SJTU 1

Chapter 11

AC power analysis

SJTU 2

rms value

The rms value of a periodic function is defined as the square root of the mean value of the squared function.

effective value or DC-equivalent value

The RMS value is the effective value of a varying voltage or current. It is the equivalent steady DC (constant) value which gives the same effect.

T

rms dttvT

V0

2 )(1

SJTU 3

If the periodic function is a sinusoid, then

What do AC meters show, is it the RMS or peak voltage?

AC voltmeters and ammeters show the RMS value of the voltage or current.

What does '6V AC' really mean, is it the RMS or peak voltage?

If the peak value is meant it should be clearly stated, otherwise assume it is the RMS value.

mm

T

mrms

VV

tVT

V

707.02

1

)(cos1

0

22

SJTU 4

AC power analysisInstantaneous Power

v(t)

i(t)

N )cos(2)(

cos2)(

tIti

tVtv

rms

rms

Suppose:

)2cos(cos

)cos(2cos2

tIVIV

tItVvip

rmsrmsrmsrms

rmsrms

Invariable part Sinusoidal part

SJTU 5

E page415 figure 10.2

SJTU 6

Stored energy

In the sinusoidal steady state an inductor operates with a current iL(t)=IAcos(wt).

The corresponding energy stored in the element is

Average stored energy

22

2

1

4

1rmsALav LILIW

WLav

SJTU 7

Stored energyIn the sinusoidal steady state the voltage across a capacitor is vc(t)=VA

cos(wt). The energy stored in the element is

Average stored energy

22

2

1

4

1rmsACav CVCVW

WCav

SJTU 8

Average power

The average power is the average of the instantaneous power over one period.-------real power

cos)(1

0 rsmrsm

TIVdttp

TP

Note : There are other methods to calculate P.

)Re(Re

,Recos

cos,

22

2

YUPorZIP

ZZ

ZIPIZV

1)1)

2) kPP

SJTU 9

Instantaneous power, real power

0 1 2 3 42

1

0

1

22

2

v t n i t n P t n

40 t n

Instantaneous power waveforms for a voltage of 2V peak and a current of 1.5A peakFlowing separately in a resistor, a capacitor and an inductor

0 1 2 3 42

0

2

43

2

v t n i t n P t n

40 t n

Resistor case

Average powerPav=0.5Vm*Im

Pav=vrms*irms

0 1 2 3 42

1

0

1

22

2

v t n i t n P t n

40 t n

Inductor case

Pav = 0

Capacitor case Pav = 0

SJTU 10

Apparent power

S=VrmsIrms (VA)

cos

cos

SP

IVP rmsrms

S

P cos

Power factor

)()( legorlead current leads voltage or current lags voltage

<0 or >0

SJTU 11

Reactive power

sinrmsrms IVQ Resistor: Q=0

Inductor: Q=VrmsIrms

Capacitor: Q=-VrmsIrms

(VAR)

tVItVI

VItVItp

2sinsin)2cos1cos(

cos)2cos()(

CavC

LavL

WQ

WQ

2

2

)(2 CL WWQ To any passive single port network

SJTU 12

The power triangle

P

Qtg

QPS

222

QS

P

SJTU 13

EXAMPLEFind the average power delivered to the load to the right of the interface in Figure 8-64.

Fig. 8-64

SOLUTION:

SJTU 14

Complex powerComplex power is the complex sum of real power and reactive power

=P+jQ

So =VI*Where V is the voltage phsor across the system and I* is the complex conjugate of the current phasor.

S~

S~

The magnitude of complex power is just apparent power

22~QPS S

SJTU 15

Are these equations right?

)(~~

)(

)(

)(

k

k

k

k

SS

SS

QQ

PP

SJTU 16

Maximum power transfer

Fig. 8-66: A source-load interface in the sinusoidal steady state.

SJTU 17

Let XL=-XT then

we know P is maximized when RL=RT

the maximum average power

where |VT| is the peak amplitude of the Thevenin equivalent voltage

SJTU 18

EXAMPLE                                                                                  

(a) Calculate the average power delivered to the load in the circuit shown in Figure 8-67 for Vs(t)=5cos106t, R=200 ohm, and

RL=200 ohm.

(b) Calculate the maximum average power available at the interface and specify the load required to draw the maximum power.

SOLUTION:

(a)

SJTU 19

SJTU 20

(b)

Question:

If the load must be a resistor, how get the maximum power on it?

SJTU 21

Maximum power transfer when ZL is

restricted 1) RL and XL may be restricted to a limited range of values.

In this situation, the optimum condition for RL and XL is to adjust XL as near to –XT as possible and then adjust RL as close to as possible

22 )( TLT XXR

2) the magnitude of ZL can be varied but its phase angle cannot.

Under this restriction, the greatest amount of power is transferred to the load when the magnitude of ZL is set equal to the magnitude of ZT

TL ZZ

SJTU 22

Note:

1. If the load is a resistor, then what value of R results in maximum average-power transfer to R? what is the maximum power then?

2. If ZL cannot be varied but ZT can, what value of ZT results in maximum average-power transfer to ZL?