Chapter 12
Phasors and
Complex Numbers
Objectives
• Use a phasor to represent a sine wave
• Use complex numbers to express phasor
quantities
• Represent phasors in two complex forms
• Do mathematical operations with complex
numbers
Introduction to Phasors
• Phasors provide a graphic means for
representing quantities that have both
magnitude and direction (angular position).
• Phasors are especially useful for
representing sine waves in terms of their
magnitude and phase angle and also for
analysis of reactive circuits.
Phasor Representation of a Sine
Wave
The instantaneous value of the sine wave at any
point is equal to the vertical distance from the tip
of the phasor to the horizontal axis.
Phasors and the Sine Wave
Formula
The instantaneous value of the sine wave at any
point is related to both the position and the length
of the phasor.
v = Vpsin
Positive and Negative Phasor
Angles
• The position of a phasor at any instant can
be expressed as a positive angle, measured
counterclockwise from 0.
• Or as a negative angle, measured clockwise
from 0.
• For a given positive angle , the
corresponding negative angle is  - 360.
Phasor Diagrams
• A phasor diagram can be used to show the relative
relationship of two or more sine waves of the
same frequency.
• A phasor in a fixed position is used to represent a
complete sine wave because once the phase angle
between two or more sine waves of the same
frequency is established, the phase angle remains
constant throughout the cycle.
Phasor Diagrams
In the example below, sine wave B leads sine
wave A by 30 and has less amplitude than sine
wave A, as indicated by the the lengths of the
phasors.
Angular Velocity of a Phasor
• When a phasor rotates through 360 or 2
radians, one complete cycle is traced out.
• The velocity of rotation is called the
angular velocity ().
 = 2f
• Instantaneous voltage at any point in time:
v = Vpsin 2f
The Complex Number System
• Complex numbers allow mathematical
operations with phasor quantities and are
useful in analysis of ac circuits.
• With the complex number system, you can
add, subtract, multiply, and divide
quantities that have both magnitude and
angle.
The Complex Plane
• In the complex plane, the horizontal axis is
called the real axis, and the vertical axis is
called the imaginary axis.
• In electrical work, a j prefix is used to
distinguish numbers that lie on the
imaginary axis from those lying on the real
axis.
• This prefix is known as the j operator.
Representing a Point on the
Complex Plane
• Points on the complex plane can be
classified as real, imaginary, or a
combination of the two.
• Real numbers lie on the horizontal axis.
• Imaginary numbers lie on the vertical axis.
Representing a Point on the
Complex Plane
• Combinations lie on the complex plane, with real
and imaginary components.
Value of j
• Mathematically, the value of the j operator
is -1 .
• Therefore, j2 = -1, which is a real number.
• Multiplication of a real number by j2
converts it to a negative real number.
Rectangular and Polar Forms
• Rectangular form and polar form are two
forms of complex numbers that are used to
represent phasor quantities.
• Each form has certain advantages when
used in circuit analysis, depending on the
particular application.
Rectangular Form
A phasor quantity is
represented in
rectangular form as
the sum of real (A)
and imaginary ( jB)
parts as:
A + jB
Polar Form
A Phasor quantity can
be expressed in polar
form, which consists
of the phasor
magnitude (C) and the
angular position (), in
the general form:
C
Conversion from Rectangular to
Polar Form
Converting from rectangular form (A+jB), to polar
form:
C = A2 + B2
 = tan-1(  B/A )
Conversion from Polar to
Rectangular Form
Converting from polar form (C) to rectangular
form:
A = C cos 
B = C sin 
Mathematical Operations
Complex numbers can be added, subtracted,
multiplied, and divided.
Addition of Complex Numbers
Add the real parts of each complex number
to get the real part of the sum. Then add the
j part of each complex number to get the j
part of the sum.
Subtraction of Complex Numbers
Subtract the real parts of the numbers to get
the real part of the difference. Then
subtract the j parts of the numbers to get the
j part of the difference.
Multiplication of Complex
Numbers
• In rectangular form, multiply each term in
one number by both terms in the other
number and then combine the resulting real
and imaginary terms.
• When both numbers are in polar form:
multiply the magnitudes and add the angle
algebraically.
Division of Complex Numbers
• In rectangular form, multiply both the numerator
and the denominator by the complex conjugate of
the denominator, combine the terms and simplify.
• When both numbers are in polar form: Divide the
magnitude of the numerator by the magnitude of
the denominator to get the magnitude of the
quotient. Then subtract the denominator angle
from the numerator angle to get the angle of the
quotient.
Summary
• The angular position of a phasor represents
the angle of the sine wave with respect to a
reference, and the length of a phasor
represents the amplitude.
• A complex number represents a phasor
quantity.
• Complex numbers can be added, subtracted,
multiplied and divided.
Summary
• The rectangular form of a complex number
consists of a real part and an imaginary part
( j ) of the form A + jB.
• The polar form of a complex number
consists of a magnitude and an angle of the
form C.