Chapter 12Phasors 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 2radians, one complete cycle is traced out.

• The velocity of rotation is called the angular velocity ().

= 2f• Instantaneous voltage at any point in time:

v = Vpsin 2f

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 jpart 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.