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AC power This worksheet and all related les are licensed under the Creative Commons Attribution License, AC power
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1 Questions
Question 1
Power is easy to calculate in DC circuits. Take for example this DC light bulb circuit:
110 V
I = 2.4 A
Calculate the power dissipation in this circuit, and describe the transfer of energy from source to load:
where does the energy come from and where does it go to?
le 02171
Question 2
A generator is coupled to a bicycle mechanism, so that a person can generate their own electricity:
Generator
Load
The person pedaling this bicycle/generator notices that it becomes more dicult to pedal when the
generator is connected to a load such as a light bulb, or when it is charging a battery. When the generator
is open-circuited, however, it is very easy to spin. Explain why this is, in terms of work and energy transfer.
le 02172
2 Question 3
If a sinusoidal voltage is applied to an impedance with a phase angle of 0
o
, the resulting voltage and
current waveforms will look like this:
Time
e
i
e
i
Given that power is the product of voltage and current (p = ie), plot the waveform for power in this
circuit.
le 00631
Question 4
If the power waveform is plotted for a resistive AC circuit, it will look like this:
Time
e
i
e
i
p
What is the signicance of the power value always being positive (above the zero line) and never negative
(below the zero line)?
le 02174
3 Question 5
If a sinusoidal voltage is applied to an impedance with a phase angle of 90
o
, the resulting voltage and
current waveforms will look like this:
Time
e
i
e
i
Given that power is the product of voltage and current (p = ie), plot the waveform for power in this
circuit. Also, explain how the mnemonic phrase ELI the ICE man applies to these waveforms.
le 00632
Question 6
If a sinusoidal voltage is applied to an impedance with a phase angle of -90
o
, the resulting voltage and
current waveforms will look like this:
Time
e
i
e
i
Given that power is the product of voltage and current (p = ie), plot the waveform for power in this
circuit. Also, explain how the mnemonic phrase ELI the ICE man applies to these waveforms.
le 00633
4 Question 7
If the power waveform is plotted for an AC circuit with a 90 degree phase shift between voltage and
current, it will look something like this:
Time
e
i
e
i
p
What is the signicance of the power value oscillating equally between positive (above the zero line) and
negative (below the zero line)? How does this dier from a scenario where there is zero phase shift between
voltage and current?
le 02175
5 Question 8
If this circuit is built and operated, it will be found that the resistor becomes much hotter than the
inductor, even though both components drop the exact same amount of voltage and carry the exact same
amount of current:
Alternator
480 VAC
60 Hz
9.02 mH
3.4 OFF
COM
A
V
A
V
A
OFF
COM
A
V
A
V
A
Explain why there is such a remarkable dierence in heat output between these two components, given
their identical voltage drops and currents.
le 02177
Question 9
Calculate the current in this circuit, and also the amount of mechanical power (in units of horsepower)
required to turn this alternator (assume 100% eciency):
Alternator
480 VAC
3.4 60 Hz
le 00767
6 Question 10
Calculate the current in this circuit, and also the amount of mechanical power (in units of horsepower)
required to turn this alternator (assume 100% eciency):
Alternator
480 VAC
60 Hz
9.02 mH
le 00768
7 Question 11
A student is pondering the behavior of a simple series RC circuit:
5 VAC
X
C
= 4 k R = 3 k I = 1 mA
4 V
3 V
It is clear by now that the 4 k capacitive reactance does not directly add to the 3 k resistance to
make 7 k total. Instead, the addition of impedances is vectorial:
X
C
2
+ R
2
= Z
total
Z
C
+ Z
R
= Z
total
(4k
90
o
) + (3k
0
o
) = (5k
53.13
o
)
It is also clear to this student that the component voltage drops form a vectorial sum as well, so that 4
volts dropped across the capacitor in series with 3 volts dropped across the resistor really does add up to 5
volts total source voltage:
V
C
+ V
R
= V
total
(4V
90
o
) + (3V
0
o
) = (5V
53.13
o
)
What surprises the student, though, is power. In calculating power for each component, the student
arrives at 4 mW for the capacitor (4 volts times 1 milliamp) and 3 mW for the resistor (3 volts times 1
milliamp), but only 5 mW for the total circuit power (5 volts times 1 milliamp). In DC circuits, component
power dissipations always added, no matter how strangely their voltages and currents might be related. The
student honestly expected the total power to be 7 mW, but that doesnt make sense with 5 volts total voltage
and 1 mA total current.
Then it occurs to the student that power might add vectorially just like impedances and voltage drops.
In fact, this seems to be the only way the numbers make any sense:
P
C
= 4 mW
P
R
= 3 mW
P
total
= 5 mW
However, after plotting this triangle the student is once again beset with doubt. According to the Law
of Energy Conservation, total power in must equal total power out. If the source is inputting 5 mW of power
8 total to this circuit, there should be no possible way that the resistor is dissipating 3 mW and the capacitor
is dissipating 4 mW. That would constitute more energy leaving the circuit than what is entering!
What is wrong with this students power triangle diagram? How may we make sense of the gures
obtained by multiplying voltage by current for each component, and for the total circuit?
le 02176
Question 12
Calculate the current in this circuit, and also the amount of mechanical power (in units of horsepower)
required to turn this alternator (assume 100% eciency):
Alternator
480 VAC
60 Hz
9.02 mH
3.4 le 00769
Question 13
In this circuit, three common AC loads are modeled as resistances, combined with reactive components
in two out of the three cases. Calculate the amount of current registered by each ammeter, and also the
amount of power dissipated by each of the loads:
Incandescent lamp
Fluorescent lamp
Induction motor
A
A
A
120 VAC, 60 Hz
240 240 240 10
�
F
0.25 H
If someone were to read each of the ammeters indications and multiply the respective currents by the
gure of 120 volts, would the resulting power gures (P = IE) agree with the actual power dissipations?
Explain why or why not, for each load.
le 00770
9 Question 14
A very important parameter in AC power circuits is power factor. Explain what power factor is, and
dene its numerical range.
le 02173
Question 15
Dene true power, in contrast to reactive or apparent power.
le 03674
Question 16
Dene apparent power, in contrast to true or reactive power.
le 03676
Question 17
Dene reactive power, in contrast to true or apparent power.
le 03675
Question 18
The three dierent types of power in AC circuits are as follows:
S = apparent power, measured in Volt-Amps (VA)
P = true power, measured in Watts (W)
Q = reactive power, measured in Volt-Amps reactive (VAR)
Explain the names of each of these power types. Why are they called apparent, true, and reactive?
le 02178
Question 19
Power calculation in DC circuits is simple. There are three formulae that may be used to calculate
power:
P = IV
P = I
2
R
P = V
2
R
Power in DC circuits
Calculating power in AC circuits is much more complex, because there are three dierent types of power:
apparent
power (S), true power (P ), and reactive power (Q). Write equations for calculating each of these
types of power in an AC circuit:
le 02181
10 Question 20
Calculate the power factor of this circuit:
C = 0.1
�
F
400 Hz
32 V
R = 7.1 k le 02179
Question 21
Explain the dierence between a leading power factor and a lagging power factor.
le 00774
Question 22
In this circuit, three common AC loads are represented as resistances, combined with reactive
components in two out of the three cases. Calculate the amount of true power (P ), apparent power (S),
reactive power (Q), and power factor (P F ) for each of the loads:
Incandescent lamp
Fluorescent lamp
Induction motor
A
A
A
120 VAC, 60 Hz
240 240 240 10
�
F
0.25 H
Also, draw power triangle diagrams for each circuit, showing how the true, apparent, and reactive powers
trigonometrically relate.
le 00772
11 Question 23
Calculate the amount of phase shift between voltage and current in an AC circuit with a power factor
of 0.97 (lagging), and an apparent power of 3.5 kVA. Also, write the equation solving for phase shift, in
degrees.
le 00775
Question 24
A common analogy used to describe the dierent types of power in AC circuits is a mug of beer that
also contains foam:
Beer
Foam