This document provides an overview of AC fundamentals including: - Definitions of key terms like EMF, direct current, alternating current, sinusoid, angular velocity, frequency, time period, average value, effective value - How electrical power is generated using alternating current - Terminologies related to sinusoidal waveforms like instantaneous value, maximum value, phase difference - Phasor representation of sinusoidal quantities using rotating vectors - Properties of inductors and capacitors and their behavior in AC circuits - Phasor algebra and representation of sinusoidal quantities using complex numbers

1. Reviewof ACfundamentals

2. Keywords and Definitions
• Electro –Motive Force (EMF): Voltage developed
by any source of electrical energy such as
a battery or dynamo
• Direct Current (DC): Steady state value of current
• Alternating Current (AC) : Time varying nature of
current
• Sinusoid : Time varying quantity in the form of
sine function
2

3. Generation of Electrical power
3
One Cycle

4. Terminologies
4
v = Vm sin t
Instantaneous Value (v)
The value of EMF at any instant of time and is represented by lower
case letter v
Maximum Value (Vm )
The highest value that v can have is Vm , known as peak value or
maximum value. This maximum value occurs at an instant when 

5. Terminologies – contd…….
Angular Velocity (ω):
• θ = ωt, is the angle of coil rotation
• angular velocity ω represents the speed (in
rad/sec) at which the coil rotates inside the
magnetic field
• If speed is given in N Rotations Per Minute
(RPM), then ω = 2p N / 60
5

6. Terminologies – contd…….
Frequency (f )
• Frequency of a supply (f) is the number of times a cycle
appears in one second
• Frequency is measured in Hertz.
• It is proportional to the speed of rotation (N)
• Hence ω can also be represented as ω = 2pf
Time period (T)
• Time taken to complete one cycle is called time period
• It is reciprocal of frequency. T= 1/f sec
6

7. Terminologies – contd…….
Average Value
• Any time varying quantity is to be quantised by taking
average of variations
• Sine wave has identical variations in both positive and
negative magnitude in a cycle - symmetrical wave
• For symmetrical wave, as the average in one full cycle
is zero
• Average value is taken only for half a cycle
7

8. Terminologies – contd…….
Effective Value (Or) Root Mean Square Value
• Another way of quantifying the alternating nature
of sinusoidal waveform is the effective value.
• The effective value of an alternating current (AC)
is defined based on its heating effect.
• That value of AC current which would generate
the same amount of heat when it is passed over
the resistance for the given time as that of the
steady dc current through the resistance for the
same time
8

9. Effective Value (Or)
Root Mean Square Value
9

10. Keywords and Definitions
10
• Phasor – rotating vector to represent a sinusoidal signal.
• Power factor – cosine of phase angle between voltage &
current
• Resistance – the property of conductor to oppose the
flow of current
• Inductance - the property of conductor to oppose the
change in flow of current
• Capacitance – the capacity of the conductor to store
electric charges

11. Phase Difference
• The phase difference or phase shift of a
sinusoidal waveform is the angle Φ , in
degrees or radians that the waveform has
shifted from a certain reference point (t=0)
along the horizontal zero axis.
11

12. Phasor Diagram
• A “Phasor” is a rotating vector
- a scaled line whose length represents the
maximum value of the sinusoidal signal and
direction is varying from 0  to 360
12
Reference axis0

Vm

v = Vm Sin t

Vm

0

v = Vm Sin (t + )
Vm

0


v = Vm Sin (t - )
Anti-clockwise rotation (Lead) Clockwise rotation (Lag)

13. Exercise-1
Draw the phasor diagram for v1 =10 sin (t+50)
and v2 = 20 sin ((t -30).
• Solution:
The sinusoid v1 is leading the reference waveform
by 50 and v2 is lagging the reference waveform
by 30.
13

14. -continued
14
According to Ohm’s law
V & I - in phase as there is no phase angle difference
between them

15. -continued
15
Waveform representation Phasor representation
Power Factor: Cosine of the phase angle between voltage and
current .Cos  = Cos 0 = 1 UPF

16. Inductance and Inductor
16
• Property of a conductor that opposes a change in
current.
• Whenever there is current through a conductor,
then a magnetic field (flux) surrounds it.
• As per Faraday’s law, change in magnetic flux
produces induced emf
• Inductance is a measure of a conductor’s ability to
establish an induced emf as a result of a change in
flux.

17. - continued
17
• SI unit is Henry (H)
• Symbolically it is represented as
• The electrical component designed to have the
property of inductance is called an inductor, coil or
choke
A steady DC current through the inductor does not
induce voltage across it .
Inductor acts as short circuit under DC

18. Inductance in AC circuits
18
• As the inductor acts as a short circuit to DC, its
behavior in AC to be analyzed
•Coil of N turns carrying a current of I amps
•Flux linking with the coil (=N) is proportional
to current flow
  I
 = L I
L is known as the self inductance of the coil

19. Initial stored charge
in the inductor i(0) is
assumed to be zero
Voltage and current relationship in inductor
19
m
0
1
i V sinωt dt 
t
L
XL is inductive reactance in 
For sinusoidal applied voltage, v = Vm sin t, the current
flow through the inductor is
mi I sin(ωt-90 ) 
m m
0
V Vcos ωt
i cos ωt
ω ω
 
    
 
t
L L

20. -continued
20
Waveform representation Phasor representation
In a pure inductor, current lags the voltage
exactly by 90 degree. Hence the power
factor is zero lagging (ZPF-lag)

21. Capacitance and Capacitor
21
• The capability to hold an electric charge is
known as capacitance
• Computation of the amount of electrical
energy stored for a given electric potential is
also known as Capacitance
• A capacitor is a charge storing device having two
electrical conductors separated by a dielectric
(non-conducting) medium between the conductors

22. Capacitance and Capacitor
22
• SI unit is Farad and 1F= 1coulomb / volt
• Symbolically, it can be represented as
• In D.C, the Capacitor charges fully to the charging voltage
within a specific time duration
• When it is fully charged, it does not get charged further
and the voltage across the capacitor is equal and opposite
to the applied voltage
In DC, the capacitor behaves as open circuit

23. Voltage induced in Capacitive circuit
23
 
v sin
v
cos
cos
sin( 90)
sin( 90)
m
m
m
m
m
V t
d
V t
dt
i C V t
C V t
I t

 
 
 




 
  mm CVI where
c
mm
m
X
V
C
V
I 
/1 C
Xc

1

XC is capacitive reactance in 

24. -continued
24
Waveform representation Phasor representation
zz
In a pure capacitor, current leads the voltage
exactly by 90 degree. Hence the power factor is
zero leading (ZPF-lead)

25. Keywords and Definitions
25
• Complex plane – Coordinate plane with x axis as real
and y axis as imaginary (j)
• Complex number – a number with real and imaginary
parts (a+jb)
• Complex algebra – algebraic operations in complex
plane
• Cartesian form – representation of complex number as
a+jb
• Polar form - representation of complex number as r

26. Phasor Algebra
26
• We have seen that a sinusoidal quantity can
be represented as a phasor and the
relationship between current, voltage , etc
are discussed for a simple AC circuits, using
the phasor diagram
• Another easy method of analyzing the AC
circuits is using phasor algebra

27. Rotating phasor in the complex plane
27
• According to phasor algebra, a rotating phasor with
magnitude OA making an angle of  with respect to
the horizontal axis, can be resolved into two
components at right angles to each other in the
complex plane
X-axis component =OA cos 
Y-axis component = OA sin 
To represent OC along the y-axis (or) imaginary axis of
the complex plane, operator j is introduced.

28. -continued
28
• Then point A is represented as
• As per Euler’s theorem in complex algebra,
jOAsinθOAcosθ
jbajOCOB


jθ
2 2 -1
jθ
e cosθ jsinθ
a jb OAe
b
where OA a b and θ tan
a
OA θ
 
 
 
    
 
 

29. Significane of ‘j’ operator
29
• The symbol j (or i), when applied to a phasor,
changes its direction by 90 in an CCW direction
without changing its length
• By rotating the phasor A with unit
magnitude by 90 in CCW direction,
it becomes Phasor j1 of same length
along the imaginary.
• Again apply operator j to the phasor
j1, by rotating the phasor CCW by
another 90, it becomes phasor j21

30. -continued
30
• The symbol j2 signifies that we have applied the operator j
twice in successions, there by rotating the phasor through
180.
• This means that the operator j2is equivalent to multiplying the
phasor by -1. Thus j2 = -1 and j =
• Multiplying again by j makes the phasor OA to be rotated
again by 90  and made to occupy in the negative imaginary
axis.
1
When a phasor is multiplied by ‘j’operator, then
the phasor rotates in the CCW direction by 90 

31. Representation of complex number
31
• The various ways of representing the phasor
are
2 2 -1
I a jb (rectangular or cartesian)
A(cosθ jsinθ (trignometric form)
b
where A (a b )and θ tan
a
j
Ae A θ

 
 
 
    
 
  

32. Complex Algebra
32
Let us consider two sinusoidal currents are represented
as 1 m1 2 m2i = I sinωt and i = I sin(ωt + )
zz
It should be noted here that for addition & subtraction of
complex phasors , cartesian form is used and for
multiplication & division polar form is used
The actual meaning of the above is, current i2 leads current i1 by an
angle of .
The sum of two currents will be the vector addition of i1 and i2 and
can be found using the concept of complex algebra.

33. Example 1 – Representation of sinusoidal
quantity as complex number
33
Represent the given sinusoidal quantities in a) Cartesian
and b) Polar form
When a phasor is represented in Cartesian or
polar form, it’s magnitude to be represented as
RMS value
Point to be worth noted:
1 2v 10sin(ωt 50 ) and v 20sin(ωt 30 )     

34. Example 1
14.14
2
20
Vand07.7
2
10
V
2
21

 m
V
RMS value
1
2
Complex polar form V 7.07 50
V 14.14 30
  
   
j7.07-12.24
))30(sin)3014.14(cos(V
41.554.4
jsin50)7.07(cos50VformcartesianComplex
2
1




j
j

35. Example 2 – Addition of two complex
numbers
Find the sum of currents
)60sin(220iandsin210 21
 tti 
20
2
220
Iand10
2
210
I
2
21

 m
I
RMS value
For addition, two sinusoidal currents should be
represented in Cartesian form

36. Example 2
36
7.3211060sin0cos620I
10jsin0)10(cos0I
2
1
jj 

1 2
max
I I I 20 17.32
26.45 40
i I 2 26.45 2 37.43
Instantaneous value of current i 37.43sin(ωt 40 )
j   
  
  
  

37. Example 3 – Divison of two complex
numbers
Find the quotient of two complex numbers given as
V 100 50 and Z 10 15j j   
100 50
111.8 26.6
10 15
18.03 56.3
V j
Z j
 
   
 
  
For divison/multiplication, the complex numbers
should be represented in polar form
111.8
I 26.6 56.3
18.03
6.2 82.9 0.766 6.15
V
Z
j
    
    