L18 1 ph-ac

ELE 102/102 Dept of E&E MIT Manipal 1
Introduction to Phasors
v(wt) = Vm sin wt
v(α) = Vm sin α
ω
α
α ωt
v Vm
0
π 2π
Vm
Vm
-Vm
0
N S
a
b Coil
Rotation
A phasor is a graphical representation of the magnitude and angular
position of a time varying quantity such as a voltage or current in
which variation is sinusoidal.
A phasor diagram can be used to show the relative relationship of two
or more sine waves of the same frequency.

ELE 102/102 Dept of E&E MIT Manipal
The operator ‘j’ :
The operator ‘j’ rotates the given vector by 90 degrees in
anti-clockwise direction.
jAAj =)(
AAjjAj −== 2
)(
1j1;jTherefore, 2
−=−=
Ref
A
jA
Aj2
jAAj3
−=

Phasor representation
Rectangular form:
Trigonometric Form:
Exponential Form:
Polar form:
jXRA ±=
θθ SinAjCosAA |||| ±=
θj
eAA ±
= ||
θ±∠= || AA

Conversion from R ↔ P:






=
+=
−
R
X
tanθ
;XR|A|
1
22
Rectangular to Polar:
Polar to Rectangular:
θ
θ
sin||
cos||
AX
AR
=
=

Conversion from R ↔ P using calculator:
Convert the following into polar form using calculator
j6-83)
j682)
j431
+
+)
36.87-10
36.8710
53.135
∠=
∠=
∠=
°∠−
°−∠
°∠
4510)3
603)2
305)1
07.7j07.7
59.2j5.1
5.2j33.4
−−=
−=
+=
Convert the following into rectangular form using
calculator

Addition & Subtraction of Phasors
Rectangular form is used
Addition:
(R1+jX1)+(R2+jX2) = (R1+R2)+j(X1+X2)
Subtraction:
(R1+jX1)-(R2+jX2) = (R1-R2)+j(X1-X2)

Multiplication & Division of Phasors
Polar form is used
Multiplication:
Division:
[ ] [ ] )(|]|*|[|||*|| 21212211 θθθθ +∠=∠∠ AAAA
)(
||
||
||
||
21
2
1
22
11
θθ
θ
θ
−∠





=





∠
∠
A
A
A
A

A phasor diagram can be used to show the relative
relationship of two or more sine waves of the same
frequency.
Graphical Representation of Phasors
α
Real
A
B
C
O
Imaginary
A phasor can be resolved into two components at right
angles to each other.
OA is a phasor.
Horizontal component = OB
Vertical component OC = AB
OA = OB + jOC = R + jX

Rectangular form:
Polar form:
jXRA +=
θ∠= || AA

Graphical Addition of Phasors
Phasors are represented in Polar form.
Mehod 1:
Method 2:

Graphical Subtraction of Phasors
Phasors are represented in Polar form.
The Second phasor is reversed and added with the first
phasor.

Multiplication and Division are
NOT possible
using graphical method

L18 1 ph-ac

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L18 1 ph-ac