Essentials of Electrical and Computer Engineering

David V. Kerns, Jr., J. David Irwin

Chapter 5

Steady-State Power Analysis - all with Video Answers

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Chapter Questions

Problem 1

In the circuit in Figure P5.1, $v_1(t)=12 \cos$ $\left(377 t-40^{\circ}\right) \mathrm{V}$. Determine the equations for the current and instantaneous power as a function of time.
Figure P5.1 can't copy

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Problem 2

In the network in Figure P5.2, $v_1(t)=24 \cos$ $\left(377 t+20^{\circ}\right) \mathrm{V}$. Find the equations for the current and instantaneous power as a function of time.
Figure P5.2 can't copy

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Problem 3

Determine the total average power absorbed and supplied in the network in Figure P5.3.
Figure P5.3 can't copy

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Problem 4

Determine the total average power absorbed and supplied in the network in Figure P5.4.
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Problem 5

Determine the average power absorbed in the $4 \Omega$ resistor in the circuit in Figure P5.5.
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Problem 6

Determine the average power absorbed in the $6 \Omega$ resistor in the network in Figure P5.6.
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Problem 7

Find the total average power absorbed and supplied in the circuit in Figure P5.7.
Figure P5.7 can't copy

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Problem 8

Find the total average power absorbed and supplied in the network in Figure P5.8.
Figure P5.8 can't copy

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Problem 9

Find the total average power absorbed and supplied in the circuit in Figure P5.9.
Figure P5.9 can't copy

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Problem 10

Show that the conservation of power holds for the network shown in Figure P5.10.
Figure P5.10 can't copy

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Problem 11

Given the network in Figure P5.11, determine the average power absorbed or supplied by each element.
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Problem 12

Find the total average power absorbed and supplied in the circuit in Figure P5.12.
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Problem 13

Determine the average power absorbed and supplied by each element in the network in Figure P5.13.
Figure P5.13 can't copy

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Problem 14

Determine the rms value of the waveform in Figure P5.14
Figure P5.14 can't copy

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Problem 15

Calculate the rms value of the waveform shown in Figure P5.15.
Figure P5.15 can't copy

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Problem 16

Determine the rms value of the waveform shown in Figure P5.16.
Figure P5.16 can't copy

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Problem 17

Compute the rms value of the voltage waveform shown in Figure P5.17.
Figure P5.17 can't copy

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Problem 18

The current waveform in Figure P5.18 exists in a $10 \Omega$ resistor. Determine the average power delivered to the resistor.
Figure P5.18 can't copy

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Problem 19

The current in a $4 \Omega$ resistor is given by the waveform in Figure P5.19. Find the average power absorbed by the resistor.
Figure P5.19 can't copy

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Problem 20

The current in a $2 \Omega$ resistor is given by the waveform shown in Figure P5.20. Find the average power absorbed by the resistor.
Figure P5.20 can't copy

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Problem 21

The voltage across a $4 \Omega$ resistor is given by the waveform shown in Figure P5.21. Find the average power absorbed by the resistor.
Figure P5.21 can't copy

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Problem 22

The power consumed by an industrial load is $120 \mathrm{~kW}$ at $0.70 \mathrm{PF}$ lagging from a $480 \mathrm{~V}$ rms line. If the transmission line resistance between the generator and the load is $0.14 \Omega$, find the power that must be supplied by the power company.

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Problem 23

An industrial load consumes $100 \mathrm{~kW}$ at $0.8 \mathrm{PF}$ lagging. The line voltage is $480 \mathrm{~V} \mathrm{rms}$. If the line resistance between the generator and load is $0.1 \Omega$, find the transmission line losses.

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Problem 24

If the power factor in problem 5.23 is changed to 0.92 lagging, determine the effect on the line losses.

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Problem 25

A load operates at $20 \mathrm{~kW}, 0.8$ PF lagging. The load voltage is $220 \angle 0^{\circ} \mathrm{V}$ rms at $60 \mathrm{~Hz}$. The impedance of the line is $0.09+j 0.3 \Omega$. Determine the voltage and power factor at the input to the line.

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Problem 26

A particular load has a PF of 0.8 lagging. The power delivered to the load is $40 \mathrm{~kW}$ from a $220 \mathrm{~V} \mathrm{rms} 60 \mathrm{~Hz}$ line. If the transmission line resistance is $0.085 \Omega$, determine the real power that must be generated at the supply.

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Problem 27

The line voltage at an industrial load is $220 \mathrm{~V}$ rms The load consumes $50 \mathrm{~kW}$ at $0.75 \mathrm{PF}$ lagging. The line resistance from the power company's transformer to the load is $0.1 \Omega$. Determine the savings in line losses if the PF is somehow changed to 0.92 lagging.

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Problem 28

Calculate the voltage $\mathbf{V}_s$ that must be supplied to obtain $2 \mathrm{~kW}, 240 \angle 0^{\circ} \mathrm{V}$ rms, and a power factor of 0.8 leading at the load $\mathbf{Z}_L$ in the network in Figure P5.28.
Figure P5.28 can't copy

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Problem 29

An industrial plant at the end of a transmission line consumes $144 \mathrm{~kW}$ at 0.88 PF lagging. The line voltage at the plant is $480 \angle 0^{\circ} \mathrm{V}$ rms. The resistance of the transmission line is $0.13 \Omega$. Determine the voltage and power factor at the input of the transmission line.

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Problem 30

A plant, representing the load on a transmission line, consumes $110 \mathrm{~kW}$ at 0.88 PF lagging. The load line voltage is $480 \angle 0^{\circ} \mathrm{V}$ rms. If the transmission line resistance is $0.15 \Omega$, determine the complex power at both ends of the line.

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Problem 31

An industrial plant consumes $160 \mathrm{~kW}$ at 0.9 PF lagging. The line voltage at the plant is $480 \angle 0^{\circ} \mathrm{V}$ rms. The transmission line feeding the plant is 2 miles long. If the impedance of the line is $0.1+j 0.2 \Omega$, find the power factor at the sending end of the line.

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Problem 32

A small plant has a bank of induction motons that consume $64 \mathrm{~kW}$ at a PF of 0.68 lagging. The $60 \mathrm{~Hz}$ line voltage across the motors is $220 \angle 0^{\circ} \mathrm{V}$ rms. The local power company has told the plant to raise the PF to 0.92 lagging. What value of capacitance is required?

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Problem 33

What value of capacitance, placed in parallel with the load in problem 5.26 , will raise the 7 to 0.9 lagging?

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Problem 34

An industrial plant consumes $75 \mathrm{~kW}$ at $0.84 \mathrm{Pf}$ lagging when connected to a $220 \angle 0^{\circ} \mathrm{V}$ rm $60 \mathrm{~Hz}$ line. Determine the value of the capacitance, which, when placed in parallel with the load, will raise the power factor to $0<$ lagging.

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Problem 35

An industrial load is connected to a $60 \mathrm{~Hz}$ line and the load voltage is $480 \angle 0^{\circ} \mathrm{V}$ rms. The load consumes $120 \mathrm{~kW}$ at $0.77 \mathrm{PF}$ lagging. $\mathrm{Sz}$ lect the capacitor value needed to raise th power factor to 0.9 lagging.

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Problem 36

An industrial plant draws $100 \mathrm{~kW}$ at $0.7 \mathrm{Fr}$ lagging from a $480 \angle 0^{\circ} \mathrm{V} \mathrm{rms}, 60 \mathrm{~Hz}$ line. a $500 \mu \mathrm{F}$ capacitor bank is placed in paral with the load, what is the new power factor

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Problem 37

An abc-sequence three-phase voltage source connected in a balanced wye has a phase voltage of $\mathbf{V}_{a n}=120 \angle-45^{\circ} \mathrm{V}$ rms. Determitu the line voltages of the source.

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Problem 38

An abc-sequence three-phase voltage sourconnected in a balanced wye has a line vor age of $\mathbf{V}_{a b}=208 \angle 175^{\circ} \mathrm{V}$ rms. Find phase voltages of the source.

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Problem 39

A three-phase, abc-sequence, balanced wys connected source is connected to a balanoted wye-connected load. The phase voltage of tit source is $120 \mathrm{~V} \mathrm{rms}$, the load impedance : $16+j 8 \Omega$, and the line impedance is negligs ble. Determine the line currents.

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Problem 40

If the line impedance in problem 5.39 is $0.8+j 0.6 \Omega$, determine the line currents and the load voltages.

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Problem 41

An abc-sequence balanced three-phase wyeconnected source with a phase voltage of 100 $\mathrm{V}$ rms supplies power to a balanced wye-connected load. The per-phase load impedance is $40+j 10 \Omega$. Determine the line currents in the circuit if $\angle \mathbf{V}_{a n}=0^{\circ}$.

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Problem 42

An abc-sequence balanced three-phase wyeconnected source supplies power to a balanced wye-connected load. The line impedance per phase is $1+j 0 \Omega$, and the load impedance per phase is $20+j 20 \Omega$. If the source line voltage $\mathbf{V}_{a b}$ is $100 \angle 0^{\circ} \mathrm{V}$ rms, find the line currents.

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Problem 43

An abc-sequence balanced wye-connected source supplies a balanced wye-connected load. If the line voltage at the source is $\mathbf{V}_{a b}=240 \angle 0^{\circ} \mathrm{V}$ rms, the load impedance is $18+j 12 \Omega$, and the line impedance is $1.2+j 1 \Omega$, determine the line currents and the load voltages.

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Problem 44

An abc-sequence balanced wye-connected source supplies a balanced wye-connected load. If the line impedance is $1+j 1 \Omega$, the load impedance is $12+j 8 \Omega$, and the load voltage is $\mathbf{V}_{A N}=109.46 \angle 18.99^{\circ} \mathrm{V} \mathrm{rms}$, determine the source voltages.

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Problem 45

In a balanced three-phase wye-delta system, the line current is $\mathbf{I}_{a A}=8 \angle 13.13^{\circ} \mathrm{A} \mathrm{rms}$. If the impedance per phase of the delta load is $36+j 27 \Omega$, find the source voltage if the line impedance can be ignored.

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Problem 46

In a balanced three-phase wye-delta system, the phase voltage of the source is $\mathbf{V}_{a n}=120 \angle 10^{\circ} \mathrm{V}$ rms, and the load impedance per phase in the $\Delta$ is $24+j 18 \Omega$. If the line impedance is negligible, determine the line currents in the system.

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Problem 47

In a balanced three-phase wye-delta system, the line voltage is $\mathbf{V}_{a b}=120 \sqrt{3} \angle 70^{\circ} \mathrm{V} \mathrm{rms}$ and the line current is $\mathbf{I}_{a A}=24 \angle 3.13^{\circ}$ A rms. If the line impedance is negligible, find the impedance per phase of the delta load.

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Problem 48

In a balanced three-phase wye-delta system, the source has an abc phase sequence. The load impedance is $12+j 8 \Omega$. If the phase voltage at the load is $\mathbf{V}_{A B}=260 \angle 45^{\circ} \mathrm{V}$ rms, find the line currents and the phase voltages of the source.

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Problem 49

In a balanced three-phase wye-delta system, the impedance per phase of the load is $21+j 9 \Omega$ and the source voltage is $\mathbf{V}_{a b}=120 \sqrt{3} \angle 60^{\circ} \mathrm{V}$ rms. If the line impedance is $1+j 1 \Omega$, determine the line currents.

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Problem 50

A balanced three-phase system has a load that consists of a balanced wye in parallel with a balanced delta. The impedance per phase of the wye is $10+j 8 \Omega$ and the impedance per phase of the delta is $24+j 12 \Omega$. Determine both the equivalent wye load and the equivalent delta load.

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Problem 51

A three-phase load impedance consists of a balanced wye in parallel with a balanced delta, as shown in Figure P5.51. Determine the equivalent delta load.
Figure P5.51 can't copy

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Problem 52

An abc-phase-sequence three-phase voltage source connected in a balanced wye supplies power to a balanced delta-connected load. The load current $\mathbf{I}_{A B}=4 \angle 20^{\circ} \mathrm{A} \mathrm{rms}$. Determine the line currents.

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Problem 53

An abc-sequence balanced three-phase wye-connected source supplies power to a balanced delta-connected load. The load impedance per phase is $12+j 8 \Omega$. If the current $\mathbf{I}_{A B}$ in one phase of the delta is $14.42 \angle 86.31^{\circ}$ A rms, determine the line currents and phase voltages at the source.

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Problem 54

Find $I_{a y}$ in the circuit shown in Figure P5.54.
Figure P5.54 can't copy

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Problem 55

Consider the network shown in Figure P5.55. Compute the magnitude of the line voltages at the load and the magnitude of the phase currents in the delta-connected source.
Figure P5.55 can't copy

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Problem 56

A balanced three-phase delta-delta system has the following parameters: $\mathbf{V}_{a b}=207.84$ $\angle-20^{\circ} \mathrm{V}$ rms, $\mathbf{Z}_{\text {Line }}=1+j 1.2 \Omega$, and $\mathbf{Z}_{\text {Load }}$ $=18+j 12 \Omega$. Find the line currents.

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Problem 57

Two small industries located side-by-side are fed from a three-phase balanced $60 \mathrm{~Hz}$ source with a line voltage of $208 \mathrm{~V}$ rms. Load #1 consumes $24 \mathrm{~kW}$ at $0.75 \mathrm{PF}$ lagging and load #2 consumes $16 \mathrm{~kW}$ at $0.88 \mathrm{PF}$ lagging. Determine the current in the line that serves both loads.

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Problem 58

A $480 \mathrm{~V}$ rms line feeds two balanced threephase loads. If the two loads are rated as follows:
Load $1: 5 \mathrm{kVA}$ at $0.8 \mathrm{PF}$ lagging
Load 2: $10 \mathrm{kVA}$ at 0.9 PF lagging
determine the magnitude of the line current from the $480 \mathrm{~V}$ rms source.

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Problem 59

A balanced three-phase source serves the following loads:
Load $1: 48 \mathrm{kVA}$ at $0.9 \mathrm{PF}$ lagging
Load 2: $24 \mathrm{kVA}$ at $0.75 \mathrm{PF}$ lagging
The line voltage at the load is $208 \mathrm{~V}$ rms at 60 $\mathrm{Hz}$. Determine the line currents and the combined power factor at the load.

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Problem 60

Three industrial loads are supplied by a balanced three-phase source. The line voltage at the loads is $208 \mathrm{~V}$ rms at $60 \mathrm{~Hz}$. The characteristics of the loads are
Load $1: 20 \mathrm{~kW}$ at $0.8 \mathrm{PF}$ lagging
Load 2: $26 \mathrm{~kW}$ at $0.85 \mathrm{PF}$ leading
Load 3: $18 \mathrm{~kW}$ at 0.9 PF lagging
determine the total complex power and power factor of the load.

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Problem 61

A balanced three-phase wye-wye system has, two parallel loads. Load 1 is rated at $3000 \mathrm{VA}, 0.7 \mathrm{PF}$ lagging, and load 2 is rated at $2000 \mathrm{VA}, 0.75 \mathrm{PF}$ leading. If the line voltage is $208 \mathrm{~V}$ rms, find the magnitude of the line current.

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Problem 62

Two industrial plants represent balanced three-phase loads. The plants receive their power from a balanced three-phase source with a line voltage of $4.6 \mathrm{kV}$ rms. Plant 1 is rated at $300 \mathrm{kVA}, 0.8 \mathrm{PF}$ lagging, and plant 2 is rated at $350 \mathrm{kVA}, 0.84 \mathrm{PF}$ lagging. Determine the power line current.

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