circuits ii sinusoid&phasors
Post on 07-Aug-2018
221 Views
Preview:
TRANSCRIPT
-
8/20/2019 Circuits II Sinusoid&Phasors
1/38
SINUSOID & PHASORS
-
8/20/2019 Circuits II Sinusoid&Phasors
2/38
SINUSOID
– is a signal that has the form of sine orcosine function. A sinusoidal current is usually referred to as
alternating current (ac ). Such a current reverses atregular time intervals and has alternately positiveand negative values.
Sinusoids is important for a number of reasons:1. Nature itself is characteristically sinusoidal.
Example, the sinusoidal variation in the motionof a pendulum, the vibration of a string, theripples on the ocean surface, the political eventsof a nation, the economic fluctuations of thestock market
-
8/20/2019 Circuits II Sinusoid&Phasors
3/38
2. A sinusoidal signal is easy to generate andtransmit.
It is the form of voltage generated worldwideand supplied to homes, factories, laboratories,business establishment and so on. It is the
dominant form of signal in the communicationsand electric power industries.3. Any practical periodic signal can be represented
by a sum of sinusoids. Therefore, play animportant role in the analysis of periodic
signals.4. Easy to handle mathematically. The derivative
and integral of a sinusoid are themselvessinusoids.
-
8/20/2019 Circuits II Sinusoid&Phasors
4/38
GENERATING P LANT TO E ND -U SERS
-
8/20/2019 Circuits II Sinusoid&Phasors
5/38
V ARIOUS SOURCES OF AC POWER :
1. Generating Plant
Masinloc Coal-fired Power PlantLocated in Zambales (660 MW)
-
8/20/2019 Circuits II Sinusoid&Phasors
6/38
V ARIOUS SOURCES OF AC POWER :
1. Generating Plant
Geothermal Power Plant Located inMindanao (55 MW)
-
8/20/2019 Circuits II Sinusoid&Phasors
7/38
V ARIOUS SOURCES OF AC POWER :
1. Generating Plant
Binga Hydroelectric Power PlantLocated in Benguet (125.8 MW)
-
8/20/2019 Circuits II Sinusoid&Phasors
8/38
V ARIOUS SOURCES OF AC POWER :
2. Portable AC Generator
-
8/20/2019 Circuits II Sinusoid&Phasors
9/38
V ARIOUS SOURCES OF AC POWER :
3. Wind Turbine Power Station
-
8/20/2019 Circuits II Sinusoid&Phasors
10/38
V ARIOUS SOURCES OF AC POWER :
4. Solar Power System
-
8/20/2019 Circuits II Sinusoid&Phasors
11/38
V ARIOUS SOURCES OF AC POWER :
5. Function Generator
-
8/20/2019 Circuits II Sinusoid&Phasors
12/38
SINUSOIDS
180
90
0
3
2
1
2
( ) or ( )v t i t
mV
t 90 180 270
2
0
r y
-
8/20/2019 Circuits II Sinusoid&Phasors
13/38
( ) sin and
( ) sin
m
m
v t V t
v t V t
the of the sinusoid
the in radians/sthe of the sinusoid
mV amplitude
angular frequencyt argument
Mathematically, a sinusoidal voltage
where:
-
8/20/2019 Circuits II Sinusoid&Phasors
14/38
The sinusoid is shown in the figure as afunction of its argument and as a function oftime. It is evident that the sinusoid repeats itselfevery T second; T is called the period of the
sinusoid. Note that, = 2
=2
-
8/20/2019 Circuits II Sinusoid&Phasors
15/38
The fact that the v(t) repeats itself every T seconds is shown by replacing t by t + T in thefirst equation.
= sin = sin 2
= sin 2 = sin
= ( )
v has the same value at t + T as it does at tand v(t) is said to be periodic.
-
8/20/2019 Circuits II Sinusoid&Phasors
16/38
Periodic function - one that satisfies f (t ) = f (t + nT ), for all t
and for all integers n .
-
8/20/2019 Circuits II Sinusoid&Phasors
17/38
P ROBLEMS
Values in a Sine Wave
-
8/20/2019 Circuits II Sinusoid&Phasors
18/38
TWO SINUSOIDS WITH DIFFERENT PHASES
-
8/20/2019 Circuits II Sinusoid&Phasors
19/38
A sinusoid can be expressed in either sine orcosine form. When comparing two sinusoids, it isexpedient to express both as either sine or cosinewith positive amplitudes.
Using the following trigonometric identities:
sin( ωt ± 180 ◦) = -sin ωtcos( ωt ± 180 ◦) = -cos ωt
sin( ωt ± 90 ◦) = ±cos ωtcos( ωt ± 90 ◦) = ∓sin ωt
-
8/20/2019 Circuits II Sinusoid&Phasors
20/38
We can use to relate, compare or transform asinusoid from sine form to cosine form or viceversa by:
a.) Trigonometric Identitiesb.) Graphical Approach
-
8/20/2019 Circuits II Sinusoid&Phasors
21/38
The graphical technique can also be used toadd two sinusoids of the same frequency when oneis in sine form and the other is in cosine form. Toadd Acos ωt and B sin ωt , we note that A is themagnitude of cos ωt while B is the magnitude of sinωt
-
8/20/2019 Circuits II Sinusoid&Phasors
22/38
( )v t
0 2 t
3/2 π 3/2 π
= ( + )
= ( + )
( ) =
-
8/20/2019 Circuits II Sinusoid&Phasors
23/38
SINUSOID-PHASORTRANSFORMATION
-
8/20/2019 Circuits II Sinusoid&Phasors
24/38
-
8/20/2019 Circuits II Sinusoid&Phasors
25/38
COMPLEX N UMBER
where = 1 x = the real part of A
y = the imaginary part of A
The variables x and y do not represent alocation as in two-dimensional vector analysisbut rather the real and imaginary parts of A inthe complex plane.
A complex number A can be written in rectangular
form as A = x + jy
-
8/20/2019 Circuits II Sinusoid&Phasors
26/38
Note that there are someresemblances between manipulatingcomplex numbers and manipulatingtwo-dimensional vectors.
-
8/20/2019 Circuits II Sinusoid&Phasors
27/38
The complex number A can also be written in polaror exponential form as
Where:r is the magnitude of A
is the phase of A
= ∠ =
-
8/20/2019 Circuits II Sinusoid&Phasors
28/38
Note that A can be represented in three ways:
= Rectangular form
Rectangular to Polar form= = −
Polar to Rectangular form=
y =
= ∠ Polar form= Exponential form
-
8/20/2019 Circuits II Sinusoid&Phasors
29/38
Therefore,
= ∠ = ∠ −
= = ( )
-
8/20/2019 Circuits II Sinusoid&Phasors
30/38
Phasor representation is based on Euler’s identity.
± = ±
Which shows that we may regard cos andsin as the real and imaginary parts of
cos = Re( )sin = Im( )
Where: Re the real part ofIm the imaginary part of
-
8/20/2019 Circuits II Sinusoid&Phasors
31/38
SINUSOID-PHASOR TRANSFORMATION
To get the phasor corresponding to asinusoid, express the sinusoid in the cosine form sothat the sinusoid can be written as the real part ofa complex number. Take out the time factor , andwhatever is left is the phasor corresponding to thesinusoid. Suppressing the time factor, transformthe sinusoid from the time domain to the phasordomain.
-
8/20/2019 Circuits II Sinusoid&Phasors
32/38
SINUSOID-PHASOR TRANSFORMATION
Time Domain Representation Phasor Domain Representation
( ) cos( )mv t V t
( ) sin( )mv t V t
( ) cos( )mi t I t
( ) sin( )mi t I t
= mV
V
= ( 90)mV V
= m I I
= ( 90 )m I I
-
8/20/2019 Circuits II Sinusoid&Phasors
33/38
Difference between v(t) and V
1. v(t) is the instantaneous or time-domainrepresentation, while V is the frequency orphasor-domain representation.
2. v( t) is time dependent, while V is not. (Often
forgotten)3. v(t) is always real with no complex term, while
V is generally complex
Bear in mind that phasor analysis applies only whenfrequency is constant; it applies in manipulating two ormore sinusoidal signals only if they are of the samefrequency.
-
8/20/2019 Circuits II Sinusoid&Phasors
34/38
P ROBLEMS
1. Determine the frequency, their maximum values
and the phase angle between the two voltages= 12 sin 1000 60 ° and
= 6 cos 1000 30 ° . Show graphically.
2. Given the voltage = 120 cos 314 ,
determine the frequency of the voltage in Hertz andthe phase angle in degrees.
3. Three branch currents in a network are known tobe as enumerated below, determine the phaseangles by which (t) leads (t) and (t) leads ( ).
= 2 sin 377 45 °
= 0.5 cos 377 10 ° = 0.25 sin 377 60 °
-
8/20/2019 Circuits II Sinusoid&Phasors
35/38
P ROBLEMS
4. Evaluate the following complex number
a) (30∠60 ° 20∠ 20 °)
b)∠− °+( − )
( + )( − ) ∗
c) [ 5 2 1 4 5∠60 °]∗
d)+ + ∠ °
− +10∠30 ° 5
5. Transform the following sinusoids to phasorsa. = 6cos(50 40 °) A
b. = 12cos (377 30 °) V
c. = 4sin (30 50 °) A
d. = 18sin (2513 25 °) A
-
8/20/2019 Circuits II Sinusoid&Phasors
36/38
P ROBLEMS
6. Find the sinusoids represented by the following
phasors.a) = 4 −
°
b) = 8°
c) = (4 3)
d) = 30∠( 20 °) e) = ( 12 5 )f) = 40∠( 20 °)
7. Given = 6 cos 40 ° A and =8 sin 20 ° A, find their sum, theirdifference and conjugate of each.
-
8/20/2019 Circuits II Sinusoid&Phasors
37/38
P ROBLEMS
8. Given the following sinusoids find their sum,
their difference, their product and theirquotient:a) = 12 30 ° and =
5 cos 30 °
b) = 6 45°
and =8 cos 60 ° c) = 15 25 ° and =
8 sin 50 °
9. Given = 6 cos 30 ° A and =8 cos 30 ° A, find their sum, theirdifference and conjugate of each.
-
8/20/2019 Circuits II Sinusoid&Phasors
38/38
PHASOR RELATIONSHIPS FOR CIRCUIT ELEMENTS
For Resistive circuit, R
If the current through a resistor R, is = cos ( ) ,
the voltage across it is given by Ohm’s law as= = cos and the phasor form is = ∠
But the phasor representation of the current is = ∠ ,
hence =
Note that voltage and current are in phase.
top related