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02- AC Theory
02 – AC Theory
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The intent of this presentation is to present enough information to provide the reader with a
fundamental knowledge of Alternating Current and to better understand basic Michelin system
and equipment operations.
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Module 3: Alternating Current
Electromagnetism
In 1819 a Danish scientist named Oersted discovered a relation between magnetism and electric current.
He found that an electric current flowing through a conductor produced a magnetic field around that
conductor. Below we see that iron filings in a definite pattern of concentric rings around the conductor
show the magnetic field of the current in the wire.
Iron filings
Glass
+
Direction of current
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Current carrying conductor
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Module 3: Alternating Current
Electromagnetism
Every section of the wire has this magnetic field of force around it in a plane perpendicular to the wire.
The strength of the magnetic field around a conductor carrying current depends on the magnitude of the
current. A high current will produce many lines of force extending far from the wire, while a low current
will produce only a few lines close to the wire.
Large field
Small field
Low Current
High Current
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Module 3: Alternating Current
Electromagnetism
Left-Hand Rule for a Single Conductor
The left-hand rule is a convenient way to determine the relationship between the flow of current in a
conductor (wire) and the direction of the magnetic lines of force around the conductor. Grasp the
current-carrying wire in the left hand, wrapping the four fingers around the wire and extending the thumb
along the wire. If the thumb points along the wire in the direction of current flow, the fingers will be
pointed in the direction of the lines of force around the conductor.
Current flow
-
+
Flux lines
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Module 3: Alternating Current
Electromagnetism
Illustrating a Magnetic Field around a Single Conductor
It is difficult to show current flow in a third dimension without introducing two new symbols. To illustrate
current flowing toward you, or out of the page, the point of the arrow is often represented by a dot. For
current flowing away from you, or into the page, a cross represents the tail of the arrow. The magnetic
field around a single conductor does not have a north or South Pole, but it does have a direction. Note
the direction of the magnetic and the symbols for current flow in the illustrations below:
cross
dot
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Module 3: Alternating Current
Electromagnetism
Magnetic Fields Aiding or Canceling
In the illustration below the magnetic fields are shown for two parallel conductors with opposite directions
of current. By applying the left-hand rule for conductors, you determine the clockwise direction of the
field of the conductor on the left and counterclockwise field direction of the conductor on the right.
Because the magnetic lines between the conductors are in the same direction, the fields repel each other
and tend to cancel their effect.
02 – AC Theory
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Module 3: Alternating Current
Electromagnetism
In the illustration below the magnetic fields are shown for two parallel conductors with currents flowing in
the same direction. By applying the left-hand rule for conductors, you determine the clockwise
direction of the field of the conductor on the left and the conductor on the right. Because the magnetic
lines between the conductors are in the opposite direction, the fields attract each other and aid in their
effect.
02 – AC Theory
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Module 3: Alternating Current
Electromagnetism
Magnetic Fields and Polarity of a Coil
Bending a straight conductor into the form of a single loop has two results. First, the magnetic field lines
are denser inside the loop, although the total number of lines is the same as for the straight conductor.
Second, all the lines inside the loop are aiding in the same direction.
-
+
A coil of wire is formed when there is more than one loop or turn. To determine the magnetic polarity of
a coil, use the left-hand rule for a coil. If the coil is grasped with the fingers of the left hand curled in
the direction of current flow through the coil, the thumb points to the North Pole of the magnetic field
created by the coil. This is illustrated on the following page.
02 – AC Theory
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Module 3: Alternating Current
Electromagnetism
N
S
+
_
If the direction of the current flow is changed as below, you can see that the polarity of the magnetic field
changes. You could also wrap the conductors the opposite direction to change the polarity of the
magnetic field.
.
N
S
-
02 – AC Theory
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+
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Module 3: Alternating Current
Electromagnetism
Strength of an Electromagnetic Field
The strength, or intensity, of a coil's magnetic field depends on a number of factors. The major factors
are listed below.
1. The number of turns of the conductor
2. The amount of current flow through the coil
3. The ratio of the coil's length to its width
4. The type of material in the core
02 – AC Theory
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Module 3: Alternating Current
Electromagnetism
Strength of an Electromagnetic Field
The strength, or intensity, of a coil's magnetic field depends on a number of factors. The major factors
are listed below.
1. The number of turns of the conductor
2. The amount of current flow through the coil
3. The ratio of the coil's length to its width
4. The type of material in the core
Electromagnetic Applications
If a bar of iron or soft steel is placed in the magnetic field of a coil, the bar will become magnetized. If
the magnetic field is strong enough, the bar will be drawn into the coil until it is approximately centered
within the magnetic field. This attraction and centering effect is termed as solenoid action.
02 – AC Theory
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Module 3: Alternating Current
Electromagnetism
Electromagnetism is basic to the operation of many common electrical and electronic devices.
Electromagnetic coils are used in contactors, relays, and solenoid valves. These devices use this
solenoid action concept.
The solenoid is a coil of wire with a moveable iron core. When a soft iron core is placed near one end of
an energized coil, the lines of force are concentrated through the iron core. The magnetic attraction pulls
the core into the center of the coil as shown below. When the coil is de-energized, the electromagnetic
field collapses and there is no longer a magnetic force holding the core inside the coil. A spring attached
to one end of this iron core will pull it back out of the coil to its initial position. The movement of the core
in response to the magnetic pull of the coil provides us with a mechanical movement we can use. This
type of movement is known as solenoid action. This mechanical movement can be used to close a
valve, close a set of open contacts, or open a set of closed contacts.
N
-
02 – AC Theory
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+
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Module 3: Alternating Current
Electromagnetism
The relay circuit shown below is a basic illustration of the operation of a control relay.
+
RELAY CIRCUIT
-
SPRING
SMALL CURRENT FROM
AN EXTERNAL SOURCE
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Module 3: Alternating Current
Electromagnetic Induction
In 1831 Michael Faraday discovered the principle of electromagnetic induction. It states that if a
conductor cuts across lines of magnetic force, or if lines of force cut across a conductor, a voltage is
induced across the ends of the conductor. Consider a magnet with its lines of force extending from its
north pole to its south pole as below.
A conductor C, which can be moved between the poles, is connected to a sensitive voltmeter called a
galvanometer G. When the conductor is not moving, the galvanometer shows zero volts. If the
conductor is moving outside the magnetic field at position 1, the galvanometer will still show zero. When
the conductor is moved to the left to position 2, it cuts across the lines of magnetic force and the
galvanometer pointer will deflect to B.
This indicates that a voltage was induced across the conductor because lines of force were cut. In
position 3, the galvanometer pointer swings back to zero because no lines of force are being cut. Now
reverse the direction of the conductor by moving it right through the lines of force back to position 1.
During this movement, the pointer will deflect to A, showing that a voltage has again been induced
across the conductor, but in the opposite direction. If the conductor is held stationary in the middle of the
field of force at position 2, the galvanometer reads zero. If the conductor is moved up or down parallel to
the lines of force so that none is cut, no voltage will be induced.
02 – AC Theory
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Module 3: Alternating Current
Electromagnetic Induction
In summary, when a conductor cuts lines of force or lines of force cut a conductor, a voltage is induced
across the conductor. There must be relative motion between the conductor and the lines of force in
order to induce a voltage. Changing the direction of cutting will change the direction of the induced
voltage.
N
1
3
2
0
A
C
B
-
+
G
S
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Module 3: Alternating Current
Introduction to Alternating Current
The majority of electrical energy is produced in large power stations and is transmitted throughout the
country on long distribution lines. Using a D.C. voltage source would be difficult and costly. The
batteries and the conductors would have to be very big. All voltages would have to be essentially the
same. However, by using an Alternating Current (A.C.), power may be transmitted much more simply
and efficiently. Therefore, A.C. generation and transmission has become a standard practice throughout
the world.
Generating an Alternating Voltage
An A.C. voltage is one that continually changes in magnitude and periodically reverses in polarity. The
time axis or line is a horizontal line across the center. The vertical axis shows the changes in magnitude
by the variations of the voltage.
V+
270O
360O
0V
0O
90O
180O
V-
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Module 3: Alternating Current
Introduction to Alternating Current
In its simplest form the A.C. generator consists of a loop of wire rotating around an axis within a
magnetic field. The magnetic field is produced between the pole faces of two permanent magnets. As
the loop of wire rotates, the conductor is cutting lines of force of the magnetic field. Then by electromagnetic induction a current is induced into the conductor. In reality this loop of wire would be a
winding. The ends of the winding are connected to slip rings mounted on the shaft of the armature. The
slip rings rotate with the armature. Brushes make contact with the slip rings enabling an external circuit
to be connected as a load.
Let's consider a cross sectional view of the armature loop as it is situated between the two magnetic
poles. We can now use the left-hand generator rule to evaluate the currents induced into the armature
and understand how this voltage is generated. First, we know that some mechanical force, such as a
water wheel, steam turbine, or other driving mechanisms is rotating the generator shaft.
Shaft
N
S
Connected to an external load
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Module 3: Alternating Current
Introduction to Alternating Current
Use the illustration on the next page to evaluate the voltage being generated from the induced current.
One side of the conductor loop is shaded so we can evaluate it as it moves in time. The horizontal line is
related to time. Since this time is very small for one revolution of this conductor loop, we will relate the
movement to the degrees of a circle. We will use the quadrants of the circle starting with quadrant one
moving counter clockwise through quadrant two, then three and four.
At the instant the loop starts to rotate in this counter clockwise direction, the conductor is moving parallel
to the lines of force and no lines are being cut. Therefore, the current induced in the conductor is zero.
As the rotation continues, the conductor starts to cut the lines of force at varying angles. As a result the
magnitude of the induced current will vary. The maximum cutting rate is reached when the conductor is
traveling at right angles to the lines of force. At this point the induced current will be at a maximum
which generates the maximum voltage.
As you can see from the illustration on the next page, the point that this maximum induced current and
generated voltage is achieved is when the conductor is at 90° to the lines of force.
02 – AC Theory
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Module 3: Alternating Current
Introduction to Alternating Current
N
+ V o lta ge
0
0
0
30
0
0
60
90
t(time )
- V o ltage
S
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Module 3: Alternating Current
Introduction to Alternating Current
N
+ Voltage
0
0
0
0
0
0
90 120150180
t(time)
- Voltage
S
As the conductor continues to rotate between 90° and 180°, the magnitude of the voltage will
decrease back to zero.
N
+ Voltage
0
0
0
0
180 210 240 270
0
0
t(time)
S
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- Voltage
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Module 3: Alternating Current
Introduction to Alternating Current
N
+ Voltage
0
0
0
0
270300 330360
t (time)
0
0
S
- Voltage
As the conductor continues to rotate between 270° and 360°, the magnitude of the voltage will
decrease back to zero. After one complete revolution of the loop, the cycle will continue to repeat itself.
The voltage induced (or generated) is continually changing in magnitude, as well as, direction. This
graphical representation of this continually changing voltage has produced a sine wave.
For each pair of magnetic poles, one cycle of alternating current and voltage will be induced into the
loop every revolution. The number of cycles in a given period of time, usually one second, is termed as
the frequency. This waveform of voltage is sinusoidal.
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Module 3: Alternating Current
Introduction to Alternating Current
AC Voltage Generation Summary
A simple generation consists of a conductor loop turning in a magnetic field, cutting across the magnetic
lines of force. A current is induced into the loop causing a voltage across the loop. The sine wave
output is the result of one side of the generator loop cutting lines of force. In the first half of rotation, this
produces a positive voltage and in the second half of rotation produces a negative voltage. This
completes one cycle of AC generation.
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Module 3: Alternating Current
Definitions of an A.C. Sine Wave
Instantaneous Voltage Value
Now that we have generated this A.C. sine wave, we will look at the instantaneous voltage values and
how they relate to time. Given that the conductor is rotating at a steady speed, we know that the sine
wave will repeat after each revolution. We now need to be able to calculate the instantaneous value for
the voltage at any time or degrees of this revolution.
We know that the maximum number of flux lines is being cut at 90° and this is the maximum value of
voltage generated. Let's assume that the 10 volt level is the maximum value being generated for the
sake of these calculations. Since the lines of flux being cut at 30° are much less, we know that the
voltage is also much less. How do we determine this value?
The illustration on the next page represents the first part of the generation of the sine wave. We can use
it to help us determine the instantaneous values for voltage.
02 – AC Theory
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Module 3: Alternating Current
Definitions of an A.C. Sine Wave
Instantaneous Voltage Value
Using the laws of geometry, we know that the length of a radius of a circle is the same at 30° and 90°.
The radius for 90° is equal to the maximum voltage (VMax). We now can draw a triangle and use
trigonometry to solve for the instantaneous voltage value at 30°. VMax will be the hypotenuse, Vi will be
the opposite side, and the angle at the center will be used for the sin function calculations. When we
substitute the value of 10 volts for VMax and 30° for the angle and solve the equation as below:
Vi  VMax  sin 
Vi  10V  sin 30
Vi  10V  0.5
Vi  5V
This instantaneous voltage relationship can be used for any value of degrees through the cycle. If we
were to plot the instantaneous voltage values for each 30° increment, the result would be a sinusoidal
waveform.
02 – AC Theory
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Module 3: Alternating Current
Definitions of an A.C. Sine Wave
N
+ Voltage
VMax
VMax
0
0
0
30
0
60
0
90
- Voltage
S
VMax
30
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0
Vinst.
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Module 3: Alternating Current
Definitions of an A.C. Sine Wave
Peak Voltage Value
The peak value of voltage (Vpk) is the maximum value of voltage. It applies to the positive or negative
peak and is sometimes called VMax. The peak-to-peak value of voltage (Vpk-pk) is double the peak value.
V+
Vpeak
360o
Vpeak to peak
0V
90o
180o
270o
Half Cycle
VOne Cycle
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Module 3: Alternating Current
Definitions of an A.C. Sine Wave
Average Voltage Value
The average value of voltage is the arithmetic average of all values in a sine wave for one half-cycle.
The half-cycle is used for the average because over a full cycle the average value is zero. If we
calculated all the instantaneous voltage values for any given half-wave of the sine wave, added all the
values and divided the result by the number of calculations, we would derive the average value of
voltage for that half-cycle. The average voltage value is used when the rectification of an A.C. voltage is
necessary. If we assume the sine wave above has a maximum instantaneous voltage of 170 volts. The
relationship established to derive the average value is as follows:
V av = V Max  0.637
V av = 170v  0.637
V av = 108.29v
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Module 3: Alternating Current
Definitions of an A.C. Sine Wave
Effective Voltage Value
If one A.C. rms ampere passes through a resistance of 1 ohm, the rms voltage drop across the resistor
is one volt rms. Therefore, the rms voltage is 0.707 of the instantaneous maximum voltage and is called
the effective voltage. The conventional A.C. voltmeter measures this effective value of voltage. The
same relationship exists between maximum and effective volts as between maximum and effective
current. In all A.C. calculations, the V without a subscript will equal the Vrms value unless otherwise
specified as with VMax or VPk.
V rms =
V rms
=
V Max
2
V Max
1.414
V rms = 0.707 V Max
V rms = 0.707 (10volts)
V rms = 7.07volts
V rms = 0.707 V Max
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Module 3: Alternating Current
Definitions of an A.C. Sine Wave
I+
Ipeak
360o
Ipeak to peak
0
90o
180o
270o
Half Cycle
IOne Cycle
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Module 3: Alternating Current
Definitions of an A.C. Sine Wave
Instantaneous Current Value
The instantaneous current value is calculated just as the instantaneous voltage was calculated.
Ii  IMax  sin 
Ii  10V  sin 30
Ii  10V  0.5
Ii  5V
This instantaneous current relationship can be used for any value of degrees through the cycle. If we
were to plot the instantaneous current values for each 30° increment, the result would be a sinusoidal
waveform.
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Module 3: Alternating Current
Definitions of an A.C. Sine Wave
Effective Current Value
To derive the effective current value of an A.C. sine wave, we will use the sine wave shown above with
an instantaneous maximum current of 10 amps. This waveform represents the induced current
generated for a given circuit.
I+
Ipeak
360o
Ipeak to peak
0
90o
180o
270o
Half Cycle
IOne Cycle
If the current waveform is considered for one complete cycle, the average value would be zero since the
negative alternation is equal to the positive alternation. If a typical D.C. ammeter were used to measure
the current of this circuit, it would indicate zero. Therefore, to measure the alternating current, an A.C.
ammeter that measures the effective value of current should be used.
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Module 3: Alternating Current
Definitions of an A.C. Sine Wave
Effective Current Value
The effective value of alternating current is based on its heating effect and not on the average value of
the sine wave. An alternating current with an effective value of ten amperes is that current that will
produce heat in a given resistance at the same rate, as does ten amperes of direct current.
In D.C. theory, it is shown that the heating effect directly relates to the square of the current for a given
resistance (P = I2R). The unit of measure of Power is watts and relates directly to heat.
To derive the effective current value of the sine wave with an IMax of 10 amps, we would first decide the
interval of time for our calculations. This interval of time could be 1° or any interval small enough to
give us a very accurate calculation. We will use 15° increments in this derivation, since we are not
going into detail for all the calculations.
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Module 3: Alternating Current
Definitions of an A.C. Sine Wave
Effective Current Value
We can calculate the instantaneous current values for each of the 15° increments starting with 0° up
to 360°. Once we've calculated the instantaneous currents for each increment, we will square each
value which relates to the heating effect. Then we can take the average of all those values. Finally, we
will get the square root of the average value of the squared instantaneous values. This value is called
the effective value or root-mean-square (rms) value. Root-mean-square current is the abbreviation of
the square root of the mean of the square of the instantaneous values.
A conventional A.C. ammeter measures the rms value of current. In all A.C. calculations, the I without a
subscript will equal the Irms value unless otherwise specified as with IMax or IPk. The relationship of
effective (rms) and maximum current is:
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Module 3: Alternating Current
Definitions of an A.C. Sine Wave
Effective Current Value
+10 Amps
7.07 Amps Effective
360o
0
90o
180o
270o
-10 Amps
I rms =
I Max
2
I rms = 0.707 I Max
I rms =
I Max
1.414
I rms = 0.707 (10amps)
I rms = 0.707 I Max
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I rms = 7.07amps
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Module 3: Alternating Current
Frequency and Period
The number of cycles per second is called frequency. It is indicated by the symbol f and is expressed in
hertz (Hz). One cycle per second equals one hertz. Thus 60 cycles per second equals 60Hz. The
amount of time for the completion of 1 cycle is the period. It is indicated by the symbol T for time and is
expressed in seconds (s). Frequency and period are reciprocals of each other.
f 
1
T
T
1
f
If 60 cycles per second equals 60Hz, how much time does it take for one cycle to happen?
T
1
f
T
1
60 Hz
T  16.667ms
How much time does it take for one-half of a cycle?
If 360  16.667 ms,
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180  8.333ms
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Module 3: Alternating Current
Phase Relationships
In phase relationship
The waveforms below show that the voltage and current are at minimum and maximum values at the
same time. These waveforms are said to be in phase with each other. The phase angle (q) between
two waveforms of the same frequency is the angular difference at a given instant of time. When two
waveforms are in phase, the phase angle (q) is zero. In AC circuit analysis, all waveforms will have a
phase angle (q) relationship with 0° being the simplest.
+
V (Voltage)
I (Current)
360o
0
0o
90o
180o
270o
-
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Module 3: Alternating Current
Phase Relationships
Vectors
To compare phase angles or phases of alternating voltages and currents, it is more convenient to use
vector diagrams corresponding to the voltage and current waveforms. A vector quantity is one that has
magnitude, as well as, direction. A vector is represented graphically, by an arrow type line segment
whose length is proportional to the magnitude of the vector. The arrow points in the direction of the
vector quantity.
The length of the arrow in a vector diagram indicates the magnitude of the alternating voltage or current.
The angle of the arrow with respect to the horizontal axis indicates the phase angle. One waveform is
chosen as the reference. Then the second waveform can be compared with the reference by means of
the angle between the vector arrows. Generally, the reference vector is horizontal, corresponding to
0°. The angles would then be measured in a counterclockwise direction.
The following vector diagram shows the relationship between voltage and current from the previous
waveforms. Since the voltage and current are in phase with each other, the vectors are in the same
direction but have different magnitudes. The magnitudes are not to scale but do relate the difference in
their value.
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Module 3: Alternating Current
Phase Relationships
Vectors
The following vector diagram shows the relationship between voltage and current from the previous
waveforms. Since the voltage and current are in phase with each other, the vectors are in the same
direction but have different magnitudes. The magnitudes are not to scale but do relate the difference in
their value.
90o
I
180o
V
0o
270o
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Module 3: Alternating Current
Phase Relationships
Resistance in AC Circuits
In an AC circuit with only resistance, the current variations are in phase with the applied voltage. The
vector diagram would be represented just as above. This in phase relationship between V and I means
that such an AC circuit can be analyzed by the same methods used for DC circuits. Therefore, Ohm's
laws for DC circuits are applicable also to AC circuits with resistance only. The calculations in AC
circuits are generally in rms values, unless otherwise specified. For the series circuit below, let's look at
the vector relationship between the applied voltage (VT) and the total current (IT).
VT = 110 volts and R1 = 10Ω, find: IT
IT
VT
IT
VT
R1
Vector Diagram
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VT
R1
110V
IT 
10
I T  11Amps
IT 
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PT  VT  I T  cos 
PT  110volts  11amps  cos 0
PT  110volts  11amps  (1)
PT  1210Watts
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Module 3: Alternating Current
Phase Relationships
Out of phase relationship
As mentioned previously, the phase angle (q) between two waveforms of the same frequency is the
angular difference at a given instant of time. When two waveforms are in phase, the phase angle (q) is
zero. When two waveforms are out of phase, the phase angle (q) is some value less than or greater
than zero. As shown in the example below, the phase angle between waves V and I is 90°.
The horizontal axis is shown in angular units of time, measured in degrees. Voltage waveform V starts
at 0° at a value of 0 volts and reaches maximum value at 90°. Current waveform (I) is at 0 amps at
90° and a maximum value at 180°. Using 0° as a reference point to start, the voltage waveform is
said to lead the current waveform. In this case the voltage leads the current by 90°.
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Module 3: Alternating Current
Phase Relationships
Out of phase relationship
The vector diagram for the out of phase condition is also illustrated below. As you can see the voltage
and current are not happening at the same time with this out of phase condition. The relationship
between voltage and current now has to be evaluated using the phase angle (q). In the next section, we
will understand more about what causes this out of phase condition.
90o
+
V (Voltage)
I (Current)
V
360o
0
0o
90o
180o
I
180o
270o
0o
-
270o
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Module 3: Alternating Current
Written Exercises
.
IT
1. Given: RT = 100Ω
VT = 75V
Find: Ii @ 64˚ =
VT
3. Given: RT = 50Ω
Vi = 100V @ 75˚
RT
Find: IT =
4. Given: RT = 45Ω
IT = 2A
2. Given: Vi = 75V @ 65˚
Ii = 1.5A @ 50˚
Find: Vi @ 295˚ =
Find: RT =
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Module 3: Alternating Current
Reactance and Impedance Concepts
Inductance in AC
Inductance is the characteristic of an electrical circuit that opposes the starting, stopping, or changing of
current flow. The above statement is of such importance to the study of inductance that it bears
repeating in a simplified form.
Inductance is the characteristic of an electrical conductor or circuit, which opposes a CHANGE
in current flow.
Michael Faraday started to experiment with electricity around 1805 while working as an apprentice
bookbinder. It was in 1831 that Faraday performed experiments on magnetically coupled coils. A
voltage was induced in one of the coils due to a magnetic field created by current flow in the other coil.
From this experiment came the induction coil. In performing this experiment Faraday also invented the
first transformer, but since alternating current had not yet been discovered the transformer had few
practical applications. Two months later, based on these experiments, Faraday constructed the first
direct current generator. At the same time Faraday was doing his work in England, Joseph Henry was
working independently along the same lines in New York. The discovery of the property of self-induction
of a coil was actually made by Henry a little in advance of Faraday and it is in honor of Joseph Henry
that the unit of inductance is called the henry.
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Module 3: Alternating Current
Reactance and Impedance Concepts
Inductance in AC
Unit of Inductance
The unit for measuring inductance (L) is the henry (h). The henry is a large unit of inductance and is
used with relatively large inductors. The unit employed with small inductors is the milli-henry, mh. For
still smaller inductors the unit of inductance is the micro-henry, h.
Self-Inductance
Even a perfectly straight length of conductor has some inductance. As previously explained, current in a
conductor always produces a magnetic field surrounding, or linking with, the conductor. If the current
changes in the conductor, then magnetic field around the conductor also changes. Then an emf is
induced in the conductor. This emf is called a Self-Induced emf because it is induced in the conductor
carrying the current. The direction of the induced emf has a definite relation to the direction in which the
field that induces the emf varies. When the current in a circuit is increasing, the flux linking with the
circuit is increasing. This flux cuts across the conductor and induces an emf in the conductor in such a
direction as to oppose the increase in current and flux. This emf is sometimes referred to as counterelectro-motive-force (Cemf) or back emf. Likewise, when the current is decreasing, an emf is induced in
the opposite direction and opposes the decrease in current. These effects are summarized by Lenz's
Law, which states that the induced emf in any circuit is always in a direction to oppose the effect
that produced it.
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Module 3: Alternating Current
Reactance and Impedance Concepts
Self-Inductance
Shaping a conductor so that the electromagnetic field around each portion of the conductor cuts across
some other portion of the same conductor increases the inductance (as illustrated on the next page). A
length of conductor is looped so those two portions of the conductor lie adjacent and parallel to each
other. This conductor is in the form of a coil or inductor.
When an alternating voltage is applied to the loops of the conductor and a current starts to flow through
the conductor loops, a concentric magnetic field starts to expand around all portions of the looped
conductor. With increasing current, the field expands outward, cutting across a portion of the adjacent
conductor. The resultant induced emf in that conductor opposes the main current. The direction of this
induced voltage and current may be determined by applying the Left-Hand-Rule for Generators. To do
this we will reference the middle conductor and evaluate what effects the expanding magnetic field
around the left conductor has on relative motion, flux, and direction of the induced current in the
middle conductor.
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Module 3: Alternating Current
Reactance and Impedance Concepts
Self-Inductance
In applying this rule we will need to use the thumb, the index finger and the middle finger. The thumb of
the left hand points in the direction a conductor is moved through a magnetic field (F - relative motion).
With the conductor stationary, the magnetic field is expanding or moving. This would be the same as if
the conductor were moving in the opposite direction. The index finger points in the direction of the
magnetic field from North to South (B). The middle finger will now indicate the direction of the induced or
generated current (I).
Direction of the Current through the Conductor
The relationship of the direction of the generated current flow in a conductor and the direction of the
relative motion of that conductor in the magnetic field may be briefly described as the Left hand
Generator Rule:
F
B
I
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Module 3: Alternating Current
Reactance and Impedance Concepts
Self-Inductance
As shown below, the thumb of the left hand is pointing up and represents the direction of the
relative motion of the conductor. The forefinger points out at a ninety-degree angle to the thumb
and gives the direction of the magnetic field. The middle finger is also at ninety-degree angle to
the forefinger and gives the direction of the generated or induced current into the conductor.
This rule does not explain anything directly. It is merely one of the ways of determining one of the
directions when the other two are known.
F = Direction of the Relative Motion
B = Direction of the Magnetic Field
I = Direction of the Current
F
B
I
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Module 3: Alternating Current
Reactance and Impedance Concepts
B = Flux
I = Current
-
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Main Current
Main Current
cemf
F = Relative Motion
+
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Module 3: Alternating Current
Reactance and Impedance Concepts
Inductive Reactance of a Coil
When an AC voltage is applied across a coil:
The AC voltage will produce an alternating current.
When a current flows in a conductor, magnetic flux lines are produced around the conductor.
Large currents produce many lines of flux and small currents produce only a few lines of flux.
As the current changes, the number of lines of flux will change. The magnetic field will expand and
contract as the current increases and decreases.
As the magnetic field expands and contracts around one conductor, these lines of flux cut across the
conductors which form the turns of the coil.
When a conductor cuts lines of magnetic flux, a voltage and current is induced in the conductor.
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Module 3: Alternating Current
Inductive Reactance
Inductive reactance, XL is the opposition to AC current due to the inductance in the circuit. The unit of
inductive reactance is the ohm. The formula for inductive reactance is
X L = 2fL
where X L = inductive reactance, measured in 
f = frequency, measured in Hz
L = inductance, measured in henrys(h)
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Module 3: Alternating Current
Inductive Reactance
In a circuit containing only inductance, Ohm's Law can be used to find current and voltage by substituting
XL for R. The circuit below would be considered a purely inductive circuit in theory only. This
purely inductive circuit is used only to help us understand more about the out of phase
relationship. In the circuit below the supply voltage leads the current by 90 due to the counter-emf
caused by the pure inductor. This 90 phase angle (θ) is only theoretical since there is no actual pure
inductor.
+
IT
I XL 
V (Voltage)
I (Current)
0
VT
L
270
VXL
0
0
90
0
180
0
360
0
Since V X L  VT and I X L  I T
0
0
0
Vector Diagram
02 – AC Theory
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VT
XL
The equation above will only
be true for a purely inductive
circuit.
V
I
XL
then I T 
-
90
t
VXL
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Module 3: Alternating Current
Inductive Reactance
Inductive Reactance and Series Internal Resistance for a Coil
Because any practical inductor must be wound with wire that has resistance, it is impossible to obtain
a coil without some internal DC resistance. The internal DC resistance associated with a coil (L), may
be considered as a separate resistor (r) in series with the inductive reactance (XL). This internal DC
resistance would be the actual resistance of the coil if you connected an ohmmeter across its terminals.
For clarity and consistency, we will show the inductor (L) with an internal resistance (r) and an
inductive reactance (XL) inside a dotted line. This will show that the inductor is one device with
two internal components in series as shown below.
IT
VT
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r
L
XL
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Vector Diagram
VXL
900
IT
Vr
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Module 3: Alternating Current
Inductive Reactance
Since the circuit below is a series circuit, the total current (IT) is equal to the current through the internal
resistance (Ir) and is equal to the current through the inductive reactance (IXL). With this series internal
resistance added, the rms current (IT) is limited by both the inductive reactance (XL) and the internal
resistance (r). The current, IT is the reference since it is common through XL and r for a series circuit.
The voltage drop across the internal resistance (r) is Vr = Ir • r. The current Ir through r and its voltage
drop happen at the same time and are in phase so the phase angle (θ) is 0.
The voltage drop across XL is VXL = IXL • XL. The voltage across XL leads the current through XL by 90
and they do not happen at the same time. This is the phase angle (θ) between the current through the
inductive reactance and voltage across the inductive reactance.
IT
VT
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r
L
XL
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Vector Diagram
VXL
900
IT
Vr
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Module 3: Alternating Current
Voltage Vector Relationship for a Coil
T
V
Since Ohm's Law applies to Vr and VXL, then VT should be the sum of the two. However, the voltage
drops don’t happen at the same time because they are not in phase with each other. Therefore they
cannot be added directly. Their sums must be added vectorially. This can be done graphically by using
a vector diagram and projecting the vectors to an intersection point or coordinate. Then the resultant
vector can be drawn from the origin to this coordinate. The resultant vector VT is the vector sum of Vr
and VXL.
VXL
T
Vr
IT
Voltage Vector Diagram
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Module 3: Alternating Current
Voltage Vector Relationship for a Coil
VT =
2
2
V r + V XL
T
V
This voltage relationship or triangle can also be expressed from the geometry of a right triangle using
the Pythagorean Theorem which states, the hypotenuse is equal to the square root of the sum of the
squares of the other two sides.
VXL
T
Vr
IT
Voltage Vector Diagram
The phase angle (θ) can also be calculated by trigonometric functions.
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opposite
= V XL
hypotenuse V T
adjacent V r
cos =
=
hypotenuse V T
opposite V XL
tan  =
=
adjacent V r
sin  =
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Module 3: Alternating Current
Impedance Vector Relationship for a Coil
The total opposition to current flow in a purely resistive AC series circuit is the resistance (R). In a
theoretical purely inductive AC series circuit the total opposition to current flow is the inductive reactance
(XL). In an AC series circuit containing both resistance and inductive reactance, the total opposition to
current flow is called impedance (Z).
Just as with the AC voltage relationship, the impedance relationship will use the total current (IT) in the
circuit as the reference since it is a series circuit and is common to both components. The internal
resistance will be drawn at 0 with reference to the series current. This is due to the current through it
and the voltage across it being in phase with each other. The inductive reactance, however, is drawn at
90 with reference to the series current because the voltage across it is out of phase with the current
through it. The total impedance (ZT) for this circuit is the vector sum of the internal resistance (r) and the
inductive reactance (XL) of the coil. . The internal resistance (r) is the total resistance in the circuit
above. Later we will see how the addition of external resistance (R) affects the circuit.
r
L
XL
XL
T
VT
Z
IT
T
r
IT
Impedance Vector Diagram
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Module 3: Alternating Current
RL - Impedance Vector Relationship for a Coil and a Resistor
In most AC circuit applications, there would be more resistance in the circuit than just the internal DC
resistance (r) of the coil. This external resistance is sometimes shown as only one resistor that includes
the internal resistance. For our applications, we will show these two resistances as two separate
resistors. The total resistance in the circuit is equal to the sum of all resistances in the circuit.
RT = r + R1 + R2 + R3 
For the circuit below, there is a series resistance (R) added to the circuit. The inductor (L) still has
internal resistance (r) and inductive reactance (XL). The impedance (ZL not ZTotal) and phase angle (qL
not qTotal) of the inductor or coil would be the same as discussed before and is illustrated with the dotted
vector and angle. The external resistance (R) is added to the x-axis along with the internal resistance
(r). These two resistances would equal RT and could be represented as one vector. Once the external
resistance has been added to the x-axis, the phase angle (qTotal) and the impedance vector for the total
impedance in the circuit (ZTotal) will change. The inductive reactance (XL) has not changed in the circuit.
IT
R
ZT
VT
L
L
XL
XL
L
Z
r
r
T
R
IT
Impedance Vector Diagram
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Module 3: Alternating Current
Power Vector Relationship in RL Circuits
Real Power
In an AC circuit with inductive reactance, the applied voltage leads the line current. The real power, P, is
equal to the voltage multiplied by only that portion of the line current that is in phase with the voltage.
The current and voltage are only in phase for the resistive components. The inductive or reactive
components are always out of phase. Real Power is measured in Watts and is equal to:
P Real = VT  I T  cos T
Power Factor
The cos of angle (q) is equal to the power factor of the circuit and is sometimes represented by PF.
PF = cos T
The real power can also be expressed using current and resistance. However, all resistances in the
circuit must be taken into account.
2
P Real = I T  RT
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Module 3: Alternating Current
Power Vector Relationship in RL Circuits
Reactive Power
Reactive power in an AC circuit is the product of voltage and the out-of-phase current. It represents the
stored energy due to the expansion of the magnetic field. The reactive power, Q, is measured in voltamperes reactive (VARS) and is equal to:
Q Reactive= VT  IT  sin T
Apparent Power
Apparent power in an AC circuit is the product of the rms voltage and rms current. It is the vector sum of
real and reactive power. The apparent power, S, is measured in volt amperes (VA) and is equal to:
S Apparent = VT  I T
Q
S
In all power formulas, the V and I are rms values. The power vector
diagram can illustrate the relationships of real, reactive, and apparent
power.
T
P
IT
Power Vector Diagram
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Module 3: Alternating Current
Summary of AC Voltage, Impedance, and Power in RL Circuits
In the three vector relationships discussed in this section, the circuit current was always the common
factor. The triangles formed by these relationships are similar. The phase angles ( ) of the three
triangles are equal even though the magnitudes of the sides are different.
VT
V
L
VXL
L
T
Vr
VR
IT
Voltage Vector Diagram
XL
L
Z
ZT
L
T
r
R
IT
Impedance Vector Diagram
Q
S
T
P
IT
Power Vector Diagram
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Module 3: Alternating Current
Definitions for RL Circuits
Inductor (L) - A device that opposes the change in current
Inductance (L) - The characteristic of an electrical conductor which opposes a CHANGE in current flow
The measure of a coils ability to oppose a change in current, measured in henrys
Inductive Reactance (Xl) - The opposition to the flow of alternating current caused by the inductance of
a circuit, measured in ohms.
Impedance (Z) - The total opposition to current flow in an AC circuit . It is the vector sum of resistance
and reactance, measured in ohms.
henry (h) - The unit of measure for inductance
Real Power (P) - Equal to the voltage multiplied by only that portion of the line current which is in phase
with the voltage, measured in watts.
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Classification : D3
Classification : D3
Conservation :
Page : ‹#›
Module 3: Alternating Current
Reactance and Impedance Calculations
IT
L
VT
r
XL
Given: VT =
IT =
XL =
R =
Solve for:
120 Volts
595.367 mAmps
200 
25 
Vr =
VXL =
ØT =
Draw the voltage vector diagram.
02 – AC Theory
Presentation : IMS – Tech Managers Conference
Author : IMS Stafff
Author : IMS Staff
Creation date : 31 Oct 2012
Creation date : 08 March 2012
Classification : D3
Classification : D3
Conservation :
Page : ‹#›
Module 3: Alternating Current
Reactance and Impedance Calculations
IT
L
VT
r
XL
Given: VT =
L=
R=
F=
Solve for:
110 Volts
40 mh
30 
60 Hz
IT =
ZT =
ØT =
Draw the voltage vector diagram.
02 – AC Theory
Presentation : IMS – Tech Managers Conference
Author : IMS Stafff
Author : IMS Staff
Creation date : 31 Oct 2012
Creation date : 08 March 2012
Classification : D3
Classification : D3
Conservation :
Page : ‹#›
Module 3: Alternating Current
Reactance and Impedance Calculations
IT
L
VT
r
XL
Given: VT =
XL =
R =
f =
Solve for:
110 Volts
10 
3
60 Hz
L=
IT =
ØT =
Draw the voltage vector diagram.
02 – AC Theory
Presentation : IMS – Tech Managers Conference
Author : IMS Stafff
Author : IMS Staff
Creation date : 31 Oct 2012
Creation date : 08 March 2012
Classification : D3
Classification : D3
Conservation :
Page : ‹#›
Module 3: Alternating Current
Reactance and Impedance Calculations
IT
L
VT
r
XL
Given: VT =
IT =
R=
f =
Solve for:
110 Volts
2A
10 
60 Hz
L=
XL =
ZT =
ØT =
Draw the voltage vector diagram.
02 – AC Theory
Presentation : IMS – Tech Managers Conference
Author : IMS Stafff
Author : IMS Staff
Creation date : 31 Oct 2012
Creation date : 08 March 2012
Classification : D3
Classification : D3
Conservation :
Page : ‹#›
Module 3: Alternating Current
Reactance and Impedance Calculations
IT
R
r
VT
L
XL
Given: VT = 110 Volts
L = 100 mh
r = 10 
R = 30 
f = 60 Hz
Solve for: IT =
ZL =
ZT =
ØT =
Draw the voltage and impedance vector diagrams.
02 – AC Theory
Presentation : IMS – Tech Managers Conference
Author : IMS Stafff
Author : IMS Staff
Creation date : 31 Oct 2012
Creation date : 08 March 2012
Classification : D3
Classification : D3
Conservation :
Page : ‹#›
Module 3: Alternating Current
Reactance and Impedance Calculations
IT
R
r
VT
L
XL
Given: L =
r=
R=
VR =
f=
Solve for:
20 mh
20 
50 
50 Volts
60 Hz
VT =
IT =
Draw the voltage and impedance vector diagrams.
02 – AC Theory
Presentation : IMS – Tech Managers Conference
Author : IMS Stafff
Author : IMS Staff
Creation date : 31 Oct 2012
Creation date : 08 March 2012
Classification : D3
Classification : D3
Conservation :
Page : ‹#›
Module 3: Alternating Current
Reactance and Impedance Calculations
IT
R
r
VT
L
XL
Given: VL =
L=
r=
R=
f=
Solve for:
230 Volts
600 mh
30 
60 
60 Hz
VT =
IT =
Draw the voltage and impedance vector diagrams.
02 – AC Theory
Presentation : IMS – Tech Managers Conference
Author : IMS Stafff
Author : IMS Staff
Creation date : 31 Oct 2012
Creation date : 08 March 2012
Classification : D3
Classification : D3
Conservation :
Page : ‹#›
Module 3: Alternating Current
Reactance and Impedance Calculations
IT
R
r
VT
L
XL
Given: VL =
L=
r=
R=
f=
Solve for:
230 Volts
600 mh
30 
60 
60 Hz
P=
Q=
S=
Draw the voltage and impedance vector diagrams.
02 – AC Theory
Presentation : IMS – Tech Managers Conference
Author : IMS Stafff
Author : IMS Staff
Creation date : 31 Oct 2012
Creation date : 08 March 2012
Classification : D3
Classification : D3
Conservation :
Page : ‹#›