11Week © Vocational Training Council, Hong Kong.
EEE3405 EEPII-Single Phase AC circuits_01
Example
An alternating voltage is represented by v = 170sin2450t V, find
a) its peak value
b) its frequency
c) its period
d) its values at t = 3.65ms.
Solution:
a) Vm = 170 V
b) 2πf = 2450, f = 390 Hz
c) T = 1/f = 1/390 = 2.56 ms
d) v = 170sin(2450x0.00365) = 78.85 V
12Week © Vocational Training Council, Hong Kong.
EEE3405 EEPII-Single Phase AC circuits_01
Introduction to Phasors
A phasor is a rotating line whose projection on a vertical axis can be used to represent sinusoidal varying quantities.
Complex notation of a.c. quantities
b. Sinusoidal voltage waveform
13Week © Vocational Training Council, Hong Kong.
EEE3405 EEPII-Single Phase AC circuits_01
Shifted Sine Wave
Phasors may be used to represent shifted waveforms.
Angle θ is the position of the phasor at t = 0
Wave MovementWave
Movement
14Week © Vocational Training Council, Hong Kong.
EEE3405 EEPII-Single Phase AC circuits_01
Phase DifferenceIt refers to the angular displacement between different waveforms of the
same frequency.
Vm
Im
ω
Vm
Im
ω
θ
ω
VmIm
θ
(a) v and i in phase
(b) i leads v (c) i lags v
15Week © Vocational Training Council, Hong Kong.
EEE3405 EEPII-Single Phase AC circuits_01
Example 15-19
Given v = 20sin(ωt + 30o) and i = 18sin(ωt - 40o), draw the phasor diagram, determine phase relationships, and sketch the waveforms
Solution:
16Week © Vocational Training Council, Hong Kong.
EEE3405 EEPII-Single Phase AC circuits_01
Complex Number for AC calculation
A Complex number is a number of the form C = a + jb, where
a and b are real numbers and
Complex numbers may be represented geometrically, either in rectangular form or in polar form. (Fig 16-1&2)
°∠= 901j
17Week © Vocational Training Council, Hong Kong.
EEE3405 EEPII-Single Phase AC circuits_01
Conversion between rectangular and polar forms
Since C = a + jb = C∠θ
To convert rectangular to polar form
a
b
baC
1
22
tan−=
+=
θ
To convert polar to rectangular form
a = C cosθ
b = C sinθ
18Week © Vocational Training Council, Hong Kong.
EEE3405 EEPII-Single Phase AC circuits_01
Algebra of complex numbers
� Addition and subtraction – use rectangular form
Suppose C1 = a1 + jb1
C2 = a2 + jb2
C1 ± C2 = (a1 ± a2) + j(b1 ± b2)
� Multiplication and division – use polar form
Suppose C1 = C1∠θ1
C2 = C2 ∠θ2
C1.C2 = C1C2 ∠(θ1+ θ2)
)( 212
1
2
1 θθ −∠=C
CCCCC
CCCC
19Week © Vocational Training Council, Hong Kong.
EEE3405 EEPII-Single Phase AC circuits_01
Example 16-2
Given A = 2 + j1 and B = 1 + j3. Determine their sum and difference.
Solution:
A + B = (2+1) + j (1+3) = 3 + j4
A – B = (2-1) + j(1-3) = 1 – j2
Example 16-3
Given A = 3∠35o and B = 2∠-20o. Determine the product A.B and the quotient A/B.
Solution:
A.B = (3)(2) ∠(35o – 20o) = 6∠15o
A/B = (3/2) ∠[35o – (-20o)] = 1.5∠55o
20Week © Vocational Training Council, Hong Kong.
EEE3405 EEPII-Single Phase AC circuits_01
Representing AC voltages and currents by complex numbers
AC voltages and currents can be represented as phasors and can be viewed as complex numbers. (Fig 16-9)
Fig 16-7 Representation of a sinusoidal source voltage as a complex number.
θ∠=2
EE)b( m
rms
Erms
21Week © Vocational Training Council, Hong Kong.
EEE3405 EEPII-Single Phase AC circuits_01
Transforming source from time domain to phasor domain
E
e(t) = 200sin(ωt + 40o) V
V404.141402
200E °∠=°∠=
22Week © Vocational Training Council, Hong Kong.
EEE3405 EEPII-Single Phase AC circuits_01
Important Notes
1. When voltage and current waveforms are expressed in wave equation, Vm and Im are the peak values.
2. When represents the voltages and currents in phasor form, the magnitudes of them are in rms values.
3. Thus, the phasor V = 120∠0o V means a voltage of 120 V rms at an angle of 0o.
4. To add or subtract sinusoidal voltages and currents, follow the three steps:
• convert sine waves to phasors and express them in complex numbers forms,
• add of subtract the complex numbers,• convert back to time functions if desired.
23Week © Vocational Training Council, Hong Kong.
EEE3405 EEPII-Single Phase AC circuits_01
Summing AC voltages and currents
Sinusoidal quantities can be added or subtracted as phasor sum of transformed sources.
Example 16-6
Given e1 = 10sinωt V and e2 = 15sin(ωt +60o) V. Determine
v = e1 + e2.
Solution:
°∠+°∠=+= 602
150
2
10EEV 21
V6.3641.15
V6061.10007.7
°∠=°∠+°∠=
Thus, v V6.368.21)6.36tsin()41.15(2 °∠=°+ω=