Figure For Review Questions 11.7 and PROBLEMS. Figure For Prob Figure For Prob Figure For Prob

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1 466 PART 2 AC Circuits (a) 2000 VA (c) 866 VAR (b) 1000 VAR (d) 500 VAR VAR 11.9 A source is connected to three loads Z 1, Z 2, and Z 3 in parallel. Which of these is not true? (a) P = P 1 P 2 P 3 (b) Q = Q 1 Q 2 Q 3 (c) S = S 1 S 2 S 3 (d) S = S 1 S 2 S W (a) (b) Figure For Review Questions 11.7 and The instrument for measuring average power is the: (a) voltmeter (b) ammeter (c) wattmeter (d) varmeter (e) kilowatt-hour meter 11.8 For the power triangle in Fig (b), the apparent power is: Answers: 11.1a, 11.2c, 11.3c, 11.4d, 11.5e, 11.6c, 11.7d, 11.8a, 11.9c, 11.10c. PROBLEMS Section 11.2 Instantaneous and Average Power 11.1 If = 160 cos 50tV and i(t) = 20 sin(50t 30 ) A, calculate the instantaneous power and the average power At t = 2s,find the instantaneous power on each of the elements in the circuit of Fig Ω 10 Ω 50 0 V j10 Ω Figure For Prob j5 Ω 20 mf 30 cos 500t A 0.3 H 200 Ω 11.5 Compute the average power absorbed by the 4- resistor in the circuit of Fig j1 Ω Figure For Prob Ω V o 4V o 4 60 Α j2 Ω 11.3 Refer to the circuit depicted in Fig Find the average power absorbed by each element. Figure For Prob Ω 11.6 Given the circuit of Fig , find the average power absorbed by the 10- resistor. 10 cos(2t 30 ) V 1 H 0.25 F I o j5 Ω Figure For Prob Given the circuit in Fig , find the average power absorbed by each of the elements V 0.1V o 8I o j5 Ω 10 Ω V o Figure For Prob

2 CHAPTER 11 AC Power Analysis In the circuit of Fig , determine the average power absorbed by the 40- resistor. I o j20 Ω Section 11.3 Maximum Average Power Transfer For each of the circuits in Fig , determine the value of load Z for maximum power transfer and the maximum average power transferred. 6 0 A j10 Ω 0.5I o 40 Ω 8 Ω Z j2 Ω 4 0 Α Figure For Prob Calculate the average power absorbed by each resistor in the op amp circuit of Fig if the rms value of v s is2v. 5 Ω (a) j3 Ω v s j2 Ω V 6 kω 10 kω Z 2 kω Figure For Prob In the op amp circuit in Fig , find the total average power absorbed by the resistors. R (b) Figure For Prob For the circuit in Fig , find: (a) the value of the load impedance that absorbs the maximum average power (b) the value of the maximum average power absorbed V o cos vt V R R j100 Ω 3 20 Α 80 Ω j40 Ω Load Figure For Prob For the network in Fig , assume that the port impedance is R Z ab = tan 1 ωrc 1 ω2 R 2 C 2 Find the average power consumed by the network when R = 10 k,c = 200 nf, and i = 2 sin(377t 22 ) ma. Linear network i a v b Figure For Prob Figure For Prob In the circuit of Fig , find the value of Z L that will absorb the maximum power and the value of the maximum power. 1 Ω j1 Ω 12 0 V V o j1 Ω 2V o Z L Figure For Prob

3 468 PART 2 AC Circuits Calculate the value of Z L in the circuit of Fig in order for Z L to receive maximum average power. What is the maximum average power received by Z? j10 Ω Z L 30 Ω 5 90 A a-b of the circuits in Fig so that the maximum power is transferred to the load? 100 Ω j10 Ω 40 Ω V 50 Ω 2 90 A j30 Ω a 40 Ω j20 Ω b Figure For Prob Find the value of Z L in the circuit of Fig for maximum power transfer. 40 Ω 60 0 V j10 Ω 40 Ω 80 Ω Figure For Prob Section 11.4 Effective or RMS Value Find the rms value of the periodic signal in Fig j20 Ω 5 0 A Z L t Figure For Prob The variable resistor R in the circuit of Fig is adjusted until it absorbs the maximum average power. Find R and the maximum average power absorbed. 3 Ω j2 Ω Figure For Prob Determine the rms value of the waveform in Fig j1 Ω 4 0 A 6 Ω R t 5 Figure For Prob The load resistance R L in Fig is adjusted until it absorbs the maximum average power. Calculate the value of R L and the maximum average power. Figure For Prob Find the effective value of the voltage waveform in Fig I o 40 Ω 4I o V j20 Ω j10 Ω j10 Ω R L t Figure For Prob Assuming that the load impedance is to be purely resistive, what load should be connected to terminals Figure For Prob Calculate the rms value of the current waveform of Fig

4 CHAPTER 11 AC Power Analysis 469 i(t) 5 i(t) 10 10t t Figure For Prob Find the rms value of the voltage waveform of Fig as well as the average power absorbed by a 2- resistor when the voltage is applied across the resistor t Figure For Prob Determine the effective value of the periodic waveform in Fig i(t) t Figure For Prob Calculate the effective value of the current waveform in Fig and the average power delivered to a 12- resistor when the current runs through the resistor. i(t) t Figure For Prob One cycle of a periodic voltage waveform is depicted in Fig Find the effective value of the voltage t Figure For Prob Compute the rms value of the waveform depicted in Fig Figure For Prob t Obtain the rms value of the current waveform shown in Fig Section t Figure For Prob Apparent Power and Power Factor A relay coil is connected to a 210-V, 50-Hz supply. If it has a resistance of 30 and an inductance of 0.5 H, calculate the apparent power and the power factor A certain load comprises 12 j8 in parallel with j4. Determine the overall power factor Obtain the power factor for each of the circuits in Fig Specify each power factor as leading or lagging.

5 470 PART 2 AC Circuits Section 11.6 j1 Ω 1 Ω j2 Ω (a) (b) j5 Ω j2 Ω Figure For Prob Complex Power j2 Ω j1 Ω A load draws 5 kvar at a power factor of 0.86 (leading) from a 220-V rms source. Calculate the peak current and the apparent power supplied to the load For the following voltage and current phasors, calculate the complex power, apparent power, real power, and reactive power. Specify whether the pf is leading or lagging. (a) V = V rms, I = A rms (b) V = V rms, I = A rms (c) V = V rms, I = A rms (d) V = V rms, I = A rms For each of the following cases, find the complex power, the average power, and the reactive power: (a) = 112 cos(ωt 10 ) V, i(t) = 4 cos(ωt 50 ) A (b) = 160 cos 377t V, i(t) = 4 cos(377t 45 ) A (c) V = V rms, Z = (d) I = V rms, Z = Determine the complex power for the following cases: (a) P = 269 W, Q = 150 VAR (capacitive) (b) Q = 2000 VAR, pf = 0.9 (leading) (c) S = 600 VA, Q = 450 VAR (inductive) (d) V rms = 220 V, P = 1kW, Z =40 (inductive) Find the complex power for the following cases: (a) P = 4kW,pf= 0.86 (lagging) (b) S = 2kVA,P = 1.6 kw (capacitive) (c) V rms = V, I rms = A (d) V rms = V, Z = 40 j Obtain the overall impedance for the following cases: (a) P = 1000 W, pf = 0.8 (leading), V rms = 220 V (b) P = 1500 W, Q = 2000 VAR (inductive), I rms = 12 A (c) S = VA, V = V For the entire circuit in Fig , calculate: (a) the power factor (b) the average power delivered by the source (c) the reactive power (d) the apparent power (e) the complex power V Section Ω j5 Ω j6 Ω 10 Ω Figure For Prob Ω Conservation of AC Power For the network in Fig , find the complex power absorbed by each element V j3 Ω Figure For Prob j5 Ω Find the complex power absorbed by each of the five elements in the circuit of Fig V rms j20 Ω j10 Ω 20 Ω Figure For Prob V rms Obtain the complex power delivered by the source in the circuit of Fig

6 CHAPTER 11 AC Power Analysis Ω j Given the circuit in Fig , find I o and the overall complex power supplied Α 5 Ω j2 Ω 6 Ω I o 1.2 kw 0.8 kvar (cap) Figure For Prob For the circuit in Fig , find the average, reactive, and complex power delivered by the dependent voltage source V j1 Ω 2 Ω 1 Ω V o j2 Ω 2V o V 2 kva pf leading Figure For Prob For the circuit in Fig , find V s. 0.2 Ω j Ω j0.15 Ω 4 kw 0.9 pf lagging Figure For Prob V s 10 W 0.9 pf lagging 15 W 0.8 pf leading 120 V rms Obtain the complex power delivered to the 10-k resistor in Fig below Calculate the reactive power in the inductor and capacitor in the circuit of Fig Ω j30 Ω V j20 Ω 4 0 A 40 Ω For the circuit in Fig , find V o and the input power factor. Figure For Prob Figure For Prob Find I o in the circuit of Fig on the bottom of the next page In the op amp circuit of Fig , v s = 4 cos 10 4 t V. Find the average power delivered to the 50-k resistor. 100 kω v s 1 nf 50 kω 6 0 A rms 20 kw 0.8 pf lagging V o 16 kw 0.9 pf lagging Figure For Prob Figure For Prob Obtain the average power absorbed by the 6-k resistor in the op amp circuit in Fig Ω I o j3 kω j1 kω V rms 20I o 4 kω 10 kω Figure For Prob

7 472 PART 2 AC Circuits 4 45 V 2 kω j4 kω 4 kω j3 kω 6 kω j2 kω (b) What is the average power dissipated? (c) What is the value of the capacitance that will give a unity power factor when connected to the load? Figure For Prob V 60 Hz C Z = 10 j12 Ω Calculate the complex power delivered to each resistor and capacitor in the op amp circuit of Fig Let v s = 2 cos 10 3 t V. v s 10 kω 0.2 mf 20 kω 0.1 mf 40 kω Figure For Prob Compute the complex power supplied by the current source in the series RLC circuit in Fig I o cos vt Section 11.8 Figure For Prob R L C Power Factor Correction Refer to the circuit shown in Fig (a) What is the power factor? Figure For Prob An 880-VA, 220-V, 50-Hz load has a power factor of 0.8 lagging. What value of parallel capacitance will correct the load power factor to unity? An 40-kW induction motor, with a lagging power factor of 0.76, is supplied by a 120-V rms 60-Hz sinusoidal voltage source. Find the capacitance needed in parallel with the motor to raise the power factor to: (a) 0.9 lagging (b) A 240-V rms 60-Hz supply serves a load that is 10 kw (resistive), 15 kvar (capacitive), and 22 kvar (inductive). Find: (a) the apparent power (b) the current drawn from the supply (c) the kvar rating and capacitance required to improve the power factor to 0.96 lagging (d) the current drawn from the supply under the new power-factor conditions A 120-V rms 60-Hz source supplies two loads connected in parallel, as shown in Fig (a) Find the power factor of the parallel combination. (b) Calculate the value of the capacitance connected in parallel that will raise the power factor to unity. I o V 12 kw pf leading 16 kw 0.85 pf lagging 20 kvar 0.6 pf lagging Figure For Prob

8 CHAPTER 11 AC Power Analysis 473 Load 1 24 kw pf = 0.8 lagging Load 2 40 kw pf = 0.95 lagging 120 cos 2t V 6 Ω 4 H 0.1 F 15 Ω Figure For Prob Consider the power system shown in Fig Calculate: (a) the total complex power (b) the power factor (c) the capacitance necessary to establish a unity power factor Figure For Prob Find the wattmeter reading of the circuit shown in Fig below The circuit of Fig portrays a wattmeter connected into an ac network. (a) Find the load current. (b) Calculate the wattmeter reading. 240 V rms, 50 Hz WM 80 j50 Ω 120 j70 Ω 110 V Z L = 6. pf = Section j0 Figure For Prob Applications Obtain the wattmeter reading of the circuit in Fig below What is the reading of the wattmeter in the network of Fig ? Figure For Prob The kilowatthour-meter of a home is read once a month. For a particular month, the previous and present readings are as follows: Previous reading: 3246 kwh Present reading: 4017 kwh Calculate the electricity bill for that month based on the following residential rate schedule: j3 Ω 12 0 V j2 Ω 8 Ω 3 30 A Figure For Prob Ω 5 Ω 1 H 20 cos 4t V 1 12 F Figure For Prob

9 474 PART 2 AC Circuits Minimum monthly charge $12.00 First 100 kwh per month at 16 cents/kwh Next 200 kwh per month at 10 cents/kwh Over 300 kwh per month at 6 cents/kwh A consumer has an annual consumption of 1200 MWh with a maximum demand of 2.4 MVA. The maximum demand charge is $30 per kva per annum, and the energy charge per kwh is 4 cents. (a) Determine the annual cost of energy. (b) Calculate the charge per kwh with a flat-rate tariff if the revenue to the utility company is to remain the same as for the two-part tariff. COMPREHENSIVE PROBLEMS A transmitter delivers maximum power to an antenna when the antenna is adjusted to represent a load of 75- resistance in series with an inductance of 4 µh. If the transmitter operates at 4.12 MHz, find its internal impedance In a TV transmitter, a series circuit has an impedance of 3 k and a total current of 50 ma. If the voltage across the resistor is 80 V, what is the power factor of the circuit? A certain electronic circuit is connected to a 110-V ac line. The root-mean-square value of the current drawn is 2 A, with a phase angle of 55. (a) Find the true power drawn by the circuit. (b) Calculate the apparent power An industrial heater has a nameplate which reads: 210 V 60 Hz 12 kva 0.78 pf lagging Determine: (a) the apparent and the complex power (b) the impedance of the heater A 2000-kW turbine-generator of 0.85 power factor operates at the rated load. An additional load of 300 kw at 0.8 power factor is added. What kvar of capacitors is required to operate the turbine -generator but keep it from being overloaded? The nameplate of an electric motor has the following information: Line voltage: 220 V rms Line current: 15 A rms Line frequency: 60 Hz Power: 2700 W Determine the power factor (lagging) of the motor. Find the value of the capacitance C that must be connected across the motor to raise the pf to unity As shown in Fig , a 550-V feeder line supplies an industrial plant consisting of a motor drawing 60 kw at 0.75 pf (inductive), a capacitor with a rating of 20 kvar, and lighting drawing 20 kw. (a) Calculate the total reactive power and apparent power absorbed by the plant. (b) Determine the overall pf. (c) Find the current in the feeder line. 60 kw 550 V 20 kvar 10 kw pf = 0.75 Figure For Prob A factory has the following four major loads: A motor rated at 5 hp, 0.8 pf lagging (1 hp = kw). A heater rated at 1.2 kw, 1.0 pf. Ten 120-W lightbulbs. A synchronous motor rated at 1.6 kva, 0.6 pf leading. (a) Calculate the total real and reactive power. (b) Find the overall power factor A 1-MVA substation operates at full load at 0.7 power factor. It is desired to improve the power factor to 0.95 by installing capacitors. Assume that new substation and distribution facilities cost $120 per kva installed, and capacitors cost $30 per kva installed. (a) Calculate the cost of capacitors needed. (b) Find the savings in substation capacity released. (c) Are capacitors economical for releasing the amount of substation capacity? A coupling capacitor is used to block dc current from an amplifier as shown in Fig (a). The amplifier and the capacitor act as the source, while the speaker is the load as in Fig (b). (a) At what frequency is maximum power transferred to the speaker? (b) If V s = 4.6 V rms, how much power is delivered to the speaker at that frequency? *An asterisk indicates a challenging problem.

10 CHAPTER 11 AC Power Analysis 475 V in Amplifier Coupling capacitor Speaker (a) Determine the impedance of the load that achieves maximum power transfer. (b) Calculate the load power under this matching condition. (a) 10 Ω 40 nf A power transmission system is modeled as shown in Fig If V s = rms, find the average power absorbed by the load. 0.1 Ω j1 Ω v s Amplifier (b) Figure For Prob mh Speaker A power amplifier has an output impedance of 40 j8. It produces a no-load output voltage of 146 V at 300 Hz. V s 0.1 Ω j1 Ω Source Line Load Figure For Prob Ω j1 Ω