Module 3 - Phasors-V3

8/10/2019 Module 3 - Phasors-V3

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Phasors

EE 102 Circuits 2

Module 3

by:

Cesar G. Manalo, Jr.


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Rotating Vectors

t

v )45sin(1 ot

45o

45o

1

2

145sin o2

1Complex plane


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Rotating Vectors

t

v )45sin(1 ot

60o

60o

1

y

The value of y(the vertical component of the vector) always equals to the value

of the sinusoid (v).

v1


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Rotating Vectors

t

v )45sin(1 ot

45o

45o

1o45sin

ttt o

cossin)45sin(12

1

2

1By trigonometric identities

tcos2

1

tsin2

1

2

1

t

v

tcos2

1 tsin2

1

1

Thus sinusoids behaves like vectors (in as far a addition of sinusoids is

concerned) with directions taken from their phase angles. A sinusoid withvector-like behavior is called a phasor.

2

1

2

1

Complex plane

Complex plane


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Illustrative Problem 1

Draw the phasor representation of the following sinusoids. Use RMS

values of the sinusoids as the magnitude.1.

2.

3.

4.

e = 100 sin (t+75o)

e = 100 sin (t+120o)

e = - 100 cos (t-50o)

e = 100 sin t + 50 cos (t -30o

)


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Polar Form of Phasor

the phasor representation of vgiven in polar form is written like

this;

V (the angle inside the symbol )

where (the effective value of v) and is the phase angle.2

mV

V

Given a sinusoid;

)sin( tVvm


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Phasor Manipulation

Given two phasors and , + is a third

phasor equal to the vector addition of the two phasors. Later, another

form of phasor called the Rectangular Form useful in phasor addtion,

will be discussed.

11 V

Addition of Phasors

22 V 11 V 22 V

Given two phasors and , ( ) ( ) is a

third phasor Vwhere,11 V

Multiplication of Phasors

22 V 11 V 22 V

)())(( 2121 VVV


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Phasor Manipulation

Given two phasors and , ( ) / ( ) is a

third phasor Vwhere,11 V

Division of Phasors

22 V 11 V 22 V

)( 21

2

1 V

VV


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Checkpoint 1

i

e = Emsin (t+50o) L

R

Draw the phasor representation of the circuit below.

L

R

oE 50

I

R

XLtan

Answer:


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1. Draw the waveform and corresponding phasor

diagram and polar form of

e = 100 sin (t+20o) and i = 50 sin (t+80o).

2. Solve for e = 100 sin t + 50 cos (t -30o) using

phasors.


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Phasor Representation of Circuit Components

In a purely resistive circuit, the

resulting current is in-phase with

the driving source voltage. That

would mean that if , the

phasor representation of e, the

Resistor i

Re

= Emsin t

vR

E

resulting current isphasor will be whereEandIare the

effective values of eand i, respectively. Since Im= Em/R, likewise,

I = E/R, or using phasor representation,

. Solving for R:

I

R

EI

o

I

E

I

ER 0

Which means that in circuit analysis, Rcan be represented in phasor

form with an angle of 0 degrees.


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Resistor i

Re

= Emsin t

vRo

I

E

I

ER 0

Based on the results, R can also be treated like a phasor graphically

represented as shown below:

R


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A purely resistance AC circuit has resistance R = 200and the

sinusoidal alternating emf (voltage) device operates at amplitudeEm= 36.0 V.Find;

a) Phasor representation of the emf.

b) Phasor representation of the resulting current

c) The time domain representation of the resulting current


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In a purely capacitive circuit, the

resulting current is leading the

driving source voltage by 90o.

That would mean that if ,

the phasor representation of e, the

Capacitive Reactance (XC)

E

resulting current isphasor will be whereEandIare the

effective values of eand i, respectively. Since Im= Em/XC, likewise,

I = E/XC, or using phasor representation,

.Solving for XC:

oI 90

C

o

X

EI

90 o

oC

I

E

I

EX 90

)90(

Which means that in circuit analysis, XCcan be represented in phasor

form with an angle of -90 degrees.

i

Ce

= Emsin t

vC


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Capacitive Reactance (XC)

o

oC

I

E

I

EX 90

)90(

i

Ce

= Emsin t

vC

Based on the results, XCcan also be treated like a phasor graphically

represented as shown below:

XC

-90o


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An AC circuit has resistance R = 200and capacitance C = 100 uF.

What is the total impedance of the circuit at f = 50 Hz?


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In a purely inductive circuit, the

resulting current is lagging the

driving source voltage by 90o.

That would mean that if ,

the phasor representation of e, the

Inductive Reactance (XL)

E

oI 90

o

oL

I

E

I

EX 90

)90(

Thus, using similar analysis as the capacitive reactance, the inductive

reactance can be represented by the phasor with phase angle of 90o.

i

L e

= Emsin t

vL


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Inductive Reactance (XL)

o

oL

I

E

I

EX 90

)90(

i

L e

= Emsin t

vL

Based on the results, XLcan also be treated like a phasor graphically

represented as shown below:XL

+90o


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A purely inductive AC circuit has an inductance of 230 henrys and

the sinusoidal alternating emf (voltage) device with an effectivevalue of 70.7 volts at f = 60 Hz. Using phasor, solve for the RMS

current.

i

L=23070.7 V


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In an RC circuit, the combined effects

of the resistance and the capacitance

form what is called the impedance Z

of the circuit defined as;

Resitance + Capacitive Reactance (XC)

)](tan[)(tan

1

1 R

X

R

X

C

C Z

E

Z

EI

i

e = Emsin t vCC

vRsw.

22

CXRZ

This impedance will produce a current that will lead the source

voltage by an angle of less than 90o. computed as;

R


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In an RL circuit, the combined effects

of the resistance and inductance form

what is called the impedance Z of the

circuit defined as;

Resitance + Inductive Reactance (XL)

)](tan[)(tan

1

1 R

X

R

X

L

L Z

E

Z

EI

i

e = Emsin t vL

vRsw.

22

LXRZ

This impedance will produce a current that will lag the source

voltage by an angle of less than 90o, computed as;

R

L


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In an RLC circuit, the combined

effects of the resistance, capacitance

inductance give a Z of;

Resitance + Capacitive and Inductive Reactance

)()](tan[)(tan

1

1 LCR

XX

R

XX XX

Z

E

Z

EI

LC

LC

i

e = Emsin t vL

vRsw.

)()(

)()(

22

22

LCLC

CLCL

XXXXRZ

XXXXRZ

This impedance will produce a current that will lead the source

voltage (XC> XL) or lag the source voltage (XL> XC).

R

L

)()](tan[)(tan

1

1 CLR

XX

R

XX XX

Z

E

Z

EI

CL

CL


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Admittance

The admittance Y of a circuit is defined as the reciprocal of its

impedance Z.

ZY

1

Admittance has SI units of Siemens or mho.


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Thank You

Documents

Module 3 - Phasors-V3