Circuits II Sinusoid&Phasors

8/20/2019 Circuits II Sinusoid&Phasors

1/38

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


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.


4/38

GENERATING P LANT TO E ND -U SERS


5/38

V ARIOUS SOURCES OF AC POWER :

1. Generating Plant

Masinloc Coal-fired Power PlantLocated in Zambales (660 MW)


6/38


1. Generating Plant

Geothermal Power Plant Located inMindanao (55 MW)


7/38


1. Generating Plant

Binga Hydroelectric Power PlantLocated in Benguet (125.8 MW)


8/38


2. Portable AC Generator


9/38


3. Wind Turbine Power Station


10/38


4. Solar Power System


11/38


5. Function Generator


12/38

SINUSOIDS

180

90

0

3

2

1

2

( ) or ( )v t i t

mV

t 90 180 270

2

0

r y


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:


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


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.


16/38

Periodic function - one that satisfies f (t ) = f (t + nT ), for all t

and for all integers n .


17/38

P ROBLEMS

Values in a Sine Wave


18/38

TWO SINUSOIDS WITH DIFFERENT PHASES


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


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


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


22/38

( )v t

0 2 t

3/2 π 3/2 π

= ( + )

= ( + )

( ) =


23/38

SINUSOID-PHASORTRANSFORMATION


24/38


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


26/38

Note that there are someresemblances between manipulatingcomplex numbers and manipulatingtwo-dimensional vectors.


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

= ∠ =


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


29/38

Therefore,

= ∠ = ∠ −

= = ( )


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


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.


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


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.


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 °


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


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.


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.


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.

Documents

Circuits II Sinusoid&Phasors