8. AC POWER

CIRCUITS by Ulaby & Maharbiz

Overview

Linear Circuits at ac

Instantaneous power Average power

)()()( tittp

Power at any instant of time Average of instantaneous power over one period

T

dttpT

P0

)(1

Power delivery (utilities) Electronics (laptops, mobile phones, etc.) Logic circuits

Power is critical for many reasons:

Note: Power is not a linear function, cannot apply superposition

Instantaneous Power for Sinusoids

Power depends on phases of voltage and current

i

i

ttIVtp

tittp

tIti

tVt

coscos)(

)()()(

cos)(

cos)(

mm

m

m

BABABA coscos2

1coscos

ii tIVtp 2coscos2

1)( mm

Trig. Identity:

Constant in time (dc term)

ac at 2w

Average Value

Sine wave

Truncated sawtooth

Average Value for

These properties hold true for any values of φ1 and φ2

Effective or RMS Value

Equivalent Value That Delivers Same Average Power to Resistor as in dc case

For current given by

Effective value is the (square) Root of the Mean of the Square of the periodic signal, or RMS value

TTdti

T

RdtRi

TP

0

2

0

21

RIP 2eff

Tdti

TI

0

2eff

1 T

dtT

V0

2eff

1

tIti cosm 2

cos1 m

0

22mrms

IdttI

TI

T

Hence:

Similarly,

Average Power

i

i

ttIVtp

tittp

tIti

tVt

coscos)(

)()()(

cos)(

cos)(

mm

m

m

ii tIVtp 2coscos2

1)( mm

BABABA coscos2

1coscos

Note dependence on phase difference

Average Power

Since and a similar relationship applies to I,

Power factor angle:

0 for a resistor= 90 degrees for inductor ‒90 degrees for capacitor

ac Power Capacitors

2

i

dt

dCi C

C

CC CVjI

2CC CVI

Capacitors (ideal) dissipate zero average power

222cos2

1

2coscos2

1

mm

mm

tIVtp

tIVtp ii

= 0

ac Power Inductors

2

i

dt

diL L

L

LL LIjV

2LL iLIV

Inductors (ideal) dissipate zero average power

222cos2

1

2coscos2

1

mm

mm

i

ii

tIVtp

tIVtp

= 0

Complex Power

Phasor form defining “real” and “reactive” power

Power Factor for Complex LoadInductive/capacitive loads will require more from the power supply than the average power being consumed Power supply needs to

supply S in order to deliver Pav to load

Power factor relates S to Pav

Power Factor

Power Factor Compensation

Introduces reactive elements to increase Power Factor

Example 8-6: pf Compensation

Maximum Power Transfer

Max power is delivered to load if load is equal to Thévenin equivalent

*sssLLL ZjXRjXRZ Max power

transfer when

Set derivatives equal to zero 0L

X

P0

L

R

P

s

2

Thmax 8R

VP

Example 8-7: Maximum Power

Cont.

Example 8-7: Maximum Power

Three Phase

Y & Delta

Y-Source Connected to a Y-Load

Multisim Measurement of Power

Multisim Measurement of Complex Power

Complex Power S

Summary