Calculate the power factor of the combined loads of the circuit depicted in Fig. 11.39

A current source of 12 cos 2000t A, a \(200\ \Omega\) resistor, and a 0.2 H inductor are in parallel. Assume steady-state conditions exist. At t = 1 ms, find the power being absorbed by the (a) resistor; (b) inductor; (c) sinusoidal source.

Given the phasor voltage \(\mathbf{V}=115\sqrt{2}\angle45^{\circ}\mathrm{\ V}\) across an impedance \(\mathbf{Z}=16.26\angle19.3^{\circ}\ \Omega\), obtain an expression for the instantaneous power, and compute the average power if \(\omega = 50\ rad/s\).

Calculate the average power delivered to the impedance \(6\angle25^{\circ}\ \Omega\) by the current I = 2 + j5 A.

If the 30 mH inductor of Example 11.5 is replaced with a \(10\ \mu F\) capacitor, what is the value of the inductive component of the unknown impedance \(\mathbf{Z}_{?}\) if it is known that \(\mathbf{Z}_{?}\) is absorbing maximum power?

A voltage source \(v_s\) is connected across a \(4\ \Omega\) resistor. Find the average power absorbed by the resistor if \(v_s\) equals (a) \(8\ sin\ 200t\ V\); (b) \(8\ sin\ 200t - 6\ cos(200t - 45^{\circ})\ V\); (c) \(8\ sin\ 200t - 4\ sin\ 100t\ V\); (d) \(8\ sin\ 200t - 6\ cos(200t - 45^{\circ}) - 5\ sin\ 100t + 4\ V\).

Calculate the effective value of each of the periodic voltages: (a) \(6\ cos\ 25t\); (b) \(6\ cos\ 25t + 4\ sin(25t + 30^{\circ})\); (c) \(6\ cos\ 25t + 5\ cos^2\ (25t)\); (d) \(6\ cos\ 25t + 5\ sin\ 30t + 4\ V\).

For the circuit of Fig. 11.14, determine the power factor of the combined loads if \(Z_L = 10\ \Omega\).

For the circuit shown in Fig. 11.22, find the complex power absorbed by the (a) \(1\ \Omega\) resistor; (b) \(?j10\ \Omega\) capacitor; (c) \(5 + j10\ \Omega\) impedance; (d) source.

A 440 V rms source supplies power to a load \(Z_L = 10 + j2\ \Omega\) through a transmission line having a total resistance of \(1.5\ \Omega\). Find (a) the average and apparent power supplied to the load; (b) the average and apparent power lost in the transmission line; (c) the average and apparent power supplied by the source; (d) the power factor at which the source operates.

Determine the instantaneous power delivered to the \(1\ \Omega\) resistor of Fig. 11.23 at t = 0, t = 1 s, and t = 2 s if \(v_s\) is equal to (a) 9 V; (b) \(9\ sin\ 2t\ V\); (c) \(9\ sin\ (2t + 13^{\circ})\ V\); (d) \(9e^{-t}\ V\).

Determine the power absorbed at t = 1.5 ms by each of the three elements of the circuit shown in Fig. 11.24 if \(v_s\) is equal to (a) 30u(?t) V; (b) 10 + 20u(t) V.

Calculate the power absorbed at \(t = 0^-\), \(t = 0^+\), and t = 200 ms by each of the elements in the circuit of Fig. 11.25 if \(v_s\) is equal to (a) -10u(-t) V; (b) 20 + 5u(t) V.

Three elements are connected in parallel: a \(1\ k \Omega\) resistor, a 15 mH inductor, and a \(100\cos\left(2\times10^5t\right)\mathrm{\ mA}\) sinusoidal source. All transients have long since died out, so the circuit is operating in steady state. Determine the power being absorbed by each element at \(t = 10\ \mu s\).

Let \(i_s = 4u(?t)\ A\) in the circuit of Fig. 11.26. (a) Show that, for all t > 0, the instantaneous power absorbed by the resistor is equal in magnitude but opposite in sign to the instantaneous power absorbed by the capacitor. (b) Determine the power absorbed by the resistor at t = 60 ms.

The current source in the circuit of Fig. 11.26 is given by \(i_s = 8 ? 7u(t)\ A\). Compute the power absorbed by all three elements at \(t = 0^-\), \(t = 0^+\), and t = 75 ms.

Assuming no transients are present, calculate the power absorbed by each element shown in the circuit of Fig. 11.27 at t = 0, 10, and 20 ms.

Calculate in Fig. 11.28 the power absorbed by the inductor at t = 0 and t = 1 s if \(v_s = 10u(t)\ V\).

A 100 mF capacitor is storing 100 mJ of energy up until the point when a conductor of resistance \(1.2\ \Omega\) falls across its terminals. What is the instantaneous power dissipated in the conductor at t = 120 ms? If the specific heat \(capacity^3\) of the conductor is 0.9 kJ/kg · K and its mass is 1 g, estimate the increase in temperature of the conductor in the first second of the capacitor discharge, assuming both elements are initially at \(23^{\circ}C\).

If we take a typical cloud-to-ground lightning stroke to represent a current of 30 kA over an interval of \(150\ \mu s\), calculate (a) the instantaneous power delivered to a copper rod having resistance \(1.2\ m \Omega\) during the stroke; (b) the total energy delivered to the rod.

The phasor current \(\mathbf{I}=9\angle15^{\circ}\mathrm{\ mA}\) (corresponding to a sinusoid operating at 45 rad/s) is applied to the series combination of a \(18\ k \Omega\) resistor and a \(1\ \mu F\) capacitor. Obtain an expression for (a) the instantaneous power and (b) the average power absorbed by the combined load.

A phasor voltage \(\mathrm{V}=100\angle45^{\circ}\mathrm{\ V}\) (the sinusoid operates at 155 rad/s) is applied to the parallel combination of a \(1\ \Omega\) resistor and a 1 mH inductor. (a) Obtain an expression for the average power absorbed by each passive element. (b) Graph the instantaneous power supplied to the parallel combination, along with the instantaneous power absorbed by each element separately. (Use a single graph.)

Calculate the average power delivered by the current 4 - j2 A to (a) \(\mathbf{Z}=9 \Omega\); (b) \(\mathbf{Z}=-j1000\ \Omega\); (c) \(\mathbf{Z}=1-j2+j3\ \Omega\); (d) \(\mathbf{Z}=6\angle32^{\circ}\ \Omega\); (e) \(\mathbf{Z}=\frac{1.5 \angle -19^{\circ}}{2+j}\ \mathrm{k}\Omega\).

With regard to the two-mesh circuit depicted in Fig. 11.29, determine the average power absorbed by each passive element, and the average power supplied by each source, and verify that the total supplied average power = the total absorbed average power.

(a) What load impedance \(\mathbf{Z}_{L}\) will draw the maximum average power from the source shown in Fig. 11.31? (b) Calculate the maximum average power supplied to the load.

The inductance of Fig. 11.31 is replaced by the impedance \(9 ?j8\ k \Omega\). Repeat Exercise 16.

Determine the average power supplied by the dependent source in the circuit of Fig. 11.32.

(a) Compute the average value of each waveform shown in Fig. 11.34. (b) Square each waveform, and determine the average value of each new periodic waveform.

Calculate the average power delivered to a \(2.2\ \Omega\) load by the voltage \(v_s\) equal to (a) \(5\ V\); (b) \(4\ cos\ 80t - 8\ sin\ 80t\ V\); (c) \(10\ cos\ 100t + 12.5\ cos\ (100t + 19^{\circ})\ V\).

Calculate the effective value of the following waveforms: (a) \(7\ sin\ 30t\ V\); (b) \(100\ cos\ 80t\ mA\); (c) \(120 \sqrt{2}\ cos\ (5000t - 45^{\circ})\ V\); (d) \(\frac{100}{\sqrt{2}}\ sin\ (2t + 72^{\circ})\ A\).

Determine the effective value of the following waveforms: (a) \(62.5\ cos\ 100t\ mV\); (b) \(1.95\ cos\ 2t\ A\); (c) \(208 \sqrt{2}\ cos\ (100 \pi t + 29^{\circ})\ V\); (d) \(\frac{400}{\sqrt{2}}\ sin\ (2000t - 14^{\circ})\ A\).

Compute the effective value of (a) \(i(t) = 3\ sin\ 4t\ A\); (b) \(v(t) = 4\ sin\ 20t\ cos\ 10t\); (c) \(i(t) = 2 ? sin\ 10t\ mA\); (d) the waveform plotted in Fig. 11.35.

For each waveform plotted in Fig. 11.34, determine its frequency, period, and rms value.

Determine both the average and rms value of each waveform depicted in Fig. 11.36.

The series combination of a \(1\ k \Omega\) resistor and a 2 H inductor must not dissipate more than 250 mW of power at any instant. Assuming a sinusoidal current with \(\omega = 500\ rad/s\), what is the largest rms current that can be tolerated?

For each of the following waveforms, determine its period, frequency and effective value: (a) \(5\ V\); (b) \(2\ sin\ 80t ? 7\ cos\ 20t + 5\ V\); (c) \(5\ cos\ 50t + 3\ sin\ 50t\ V\); (d) \(8\ cos^2\ 90t\ mA\). (e) Verify your answers with an appropriate simulation.

With regard to the circuit of Fig. 11.37, determine whether a purely real value of R can result in equal rms voltages across the 14 mH inductor and the resistor R. If so, calculate R and the rms voltage across it; if not, explain why not.

Calculate both the average and rms values of the waveform plotted in Fig. 11.38. (b) Verify your solutions with appropriate PSpice simulations (Hint: you may want to employ two pulse waveforms added together).

For the circuit of Fig. 11.39, compute the average power delivered to each load, the apparent power supplied by the source, and the power factor of the combined loads if (a) \(\mathbf{Z}_1=14 \angle 32^{\circ}\ \Omega\) and \(\mathbf{Z}_2=22\ \Omega\); (b) \(\mathbf{Z}_1=2 \angle 0^{\circ}\ \Omega\) and \(\mathbf{Z}_2=6-j\ \Omega\); (c) \(\mathbf{Z}_1=100 \angle 70^{\circ}\ \Omega\) and \(\mathbf{Z}_2=75 \angle 90^{\circ}\ \Omega\).

Calculate the power factor of the combined loads of the circuit depicted in Fig. 11.39 if (a) both loads are purely resistive; (b) both loads are purely inductive and \(\omega = 100\ rad/s\); (c) both loads are purely capacitive and \(\omega = 200\ rad/s\); (d) \(\mathbf{Z}_1=2\mathbf{Z}_2=5-j8\ \Omega\).

A given load is connected to an ac power system. If it is known that the load is characterized by resistive losses and either capacitors, inductors, or neither (but not both), which type of reactive element is part of the load if the power factor is measured to be (a) unity; (b) 0.85 lagging; (c) 0.221 leading; (d) \(cos\ (-90^{\circ})\)?

An unknown load is connected to a standard European household outlet (240 V rms, 50 Hz). Determine the phase angle difference between the voltage and current, and whether the voltage leads or lags the current, if (a) \(\mathbf{V}=240 \angle 243^{\circ}\mathrm{\ V}\mathrm{\ rms}\) and \(\mathbf{I}=3 \angle 9^{\circ}\mathrm{\ A}\mathrm{\ rms}\); (b) the power factor of the load is 0.55 lagging; (c) the power factor of the load is 0.685 leading; (d) the capacitive load draws 100 W average power and 500 volt-amperes apparent power.

(a) Design a load which draws an average power of 25 W at a leading PF of 0.88 from a standard North American household outlet (120 V rms, 60 Hz). (b) Design a capacitor-free load which draws an average power of 150 W and an apparent power of 25 W from a household outlet in eastern Japan (110 V rms, 50 Hz).

Assuming an operating frequency of 40 rad/s for the circuit shown in Fig. 11.40, and a load impedance of \(50 \angle -100^{\circ}\ \Omega\), calculate (a) the instantaneous power separately delivered to the load and to the \(1\ k \Omega\) shunt resistance at t = 20 ms; (b) the average power delivered to both passive elements; (c) the apparent power delivered to the load; (d) the power factor at which the source is operating.

Calculate the power factor at which the source in Fig. 11.40 is operating if the load is (a) purely resistive; (b) \(1000 + j900\ \Omega\); (c) \(500\angle-5^{\circ}\ \Omega\).

Determine the load impedance for the circuit depicted in Fig. 11.40 if the source is operating at a PF of (a) 0.95 leading; (b) unity; (c) 0.45 lagging.

For the circuit of Fig. 11.41, find the apparent power delivered to each load, and the power factor at which the source operates, if (a) \(\mathbf{Z}_A=5-j2\ \Omega\), \(\mathbf{Z}_B=3\ \Omega\), \(\mathbf{Z}_C=8+j4\ \Omega\), and \(\mathbf{Z}_D=15 \angle -30^{\circ}\ \Omega\); (b) \(\mathbf{Z}_A=2 \angle {-15^{\circ}}\ \Omega\), \(\mathbf{Z}_B=1\ \Omega\), \(\mathbf{Z}_C=2+j\ \Omega\), and \(\mathbf{Z}_D=4 \angle 45^{\circ}\ \Omega\).

Compute the complex power S (in polar form) drawn by a certain load if it is known that (a) it draws 100 W average power at a lagging PF of 0.75; (b) it draws a current I = 9 + j5 A rms when connected to the voltage \(120\angle32^{\circ}\ \mathrm{V\ }\mathrm{rms}\); (c) it draws 1000 W average power and 10 VAR reactive power at a leading PF; (d) it draws an apparent power of 450 W at a lagging PF of 0.65.

Calculate the apparent power, power factor, and reactive power associated with a load if it draws complex power S equal to (a) 1 + j0.5 kVA; (b) 400 VA; (c) \(150 \angle -21^{\circ}\mathrm{\ VA}\); (d) \(75 \angle 25^{\circ}\mathrm{\ VA}\).

For each power triangle depicted in Fig. 11.42, determine S (in polar form) and the PF.

Referring to the network represented in Fig. 11.21, if the motor draws complex power \(150 \angle 24^{\circ}\mathrm{\ VA}\). (a) determine the PF at which the source is operating; (b) determine the impedance of the corrective device required to change the PF of the source to 0.98 lagging. (c) Is it physically possible to obtain a leading PF for the source? Explain.

Determine the complex power absorbed by each passive component in the circuit of Fig. 11.43, and the power factor at which the source is operating.

What value of capacitance must be added in parallel to the \(10\ \Omega\) resistor of Fig. 11.44 to increase the PF of the source to 0.95 at 50 Hz?

The kiln operation of a local lumberyard has a monthly average power demand of 175 kW, but associated with that is an average monthly reactive power draw of 205 kVAR. If the lumberyard’s utility company charges $0.15 per kVAR for each kVAR above the benchmark value (0.7 times the peak average power demand), (a) estimate the annual cost to the lumberyard from PF penalties; (b) calculate the money saved in the first and second years, respectively, if 100 kVAR compensation capacitors are available for purchase at $75 each (installed).

Calculate the complex power delivered to each passive component of the circuit shown in Fig. 11.45, and determine the power factor of the source.

Replace the \(10\ \Omega\) resistor in the circuit of Fig. 11.45 with a 200 mH inductor, assume an operating frequency of 10 rad/s, and calculate (a) the PF of the source; (b) the apparent power supplied by the source; (c) the reactive power delivered by the source.

Instead of including a capacitor as indicated in Fig. 11.45, the circuit is erroneously constructed using two identical inductors, each having an impedance of j30 W at the operating frequency of 50 Hz. (a) Compute the complex power delivered to each passive component. (b) Verify your solution by calculating the complex power supplied by the source. (c) At what power factor is the source operating?

Making use of the general strategy employed in Example 11.9, derive Eq. [28], which enables the corrective value of capacitance to be calculated for a general operating frequency.

A load is drawing 10 A rms when connected to a 1200 V rms supply running at 50 Hz. If the source is operating at a lagging PF of 0.9, calculate (a) the peak voltage magnitude; (b) the instantaneous power absorbed by the load at t = 1 ms; (c) the apparent power supplied by the source; (d) the reactive power supplied to the load; (e) the load impedance; and (f) the complex power supplied by the source (in polar form).

Remove the \(50\ \Omega\) resistor in Fig. 11.46, assume an operating frequency of 50 Hz, and (a) determine the power factor at which the load is operating; (b) compute the average power delivered by the source; (c) compute the instantaneous power absorbed by the inductance at t = 2 ms; (d) determine the capacitance that must be connected between the terminals marked a and b to increase the PF of the source to 0.95.

A source \(45\ sin\ 32t\ V\) is connected in series with a \(5\ \Omega\) resistor and a 20 mH inductor. Calculate (a) the reactive power delivered by the source; (b) the apparent power absorbed by each of the three elements; (c) the complex power S absorbed by each element; (d) the power factor at which the source is operating.

For the circuit of Fig. 11.37, (a) derive an expression for the complex power delivered by the source in terms of the unknown resistance R; (b) compute the necessary capacitance that must be added in parallel to the 28 mH inductor to achieve a unity PF.

A 440 V rms source supplies power to a load \(Z_L = 10 + j2\ \Omega\) through a transmission line having a total resistance of \(1.5\ \Omega\). Find (a) the average and apparent power supplied to the load; (b) the average and apparent power lost in the transmission line; (c) the average and apparent power supplied by the source; (d) the power factor at which the source operates.

(a) Calculate the average power absorbed by each passive element in the circuit of Fig. 11.30, and verify that it equals the average power supplied by the source. (b) Check your solution with an appropriate PSpice simulation.

(a) Calculate the average power supplied to each passive element in the circuit of Fig. 11.33. (b) Determine the power supplied by each source. (c) Replace the \(8\ \Omega\) resistive load with an impedance capable of drawing maximum average power from the remainder of the circuit. (d) Verify your solution with a PSpice simulation.

For the circuit of Fig. 11.46, assume the source operates at a frequency of 100 rad/s. (a) Determine the PF at which the source is operating. (b) Calculate the apparent power absorbed by each of the three passive elements. (c) Compute the average power supplied by the source. (d ) Determine the Thévenin equivalent seen looking into the terminals marked a and b, and calculate the average power delivered to a \(100\ \Omega\) resistor connected between the same terminals.

(a) Calculate the average power absorbed by each passive element in the circuit of Fig. 11.30, and verify that it equals the average power supplied by the source. (b) Check your solution with an appropriate PSpice simulation.

(a) Calculate the average power supplied to each passive element in the circuit of Fig. 11.33. (b) Determine the power supplied by each source. (c) Replace the \(8\ \Omega\) resistive load with an impedance capable of drawing maximum average power from the remainder of the circuit. (d) Verify your solution with a PSpice simulation.

For the circuit of Fig. 11.46, assume the source operates at a frequency of 100 rad/s. (a) Determine the PF at which the source is operating. (b) Calculate the apparent power absorbed by each of the three passive elements. (c) Compute the average power supplied by the source. (d ) Determine the Thévenin equivalent seen looking into the terminals marked a and b, and calculate the average power delivered to a \(100\ \Omega\) resistor connected between the same terminals.