COMPLEX NUMBERS and PHASORS

OBJECTIVES Use a phasor to represent a sine wave. Illustrate phase relationships of waveforms using phasors. Explain what is meant by a complex number. Write complex numbers in rectangular or polar form, and

convert between the two. Perform addition, subtraction, multiplication and division

using complex numbers. Convert between the phasor form and the time domain form

of a sinusoid. Explain lead and lag relationships with phasors and

sinusoids.

Ex.

• For the sinusoid given below, find:

a) The amplitudeb) The phase anglec) The period, and d) The frequency

1050cos12)( ttv

Ex.

• For the sinusoid given below, calculate:

a) The amplitude (Vm)

b) The phase angle () c) Angular frequency ()d) The period (T), and e) The frequency (f)

604sin5)( tti

PHASORS

INTRODUCTION TO PHASORS

• PHASOR:– a vector quantity with:

• Magnitude (Z): the length of vector. • Angle () : measured from (0o)

horizontal. • Written form:

Z

Ex: A<

A

270

0180

90

PHASORS & SINE WAVES

• If we were to rotate a phasor and plot the vertical component, it would graph a sine wave.

• The frequency of the sine wave is proportional to the angular velocity at which the phasor is rotated. ( =2f)

PHASORS & SINE WAVES

• One revolution of the phasor ,through 360°, = 1 cycle of a sinusoid.

t

Z

Z

270

0180

90

ddt

• Thus, the vertical distance from the end of a rotating phasor represents the instantaneous value of a sine wave at any time, t.

INSTANTANEOUS VALUES

Z

270

0180

90

V inst

sin( )instv Z t

t

Z

USE OF PHASORS in EE

• Phasors are used to compare phase differences

• The magnitude of the phasor is the Amplitude (peak)

• The angle measurement used is the PHASE ANGLE,

Ex.

1. i(t) = 3A sin (2ft+30o) 3A<30o

2. v(t) = 4V sin (-60o) 4V<-60o

3. p(t) = 1A +5A sin (t-150o) 5A<-150o

DC offsets are NOT represented.Frequency and time are NOT

represented unless the phasor’s is specified.

GRAPHING PHASORS

• Positive phase angles are drawn counterclockwise from the axis;

• Negative phase angles are drawn clockwise from the axis.

GRAPHING PHASORS

270

0180

90

3A 30

5A -150 4V -60

A

BC

Note:A leads BB leads CC lags Aetc

PHASOR DIAGRAM• Represents one or more sine waves (of the

same frequency) and the relationship between them.

• The arrows A and B rotate together. A leads B or B lags A.

A

270

0

180

90

B

Ex:– Write the phasors for A and B, if wave A is the reference

wave.

B

A

4 V

-4 V

t = 5ms per divisiont = 5ms per division

6.57 VA 04 VB 6.575.2

Ex.

1. What is the instantaneous voltage at t = 3 s, if: Vp = 10V, f = 50 kHz, =0o (t measured from the “+” going zero crossing)

2. What is your phasor?

COMPLEX NUMBERS

COMPLEX NUMBER SYSTEM• COMPLEX PLANE:

X-AxisX-Axis

Re-Re

-j

j

0

90

180

270

FORMS of COMPLEX NUMBERS

• Complex numbers contain real and imaginary (“j”) components.– imaginary component is a real number that has been

rotated by 90o using the “j” operator.• Express in:

– Rectangular coordinates (Re, Im) – Polar (A<) coordinates - like phasors

COORDINATE SYSTEMS– RECTANGULAR:– addition of the real and

imaginary parts:– V R = A + j B

– POLAR:– contains a magnitude and

an angle: – V P = Z<

– like a phasor!

Y-A

xis

X-AxisX-Axis

Y-A

xis

B

A

Z

j

-j

Re-Re

CONVERTING BETWEEN FORMS

• Rectangular to Polar:V R = A + j B to V P = Z<

22 BAZ

AB1tan

Y-A

xis

X-AxisX-Axis

Y-A

xisB

A

Z

j

-j

Re-Re

POLAR to RECTANGULAR

• V P = Z< to V R = A + j B

cosZA

sinZB

Y-A

xis

X-AxisX-Axis

Y-A

xis

B

A

Z

j

-j

Re-Re

MATH OPERATIONS

• ADDITION/ SUBTRACTION - use Rectangular form add real parts to each other, add imaginary parts to

each other; subtract real parts from each other, subtract imaginary

parts from each other • ex:

(4+j5) + (4-j6) = 8-j1 (4+j5) - (4-j6) = 0+j11 = j11

• OR use calculator to add/subtract phasors directly

• MULTIPLICATION/ DIVISION - use Polar form

• Multiplication: multiply magnitudes, add angles;

• Division: divide magnitudes, subtract angles

702)20(5024

202504

308)20(502.4202304

Ex.

• Evaluate these complex numbers:

5342433010 b)

30205040 a) 1/2

jjj