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
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?