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ECE 1311

Chapter 11AC PowerAnalysis

1

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Outlines

11.1 Instantaneous and Average Power11.2 Maximum Average Power Transfer

11.3 Effective or RMS Value11.4 Apparent Power and Power Factor11.5 Complex Power

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Sinusoidal power at 2tConstant power

)2(cos2

1

)(cos2

1

)(cos)(cos)()()(

ivmmivmm

ivmm

tIVIV

ttIVtitvtp

w

ww

11.1 Instantaneous andAverage Power (1)

The instantaneous power (p(t))

p(t) > 0: power is absorbed by the circuit; p(t) < 0: power is supplied by the circuit.

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)(cos2

1)(

1

0ivmm

T

IVdttpT

P

11.1 Instantaneous andAverage Power (2)

The average power (P) is the average of the instantaneouspower over one period.

1. P is not time dependent.

2. When v= i , it is a purely

resistiveload case.

3. When vi= 90o, it is a

purely reactiveload case.

4. P = 0 means that the circuit

absorbs no average power.

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)6010(sin15)(

)2010(cos80)(

tti

ttv

11.1 Instantaneous andAverage Power (3)

Example 1

Calculate the instantaneous power and average

power absorbed by a passive linear network if:

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Example 2

A current flows through an

impedance . Find the average

power delivered to the impedance.

11.1 Instantaneous andAverage Power (4)

3010I

2220 Z

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Example 3

Calculate the average power absorbed by the resistor and

inductor. Find the average power supplied by the voltage

source.

11.1 Instantaneous andAverage Power (5)

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11.2 Maximum Average PowerTransfer (1)

The maximum average power

can be transferred to the load if:

TH

2

TH

maxR8

VP

If the load is purely real, then:TH

2

TH

2

THL ZXRR

*ThL ZZ

L

2

max RI

2

1P

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11.2 Maximum Average PowerTransfer (2)

Example 4

For the circuit shown below, find the load impedance ZLthatabsorbs the maximum average power. Calculate that maximum

average power.

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11.3 Effective or RMS Value (1)

2

II mrms

The average power can be written in terms of

the rms values:

)(cosIV)(cosIV2

1

ivrmsrmsivmm P

Note: If you express amplitude of a phasor source(s) in rms, then all the

answers as a result of this phasor source(s) must also be in rms value.

The rms value of a sinusoid i(t) = Imcos(wt)is given by:

rmseff II--Note

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11.4 Apparent Power andPower Factor (1)

Apparent Power (S)is the product of the rms values ofvoltage and current.

It is measured in volt-amperesor VA to distinguish it fromthe average or real power which is measured in watts.

Power factor is: the cosine of the phase difference between the voltage and current

or

The cosine of the angle of the load impedance.

)(cosS)(cosIVP ivivrmsrms

Apparent Power, S Power Factor, pf

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11.4 Apparent Power andPower Factor (2)

Purely resistive

load (R)vi = 0, pf = 1 P/S = 1, all power are

consumed

Purely reactive

load (L or C)vi = 90

o,

pf = 0

P = 0, no real power

consumption

Resistive and

reactive load

(R and L/C)

vi > 0

vi < 0

Lagging- inductive

load

Leading- capacitive

load

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11.4 Apparent Power andPower Factor (3)

Example 5

For the circuit shown below, calculate the power factor as seenby the voltage source. What is the average power supplied by

the voltage source?

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11.5 Complex Power (1)

Complex power (S) is the product of the voltage and the

complex conjugate of the current:

*IVIV2

1S

rmsrms

*Z

VZIS

2

rms2

rms

imvm IV IV

irmsvrms IV IV

Note:

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11.5 Complex Power (2)

ivrmsrms IVIV2

1S

)(sinIVj)(cosIVS ivrmsrmsivrmsrms

P: is the average power in wattsdelivered to a load and it is the only useful

power.

Q: is the reactive power exchangebetween the source and

the reactive part of the load. It is measured in volt-ampere-reactive (VAR).

Q = 0 for resistive loads(unity pf).

Q < 0 for capacitive loads(leading pf).

Q > 0 for inductive loads(lagging pf).

S = P + j Q

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11.5 Complex Power (3)

)(sinIVj)(cosIVS ivrmsrmsivrmsrms

Apparent Power:S = |S| = Vrms*Irms=

Real power: P = Re(S) = S cos(vi)

Reactive Power: Q = Im(S) = S sin(vi)

Power factor: pf = P/S = cos(vi)

S = P + j Q

22 QP

S

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11.5 Complex Power (4)

)(sinIVj)(cosIVS ivrmsrmsivrmsrms

S = P + j Q

Impedance TrianglePower Factor

Power Triangle

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11.5 Complex Power (5)

Example 6:

Given:

Determine:

The complex and apparent powers.

The real and reactive powers.

The power factor and the load impedance.

AI

VV

rms

rms

154.0

85110