2. ALTERNATING
CURRENT AND VOLTAGE
The sinusoidal waveform
• The sinusoidal waveform or sine wave is the fundamental type of altern
ating current (ac) and alternating voltage. It is also referred to as a sinu
soidal wave, or, simply, sinusoid.
• Sine waves, or sinusoids, are produced by two types of sources: rotatin
g electrical machines (ac generators) or electronic oscillator circuits, whi
ch are used in instruments commonly known as electronic signal gener
ators.
5. ALTERNATING
CURRENT AND VOLTAGE
Frequency of a sine wave
Frequency is the number of cycles that a sine wave completes in one second.
The more cycles completed in one second, the higher the frequency. Frequency (f )
is measured in units of hertz (Hz). One hertz is equivalent to one cycle per second;
for example, 60 Hz is 60 cycles per second.
The formulas for th
e relationship betw
een frequency and
period are
6. ALTERNATING
CURRENT AND VOLTAGE
Voltage and current values of sine waves
There are five ways to express and measure the value of a sine wave in te
rms of its voltage or its current magnitude:
• instantaneous,
• peak,
• peak-to-peak,
• rms,
• average values
7. ALTERNATING
CURRENT AND VOLTAGE
Voltage and current values of sine waves
Instantaneous Value
Figure 11 illustrates that at any point in time on a sine wave, the voltage (or current) has an
instantaneous value. This instantaneous value is different at different points along the
curve. Instantaneous values are positive during the positive alternation and negative duringthe
negative alternation. Instantaneous values of voltage and current are symbolized by
lowercase v and i, respectively.
8. ALTERNATING
CURRENT AND VOLTAGE
Peak Value
The peak value of a sine wave is the value of voltage (or current) at the positive or
the negative maximum (peaks) with respect to zero. Since positive and negative pe
ak values are equal in magnitude, a sine wave is characterized by a single peak val
ue, as is illustrated in Figure 12.
For a given sine wave, the peak
value is constant and is represented
by Vp or Ip. The maximum or peak
value of a sine wave is also called its
amplitude. The amplitude is measured
from the 0 V line to the peak. In the
figure, the peak voltage is 8 V, which is
also its amplitude.
9. ALTERNATING
CURRENT AND VOLTAGE
Peak-to-Peak Value
The peak-to-peak value of a sine wave, as illustrated in Figure 13, is the voltage (o
r current) from the positive peak to the negative peak. It is always twice the peak
value as expressed by the following equations. Peak-to-peak values are represente
d by Vpp or Ipp.
10. ALTERNATING
CURRENT AND VOLTAGE
RMS Value
The term rms stands for root mean square. The rms value, also referred to as the effective
value, of a sinusoidal voltage is actually a measure of the heating effect of the sine wave.
The rms value of a sinusoidal voltage is equal to the dc voltage that produces the
same amount of heat in a resistance as does the sinusoidal voltage.
11. ALTERNATING
CURRENT AND VOLTAGE
Average value
The average value of a sine wave when taken over one complete cycle is always ze
ro because the positive values (above the zero crossing) offset the negative values
(below the zero crossing). To be useful for comparison purposes and in determinig
the average value of a rectified voltage such as found in power supplies, the avera
ge value of a sine wave is defined over a half-cycle rather than over a full cycle. Th
e average value is expressed in terms of the peak value as follows for both voltage
and current sine waves:
12. ALTERNATING
CURRENT AND VOLTAGE
Angular measurement of a sine wave
A degree is an angular measurement corresponding to 1/360 of a circle or a complete revoluti
on. A radian (rad) is the angle formed when the distance along the circumference of a circle is
equal to the radius of the circle. One radian is equivalent to 57.3°, as illustrated in Figure 17. In
a 360° revolution, there are 2p radians.
The Greek letter (pi) represents the ratio of the circumference of any circle to its
diameter and has a constant value of approximately 3.1416.
15. ALTERNATING
CURRENT AND VOLTAGE
Phase of a sine wave
The phase of a sine wave is an angular measurement that specifies the position of
that sine wave relative to a reference. When the sine wave is shifted left or right w
ith respect to this reference, there is a phase shift.
16. ALTERNATING
CURRENT AND VOLTAGE
Polyphase power
One important application of phase-shifted sine waves is in electrical power syste
ms. Electrical utilities generate ac with three phases that are separated by 120° as
shown in Figure 23. The reference is called neutral. Normally, three-phase power is
delivered to the user with Four lines—three hot lines and neutral. There are import
ant advantages to three-phase power for ac motors. Three-phase motors are more
efficient and simpler than an equivalent singlephase motor.
17. ALTERNATING
CURRENT AND VOLTAGE
The sine wave formula
A-amplitude, is the maximum value of the voltage or current on the vertical axis.
y-is an instantaneous value representing either voltage or current.
θ-is a angle.
18. ALTERNATING
CURRENT AND VOLTAGE
The sine wave formula
For example, a certain voltage sine wave has a peak value of 10 V. The instantaneo
us voltage at a point 60° along the horizontal axis can be calculated as follows, wh
ere y = v and A= Vp:
19. ALTERNATING
CURRENT AND VOLTAGE
Expressions for phase-shifted sine waves
When a sine wave is shifted to the right of the reference (lags) by a certain angle φ, (Greek
letter phi) as illustrated in Figure 29(a), the general expression is
When a sine wave is shifted to the left of the reference (leads) by a certain angle φ, as shown
in Figure 29(b), the general expression is
21. ALTERNATING
CURRENT AND VOLTAGE
Analysis of ac circuits
If VDC is greater than the peak value of the sinusoidal voltage, the combined ac and dc
voltage is a sine wave that never reverses polarity and is therefore nonalternating. That is,
the sine wave is riding on a dc level, as shown in Figure 36(a).
22. ALTERNATING
CURRENT AND VOLTAGE
Single and three phase systems
A single phase system consists of just two conductors (wires): one is called the pha
se (sometimes line, live or hot), through which the current flows and the other is c
alled neutral, which acts as a return path to complete the circuit.
In a three – phase system, we have a minimum of three conductors or wires carryi
ng AC voltages. It is more economical to transmit power using a 3 – phase power
supply when compared to a single phase power supply as a three – phase supply
can transmit three times the power with just three conductors when compared to
a two – conductor single – phase power supply.
Single-phase is 230 to 240 volts
Three-phase 400 or 415 volts
23. ALTERNATING
CURRENT AND VOLTAGE
Star Connection
The star connection requires four wires in which there are three phase conductors
and one neutral. Due to its shape, the star connection is sometimes also called as
Y or Wye connection.
The star connected three phase systems gives two different voltages, i.e., the 230
V and 440V. The voltage between the single phase and the neutral is 230V, and th
e voltage between the two phases is equal to the 440V.
24. ALTERNATING
CURRENT AND VOLTAGE
Delta Connection
The delta connection has three wires, and there is a no neutral point. The delta co
nnection is shown in the figure below. The line voltage of the delta connection is e
qual to the phase voltage.
25. ALTERNATING
CURRENT AND VOLTAGE
Connection of Loads in Three Phase System
The three phase load may be balanced or unbalanced. If the three loads (impedances) Z1, Z2 a
nd Z3 has the same magnitude and phase angle then the three phase load is said to be a bal
anced load. The balance system is one in which the load are equally distributed in all the three
phases of the system. The magnitude of voltage remains same in all the three phases and it is
separated by an angle of 120º.
In the unbalance system the magnitude of voltage in all the three phases becomes different.
26. ALTERNATING
CURRENT AND VOLTAGE
Difference Between Star and Delta Connection
1. The terminals of the three branches are connected to a common point. The net
work formed is known as Star Connection. The three branches of the network a
re connected in such a way that it forms a closed loop known as Delta Connect
ion.
2. In a star connection, the starting and the finishing point ends of the three coils
are connected together to a common point known as the neutral point. But in
Delta connection, there is no neutral point. The end of each coil is connected t
o the starting point of the other coil that means the opposite terminals of the c
oils are connected together.
3. In star connection, the line current is equal to the phase current, whereas in del
ta connection the line current is equal to root three times of the phase current.
4. In Star connection, the line voltage is equal to root three times of the phase vol
tage, whereas in delta connection line voltage is equal to the phase voltage.
27. ALTERNATING
CURRENT AND VOLTAGE
Difference Between Star and Delta Connection
5. The speed of the star-connected motors is slow as they receive 1/√3 of the volt
age but the speed of the delta connected motors is high because each phase gets
the total of the line voltage.
6. In star connection, phase voltage is low as 1/√3 times the line voltage, whereas
in delta connection phase voltage is equal to the line voltage.
7. Star connections are mainly required for the Power Transmission Network for lo
nger distances, whereas in delta connection mainly in Distribution networks and is
used for shorter distances.
8. In star connection, each winding receives 230 volts and in delta connection, eac
h winding receives 415 volts.
9. Both 3 phase 4 wire and 3 phase 3 wire systems can be derived in the star conn
ection, whereas in Delta Connection only 3 phase 4 wire systems can be derived.
10. The amount of insulation required in star connection is low and in delta conne
ction high insulation level is required.
28. ALTERNATING
CURRENT AND VOLTAGE
Alternators (ac generators)
An alternator is an ac generator that converts energy of motion into electrical energy.
Alternator, which generates ac voltage based on the principle of electromagnetic induction tha
t produces a voltage when there is relative motion between a magnetic field and a conductor.
29. ALTERNATING
CURRENT AND VOLTAGE
Alternators (ac generators)
Frequency
where f is the frequency in hertz, N is the number of poles, and s is the rotational
speed in revolutions per minute.
30. ALTERNATING
CURRENT AND VOLTAGE
Practical alternators
Rotating-Armature Alternators
In a rotating-armature alternator, the magnetic field is
stationary and is supplied by permanent magnets or electromagnets operated from dc. With
electromagnets, field windings are used instead of permanent magnets and provide a fixed
magnetic field that interacts with the rotor coils. Power is generated in the rotating assembly
and supplied to the load through the slip rings.
In a rotating-armature alternator, the rotor is the component from which power is taken.
In addition to hundreds of windings, the practical rotating-armature alternator usually has
many pole pairs in the stator that alternate as north and south poles, which serve to increase
the output frequency.
31. ALTERNATING
CURRENT AND VOLTAGE
Practical alternators
Rotating-Field Alternator
The rotating-armature alternator is generally limited to low power applications because all out
put current must pass through the slip rings and brushes.
To avoid this problem, rotating-field alternators take the output from the stator coils and use
A rotating magnet, hence the name. Small alternators may have a permanent magnet for a
rotor, but most use an electromagnet formed by a wound rotor. A relatively small amount of
dc is supplied to the rotor (through the slip rings) to power the electromagnet. As the rotatin
g magnetic field sweeps by the stator windings, power is generated in the stator. The stator is
therefore the armature in this case.
33. ALTERNATING
CURRENT AND VOLTAGE
AC motors
An induction motor is so named because a magnetic field induces current in the r
otor, creating a magnetic field that interacts with the stator field. Normally, there is
no electrical connection to the rotor, so there is no need for slip rings or brushes,
which tend to wear out. The rotor current is caused by electromagnetic induction,
which also occurs in transformers, so induction motors are said to work by transfo
rmer action.
In a synchronous motor, the rotor moves in sync (at the same rate) as the rotatin
g field of the stator. Synchronous motors are used in applications where maintaini
ng constant speed is important. Synchronous motors are not self-starting and mus
t receive starting torque from an external source or from built-in starting windings.
Like alternators, synchronous motors use slip rings and brushes to provide current
to the rotor.
35. ALTERNATING
CURRENT AND VOLTAGE
Rotating stator field
How can the magnetic field in the stator rotate if the stator itself does not mo
ve?
When phase 1 is at 90°, the current in the phase 1 winding is at a maximum and c
urrent in the other windings is smaller. Therefore, the stator magnetic field will be
oriented toward the phase-1 stator winding. As the phase-1 current declines, the p
hase-2 current increases, and the field rotates toward the phase-2 winding. The m
agnetic field will be oriented toward the phase-2 winding when current in it is a m
aximum. As the phase-2 current declines, the phase-3 current increases, and the fi
eld rotates toward the phase-3 winding. The process repeats as the field returns to
the phase-1 winding. Thus, the field rotates at a rate determined by the frequency
of the applied voltage. With a more detailed analysis, it can be shown that the ma
gnitude of the field is unchanged; only the direction of the field changes.
As the stator field moves, the rotor moves in sync with it in a synchronous motor
but lags behind in an induction motor. The rate the stator field moves is called the
synchronous speed of the motor.
36. ALTERNATING
CURRENT AND VOLTAGE
Induction motors
Operation of an Induction Motor When the magnetic field from the stator moves across the squirrel cage of t
he inductor, a current is generated in the squirrel cage. This current creates a magnetic field that reacts with t
he moving field of the stator, causing the rotor to start turning. The rotor will try to “catch-up” with the movin
g field, but cannot, in a condition known as slip. Slip is defined as the difference between the synchronous sp
eed of the stator and the rotor speed. The rotor can never reach the synchronous speed of the stator field be
cause, if it did, it would not cut any field lines and the torque would drop to zero. Without torque, the rotor c
ou d not turn itself.
Initially, before the rotor starts moving, there is no back
emf, so the stator current ishigh. As the rotor speeds up,
it generates a back emf that opposes the stator current.
As themotor speeds up, the torque produced balances the
load and the current is just enough to keep the rotor
turning. The running current is significantly lower than the
initial start-up current because of the back emf. If the load
on the motor is then increased, the motor will slow down
and generate less back emf. This increases the current to the
motor and increases the torque it can apply to the load. Thus,
an induction motor can operate over a range of speeds and
torque. Maximum torque occurs when the rotor is spinning at
About 75% of the synchronous speed.
37. ALTERNATING
CURRENT AND VOLTAGE
Induction motors
What is a self-starting motor?
When the motor starts running automatically without any external force applied to
the machine, then the motor is referred to as ‘self-starting’.
Single phase induction motor is not a self-starting motor but three phase is a self-
starting motor.
A single-phase induction motor is not self-starting because it lacks a rotating mag
netic field in the stator that is essential for the motor to start and develop torque.
Unlike a three-phase induction motor, which naturally produces a rotating magneti
c field, a single-phase induction motor requires additional mechanisms to initiate r
otation.
The absence of a rotating magnetic field in a single-phase motor is primarily due t
o the nature of the single-phase AC power supply. A single-phase power supply pr
oduces a sinusoidal voltage waveform that alternates in polarity but does not crea
te a rotating magnetic field on its own. As a result, the single-phase induction mot
or requires assistance to overcome the initial inertia and initiate rotation.
38. ALTERNATING
CURRENT AND VOLTAGE
Induction motors
Why is Three Phase Induction Motor Self Starting?
In a three phase system, there are three single phase lines with a 120° phase difference. So th
e rotating magnetic field has the same phase difference which will make the rotor to move.
This inherent characteristic of a three-phase motor allows it to start without requiring any addi
tional mechanisms or external assistance.
39. ALTERNATING
CURRENT AND VOLTAGE
Types of induction motors
Single Phase Induction Motor
• Split Phase Induction Motor
• Capacitor Start Induction Motor
• Capacitor Start and Capacitor Run Induction Motor
• Shaded Pole Induction Motor
Three Phase Induction Motor
• Squirrel Cage Induction Motor
• Slip Ring Induction Motor
40. ALTERNATING
CURRENT AND VOLTAGE
Types of induction motors
Split-Phase Induction Motor: This type of motor uses a special winding configuration that cre
ates a phase shift between two windings. The main winding carries the majority of the current,
while the auxiliary winding, with higher reactance, creates a phase shift. The phase difference g
enerates a rotating magnetic field during starting, which allows the motor to overcome inertia
and initiate rotation.
Capacitor-Start Induction Motor: In this motor, an auxiliary winding and a capacitor are adde
d to create a phase shift. The capacitor provides a leading current, resulting in a phase differen
ce between the main winding and auxiliary winding. Once the motor reaches a certain speed,
a centrifugal switch disconnects the auxiliary winding and capacitor, allowing the motor to con
tinue running with the main winding alone.
Capacitor-Start Capacitor-Run Induction Motor: This motor uses two capacitors, one for start
ing and the other for running. The starting capacitor provides an initial phase shift, while the r
unning capacitor remains connected during operation to improve the motor's performance an
d efficiency.
A shaded pole induction motor: This motor utilizes a shading coil or a shaded pole to provid
e the starting torque. It is a simple and economical motor design widely used in applications r
equiring low power output.
41. ALTERNATING
CURRENT AND VOLTAGE
Synchronous motors
Essentially, the rotating stator field of the synchronous motor is identical to that of
an induction motor. The primary difference in the two motors is in the rotor. The i
nduction motor has a rotor that is electrically isolated from a supply, and the sync
hronous motor uses a magnet to follow the rotating stator field. Small synchronou
s motors use a permanent magnet for the rotor; larger motors use an electromagn
et. When an electromagnet is used, dc is supplied from an external source via slip
rings as in case of the alternator.
