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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