Assuming the passive sign convention and an operating frequency of 314 rad/s, calculate

Find the angle by which \(i_1\) lags \(v_1\) if \(v_1 = 120\ cos (120 \pi t - 40^{\circ})\ V\) and \(i_1\) equals (a) \(2.5\ cos (120 \pi t + 20^{\circ})\ A\); (b) \(1.4\sin\left(120\pi t-70^{\circ}\right)\ \mathrm{A}\); (c) \(-0.8\cos\left(120\pi t-110^{\circ}\right)\ \mathrm{A}\).

Find A, B, C, and \(\phi\) if \(40\ cos (110t - 40^{\circ}) - 20\ sin (100t + 170^{\circ}) = A\ cos\ 100t + B\ sin\ 100t = C\ cos(100t + \phi)\).

Let \(v_s = 40\ cos\ 8000t\ V\) in the circuit of Fig. 10.7. Use Thévenin’s theorem where it will do the most good, and find the value at t = 0 for (a) \(i_L\); (b) \(v_L\); (c) \(i_R\); (d) \(i_s\).

Evaluate and express the result in rectangular form: (a) \(\left[\left(2 \angle30^{\circ}\right)\left(5 \angle-110^{\circ}\right)\right](1+j 2)\); (b) \(\left(5 \angle-200^{\circ}\right)+4 \angle 20^{\circ}\). Evaluate and express the result in polar form: (c) \((2-j 7) /(3-j)\); (d) \(8-j 4+\left[\left(5 \angle 80^{\circ}\right) /\left(2 \angle 20^{\circ}\right)\right]\).

If the use of the passive sign convention is specified, find the (a) complex voltage that results when the complex current \(4e^{j800t}\ A\) is applied to the series combination of a 1 mF capacitor and a \(2\ \Omega\) resistor; (b) complex current that results when the complex voltage \(100e^{j2000t}\ V\) is applied to the parallel combination of a 10 mH inductor and a \(50\ \Omega\) resistor.

Let \(\omega = 2000\ rad/s\) and t = 1 ms. Find the instantaneous value of each of the currents given here in phasor form: (a) j10 A; (b) 20 + j10 A; (c) \(20+j\left(10\angle20^{\circ}\right)\ \mathrm{A}\).

Transform each of the following functions of time into phasor form: (a) \(-5 \sin \left(580 t-110^{\circ}\right)\); (b) \(3 \cos 600 t-5 \sin \left(600 t+110^{\circ}\right)\); (c) \(8 \cos \left(4 t-30^{\circ}\right)+4 \sin \left(4 t-100^{\circ}\right)\). Hint: First convert each into a single cosine function with a positive magnitude.

In the circuit of Fig. 10.17, both sources operate at \(\omega = 1\ rad/s\). If \(\mathbf{I}_{C}=2 \angle 28^{\circ} \mathrm{A}\) and \(\mathbf{I}_{L}=3 \angle 53^{\circ} \mathrm{A}\), calculate (a) \(\mathbf{I}_{s}\); (b) \(\mathbf{V}_{s}\); (c) \(i_{R_{1}}(t)\).

With reference to the network shown in Fig. 10.19, find the input impedance \(Z_{in}\) that would be measured between terminal: (a) a and g; (b) b and g; (c) a and b.

In the frequency-domain circuit of Fig. 10.21, find (a) \(I_1\); (b) \(I_2\); (c) \(I_3\).

Determine the admittance (in rectangular form) of (a) an impedance \(Z = 1000 + j400\ \Omega\); (b) a network consisting of the parallel combination of an \(800\ \Omega\) resistor, a 1 mH inductor, and a 2 nF capacitor, if \(\omega = 1\ Mrad/s\); (c) a network consisting of the series combination of an \(800\ \Omega\) resistor, a 1 mH inductor, and a 2 nF capacitor, if \(\omega = 1\ Mrad/s\).

Use nodal analysis on the circuit of Fig. 10.23 to find \(V_1\) and \(V_2\).

Use mesh analysis on the circuit of Fig. 10.25 to find \(I_1\) and \(I_2\).

If superposition is used on the circuit of Fig. 10.30, find \(V_1\) with (a) only the \(20 \angle 0^{\circ} \mathrm{\ mA}\) source operating; (b) only the \(50 \angle-90^{\circ} \mathrm{\ mA}\) source operating.

For the circuit of Fig. 10.32, find the (a) open-circuit voltage \(V_{ab}\); (b) downward current in a short circuit between a and b; (c) Thévenin equivalent impedance \(Z_{ab}\) in parallel with the current source.

Determine the current i through the \(4\ \Omega\) resistor of Fig. 10.34.

Select some convenient reference value for \(I_C\) in the circuit of Fig. 10.44; draw a phasor diagram showing \(V_R\), \(V_2\), \(V_1\), and \(V_s\); and measure the ratio of the lengths of (a) \(V_s\) to \(V_1\); (b) \(V_1\) to \(V_2\); (c) \(V_s\) to \(V_R\).

Evaluate the following: (a) \(5\ sin\ (5t - 9^{\circ})\) at t = 0, 0.01, and 0.1 s; (b) 4 cos 2t and \(4\ sin\ (2t + 90^{\circ})\) at t = 0, 1, and 1.5 s; (c) \(3.2\ cos\ (6t + 15^{\circ})\) and \(3.2\ \sin\ (6t + 105^{\circ})\) at t = 0, 0.01, and 0.1 s.

(a) Express each of the following as a single cosine function: \(5\ \sin\ 300t\), \(1.95\ \sin\ (\pi t - 92^{\circ})\), \(2.7\ \sin\ (50t + 5^{\circ}) - 10\ \cos\ 50t\). (b) Express each of the following as a single sine function: \(66\ \cos\ (9t - 10^{\circ})\), \(4.15\ \cos\ 10t\), \(10\ \cos\ (100t - 9^{\circ}) + 10\ \sin\ (100t + 19^{\circ})\).

Determine the angle by which \(v_1\) leads \(i_1\) if \(v_1 = 10\ \cos\ (10t - 45^{\circ})\) and \(i_1\) is equal to (a) \(5\ cos\ 10t\); (b) \(5\ \cos\ (10t - 80^{\circ})\); (c) \(5\ \cos\ (10t - 40^{\circ})\); (d) \(5\ \cos\ (10t + 40^{\circ})\); (e) \(5\ \sin\ (10t - 19^{\circ})\).

Determine the angle by which \(v_1\) lags \(i_1\) if \(v_1 = 34\ \cos\ (10t + 125^{\circ})\) and \(i_1\) is equal to (a) \(5\ \cos\ 10t\); (b) \(5\ \cos\ (10t - 80^{\circ})\); (c) \(5\ \cos (10t - 40^{\circ})\); (d) \(5\ \cos (10t + 40^{\circ})\); (e) \(5\ \sin (10t - 19^{\circ})\).

Determine which waveform in each of the following pairs is lagging: (a) cos 4t, sin 4t; (b) \(cos\ (4t - 80^{\circ})\), cos (4t); (c) \(cos\ (4t + 80^{\circ})\), cos 4t; (d) \(-\sin 5 t, \cos\left(5 t+2^{\circ}\right)\); (e) \(sin\ 5t + cos\ 5t,\ cos\ (5t - 45^{\circ})\).

Calculate the first three instants in time (t > 0) for which the following functions are zero, by first converting to a single sinusoid: (a) cos 3t ? 7 sin 3t; (b) \(cos\ (10t + 45^{\circ})\); (c) cos 5t? sin 5t; (d) cos 2t + sin 2t? cos 5t + sin 5t.

(a) Determine the first two instants in time (t > 0) for which each of the functions in Exercise 6 are equal to unity, by first converting to a single sinusoid. (b) Verify your answers by plotting each waveform using an appropriate software package.

The concept of Fourier series is a powerful means of analyzing periodic waveforms in terms of sinusoids. For example, the triangle wave in Fig. 10.45 can be represented by the infinite sum \(v(t)=\frac{8}{\pi^{2}}\left(\sin \pi t-\frac{1}{3^{2}} \sin 3 \pi t+\frac{1}{5^{2}} \sin 5 \pi t-\frac{1}{7^{2}} \sin 7 \pi t+\cdots\right)\) where in practice perhaps the first several terms provide an accurate enough approximation. (a) Compute the exact value of v(t) at t = 0.25 s by first obtaining an equation for the corresponding segment of the waveform. (b) Compute the approximate value at t = 0.25 s using the first term of the Fourier series only. (c) Repeat part (b) using the first three terms. (d) Plot v(t) using the first term only. (e) Plot v(t) using the first two terms only. (f) Plot v(t) using the first three terms only.

Household electrical voltages are typically quoted as either 110 V, 115 V, or 120 V. However, these values do not represent the peak ac voltage. Rather, they represent what is known as the root mean square of the voltage, defined as \(V_{\mathrm{rms}}=\sqrt{\frac{1}{T} \int_{0}^{T} V_{m}^{2} \cos ^{2}(\omega t) d t}\) where T = the period of the waveform, \(V_m\) is the peak voltage, and \(\omega =\) the waveform frequency ( f = 60 Hz in North America). (a) Perform the indicated integration, and show that for a sinusoidal voltage, \(V_{\mathrm{rms}}=\frac{V_{m}}{\sqrt{2}}\) (b) Compute the peak voltages corresponding to the rms voltages of 110, 115, and 120 V.

If the source \(v_s\) in Fig. 10.46 is equal to \(4.53\ cos\ (0.333\ \times\ 10^{-3}t + 30^{\circ})\ V\), (a) obtain \(i_s\), \(i_L\), and \(i_R\) at t = 0 assuming no transients are present; (b) obtain an expression for \(v_L (t)\) in terms of a single sinusoid, valid for t > 0, again assuming no transients are present.

Assuming there are no longer any transients present, determine the current labeled \(i_L\) in the circuit of Fig. 10.47. Express your answer as a single sinusoid.

Calculate the power dissipated in the \(2\ \Omega\) resistor of Fig. 10.47 assuming there are no transients present. Express your answer in terms of a single sinusoidal function.

Obtain an expression for \(v_C\) as labeled in Fig. 10.48, in terms of a single sinusoidal function. You may assume all transients have died out long before t = 0.

Calculate the energy stored in the capacitor of the circuit depicted in Fig. 10.48 at t = 10 ms and t = 40 ms.

Obtain an expression for the power dissipated in the \(10\ \Omega\) resistor of Fig. 10.49, assuming no transients present.

Express the following complex numbers in rectangular form: (a) \(50 \angle-75^{\circ}\); (b) \(19e^{j30^{\circ}}\); \(2.5 \angle-30^{\circ}+0.5 \angle 45^{\circ}\). Convert the following to polar form: (c) \((2 + j2) (2 - j2)\); (d) \((2+j 2)\left(5 \angle 22^{\circ}\right)\).

Express the following in polar form: (a) \(2 + e^{j35^{\circ}}\); (b) (j)(j)(-j); (c) 1. Express the following in rectangular form: (d) \(2 + e^{j35^{\circ}}\); (e) \(-j 9+5 \angle 55^{\circ}\).

Evaluate the following, and express your answer in polar form: (a) 4(8 - j8); (b) \(4 \angle 5^{\circ}-2 \angle 15^{\circ}\); (c) \((2+j 9)-5 \angle 0^{\circ}\); (d) \(\frac{-j}{10+5 j}-3 \angle 40^{\circ}+2\); (e) \((10+j 5)(10-j 5)\left(3 \angle 40^{\circ}\right)+2\).

Evaluate the following, and express your answer in rectangular form: (a) \(3\left(3 \angle 30^{\circ}\right)\); (b) \(2 \angle 25^{\circ}+5 \angle-10^{\circ}\); (c) \((12+j 90)-5 \angle 30^{\circ}\); (d) \(\frac{10+5 j}{8-j}+2 \angle 60^{\circ}+1\); (e) \((10+5 j)(10-5 j)\left(3 \angle 40^{\circ}\right)+2\).

Perform the indicated operations, and express the answer in both rectangular and polar forms: (a) \(\frac{2+j 3}{1+8 \angle 90^{\circ}}-4\); (b) \(\left(\frac{10 \angle 25^{\circ}}{5 \angle-10^{\circ}}+\frac{3 \angle 15^{\circ}}{3-j 5}\right)\ j 2\) (c) \(\left[\frac{(1-j)(1+j)+1 \angle 0^{\circ}}{-j}\right]\left(3 \angle-90^{\circ}\right)+\frac{j}{5 \angle-45^{\circ}}\).

Insert an appropriate complex source into the circuit represented in Fig. 10.50, and use it to determine steady-state expressions for \(i_C(t)\) and \(v_C(t)\).

For the circuit of Fig. 10.51, if \(i_s = 5\ cos\ 10t\ A\), use a suitable complex source replacement to obtain a steady-state expression for \(i_L(t)\).

In the circuit depicted in Fig. 10.51, \(i_s\) is modified such that the \(2\ \Omega\) resistor is replaced by a \(20\ \Omega\) resistor. If \(i_{L}(t)=62.5 \angle 31.3^{\circ} \mathrm{\ mA}\), determine \(i_s\).

Employ a suitable complex source to determine the steady-state current \(i_L\) in the circuit of Fig. 10.52.

Transform each of the following into phasor form: (a) 75.928 cos (110.1t); (b) \(5\ cos\ (55t - 42^{\circ})\); (c) \(-sin\ (8000t + 14^{\circ})\); (d) \(3\ \cos\ 10t - 8\ cos\ (10t + 80^{\circ})\).

Transform each of the following into phasor form: (a) 11 sin 100t; (b) 11 cos 100t; (c) \(11\ \cos\ (100t - 90^{\circ})\); (d) \(3\ cos\ 100t - 3\ sin\ 100t\).

Assuming an operating frequency of 1 kHz, transform the following phasor expressions into a single cosine function in the time domain: (a) \(9 \angle 65^{\circ} \mathrm{V}\); (b) \(\frac{2 \angle 31^{\circ}}{4 \angle 25^{\circ}} \mathrm{A}\); (c) \(22 \angle 14^{\circ}-8 \angle 33^{\circ} \mathrm{V}\)

The following complex voltages are written in a combination of rectangular and polar form. Rewrite each, using conventional phasor notation (i.e., a magnitude and angle): (a) \(\frac{2-j}{5 \angle 45^{\circ}} \mathrm{V}\) (b) \(\frac{6 \angle 20^{\circ}}{1000}-j \mathrm{V}\); (c) \((j)\left(52.5\angle-90^{\circ}\right)\ \mathrm{V}\).

Assuming an operating frequency of 50 Hz, compute the instantaneous voltage at t = 10 ms and t = 25 ms for each of the quantities represented in Exercise 26.

Assuming an operating frequency of 50 Hz, compute the instantaneous voltage at t = 10 ms and t = 25 ms for each of the quantities represented in Exercise 27.

Assuming the passive sign convention and an operating frequency of 5 rad/s, calculate the phasor voltage which develops across the following when driven by the phasor current \(\mathbf{I}=2\angle0^{\circ}\mathrm{\ mA}\); (a) a \(1\ k \Omega\) resistor; (b) a 1 mF capacitor; (c) a 1 nH inductor.

(a) A series connection is formed between a \(1\ \Omega\) resistor, a 1 F capacitor, and a 1 H inductor, in that order. Assuming operation at \(\omega = 1\ rad/s\), what are the magnitude and phase angle of the phasor current which yields a voltage of \(1 \angle 30^{\circ} \mathrm{V}\) across the resistor (assume the passive sign convention)? (b) Compute the ratio of the phasor voltage across the resistor to the phasor voltage which appears across the capacitor-inductor combination. (c) The frequency is doubled. Calculate the new ratio of the phasor voltage across the resistor to the phasor voltage across the capacitor-inductor combination.

Assuming the passive sign convention and an operating frequency of 314 rad/s, calculate the phasor voltage V which appears across each of the following when driven by the phasor current \(\mathbf{I}=10 \angle 0^{\circ} \mathrm{\ mA}\): (a) a \(2\ \Omega\) resistor; (b) a 1 F capacitor; (c) a 1 H inductor; (d) a \(2\ \Omega\) resistor in series with a 1 F capacitor; (e) a \(2\ \Omega\) resistor in series with a 1 H inductor. (f) Calculate the instantaneous value of each voltage determined in parts (a) to (e) at t = 0.

The circuit of Fig. 10.53 is shown represented in the phasor (frequency) domain. If \(\mathbf{I}_{10}=4 \angle 35^{\circ} \mathrm{A}\), \(\mathbf{V}=10 \angle 35^{\circ}\), and \(\mathbf{I}=2 \angle 35^{\circ} \mathrm{A}\), (a) across what type of element does V appear, and what is its value? (b) Determine the value of \(\mathbf{V}_{s}\).

(a) Obtain an expression for the equivalent impedance \(\mathbf{Z}_{\text {eq }}\) of a \(1\ \Omega\) resistor in series with a 10 mH inductor as a function of \(\omega\). (b) Plot the magnitude of \(\mathbf{Z}_{\text {eq }}\) as a function of \(\omega\) over the range \(1 < \omega < 100\ krad/s\) (use a logarithmic scale for the frequency axis). (c) Plot the angle (in degrees) of \(\mathbf{Z}_{\text {eq }}\) as a function of \(\omega\) over the range \(1 < \omega < 100\ krad/s\) (use a logarithmic scale for the frequency axis). [Hint: semilogx() in MATLAB is a useful plotting function.]

Determine the equivalent impedance of the following, assuming an operating frequency of 20 rad/s: (a) \(1\ k \Omega\) in series with 1 mF; (b) \(1\ k \Omega\) in parallel with 1 mH; (c) \(1\ k \Omega\) in parallel with the series combination of 1 F and 1 H.

(a) Obtain an expression for the equivalent impedance \(\mathbf{Z}_{\mathrm{eq}}\) of a \(1\ \Omega\) resistor in series with a 10 mF capacitor as a function of \(\omega\). (b) Plot the magnitude of \(\mathbf{Z}_{\mathrm{eq}}\) as a function of \(\omega\) over the range \(1 < \omega < 100\ krad/s\) (use a logarithmic scale for the frequency axis). (c) Plot the angle (in degrees) of \(\mathbf{Z}_{\mathrm{eq}}\) as a function of \(\omega\) over the range \(1 < \omega < 100\ krad/s\) (use a logarithmic scale for the frequency axis). [Hint: semilogx() in MATLAB is a useful plotting function.]

Determine the equivalent admittance of the following, assuming an operating frequency of 1000 rad/s: (a) \(25\ \Omega\) in series with 20 mH; (b) \(25\ \Omega\) in parallel with 20 mH; (c) \(25\ \Omega\) in parallel with 20 mH in parallel with 20 mF; (d) \(1\ \Omega\) in series with 1 F in series with 1 H; (e) \(1\ \Omega\) in parallel with 1 F in parallel with 1 H.

Consider the network depicted in Fig. 10.54, and determine the equivalent impedance seen looking into the open terminals if (a) \(\omega = 1\ rad/s\); (b) \(\omega = 10\ rad/s\); (c) \(\omega = 100\ rad/s\).

Exchange the capacitor and inductor in the network shown in Fig. 10.54, and calculate the equivalent impedance looking into the open terminals if \(\omega = 25\ rad/s\).

Find V in Fig. 10.55 if the box contains (a) \(3\ \Omega\) in series with 2 mH; (b) \(3\ \Omega\) in series with \(125\ \mu F\); (c) \(3\,\Omega\), 2 mH, and \(125\ \mu F\) in series; (d) \(3\ \Omega\), 2 mH, and \(125\ \mu F\) in series, but \(\omega = 4\ krad/s\).

Calculate the equivalent impedance seen at the open terminals of the network shown in Fig. 10.56 if f is equal to (a) 1 Hz; (b) 1 kHz; (c) 1 MHz; (d) 1 GHz; (e) 1 THz.

Employ phasor-based analysis to obtain an expression for i(t) in the circuit of Fig. 10.57.

Design a suitable combination of resistors, capacitors, and/or inductors which has an equivalent impedance at \(\omega = 100\ rad/s\) of (a) \(1\ \Omega\) using at least one inductor; (b) \(7 \angle 10^{\circ} \Omega\); (c) \(3 - j4\ \Omega\).

Design a suitable combination of resistors, capacitors, and/or inductors which has an equivalent admittance at \(\omega = 10\ rad/s\) of (a) 1 S using at least one capacitor; (b) \(12 \angle -18^{\circ} \ \mathrm{S}\); (c) 2 + j mS.

For the circuit depicted in Fig. 10.58, (a) redraw with appropriate phasors and impedances labeled; (b) employ nodal analysis to determine the two nodal voltages \(v_1(t)\) and \(v_2 (t)\).

For the circuit illustrated in Fig. 10.59, (a) redraw, labeling appropriate phasor and impedance quantities; (b) determine expressions for the three time-domain mesh currents.

Referring to the circuit of Fig. 10.59, employ phasor-based analysis techniques to determine the two nodal voltages.

In the phasor-domain circuit represented by Fig. 10.60, let \(\mathbf{V}_{1}=10 \angle {-80^{\circ}} \mathrm{V}\), \(\mathbf{V}_{2}=4 \angle-0^{\circ} \mathrm{V}\) and \(\mathbf{V}_{3}=2 \angle-23^{\circ} \mathrm{V}\). Calculate \(\mathbf{I}_{1}\) and \(\mathbf{I}_{2}\).

With regard to the two-mesh phasor-domain circuit depicted in Fig. 10.60, calculate the ratio of \(\mathbf{I}_{1}\) to \(\mathbf{I}_{2}\) if \(\mathbf{V}_{1}=3 \angle 0^{\circ} \mathrm{\ V}\), \(\mathbf{V}_2=5.5\angle-130^{\circ}\mathrm{\ V}\), and \(\mathbf{V}_{3}=1.5\angle 17^{\circ}\mathrm{\ V}\).

Employ phasor analysis techniques to obtain expressions for the two mesh currents \(i_1\) and \(i_2\) as shown in Fig. 10.61.

Determine \(\mathbf{I}_{B}\) in the circuit of Fig. 10.62 if \(\mathbf{I}_{1}=5 \angle -18^{\circ} \mathrm{A}\) and \(\mathbf{I}_{2}=2 \angle 5^{\circ} \mathrm{A}\).

Determine \(\mathbf{V}_{2}\) in the circuit of Fig. 10.62 if \(\mathbf{I}_{1}=15 \angle 0^{\circ} \mathrm{A}\) and \(\mathbf{I}_{2}=25 \angle 131^{\circ} \mathrm{A}\).

Employ phasor analysis to obtain an expression for \(v_x\) as labeled in the circuit of Fig. 10.63.

Determine the current \(i_x\) in the circuit of Fig. 10.63.

Obtain an expression for each of the four (clockwise) mesh currents for the circuit of Fig. 10.64 if \(v_1 = 133\ cos\ (14t + 77^{\circ})\ V\) and \(v_2 = 55\ cos\ (14t + 22^{\circ})\ V\).

Determine the nodal voltages for the circuit of Fig. 10.64, using the bottom node as the reference node, if \(v_1 = 0.009\ cos\ (500t + 0.5^{\circ})\ V\) and \(v_2 = 0.004\ cos\ (500t + 1.5^{\circ})\ V\).

The op amp shown in Fig. 10.65 has an infinite input impedance, zero output impedance, and a large but finite (positive, real) gain, \(A=-\mathbf{V}_{o} / \mathbf{V}_{i}\). (a) Construct a basic differentiator by letting \(\mathbf{Z}_{f}=R_{f}\), find \(\mathbf{V}_{o} / \mathbf{V}_{s}\), and then show that \(\mathbf{V}_{o} / \mathbf{V}_{s} \rightarrow-j \omega C_{1} R_{f}\) as \(\mathbf{A} \rightarrow \infty\). (b) Let \(\mathbf{Z}_{f}\) represent \(C_f\) and \(R_f\) in parallel, find \(\mathbf{V}_{o} / \mathbf{V}_{s}\), and then show that \(\mathbf{V}_{o} / \mathbf{V}_{s} \rightarrow-j \omega C_{1} R_{f} /\left(1+j \omega C_{f} R_{f}\right)\) as \(\mathbf{A} \rightarrow \infty\).

Obtain an expression for each of the four mesh currents labeled in the circuit of Fig. 10.66.

Determine the individual contribution each current source makes to the two nodal voltages \(\mathbf{V}_{1}\) and \(\mathbf{V}_{2}\) as represented in Fig. 10.67.

Determine \(\mathbf{V}_{1}\) and \(\mathbf{V}_{2}\) in Fig. 10.68 if \(\mathbf{I}_1=33\angle3^{\circ}\mathrm{\ mA}\) and \(\mathbf{I}_2=51\angle-91^{\circ}\mathrm{\ mA}\).

The phasor domain circuit of Fig. 10.68 was drawn assuming an operating frequency of 2.5 rad/s. Unfortunately, the manufacturing unit installed the wrong sources, each operating at a different frequency. If \(i_1 (t) = 4\ cos\ 40t\ mA\) and \(i_2 (t) = 4\ sin\ 30t\ mA\), calculate \(v_1 (t)\) and \(v_2 (t)\).

The \((2 ? j)\ \Omega\) impedance in the circuit of Fig. 10.69 is replaced with a \((1 + j)\ \Omega\) impedance. Perform a source transformation on each source, simplify the resulting circuit as much as possible, and calculate the current flowing through the \((1 + j)\ \Omega\) impedance.

With regard to the circuit depicted in Fig. 10.70, (a) calculate the Thévenin equivalent seen looking into the terminals marked a and b; (b) determine the Norton equivalent seen looking into the terminals marked a and b; (c) compute the current flowing from a to b if a \((7 – j2)\ \Omega\) impedance is connected across them.

In the circuit of Fig. 10.71, \(i_{s1} = 8\ cos\ (4t - 9^{\circ})\ mA\), \(i_{s2} = 5\ cos\ 4t\) and \(v_{s3} = 2\ sin\ 4t\). (a) Redraw the circuit in the phasor domain; (b) reduce the circuit to a single current source with the assistance of a source transformation; (c) calculate \(v_L(t)\). (d) Verify your solution with an appropriate PSpice simulation.

Determine the individual contribution of each source in Fig. 10.72 to the voltage \(v_1(t)\).

Determine the power dissipated by the \(1\ \Omega\) resistor in the circuit of Fig. 10.73. Verify your solution with an appropriate PSpice simulation.

Use \(\omega = 1\ rad/s\), and find the Norton equivalent of the network shown in Fig. 10.74. Construct the Norton equivalent as a current source \(\mathbf{I}_{N}\) in parallel with a resistance \(R_N\) and either an inductance \(L_N\) or a capacitance \(C_N\).

The source \(\mathbf{I}_{s}\) in the circuit of Fig. 10.75 is selected such that \(\mathbf{V}=5 \angle 120^{\circ}\mathrm{\ V}\). (a) Construct a phasor diagram showing \(\mathbf{I}_{R}\), \(\mathbf{I}_{L}\), and \(\mathbf{I}_{C}\). (b) Use the diagram to determine the angle by which \(\mathbf{I}_{s}\) leads \(\mathbf{I}_{R}\), \(\mathbf{I}_{C}\), and \(\mathbf{I}_{s}\).

Let \(\mathbf{V}_{1}=100 \angle 0^{\circ} \mathrm{V}\), \(\left|\mathbf{V}_2\right|=140\mathrm{\ V}\), and \(\left|\mathbf{V}_1+\mathbf{V}_2\right|=120\mathrm{\ V}\). Use graphical methods to find two possible values for the angle of \(\mathbf{V}_{2}\).

(a) Calculate values for \(\mathbf{I}_{L}\), \(\mathbf{I}_{R}\), \(\mathbf{I}_{C}\), \(\mathbf{V}_{L}\), \(\mathbf{V}_{R}\), and \(\mathbf{V}_{C}\) for the circuit shown in Fig. 10.76. (b) Using scales of 50 V to 1 in and 25 A to 1 in, show all seven quantities on a phasor diagram, and indicate that \(\mathbf{I}_{L}=\mathbf{I}_{R}+\mathbf{I}_{C}\) and \(\mathbf{V}_{s}=\mathbf{V}_{L}+\mathbf{V}_{R}\).

In the circuit of Fig. 10.77, (a) find values for \(\mathbf{I}_{1}\), \(\mathbf{I}_{2}\), and \(\mathbf{I}_{3}\). (b) Show \(\mathbf{V}_{s}\), \(\mathbf{I}_{1}\), \(\mathbf{I}_{2}\), and \(\mathbf{I}_{3}\) on a phasor diagram (scales of 50 V/in and 2 A/in work fine). (c) Find \(\mathbf{I}_{s}\) graphically and give its amplitude and phase angle.

For the circuit shown in Fig. 10.79, (a) draw the phasor representation of the circuit; (b) determine the Thévenin equivalent seen by the capacitor, and use it to calculate \(v_C(t)\). (c) Determine the current flowing out of the positive reference terminal of the voltage source. (d) Verify your solution with an appropriate PSpice simulation.

The circuit of Fig. 10.79 is unfortunately operating differently than specified; the frequency of the current source is only 19 rad/s. Calculate the actual capacitor voltage, and compare it to the expected voltage had the circuit been operating correctly.

For the circuit shown in Fig. 10.80, (a) draw the corresponding phasor representation; (b) obtain an expression for \(\mathbf{V}_{o} / \mathbf{V}_{s}\); (c) Plot \(\left|\mathbf{V}_{o} / \mathbf{V}_{s}\right|\), the magnitude of the phasor voltage ratio, as a function of frequency \(\omega\) over the range \(0.01\ \leq\ \omega\ \leq\ 100\ rad/s\) (use a logarithmic x axis). (d) Does the circuit transfer low frequencies or high frequencies more effectively to the output?

(a) Replace the inductor in the circuit of Fig. 10.80 with a 1 F capacitor and repeat Exercise. 10.78. (b) If we design the “corner frequency” of the circuit as the frequency at which the output is reduced to \(1 / \sqrt{2}\) times its maximum value, redesign the circuit to achieve a corner frequency of 2 kHz.

Design a purely passive network (containing only resistors, capacitors, and inductors) which has an impedance of \((22-j7)/5\angle8^{\circ}\ \Omega\) at a frequency of f = 100 MHz.

In the frequency-domain circuit of Fig. 10.21, find (a) \(I_1\); (b) \(I_2\); (c) \(I_3\).

In the circuit of Fig. 10.53, which is shown in the phasor (frequency) domain, \(I_{10}\) is determined to be \(2 \ang {42^{\circ}}\ mA\). If \(V = 40 \ang {132^{\circ}}\ mV\): (a) what is the likely type of element connected to the right of the \(10\ \Omega\) resistor and (b) what is its value, assuming the voltage source operates at a frequency of 1000 rad/s?

Obtain the Thévenin equivalent seen by the \((2 - j)\ \Omega\) impedance of Fig. 10.69, and employ it to determine the current \(I_1\).

The voltage source \(V_s\) in Fig. 10.78 is chosen such that \(I_C = 1 \ang {0^{\circ}}\ A\). (a) Draw a phasor diagram showing \(V_1\), \(V_2\), \(V_S\), and \(V_R\). (b) Use the diagram to determine the ratio of \(V_2\) to \(V_1\).

In the circuit of Fig. 10.53, which is shown in the phasor (frequency) domain, \(I_{10}\) is determined to be \(2 \ang {42^{\circ}}\ mA\). If \(V = 40 \ang {132^{\circ}}\ mV\): (a) what is the likely type of element connected to the right of the \(10\ \Omega\) resistor and (b) what is its value, assuming the voltage source operates at a frequency of 1000 rad/s?

Obtain the Thévenin equivalent seen by the \((2 - j)\ \Omega\) impedance of Fig. 10.69, and employ it to determine the current \(I_1\).

The voltage source \(V_s\) in Fig. 10.78 is chosen such that \(I_C = 1 \ang {0^{\circ}}\ A\). (a) Draw a phasor diagram showing \(V_1\), \(V_2\), \(V_S\), and \(V_R\). (b) Use the diagram to determine the ratio of \(V_2\) to \(V_1\).