\ mH\), \(L_2 = 8\ mH\), and \(v_1 = cos\ 8t\ V\). (a) Obtain an equation for \(v_2(t)\). (b) Plot \(\mathbf{V}_{2}\) as a function of k. (c) Plot the phase angle (in degrees) of \(\mathbf{V}_{2}\) as a function of k.

Connect a load \(\mathbf{Z}_L=5 \angle 33^{\circ}\ \Omega\) to the right-hand terminals of Fig. 13.53. Derive an expression for the input impedance at f = 100 Hz, seen looking into the left hand terminals, if \(L_1 = 1.5\ mH\), \(L_2 = 3\ mH\), and k = 0.55.

Consider the circuit represented in Fig. 13.55. The coupling coefficient k = 0.75. If \(i_s = 5\ cos\ 200t\ mA\), calculate the total energy stored at t = 0 and t = 5 ms if (a) a-b is open-circuited (as shown); (b) a-b is short-circuited.

Compute \(v_1\), \(v_2\), and the average power delivered to each resistor in the circuit of Fig. 13.56.

Assume the following values for the circuit depicted schematically in Fig. 13.16: \(R_1 = 10\ \Omega\), \(R_2 = 1\ \Omega\), \(L_1 = 2\ \mu H\), \(L_2 = 1\ \mu H\), and \(M = 500\ nH\). Calculate the input impedance for \(\omega = 10\ rad/s\) if \(\mathbf{Z}_{L}\) is equal to (a) \(1\ \Omega\); (b) \(j\ \Omega\); (c) \(–j\ \Omega\); (d) \(5\angle33^{\circ}\ \Omega\).

Determine the T equivalent of the linear transformer represented in Fig. 13.57 (draw and label an appropriate diagram).

Represent the T network shown in Fig. 13.59 as an equivalent linear transformer if (a) \(L_x = 1\ H\), \(L_y = 2\ H\), and \(L_z = 4\ H\); (b) \(L_x = 10\ mH\), \(L_y = 50\ mH\), and \(L_z = 22\ mH\).

Assuming zero initial currents, obtain an equivalent \(\Pi\) network of the transformer depicted in Fig. 13.57.

(a) Draw and label a suitable equivalent \(\Pi\) network of the linear transformer shown in Fig. 13.58, assuming zero initial currents. (b) Verify their equivalence with an appropriate simulation.

For the circuit of Fig. 13.61, determine an expression for (a) \(\mathbf{I}_{L} / \mathbf{V}_{s}\); (b) \(\mathbf{V}_{1} / \mathbf{V}_{s}\).

(a) For the circuit of Fig. 13.62, if \(v_s = 8\ cos\ 1000t\ V\), calculate \(v_o\). (b) Verify your solution with an appropriate PSpice simulation.

With respect to the network shown in Fig. 13.63, derive an expression for \(\mathbf{Z}(j \omega)\) if \(M_1\) and \(M_2\) are set to their respective maximum values.

Calculate \(\mathbf{I}_{2}\) and \(\mathbf{V}_{2}\) for the ideal transformer circuit of Fig. 13.64 if (a) \(\mathbf{V}_1=4 \angle 32^{\circ}\mathrm{\ V}\) and \(\mathbf{Z}_{L}=1-j \Omega\); (b) \(\mathbf{V}_1=4 \angle 32^{\circ}\mathrm{\ V}\) and \(\mathbf{Z}_{L}=0\); (c) \(\mathbf{V}_1=2 \angle 118^{\circ}\mathrm{\ V}\) and \(\mathbf{Z}_L=1.5 \angle 10^{\circ}\ \Omega\).

With respect to the ideal transformer circuit depicted in Fig. 13.64, calculate \(\mathbf{I}_{2}\) and \(\mathbf{V}_{2}\) if (a) \(\mathbf{I}_1=244 \angle 0^{\circ}\mathrm{\ mA}\) and \(\mathbf{Z}_L=5-j2\ \Omega\); (b) \(\mathbf{I}_1=100 \angle 10^{\circ}\mathrm{\ mA}\) and \(\mathbf{Z}_L=j2\ \Omega\).

Calculate the average power delivered to the \(400\ m \Omega\) and \(21\ \Omega\) resistors, respectively, in the circuit of Fig. 13.65.

With regard to the ideal transformer circuit represented in Fig. 13.65, determine an equivalent circuit in which (a) the transformer and primary circuit are replaced, so that \(\mathbf{V}_{2}\) and \(\mathbf{I}_{2}\) are unchanged; (b) the transformer and secondary circuit are replaced, so that \(\mathbf{V}_{1}\) and \(\mathbf{I}_{1}\) are unchanged.

Calculate the average power delivered to each resistor shown in Fig. 13.66.

Calculate \(\mathbf{I}_{x}\) and \(\mathbf{V}_{2}\) as labeled in Fig. 13.68.

The ideal transformer of the circuit in Fig. 13.68 is removed, flipped across its vertical axis, and reconnected such that the same terminals remain connected to the negative terminal of the source. (a) Calculate \(\mathbf{I}_{x}\) and \(\mathbf{V}_{2}\). (b) Repeat part (a) if both dots are placed at the bottom terminals of the transformer.

For the circuit of Fig. 13.69, \(v_s = 117\ sin\ 500t\ V\). Calculate \(v_2\) if the terminals marked a and b are (a) left open-circuited; (b) short-circuited; (c) bridged by a \(2\ \Omega\) resistor.

The turns ratio of the ideal transformer in Fig. 13.69 is changed from 30:1 to 1:3. Take \(v_s = 720\ cos\ 120 \pi t\ V\), and calculate \(v_2\) if terminals a and b are (a) short-circuited; (b) bridged by a \(10\ \Omega\) resistor; (c) bridged by a \(1\ M \Omega\) resistor.

For the circuit of Fig. 13.70, \(R_1 = 1\ \Omega\), \(R_2 = 4\ \Omega\), and \(R_L = 1\ \Omega\). Select a and b to achieve a peak voltage of 200 V magnitude across \(R_L\).

Calculate \(v_x\) for the circuit of Fig. 13.70 if \(a = 0.01b = 1\), \(R_1 = 300\ \Omega\), \(R_2 = 14\ \Omega\), and \(R_L = 1\ k \Omega\).

(a) Referring to the ideal transformer circuit in Fig. 13.70, determine the load current \(i_L\) if \(b = 0.25a = 1\), \(R_1 = 2.2\ \Omega\), \(R_2 = 3.1\ \Omega\), and \(R_L = 200\ \Omega\). (b) Verify your solution with an appropriate PSpice simulation.

Determine the Thévenin equivalent of the network in Fig. 13.71 as seen looking into terminals a and b.

Calculate \(\mathbf{V}_{2}\) and the average power delivered to the \(8\ \Omega\) resistor of Fig. 13.72 if \(\mathbf{V}_s=10 \angle 15^{\circ}\mathrm{\ V}\), and the control parameter c is equal to (a) 0; (b) 1 mS.

(a) For the circuit of Fig. 13.72, take \(c = –2.5\ mS\) and select values of a and b such that 100 W average power is delivered to the \(8\ \Omega\) load when \(\mathbf{V}_s=5 \angle -35^{\circ}\mathrm{\ V}\), (b) Verify your solution with an appropriate PSpice simulation.

A transformer whose name plate reads “2300/230 V, 25 kVA” operates with primary and secondary voltages of 2300 V and 230 V rms, respectively, and can supply 25 kVA from its secondary winding. If this transformer is supplied with 2300 V rms and is connected to secondary loads requiring 8 kW at unity PF and 15 kVA at 0.8 PF lagging, (a) what is the primary current? (b) How many kilowatts can the transformer still supply to a load operating at 0.95 PF lagging? (c) Verify your answers with PSpice.

A friend brings a vintage stereo system back from a recent trip to Warnemünde, unaware that it was designed to operate on twice the supply voltage (240 VAC) available at American household outlets. Design a circuit to allow your friend to listen to the stereo in the United States, assuming the operating frequency (50 Hz in Germany, 60 Hz in the United States) difference can be neglected.

The friend referred to in Exercise 57 attempts to justify the erroneous assumption made regarding the stereo by pointing out that the wall outlet in the W.C. (bathroom) had a socket for his U.S. electric razor, clearly marked 120 VAC. He failed to notice that the small sign below the outlet clearly stated “Razors only.” With the knowledge that all power lines running into the room operated at 240 VAC, draw the likely circuit built into the bathroom wall outlet, and explain why it is limited to “razors only.”

Obtain an expression for \(\mathbf{V}_{2} / \mathbf{V}_{s}\) in the circuit of Fig. 13.73 if (a) \(L_1 = 100\ mH\), \(L_2 = 500\ mH\), and M is its maximum possible value; (b) \(L_1 = 5L_2 = 1.4\ H\) and \(k=87 \%\) of its maximum possible value; (c) the two coils can be treated as an ideal transformer, the left-hand coil having 500 turns and the right-hand coil having 10,000 turns.

For the circuit of Fig. 13.11, write an appropriate mesh equation in terms of the phasor currents \(I_1\) and \(I_2\) for the (a) left mesh; (b) right mesh.

(a) If the two networks shown in Fig. 13.20 are equivalent, specify values for \(L_x\), \(L_y\), and \(L_z\). (b) Repeat if the dot on the secondary in Fig. 13.20b is located at the bottom of the coil.

For the circuit of Fig. 13.11, write an appropriate mesh equation in terms of the phasor currents \(I_1\) and \(I_2\) for the (a) left mesh; (b) right mesh.

(a) If the two networks shown in Fig. 13.20 are equivalent, specify values for \(L_x\), \(L_y\), and \(L_z\). (b) Repeat if the dot on the secondary in Fig. 13.20b is located at the bottom of the coil.

Calculate \(v_1\) and \(v_2\) if \(i_1 = 5\ sin\ 40t\ mA\) and \(i_2 = 5\ cos\ 40t\ mA\), \(L_1 = 1\ mH\), \(L_2 = 3\ mH\), and \(M = 0.5\ mH\), for the coupled inductances shown in (a) Fig. 13.37; (b) Fig. 13.38.

The circuit of Fig. 13.45 is designed to drive a simple \(8\ \Omega\)speaker. What value of M results in 1 Wof average power being delivered to the speaker?

For the circuit of Fig. 13.47, find the currents \(i_1(t)\), \(i_2(t)\), and \(i_3(t)\) if \(f = 60\ Hz\).

Note that there is no mutual coupling between the 5 H and 6 H inductors in the circuit of Fig. 13.50. (a) Write a set of equations in terms of \(I_1\ ( j \omega)\), \(I_2\ ( j \omega)\), and \(I_3\ ( j \omega)\). (b) Find \(I_3\ ( j \omega)\) if \(\omega = 2\ rad/s\).

(a) Draw and label an appropriate diagram of a T equivalent network for the linear transformer shown in Fig. 13.58. (b) Verify the two are equivalent by connecting a voltage \(v_{AB} = 5\ sin\ 45t\ V\) and calculating the open-circuit voltage \(v_{CD}\).

Represent the \(\Pi\) network of Fig. 13.60 as an equivalent linear transformer with zero initial currents if (a) \(L_A = 1\ H\), \(L_B = 2\ H\), and \(L_C = 4\ H\); (b) \(L_A = 10\ mH\), \(L_B = 50\ mH\), and \(L_C = 22\ mH\).

With respect to the circuit depicted in Fig. 13.67, calculate (a) the voltages \(v_1\) and \(v_2\); (b) the average power delivered to each resistor.

You notice your neighbor has installed a large coil of wire in close proximity to the power line coming into your house (underground cables are not available in your neighborhood). (a) What is the likely intention of your neighbor? (b) Is the plan likely to succeed? Explain. (c) When confronted, your neighbor simply shrugs and claims there’s no way it can cost you anything, anyway, since nothing of his is touching anything on your property. True or not? Explain.