ECE 1311 Ch9
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Transcript of ECE 1311 Ch9
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ECE 1311
Chapter 9 Sinusoids and Phasors
1
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Outlines
Sinusoids
Phasors
Phasor relationships for circuit elements
Impedance and admittance
Kirchoffslaws in the frequency domain
Impedance and combinations
2
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Introduction
3
Thus far our analysis only concentrates on dc circuits.
Now we begin the analysis in which the source is time
varying (ac circuits).
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Sinusoids
4
A sinusoid is a signal that has the form of the sine andcosine function
Vm= the amplitude of the sinusoid
= the angular frequency of the sinusoid
t= the argument of the sinusoid
tVtv m sin)( Sinusoidal voltage
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Sinusoids
5
Period of the sinusoid (T):
Cyclic frequency (f):
And:
2T Measured in second
2
1
Tf
f 2
Measured in herts (Hz)
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General Expression - Sinusoid
6
Expressed insine
form:
)sin()( tVtv m
t Argument of the sinusoid phase
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Example 1
7
Given the sinusoid, calculate its amplitude, phase,
,period and frequency.
)604sin(5 t
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Sinusoids
8
If two sinusoids are given:
v2starts first in time.
v2leads v1by .
0.
v2and v1are out of
phase.
)sin()(2 tVtv m)sin()(1 tVtv m
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Sinusoids
9
A sinusoid can be expressed in either sine or cosineform.
When comparing two sinusoids, it is easier if both are in
sine or both in cosine forms with positive amplitudes.
This can be achieved by using the Trigonometric
identities:
BABABA
BABABA
sinsincoscos)cos(
sincoscossin)sin(
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Trigonometric Identities
10
It is easy to show using the trigonometric identities that:
tt
tt
tt
tt
sin)90cos(
cos)90sin(
cos)180cos(
sin)180sin(
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Graphical Approach for Sinusoids
11
Alternative approach to trigonometric identities.
Eliminates memorization.
Do not confuse the sine and cosine axes with the axes for complex numberstobe discuss later in this chapter.
.
)90sin(cos tt )180sin(sin tt
Angle:
clockwise+ve counterclockwise-ve
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Example 2
12
Calculate the phase angle between v1and v2.
State which sinusoid is leading.
)50cos(101 tv )10sin(122 tv
Note: when comparing two sinusoids, express them in the same form.
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Graphical Approach for Sinusoids
13
The approach can be used to add two sinusoids of the samefrequency when one is in sine form and the other in cosineform.
For example:
Where:
.
22 BAC
A
B1tan
)cos(sincos tCtBtA
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Example 3
14
Add the two sinusoids
Hence:
Answer:
tt sin4cos3
543 22 C
1.5334tan 1
)13.53cos(5 t
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Phasors
15
Sinusoids are easily expressed in terms of phasors.
A phasor is a complex number that represents the
amplitudeand phaseof a sinusoid,
Phasors are written in boldface.
Before completely define and apply phasors to circuit
analysis, we have to be familiar with complex numbers.
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Complex Number
16
A complex number can be written in rectangular form as:
Where:
x is the real part ofz.
yis the imaginary part ofz.
Also can be written in:
Polar form
Exponential form
jyxz
11
2
jandj
rzjrez
Graphical representation of a complex
numberz.
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Complex Number Conversions
17
Rectangular form to polar form conversion:
Polar form to rectangular form conversion:
Hence:
rztojyxz
22 yxr
x
y1tan
jyxztorz
sincos ryandrx
sincos jrrjyxz
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Example 4
18
Convert the following complex numbers into (a) polarform (b) rectangular form.
ob
ja
3010.
)25.(
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Operations on Complex Numbers
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Example 5
20
Evaluate the following complex numbers.
5301043403510.
*605)41)(25(.
jj
jb
jja
oo
o
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Time Domain to Phasor Domain
Transformation (SinusoidPhasor)
21
)cos()( tVtv m
mVV
Time domain
representation of a sinusoid
Phasor domainrepresentation of a sinusoid
Note: A sinusoid should be expressed in a cosine form before it can be
transformed into a phasor form.
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A Phasor Diagram
22
If two sinusoids are given: and
Then, the phasor diagram is shown below:
mVV mII
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Time Domain to Phasor Domain
Transformation (SinusoidPhasor)
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Example 6
24
Express these sinusoids as phasors.
)1010sin(4.
)402cos(7.
o
o
tib
tva
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Example 7
25
Find the sinusoids corresponding to these phasors.
)125(.
3010.
jjIb
Va o
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Example 8
26
If v1= -10sin(t-30o) V and v
2= 20cos(t+45o) V, find
v=v1+v2.
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Phasors Differentiation and Integration
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Example 9
28
Find the voltage v(t)in a circuit described by theintegrodifferential equation using phasor approach.
)305cos(501052 0 tvdtvdt
dv
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Phasor Relationships for Circuit Elements
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Example 10
30
If voltage v=10cos(100t+30)is applied to a 50F capacitor,
calculate the current through the capacitor.
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Impedance (Z)
31
It is the ratio of the phasor voltage Vto the phasor
current I.
It is measured in ohms ().
Where:
R = Real Z (resistance)
X = Imaginary Z (reactance)
jXRIVZ
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Admittance (Y)
32
It is the reciprocal of impedance.
It is measured in siemens (S).
V
I
ZY 1
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Impedance and Admittance - summary
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Equivalent Circuits at Dc and High
Frequencies
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Example 11
35
Determine v(t)and i(t)for the following circuit.
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Impedance Combinations
36
KCL and KVL both hold true in the frequency (phasor)
domain.
Series Configurations:
ImpedanceZeq=Z1+Z2
Admittance
Voltage division rule:
21
111
YYYeq
VZZ
ZVV
ZZ
ZV
21
22
21
11
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Impedance Combinations
37
Parallel Configurations:
Impedance
AdmittanceYeq=Y1+Y2
Current division rule:
21
111
ZZZeq
IZZ
ZI
IZZ
ZI
21
12
21
21
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Example 12
38
Determine the input impedance of the circuit at=10rad/s.
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Example 13
39
Determine v0.
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Impedance Combinations
40
Delta to Y transformation:
Y to delta transformation:
cba
ba
cba
ca
cba
cb
ZZZ
ZZZ
ZZZ
ZZZ
ZZZ
ZZZ
3
21
3
133221
2
133221
1
133221
Z
ZZZZZZZ
Z
ZZZZZZZ
ZZZZZZZZ
cb
a
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Example 14
41
Determine I.