This pdf provides detailed and elaborated information on Single-phase and three-phase circuits. Through these notes, you can learn about phasor diagrams, delta connections, star connections, RMS value and Average value of AC signal, RLC circuits and their power factors.
Single - Phase AC Circuits
2.1 Equation for generation of alternating induce EMF
An AC generator uses the principle of Faraday’s electromagnetic induction law. It states that
when current carrying conductor cut the magnetic field then emf induced in the conductor.
Inside this magnetic field a single rectangular loop of wire rotes around a fixed axis allowing
it to cut the magnetic flux at various angles as shown below figure 2.1.
Magnetic Pole
Magnetic Flux Where,
N =No. of turns of coil
A = Area of coil (m2)
ω=Angular velocity (radians/second)
N S m= Maximum flux (wb)
Wire
Wire
Loop(Conductor)
Axis of Rotation
Axis of Rotation Loop(Conductor)
Figure 2.2.1 Generation of EMF
When coil is along XX’ (perpendicular to the lines of flux), flux linking with coil= m. When
coil is along YY’ (parallel to the lines of flux), flux linking with the coil is zero. When coil is
making an angle with respect to XX’ flux linking with coil, = m cosωt [ = ωt].
X
ωt m cosωt
N Y’ Y S
m sinωt
X’
Figure 2.2 Alternating Induced EMF
According to Faraday’s law of electromagnetic induction,
d
e N Em N m
dt Where,
N no. of turns of the coil
( cos t )
e Nd m m Bm A
dt
e N m ( sin t ) Bm Maximum flux density (wb/m 2 )
e N m sin t A Area of the coil (m 2 )
e Em sin t 2f
Bhavesh M Jesadia -EE Department Basic Electrical Engineering (3110005) 1
,2 A.C. Circuits
e N Bm A2f sin t
Similarly, an alternating current can be express as
i I m sin t Where, Im = Maximum values of current
Thus, both the induced emf and the induced current vary as the sine function of the phase
angle t . Shown in figure 2.3.
Phase Induced
N angle emf
90 e e Em sin t
C
135 45
D B
t 00 e0
180 E A 0/360 225 270 315 360
0 45 90 135 180
ωt
t 900 e Em
225 F H 315 t 1800 e 0
G
270
S
t 2700 e Em
t 3600 e 0
Figure 2.3 Waveform of Alternating Induced EMF
2.2 Definitions
Waveform
It is defined as the graph between magnitude of alternating quantity (on Y axis) against time
(on X axis).
+V Sine Wave +V Square Wave
Amplitude
Amplitude
0 0
Time Time
-V -V
+V Triangular +V Complex
Amplitude
Amplitude
Wave Wave
0 0
Time Time
-V -V
Figure 2.4 A.C. Waveforms
Cycle
It is defined as one complete set of positive, negative and zero values of an alternating
quantity.
Bhavesh M Jesadia -EE Department Basic Electrical Engineering (3110005) 2
, 2 A.C. Circuits
Instantaneous value
It is defined as the value of an alternating quantity at a particular instant of given time.
Generally denoted by small letters.
e.g. i= Instantaneous value of current
v= Instantaneous value of voltage
p= Instantaneous values of power
Amplitude/ Peak value/ Crest value/ Maximum value
It is defined as the maximum value (either positive or negative) attained by an alternating
quantity in one cycle. Generally denoted by capital letters.
e.g. Im= Maximum Value of current
Vm= Maximum value of voltage
Pm= Maximum values of power
Average value
It is defined as the average of all instantaneous value of alternating quantities over a half
cycle.
e.g. Vave = Average value of voltage
Iave = Average value of current
RMS value
It is the equivalent dc current which when flowing through a given circuit for a given time
produces same amount of heat as produced by an alternating current when flowing through
the same circuit for the same time.
e.g. Vrms =Root Mean Square value of voltage
Irms = Root Mean Square value of current
Frequency
It is defined as number of cycles completed by an alternating quantity per second. Symbol is
f. Unit is Hertz (Hz).
Time period
It is defined as time taken to complete one cycle. Symbol is T. Unit is seconds.
Power factor
It is defined as the cosine of angle between voltage and current. Power Factor = pf = cos,
where is the angle between voltage and current.
Active power
It is the actual power consumed in any circuit. It is given by product of rms voltage and rms
current and cosine angle between voltage and current. (VI cos).
Active Power= P= I2R = VI cos.
Unit is Watt (W) or kW.
Bhavesh M Jesadia -EE Department Basic Electrical Engineering (3110005) 3
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