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51 ectr c rcu ts nter 1

Instructor: Dr. Karthi Umapathy

karthi@ee.ryerson.ca

1EES 512, Winter 2010, KU

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Previous Lecture

- Series and Parallel - C and L

- AC circuits

- Generation of AC voltage

- Parameters of AC waveforms

- Phasor representation

-

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AC Waveforms

T

Amplitude = mV

Phase Shift 1

Angular Freq. =

Period = T

zT Frequency = 1/T

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Phasor representation

mm ttv cos

(Time domain) (Phasor or frequency domain)

tjte

sincos

jtjVeVetv VVRe

Eulers Identity

mm

j Im 1

T

TTTT

t ,3

,,,0

A rotating phasor

1-1Re

424

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Phasor representation

mm VtVtv V)cos()(

t,, mmeev

Phasor diagram

mag nary x s

V

m

II Real Axis

mLeadingdirection

mI

Laggingdirection

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Examples

Express these sinusoids as phasors:

a v t = 7 cos 2t + 40 V

b) i(t) = - 4 sin ( 10t + 10) A

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Find the sinusoids corres ondin to these hasors:

a V = -10 30 V

b) I = j(5 j12) A

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Phasor relationship for circuit elementsResistor Inductor Capacitor

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Phasor relationship for circuit elements

Summary of voltage-current relationship

emen

R

L diLv IV L

C dvCi

IV

dvVj

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Examples:

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Impedance and Admittance

Resistor Inductor Capacitor

IV R IV LjC

Vj

I

CjLjR

1,,

IV

IV

IV

V I

,Impedance

Im edance of a circuit is the ratio of the hasor volta e V and the hasor

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current I, measure in Ohms

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1Y

Admittance of a circuit is the reciprocal of Impedance

Impedances and admittances of passive elements

Element Im edance Admittance

R 1

YRZ

L

L

1YLjZ

CCjY

C

1Z

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j

LR

1

,,

VVV

CCj III

V,

I

1

CCL

,

Inductive Capacitiveeactance eactance

ZZ jXR)( InductiveXvejXR Z

)( CL XXjR ZZ

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1

( CLjR Z

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ZZ jXR

R

XXR

122tan,

Z

s n,cos

Admittance

V

I

Z

1Y jBG Y

Susceptance2222

,XR

XB

XR

RG

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Series and Parallel Impedances

Volta eDivision

ZZZZ .... VZ

VVZ

V 21 eq ZZZZ 2121

Current

1....

111 ZZ 12

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N21eq ZZ,

ZZ 212

21

1

N21eq YYYY ....