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Text Book of ET-125 Principles of Electrical Engineering DAE 1st Year Electrical Technology
Chapter # 7 AC Fundamentals

UNIT - II
PAPER B
Chapter # 7
AC FUNDAMENTALS
7
...
It starts, say, from zero, grows to a
maximum value, decreases to zero, reverses in direction, reaches a maximum
in the opposite direction, returns again to the original value zero, and repeats
this cycle indefinitely
...


AC current flows with AC voltage source
...
AC voltage periodically reverses
...
In brief, voltage = pressure, and
it is denoted with “V” and measured in volts which is also denoted with “V”
...
2

PRINCIPLE OF WORKING OF AC GENERATOR:

The machines which are used to generate electrical voltages are called
generators
...

The basic principle of an alternator is the principle of electromagnetic
induction
...
It says that whenever there is a relative motion
between the conductor and the magnetic field in which it is kept, an e
...
f
...
The relative motion may exist because of
movement of conductors with respect to magnetic field or movement of
magnetic field with respect to conductor
...
m
...
then can be
used to supply the electrical load
...
3

SIMPLE LOOP ALTERNATOR:

A basic design, called elementary generator, is to have a rectangular
loop armature to cut the lines of force between the north and south poles
...
The
current is sent out of the generator unit through two sets of slip
rings and brushes, one of which is used for each end of the armature
...
To generate an AC output, the
armature is rotated at a constant speed having the number of rotations per
second to match the desired frequency (in hertz) of the AC output
...
Due to the circular motion of the armature
against the straight lines of force, a variable number of lines of force will be
cut even at a constant speed of the motion
...

As the armature arm rotates at a constant speed toward the 90° position, more
lines are cut
...
As it turns toward the
180° position, lesser number of lines of force are cut, giving out lesser voltage
until it becomes zero again at the 180° position
...
Toward
this position, the current is generated on the opposite direction, giving out the
maximum voltage on the opposite side
...
In one rotation, the AC output is produced with
one complete cycle as represented in the sine wave
...
The relation between
speed and frequency is
, where is the frequency in Hz (cycles per
second), is the number of poles (2,4,6
...
The output frequency of an alternator
depends on the number of poles and the rotational speed
...


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Text Book of ET-125 Principles of Electrical Engineering DAE 1st Year Electrical Technology
Chapter # 7 AC Fundamentals

7
...
Let time is measured from the x-axis
...
In time t
seconds, the coil rotates through an angle of θ = ωt, in this deflected position,
the component of the flux which is perpendicular to the plane of the coil is;
Φ =Φm cos ω t
...

According to the Faraday‟s laws of electromagnetic induction the emf
induced in the coil is given by the ratio of the rate of change of flux linkage of
the coil
...
dt (Φm cos ω t) volt
= -NΦmω(- sin ω t) volt
=ωNΦmsin ω t volt
=ωNΦmsinθ volt
--------------- (i)
When the coil has turned through 90º i
...
when θ=90º then sin θ=1,
hence e has maximum value, say Em
...
5

WAVE FORM:

A wave form is a representation of how alternating current (AC) varies
with time
...

There are many different types of electrical wave forms available but
generally they can all be broken down into two distinctive groups
...
Uni-directional Wave forms – these electrical wave forms are always
positive or negative in nature flowing in one forward direction only as they
do not cross the zero-axis point
...

 2
...
Bi-directional
wave forms go through periodic changes in amplitude, with the most
common by far being the Sine-wave
...
The AC in
most homes and offices has an oscillating voltage that produces a sine wave
...
When graphed over time, the
“wave” traced by this voltage of alternating polarity from an alternator takes
on a distinct shape, known as a sine wave as shown below
...

Angle (°)
0
15
30
45
60
75
90
105
120
135
150
165
180

Sin
(Angle)
0
...
2588
0
...
7071
0
...
9659
1
...
9659
0
...
7071
0
...
2588
0
...
0000
-0
...
5000
-0
...
8660
-0
...
0000
-0
...
8660
-0
...
5000
-0
...
0000

Wave
zero
-peak
zero

Square-wave are used extensively in electronic and microelectronic circuits
for clock and timing control signals as they are symmetrical waveforms of
equal and square duration representing each half of a cycle and nearly all
digital logic circuits use square wave waveforms on their input and output
gates
...
Rectangular waveforms are therefore classed as “Nonsymmetrical” waveforms as shown below
...

Although called a triangular waveform, the triangular wave is actually more of
a symmetrical linear ramp waveform because it is simply a slow rising and
falling voltage signal at a constant frequency or rate
...


Sawtooth Waveforms are another type of periodic waveform
...

Sawtooth waveforms can have a mirror image of themselves, by having either
a slow-rising but extremely steep decay, or an extremely steep almost vertical
rise and a slow-decay as shown below
...
The Sawtooth waveform is commonly available from most function
generators and consists of a fundamental frequency ( ƒ ) and all its integer
ratios of harmonics, such as: 1/2, 1/4, 1/6 1/8 … 1/n etc
...


7
...


TIME PERIOD:
The time period of a waveform is the time required for completing one
full cycle
...
In the figure shown the sinusoidal
waveform is plotted as a function of the argument ωt, and the periodic nature
of the sine wave is evident
...
The relationship between time (T) and
frequency (f) is indicated by the formulas
T = 1/ f

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Text Book of ET-125 Principles of Electrical Engineering DAE 1st Year Electrical Technology
Chapter # 7 AC Fundamentals

FREQUENCY:
The number of cycles per second is called frequency of the alternating
quantity
...
If the signal
in the Figure shown above makes one complete revolution each second, the
Generator produces one complete cycle of AC during each second (1 Hz)
...

f
=1 / T
For a generator frequency can be calculated by the given formula;
f
= PN / 120;
Where P is the number of magnetic poles, N is speed of rotation of
generator in revolution per minute
...
This is known as the peak or crest value of an AC waveform:
Figure below shows the peak voltage of a waveform
...
So peak-to-peak voltage is just
the full vertical length of a voltage waveform from the very top to the very
bottom
...
The value may be zero if the particular instant
is the time in the cycle at which the polarity of the voltage is changing
...

There are actually an infinite number of instantaneous values between zero
and the peak value
...
They are actually DC values
...
Since the voltage
increases from zero to peak value and decreases back to zero during one
alternation, the average value must be some value between those two limits
...
The
computation would show that one alternation of a sine wave has an average
value equal to 0
...
The formula for average voltage is;
Vavg =
0
...
Similarly, the formula for average current is;
Iavg
=
0
...


EFFECTIVE / RMS VALUE:
This is the value of AC signal that will have the same effect on a
resistance as a comparable value of direct voltage or current will have on the
same resistance
...
For this reason, the effective value is often called
the “root mean square” (RMS) value
...
The effective or rms value (IefforIrms) of a sine wave
of current is 0
...

Thus, Ieff/ Irms = 0
...
Importantly, all AC voltage and current
values are given as effective values
...
It identifies the ratio of
the direct current of equal power relative to the given alternating current
...
707I max
0
...
11

PEAK / CREST FACTOR:
Peak / Crest factor is the peak amplitude of the waveform divided by
the RMS value of the wave form
...
707I max

=

1
0
...
414

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Text Book of ET-125 Principles of Electrical Engineering DAE 1st Year Electrical Technology
Chapter # 7 AC Fundamentals

7
...
The length of the vector represents the magnitude (or amplitude) of the
waveform, like this: (Figure below)

Vector length represents magnitude of waveform of AC quantity, thus
greater the amplitude of the waveform, the greater the length of its
corresponding vector
...
Usually, when the phase of a waveform in a
circuit is expressed, it is referenced to the power supply voltage waveform
(arbitrarily stated to be “at” 0o)
...
(Figure below)

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Text Book of ET-125 Principles of Electrical Engineering DAE 1st Year Electrical Technology
Chapter # 7 AC Fundamentals

Vector angle is the phase with respect to another waveform
...
8

PHASE:

Phase represent the live wire / positive terminal where the
electricity passes
...


PHASE DIFFERENCE:
Sometimes when we are analyzing alternating waveforms we may
need to know the position of the Phasor, representing the Alternating
Quantity at some particular instant in time especially when we want to
compare two different waveforms on the same axis, for example voltage and
current
...
But if
a second waveform starts to the left or to the right of this zero-point or we
want to represent in Phasor notation the relationship between the two

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Text Book of ET-125 Principles of Electrical Engineering DAE 1st Year Electrical Technology
Chapter # 7 AC Fundamentals

waveforms then we will need to take into account this phase difference, Φ of
the waveform
...


The generalized mathematical expression to define these two
sinusoidal quantities will be written as:

The current i, is lagging the voltage v, by angle Φ and in our example
above this is 30o
...


IN PHASE:
The phase involves the relationship between the position of the
amplitude crests and troughs of two waveforms
...
If the peaks of two signals with the same frequency
are in exact alignment at the same time, they are said to be in phase
...


OUT OF PHASE:
The phase involves the relationship between the position of the
amplitude crests and troughs of two waveforms
...
If the peaks of two signals with the same frequency
are not in exact alignment at the same time, they are said to be out of phase
...
As both
alternating quantities rotate at the same speed, i
...
they have the same
frequency, this phase difference will remain constant for all instants in time,
then the phase difference of 30o between the two quantities is represented by
phi, Φ as shown below
...


ANTI PHASE:
If the peaks of two waves with the same frequency are 180 degree
(Half wave cycle) out of phase, they are said to be in anti-phase
...
e
...
This is shown in Figure, where there is a phase difference of 30°
between the waveforms A and B
...

When the waveform A is ahead of B (i
...
, when A reaches its maximum value
before B reaches its maximum value), it is said to be leading waveform B
...

Note that when A is leading B, it also reaches its minimum value and zero
value before B reaches those values
...

To compare the phase between two sine wave, we should follow the
conditions below,





7
...

The two signals should have the same frequency
Both the signal must be written in positive amplitude
The phase angle must always be less than 180°
...

Sinusoidal waveforms of the same frequency can have a Phase Difference
between themselves which represents the angular difference of the two
sinusoidal waveforms
...

But when presented mathematically in this way it is sometimes difficult to
visualize this angular or phasor difference between two or more sinusoidal
waveforms
...

Basically, a rotating vector, simply called a “Phasor” is a scaled line
whose length represents an AC quantity that has both magnitude (“peak
amplitude”) and direction (“phase”) which is “frozen” at some point in time
...

Generally, vectors are assumed to pivot at one end around a fixed zero
point known as the “point of origin” while the arrowed end representing the
quantity, freely rotates in an anti-clockwise direction at an angular velocity,
(ω) of one full revolution for every cycle
...
Likewise, a clockwise rotation is
considered to be a negative rotation
...
In both cases the
phase angle and direction remain the same
...
A complete sine wave can be constructed by a single
vector rotating at an angular velocity of ω = 2πƒ, where ƒ is the frequency of
the waveform
...

Generally, when constructing a phasor diagram, angular velocity of a
sine wave is always assumed to be: ω in rad/sec
...
10

COMPLEX NUMBERS:

Complex numbers are that which consists two parts; a real number and
an imaginary number
...
Technically any
real number or imaginary number can be considered as complex number
...
Real numbers are tangible values that can be
plotted on horizontal number line, such as fractions, integers or any countable
number that you can think of
...
Two forms of
complex numbers which are used frequently in calculation to solve numerical
problems by an electrical technologist or engineer are Polar and Rectangular
forms
...
Here are two examples of vectors and their polar notations:
(Figure below)

Standard orientation for vector angles in AC circuit calculations
defines 0o as being to the right (horizontal), making 90o straight up, 180o to
the left, and 270o straight down
...
For example, a vector angled ∠ 270o (straight
down) can also be said to have an angle of -90o
...
81 ∠ 230
...
81 ∠ -129
...


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Text Book of ET-125 Principles of Electrical Engineering DAE 1st Year Electrical Technology
Chapter # 7 AC Fundamentals

RECTANGULAR FORM OF AC QUANTITY:
Rectangular form, on the other hand, is where a complex number is
denoted by its respective horizontal and vertical components
...
Rather than describing a
vector's length and direction by denoting magnitude and angle, it is described
in terms of “how far left/right” and “how far up/down
...
In order to distinguish the horizontal
and vertical dimensions from each other, the vertical is prefixed with a lowercase “i” (in pure mathematics) or “j” (in electronics)
...
As a complete complex number, the horizontal
and vertical quantities are written as a sum: (Figure below)

In “rectangular” form the vector's length and direction are denoted in
terms of its horizontal and vertical span, the first number representing the
horizontal (“real”) and the second number (with the “j” prefix) representing
the vertical (“imaginary”) dimensions
...
The
vertical component is referred to as the imaginary component, since that
dimension lies in a different direction, totally alien to the scale of the real
numbers
...
The “imaginary”
axis of the graph corresponds to another number line situated at 90o to the
“real” one
...

Either method of notation is valid for complex numbers
...


7
...
To convert from polar to rectangular, find the real component
by multiplying the polar magnitude by the cosine of the angle, and the
imaginary component by multiplying the polar magnitude by the sine of the
angle
...
87º into rectangular form?
SOLUTION:

5  36
...
87º) = 4
(5) (sin 36
...
33
Real component
-5 Sin 30º = -2
...
33-j2
...
5 Real component
15 Sin (-60º) = -13 Imaginary component
7
...
87°
Z = r  = 5 36
...
87º
 
Polar form = 10 < -36
...


2
...

4
...

6
...


8
...

10
...

12
...

14
...

(a) 1
(b) 2
(c) 4
(d) None
Frequency in Pakistan is:
(a) 25 c/s
(b) 40 c/s
(c) 50 c/s
(d) 60 c/s
The sign wave which complete 50 cycles per second, its frequency
will be
(a) 25c/s
(b) 50c/s
(c) 75c/s
(d) 100c/s
One complete set of positive and negative values of an alternating
quantity is known as;
(a) Waveform
(b) Cycle
(c) Amplitude (d) Phasor
In single loop generator when coil is parallel to lines of force, then
induced emf
...
M
...
value
R
...
S
...
707Im
(b) 7
...
636Im (d) 6
...
36Im
(b) 0
...
07Im
(d) 0
...
The time period of a sine wave of frequency of 50 c/s is:
(a) 5 sec
(b) 0
...
Sine wave equation of current can be written as:
(a) i = Im sin  (b) i = Im sin t (c) I = Im sin 2ft (d) All
17
...
M
...
Form factor =
(a) Average value / RMS
...
M
...
value / R
...
S value
(d) Max
...
The ratio between R
...
S value and Average value is called:
(a) Peak factor
(b) Form factor
(c) Crest factor
(d) Amplitude factor
20
...
11
(c) 1
...
707
21
...
value
(b) R
...
S value / average value
(c) Max
...
M
...
value / Average value
22
...
M
...
C Wave is
called:
(a) Peak factor
(b) Amplitude factor
(c) Crest factor
(d) All
23
...
11
(c) 1
...
707
24
...
M
...
07
(b) 10
(c) 14
...
28
25
...
If peak value of sine wave is 10A then its average value will be:
(a) 0
...
707
(c) 6
...
07
27
...
The line which has is specific direction and length is called:
(a) Scalar
(b) Vector
(c) Phaser
(d) Both b & c
29
...
The waveform which is behind the other waveform w
...
t
...
The waveform which is ahead the other waveform w
...
t
...
This is an example of rectangular form
...
This is an example of polar form
...

6
...

16
...

26
...


c
b
b
d
c
a
b

2
...

12
...

22
...

32
...

8
...

18
...

28
...


b
a
a
b
c
c
b

4
...

14
...

24
...


b
a
b
b
a
a

5
...

15
...

25
...


c
c
b
b
a
a

PART-B
Give the short answer of the following questions
1
...

State the working principles of Ac generator
...

The speed of 2 pole generator is 3000 r
...
m
...

4
...
m
...
& current
...

Write the names of 4 types of waveforms
...

Define Square Wave
...

What is saw tooth wave?
8
...

Define cycle
...

What is time period?
11
...

12
...

13
...
C
...

Define Average value of A
...

15
...

17
...

19
...

21
...

23
...


Define the from factor
...

What is phase and phase angle?
What is difference between in phase & out of phase
...
C
...

Define Phase difference
...

Convert 3030 into rectangular form
...

1)
2)
3)
4)
5)
6)
7)

Explain the construction and working principle of simple single loop
AC Generator
...
C Generator, Number of poles
and frequency, explained
...

Derive the equation of sine wave
...

Define lagging and leading values of A
...
explain it with wave form
and vector diagrams
...


PART-D
SOLVE THE PROBLEMS
Q
...

Ans
...
2:

13  67
...

Q
...
4  51
...
42Convert it into polar form
...


47
...
9º

Q
...


Convert 14
...
57º into (Rectangular) form
...
2

Q
...
749  -32
...


25  j16
...


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Text Book of ET-125 Principles of Electrical Engineering DAE 1st Year Electrical Technology
Chapter # 7 AC Fundamentals

Q
...

Q
...
13º into Vector form
...


Ans
...
31  66
...
8:

Subtract „10 + j30‟ from 20  j20 and convert into polar form
...


50
...
69º

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