Assignment 22: AC 1 and Transformers
Graphical Analysis of AC Voltage Source Conceptual Question
The figure shows a graph of the output from an AC voltage source.
Part A
What is the maximum voltage of the source?
Hint A.1 AC voltage
Hint not displayed
ANSWER:
=
3
Correct
Part B
What is the average voltage of the source?
Hint B.1 Average voltage
Hint not displayed
ANSWER:
=
0
Correct
Part C
What is the root-mean-square voltage of the source?
Hint C.1 Root-mean-square voltage
Hint not displayed
Express your answer to three significant figures.
ANSWER:
=
2.12
Correct
Part D
What is the period of the source?
Hint D.1 Period
Hint not displayed
Express your answer in seconds to two significant figures.
ANSWER:
=
0.08
Correct
Part E
What is the frequency of the source?
Hint E.1 Frequency
Hint not displayed
Express your answer in hertz to three significant figures.
ANSWER:
=
12.5
Correct
Part F
What is the angular frequency of the source?
Hint F.1 Angular frequency
Hint not displayed
Express your answer in radians per second to three significant figures.
ANSWER:
=
78.5
Correct
Voltage and Current in AC Circuits
Learning Goal: To understand the relationship between AC voltage and current in resistors, inductors, and
capacitors, especially the phase shift between the voltage and the current.
In this problem, we consider the behavior of resistors, inductors, and capacitors driven individually by a
sinusoidally alternating voltage source, for which the voltage is given as a function of time by . The
main challenge is to apply your knowledge of the basic properties of resistors, inductors, and capacitors to these
"single-element" AC circuits to find the current through each. The key is to understand the phase difference,
also known as the phase angle, between the voltage and the current. It is important to take into account the sign of
the current, which will be called positive when it flows clockwise from the b terminal (which has positive voltage
relative to the a terminal) to the a terminal (see figure). The sign is critical in the analysis of circuits containing
combinations of resistors, capacitors, and inductors.
Part A
First, let us consider a resistor with resistance connected to an AC source (diagram 1). If the AC source
provides a voltage , what is the current through the resistor as a function of time?
Hint A.1 Ohm's law
Hint not displayed
Express your answer in terms of , , , and .
ANSWER:
= Correct
Note that the voltage and the current are in phase; that is, in the expressions for and , the arguments of
the cosine functions are the same at any moment of time. This will not be the case for the capacitor and inductor.
Part B
Now consider an inductor with inductance in an AC circuit (diagram 2). Assuming that the current in the
inductor varies as , find the voltage that must be driving the inductor.
Hint B.1 Kirchhoff's loop rule
Hint not displayed
Hint B.2 The derivative of
Hint not displayed
Hint B.3 The phase relationship between sine and cosine
Hint not displayed
Express your answer in terms of , , , and . Use the cosine function, not the sine function, in your answer.
ANSWER:
= Correct
Graphs of and are shown below. As you can see, for an inductor, the voltage leads (i.e., reaches its
maximum before) the current by ; in other words, the current lags the voltage by . This can be conceptually
understood by thinking of inductance as giving the current inertia: The voltage "tries" to push current through the
inductor, but some sort of inertia resists the change in current. This is another manifestation of Lenz's law. The
difference is called the phase angle.
Part C
Again consider an inductor with inductance connected to an AC source. If the AC source provides a voltage
, what is the current through the inductor as a function of time?
Hint C.1 Using Part B
Hint not displayed
Express your answer in terms of , , , and . Use the cosine function, not the sine function, in your answer.
ANSWER: =
Correct
For the amplitudes (magnitudes) of voltage and current, one can write (for the resistor) and (for
the inductor). If one compares these expressions, it should not come as a surprise that the quantity , measured
in ohms, is called inductive reactance; it is denoted by (sometimes ). It is called reactance rather than
resistance to emphasize that there is no dissipation of energy. Using this notation, we can write (for a
resistor) and (for an inductor). Also, notice that the current is in phase with voltage when a resistor is
connected to an AC source; in the case of an inductor, the current lags the voltage by . What will happen if we
replace the inductor with a capacitor? We will soon see.
Part D
Consider the potentials of points a and b on the inductor in diagram 2. If the voltage at point b is greater than that
at point a, which of the following statements is true?
Hint D.1 How to approach the problem
Hint not displayed
ANSWER:
The current must be positive (clockwise).
The current must be directed counterclockwise.
The derivative of the current must be negative.
The derivative of the current must be positive.
Correct
It may help to think of the current as having inertia and the voltage as exerting a force that overcomes this inertia.
This viewpoint also explains the lag of the current relative to the voltage.
Part E
Assume that at time , the current in the inductor is at a maximum; at that time, the current flows from point b to
point a. At time , which of the following statements is true?
Hint E.1 How to approach the problem
Hint not displayed
ANSWER:
The voltage across the inductor must be zero and increasing.
The voltage across the inductor must be zero and decreasing.
The voltage across the inductor must be positive and momentarily constant.
The voltage across the inductor must be negative and momentarily constant.
Correct
Part F
Now consider a capacitor with capacitance connected to an AC source (diagram 3). If the AC source provides a
voltage , what is the current through the capacitor as a function of time?
Hint F.1 The relationship between charge and voltage for a capacitor
Hint not displayed
Hint F.2 The relationship between charge and current
Hint not displayed
Hint F.3 Mathematical details
Hint not displayed
Express your answer in terms of , , and . Use the cosine function with a phase shift, not the sine function, in
your answer.
ANSWER:
= Correct
For the amplitude values of voltage and current, one can write . If one compares this expression with a
similar one for the resistor, it should come as no surprise that the quantity , measured in ohms, is called
capacitive reactance; it is denoted by (sometimes ). It is called reactance rather than resistance to emphasize
that there is no dissipation of energy. Using this notation, we can write , and voltage lags current by
radians (or 90 degrees). The notation is analogous to for a resistor, where voltage and current are in phase,
and for an inductor, where voltage leads current by radians (or 90 degrees). We see, then, that in a
capacitor, the voltage lags the current by , while in the case of an inductor, the current lags the voltage by the
same quantity . In a capacitor, where voltage lags the current, you may think of the current as driving the
change in the voltage.
Part G
Consider the capacitor in diagram 3. Which of the following statements is true at the moment the alternating
voltage across the capacitor is zero?
Hint G.1 How to approach the problem
Hint not displayed
Hint G.2 Graphs of and
Hint not displayed
ANSWER:
The current must be directed clockwise.
The current must be directed counterclockwise.
The magnitude of the current must be a maximum.
The current must be zero.
Correct
Part H
Consider the capacitor in diagram 3. Which of the following statements is true at the moment the charge of the
capacitor is at a maximum?
Hint H.1 How to approach this problem
Hint not displayed
Hint H.2 Graphs of and
Hint not displayed
ANSWER:
The current must be directed clockwise.
The current must be directed counterclockwise.
The magnitude of the current must be a maximum.
The current must be zero.
Correct
Part I
Consider the capacitor in diagram 3. Which of the following statements is true if the voltage at point b is greater
than that at point a?
Hint I.1 How to approach the problem
Hint not displayed
Hint I.2 Graphs of and
Hint not displayed
ANSWER:
The current must be directed clockwise.
The current must be directed counterclockwise.
The current may be directed either clockwise or counterclockwise.
Correct
Part J
Consider a circuit in which a capacitor and an inductor are connected in parallel to an AC source. Which of the
following statements about the magnitude of the current through the voltage source is true?
Hint J.1 Driven AC parallel circuits
Hint not displayed
ANSWER:
It is always larger than the sum of the magnitudes of the currents in the capacitor and
inductor.
It is always less than the sum of the magnitudes of the currents in the capacitor and inductor.
At very high frequencies it will become small.
At very low frequencies it will become small.
Correct
This surprising result occurs because the currents in inductor and capacitor are exactly out of phase with each
other (i.e., one lags and the other leads the voltage), and hence they cancel to some extent. At a particular
frequency, called the resonant frequency, the currents have exactly the same amplitude, and they cancel exactly;
that is, no current flows from the voltage source to the circuit. (Lots of current flows around the loop made by the
inductor and capacitor, however.) If an L-C parallel circuit like this one connects the wire between amplifier
stages in a radio, it will allow frequencies near the resonance frequency to pass easily, but will tend to short those
at other freqeuncies to ground. This is the basic mechanism for selecting a radio station.
± Transformers
Learning Goal: To understand the concepts explaining the operation of transformers.
One of the advantages of alternating current (ac) over direct current (dc) is the ease with which voltage levels can
be increased or decreased. Such a need is always present due to the practical requirements of energy distribution.
On the one hand, the voltage supplied to the end users must be reasonably low for safety reasons (depending on
the country, that voltage may be 110 volts, 220 volts, or some other value of that order). On the other hand, the
voltage used in transmitting electric energy must be as high as possible to minimize losses in the transmission
lines. A device that uses the principle of electromagnetic induction to increase or decrease the voltage by a certain
factor is called a transformer.
The main components of a transformer are two coils (windings) that are electrically insulated from each other. The
coils are wrapped around the same core, which is typically made of a material with a very large relative
permeability to ensure maximum mutual inductance. One coil, called the primary coil, is connected to a voltage
source; the other, the secondary coil, delivers the power. The alternating current in the primary coil induces the
changing magnetic flux in the core that creates the emf in the secondary coil. The magnitude of the emf induced in
the secondary coil can be controlled by the design of the transformer. The key factor is the number of turns in each
coil.
Consider an ideal transformer, that is, one in which the coils have no ohmic resistance and the magnetic flux is
the same for each turn of both the primary and secondary coils. If the number of turns in the primary coil is and
that in the secondary coil is , then the emfs induced in the coils can be written as
,
and therefore,
.
Since both emfs oscillate with the same frequency as the ac source, the formula above can be applied to the
instantaneous amplitude or the rms values of the emfs. Moreover, if the coils have zero resistance (as we
assumed), then for each coil the terminal voltage will be equal to the induced emf. Therefore, we can write
.
Note that if , then . This is a case of a step-up transformer. Conversely, if , then . This is a
case of a step-down transformer. Without energy losses, the power in the primary and secondary coils is the same:
.
If the secondary circuit is completed by a resistance , then . Combining this with the two equations above
gives
.
Dividing the first and last expressions by and then inverting gives
.
In other words, the current in the primary coil is the same as if it were connected directly to a resistance equal to
. In a way, transformers "transform" resistances as well as voltages and currents. In reality, no transformer
is ideal. There are always some energy losses. However, modern transformers have very high efficiencies, usually
well exceeding 90%.
In answering the questions below, consider the transformer ideal unless otherwise noted.
Part A
The primary coil of a transformer contains 100 turns; the secondary has 200 turns. The primary coil is connected
to a size AA battery that supplies a constant voltage of 1.5 volts. What voltage would be measured across the
secondary coil?
ANSWER:
zero
Correct
In order for an emf to be induced in the secondary coil, the flux through it must be changing; therefore, the
current in the primary coil must also be changing. If a constant voltage is supplied to the primary coil, no emf
would be induced in the secondary, and therefore, the secondary voltage would be zero.
Part B
A transformer is intended to decrease the rms value of the alternating voltage from 500 volts to 25 volts. The
primary coil contains 200 turns. Find the necessary number of turns in the secondary coil.
ANSWER:
=
10
Correct
This is a step-down transformer: The voltage decreases.
Part C
A transformer is intended to decrease the rms value of the alternating current from 500 amperes to 25 amperes.
The primary coil contains 200 turns. Find the necessary number of turns in the secondary coil.
Hint C.1 How to approach this problem
Hint not displayed
ANSWER:
=
4000
Correct
This is a step-up transformer: The voltage increases by the same factor by which the current decreases.
Part D
In a transformer, the primary coil contains 400 turns, and the secondary coil contains 80 turns. If the primary
current is 2.5 amperes, what is the secondary current ?
Express your answer numerically in amperes.
ANSWER:
=
12.5
Correct
Part E
The primary coil of a transformer has 200 turns and the secondary coil has 800 turns. The power supplied to the
primary coil is 400 watts. What is the power generated in the secondary coil if it is terminated by a 20-ohm
resistor?
Hint E.1 In the ideal world...
Hint not displayed
ANSWER:
Correct
In case of an ideal transformer, the power in the primary circuit is the same as that in the secondary circuit.
Part F
The primary coil of a transformer has 200 turns, and the secondary coil has 800 turns. The transformer is
connected to a 120-volt (rms) ac source. What is the (rms) current in the primary coil if the secondary coil is
terminated by a 20-ohm resistor?
Hint F.1 How to approach the problem
Hint not displayed
Express your answer numerically in amperes.
ANSWER: = 96
Correct
Part G
A transformer supplies 60 watts of power to a device that is rated at 20 volts (rms). The primary coil is connected
to a 120-volt (rms) ac source. What is the current in the primary coil?
Hint G.1 How to approach the problem
Hint not displayed
Express your answer in amperes.
ANSWER:
=
0.5
Correct
Part H
The voltage and the current in the primary coil of a nonideal transformer are 120 volts and 2.0 amperes. The
voltage and the current in the secondary coil are 19.4 volts and 11.8 amperes. What is the efficiency of the
transformer? The efficiency of a transformer is defined as the ratio of the output power to the input power,
expressed as a percentage: .
Express your answer as a percentage.
ANSWER:
=
95.4
Correct %
Exercise 26.52
The heating element of an electric dryer is rated at 4.1 when connected to a 240- line.
Part A
What is the current in the heating element?
Express your answer using two significant figures.
ANSWER:
=
17
Correct
Part B
Is 12-gauge wire large enough to supply this current?
ANSWER:
yes
no
Correct
Part C
What is the resistance of the dryer's heating element at its operating temperature?
Express your answer using two significant figures.
ANSWER:
=
14
Correct
Part D
At 11 cents per , how much does it cost per hour to operate the dryer?
Express your answer using two significant figures.
ANSWER:
45
Correct cents
Exercise 31.35
A transformer connected to a 120 ac line is to supply 12.0 (rms) to a portable electronic device. The load
resistance in the secondary is 4.00 .
Part A
What should the ratio of primary to secondary turns of the transformer be?
ANSWER:
=
10.0
Correct
Part B
What rms current must the secondary supply?
ANSWER:
=
3.00
Correct
Part C
What average power is delivered to the load?
ANSWER:
=
36.0
Correct
Part D
What resistance connected directly across the source line (which has a voltage of 120 ) would draw the same
power as the transformer?
ANSWER:
=
400
Correct
Exercise 31.36
A transformer connected to a 120- (rms) ac line is to supply 13000 (rms) for a neon sign. To reduce shock
hazard, a fuse is to be inserted in the primary circuit; the fuse is to blow when the rms current in the secondary
circuit exceeds 8.50 .
Part A
What is the ratio of secondary to primary turns of the transformer?
ANSWER:
=
108
Correct
Part B
What power must be supplied to the transformer when the rms secondary current is 8.50 ?
ANSWER:
=
111
Correct
Part C
What current rating should the fuse in the primary circuit have?
ANSWER: = 0.921
Correct