Fundamentals of Electric Circuits

Charles K. Alexander, Matthew N.O. Sadiku

Chapter 11

AC Power Analysis - all with Video Answers

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Chapter Questions

Problem 1

If $v(t)=160 \cos 50 t \mathrm{V}$ and $i(t)=$
$-20 \sin \left(50 t-30^{\circ}\right)$ A, calculate the instantaneous power and the average power

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Problem 2

At $t=2 \mathrm{s}$, find the instantaneous power on each of the elements in the circuit of Fig. 11.35

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Problem 3

Refer to the circuit depicted in Fig. $11.36 .$ Find the average power absorbed by each element.

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Problem 4

Given the circuit in Fig. 11.37 , find the average power absorbed by each of the elements.

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Problem 5

Compute the average power absorbed by the $4-\Omega$ resistor in the circuit of Fig. 11.38

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Problem 6

Given the circuit of Fig. $11.39,$ find the average power absorbed by the $10-\Omega$ resistor.

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Problem 7

In the circuit of Fig. 11.40 , determine the average power absorbed by the $40-\Omega$ resistor.

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Problem 8

Calculate the average power absorbed by each resistor in the op amp circuit of Fig. 11.41 if the rms value of $v_{s}$ is $2 \mathrm{V}$

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Problem 9

In the op amp circuit in Fig. $11.42,$ find the total average power absorbed by the resistors.

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Problem 10

For the network in Fig. 11.43 , assume that the port impedance is
$$\mathbf{Z}_{a b}=\frac{R}{\sqrt{1+\omega^{2} R^{2} C^{2}}} \swarrow-\tan ^{-1} \omega R C$$
Find the average power consumed by the network when $R=10 \mathrm{k} \Omega, C=200 \mathrm{nF},$ and $i=$
$2 \sin \left(377 t+22^{\circ}\right) \mathrm{mA}$

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Problem 11

For each of the circuits in Fig. 11.44 , determine the value of load $\mathbf{Z}$ for maximum power transfer and the maximum average power transferred.

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Problem 12

For the circuit in Fig. $11.45,$ find:
(a) the value of the load impedance that absorbs the maximum average power
(b) the value of the maximum average power absorbed

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Problem 13

In the circuit of Fig. $11.46,$ find the value of $\mathbf{Z}_{L}$ that will absorb the maximum power and the value of the maximum power.

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Problem 14

Calculate the value of $\mathbf{Z}_{L}$ in the circuit of Fig. 11.47 in order for $\mathbf{Z}_{L}$ to receive maximum average power What is the maximum average power received by $\mathbf{Z} ?$

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Problem 15

Find the value of $\mathbf{Z}_{L}$ in the circuit of Fig. 11.48 for maximum power transfer.

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Problem 16

The variable resistor $R$ in the circuit of Fig. 11.49 is adjusted until it absorbs the maximum average power. Find $R$ and the maximum average power absorbed.

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Problem 17

The load resistance $R_{L}$ in Fig. 11.50 is adjusted until it absorbs the maximum average power. Calculate the value of $R_{L}$ and the maximum average power

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Problem 18

Assuming that the load impedance is to be purely resistive, what load should be connected to terminals

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Problem 19

Find the rms value of the periodic signal in Fig. 11.52

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Problem 20

Determine the rms value of the waveform in Fig. 11.53

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Problem 21

Find the effective value of the voltage waveform in Fig. 11.54

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Problem 22

Calculate the rms value of the current waveform of Fig. 11.55

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Problem 23

Find the rms value of the voltage waveform of Fig. 11.56 as well as the average power absorbed by a $2-\Omega$ resistor when the voltage is applied across the resistor.

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Problem 24

Calculate the effective value of the current waveform in Fig. 11.57 and the average power delivered to a $12-\Omega$ resistor when the current runs through the resistor.

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Problem 25

Compute the rms value of the waveform depicted in Fig. 11.58

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Problem 26

Obtain the rms value of the current waveform shown in Fig. 11.59

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Problem 27

Determine the effective value of the periodic waveform in Fig. 11.60

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Problem 28

One cycle of a periodic voltage waveform is depicted in Fig. $11.61 .$ Find the effective value of the voltage

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Problem 29

A relay coil is connected to a 210 -V, 50 -Hz supply If it has a resistance of $30 \Omega$ and an inductance of $0.5 \mathrm{H},$ calculate the apparent power and the power factor.

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Problem 30

A certain load comprises $12-j 8 \Omega$ in parallel with $j 4 \Omega$ Determine the overall power factor

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Problem 31

Obtain the power factor for each of the circuits in Fig. $11.62 .$ Specify each power factor as leading or lagging.

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Problem 32

A load draws $5 \mathrm{kVAR}$ 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

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Problem 33

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) $\mathbf{V}=220 / 30^{\circ} \mathrm{V} \mathrm{rms}, \mathbf{I}=0.5 / 60^{\circ} \mathrm{Arms}$
(b) $\mathbf{V}=250 \angle-10^{\circ} \mathrm{Vrms}$
$\mathbf{I}=6.2 \angle-25^{\circ} \mathrm{A} \mathrm{rms}$
(c) $\mathbf{V}=120 / 0^{\circ} \mathrm{V} \mathrm{rms}, \mathbf{I}=2.4 /-15^{\circ} \mathrm{Arms}$
(d) $\mathbf{V}=160 / 45^{\circ} \mathrm{V} \mathrm{rms}, \mathbf{I}=8.5 / 90^{\circ} \mathrm{Arms}$

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Problem 34

For each of the following cases, find the complex power, the average power, and the reactive power:
(a) $v(t)=112 \cos \left(\omega t+10^{\circ}\right) \mathrm{V}$
$i(t)=4 \cos \left(\omega t-50^{\circ}\right) \mathrm{A}$
(b) $v(t)=160 \cos 377 t \mathrm{V}$
$i(t)=4 \cos \left(377 t+45^{\circ}\right) \mathrm{A}$
(c) $\mathbf{V}=80 / 60^{\circ} \mathrm{V} \mathrm{rms}, \mathbf{Z}=50 / 30^{\circ} \Omega$
(d) $\mathbf{I}=10 / 60^{\circ} \mathrm{V} \mathrm{rms}, \mathbf{Z}=100 / 45^{\circ} \Omega$

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Problem 35

Determine the complex power for the following
cases:
(a) $P=269 \mathrm{W}, Q=150 \mathrm{VAR}$ (capacitive)
(b) $Q=2000 \mathrm{VAR}, \mathrm{pf}=0.9$ (leading)
(c) $S=600$ VA, $Q=450$ VAR (inductive)
(d) $V_{\mathrm{rms}}=220 \mathrm{V}, P=1 \mathrm{kW}$
$|\mathbf{Z}|=40 \Omega$ (inductive)

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Problem 36

Find the complex power for the following cases:
(a) $P=4 \mathrm{kW}, \mathrm{pf}=0.86$ (lagging)
(b) $S=2 \mathrm{kVA}, P=1.6 \mathrm{kW}$ (capacitive)
(c) $\mathbf{V}_{\mathrm{rms}}=208 / 20^{\circ} \mathrm{V}, \mathbf{I}_{\mathrm{rms}}=6.5 /-50^{\circ} \mathrm{A}$
(d) $\mathbf{V}_{\mathrm{rms}}=120 / 30^{\circ} \mathrm{V}, \mathbf{Z}=40+j 60 \Omega$

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Problem 37

Obtain the overall impedance for the following cases:
(a) $P=1000 \mathrm{W}, \mathrm{pf}=0.8$ (leading) $V_{\mathrm{rms}}=220 \mathrm{V}$
(b) $P=1500 \mathrm{W}, Q=2000 \mathrm{VAR}$ (inductive) $I_{\mathrm{rms}}=12 \mathrm{A}$
(c) $\mathbf{S}=4500 / 60^{\circ}$ VA, $\mathbf{V}=120 / 45^{\circ} \mathrm{V}$

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Problem 38

For the entire circuit in Fig. $11.63,$ 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

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Problem 39

For the network in Fig. $11.64,$ find the complex power absorbed by each element.

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Problem 40

Find the complex power absorbed by each of the five elements in the circuit of Fig. 11.65

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Problem 41

Obtain the complex power delivered by the source in the circuit of Fig. 11.66

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Problem 42

For the circuit in Fig. 11.67 , find the average, reactive, and complex power delivered by the dependent voltage source.

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Problem 43

Obtain the complex power delivered to the $10-\mathrm{k} \Omega$ resistor in Fig. 11.68 below.

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Problem 44

Calculate the reactive power in the inductor and capacitor in the circuit of Fig. 11.69

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Problem 45

For the circuit in Fig. 11.70 , find $\mathbf{V}_{o}$ and the input power factor

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Problem 46

Given the circuit in Fig. 11.71 , find $\mathbf{I}_{o}$ and the overall complex power supplied.

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Problem 47

For the circuit in Fig. $11.72,$ find $\mathbf{V}_{s}$

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Problem 48

Find $\mathbf{I}_{o}$ in the circuit of Fig. 11.73 on the bottom of the next page.

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Problem 49

In the op amp circuit of Fig. $11.74, v_{s}=$ $4 \cos 10^{4} t \mathrm{V} .$ Find the average power delivered to the $50-\mathrm{k} \Omega$ resistor

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Problem 50

Obtain the average power absorbed by the $6-\mathrm{k} \Omega$ resistor in the op amp circuit in Fig. 11.75

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Problem 51

Calculate the complex power delivered to each resistor and capacitor in the op amp circuit of Fig. 11.76. Let $v_{s}=2 \cos 10^{3} t \mathrm{V}$

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Problem 52

Compute the complex power supplied by the current source in the series $R L C$ circuit in Fig. 11.77

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Problem 53

Refer to the circuit shown in Fig. 11.78
(a) What is the power factor?
(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?

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Problem 54

An 880 -VA, $220-\mathrm{V}, 50-\mathrm{Hz}$ load has a power factor of 0.8 lagging. What value of parallel capacitance will correct the load power factor to unity?

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Problem 55

An $40-\mathrm{kW}$ induction motor, with a lagging power factor of $0.76,$ is supplied by a $120-\mathrm{V} \mathrm{rms} 60-\mathrm{Hz}$ sinusoidal voltage source. Find the capacitance needed in parallel with the motor to raise the power factor to:
(a) 0.9 lagging
(b) 1.0

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Problem 56

A $240-\mathrm{V} \mathrm{rms} 60-\mathrm{Hz}$ supply serves a load that is $10 \mathrm{kW}$ (resistive), $15 \mathrm{kVAR}$ (capacitive), and
$22 \mathrm{kVAR}$ (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

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Problem 57

A $120-\mathrm{V} \mathrm{rms} 60-\mathrm{Hz}$ source supplies two loads connected in parallel, as shown in Fig. 11.79
(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.

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Problem 58

Consider the power system shown in Fig. 11.80 Calculate:
(a) the total complex power
(b) the power factor
(c) the capacitance necessary to establish a unity power factor

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Problem 59

Obtain the wattmeter reading of the circuit in Fig. 11.81 below.

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Problem 60

$\mathbf{0} \quad$ What is the reading of the wattmeter in the network of Fig. $11.82 ?$

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Problem 61

Find the wattmeter reading of the circuit shown in Fig. 11.83 below

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Problem 62

The circuit of Fig. 11.84 portrays a wattmeter connected into an ac network.
(a) Find the load current.
(b) Calculate the wattmeter reading.

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Problem 63

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: $3246 \mathrm{kWh}$ Present reading: $4017 \mathrm{kWh}$ Calculate the electricity bill for that month based on the following residential rate schedule: Minimum monthly charge- $\$ 12.00$ First $100 \mathrm{kWh}$ per month at 16 cents/kWh Next $200 \mathrm{kWh}$ per month at 10 cents/kWh Over 300 kWh per month at 6 cents/kWh

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Problem 64

A consumer has an annual consumption of $1200 \mathrm{MWh}$ with a maximum demand of $2.4 \mathrm{MVA}$

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Problem 65

A transmitter delivers maximum power to an antenna when the antenna is adjusted to represent a load of $75-\Omega$ resistance in series with an inductance of $4 \mu \mathrm{H}$. If the transmitter operates at $4.12 \mathrm{MHz}$ find its internal impedance.

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Problem 66

In a TV transmitter, a series circuit has an impedance of $3 \mathrm{k} \Omega$ and a total current of $50 \mathrm{mA}$. If the voltage across the resistor is $80 \mathrm{V}$, what is the power factor of the circuit?

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Problem 67

A certain electronic circuit is connected to a $110-\mathrm{V}$ ac line. The root-mean-square value of the current drawn is $2 \mathrm{A},$ with a phase angle of $55^{\circ}$
(a) Find the true power drawn by the circuit.
(b) Calculate the apparent power.

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Problem 68

An industrial heater has a nameplate which reads:
$210 \mathrm{V} \quad 60 \mathrm{Hz} \quad 12 \mathrm{kVA} \quad 0.78 \mathrm{pf}$ lagging
Determine:
(a) the apparent and the complex power
(b) the impedance of the heater

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Problem 69

A 2000 -kW turbine-generator of 0.85 power factor operates at the rated load. An additional load of $300 \mathrm{kW}$ at 0.8 power factor is added. What $\mathrm{kVAR}$ of capacitors is required to operate the turbine
-generator but keep it from being overloaded?

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Problem 70

The nameplate of an electric motor has the following information:
Line voltage: $220 \mathrm{V} \mathrm{rms}$ Line current: 15 A rms Line frequency: $60 \mathrm{Hz}$ 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.

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Problem 71

As shown in Fig. $11.85,$ a 550 -V feeder line supplies an industrial plant consisting of a motor drawing $60 \mathrm{kW}$ at $0.75 \mathrm{pf}$ (inductive), a capacitor with a rating of $20 \mathrm{kVAR}$, and lighting drawing $20 \mathrm{kW}$
(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.

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Problem 72

A factory has the following four major loads:
. A motor rated at 5 hp, 0.8 pf lagging $(1 \mathrm{hp}=0.7457 \mathrm{kW})$
. A heater rated at $1.2 \mathrm{kW}, 1.0 \mathrm{pf}$
. Ten 120-W lightbulbs.
. A synchronous motor rated at $1.6 \mathrm{kVA}, 0.6 \mathrm{pf}$ leading.
(a) Calculate the total real and reactive power
(b) Find the overall power factor

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Problem 73

A $1-\mathrm{MVA}$ 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 $\mathrm{kVA}$ installed, and capacitors cost $\$ 30$ per $\mathrm{kVA}$ 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?

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Problem 74

A coupling capacitor is used to block de current from an amplifier as shown in Fig. $11.86(\mathrm{a}) .$ The amplifier and the capacitor act as the source, while the speaker is the load as in Fig. $11.86(\mathrm{b})$
(a) At what frequency is maximum power transferred to the speaker?
(b) If $V_{s}=4.6 \mathrm{V} \mathrm{rms}$, how much power is delivered to the speaker at that frequency?

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Problem 75

A power amplifier has an output impedance of $40+j 8 \Omega .$ It produces a no-load output voltage of $146 \mathrm{V}$ at $300 \mathrm{Hz}$
(a) Determine the impedance of the load that achieves maximum power transfer
(b) Calculate the load power under this matching condition.

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Problem 76

A power transmission system is modeled as shown in Fig. $11.87 .$ If $\mathbf{V}_{s}=240 \underline{/} 0^{\circ} \mathrm{rms}$, find the average power absorbed by the load.

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