Download Chapter 11

Chapter 11
AC power analysis
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rms 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.
effective value or DC-equivalent value
The rms value of a periodic function is defined as the square
root of the mean value of the squared function.
Vrms
1 T 2

v (t )dt

T 0
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If the periodic function is a sinusoid, then
Vrms
1 T 2
2

V
cos
(t   )
m

0
T
1

Vm  0.707Vm
2
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.
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AC power analysis
Instantaneous Power
i(t)
N
v(t)
Suppose:
v(t )  2Vrms cos t
i (t )  2 I rms cos(t   )
p  vi  2Vrms cos t  2 I rms cos(t   )
 Vrms I rms cos   Vrms I rms cos(2t   )
Invariable part
Sinusoidal part
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E page415 figure 10.2
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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
WLav
WLav
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1 2 1 2
 LI A  LI rms
4
2
6
Stored energy
In the sinusoidal steady state the
voltage across a capacitor is
vc(t)=VAcos(wt). The energy stored
in the element is
Average stored energy
WCav
WCav
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1
2
2
 CVA  CVrms
4
2
7
Average power
The average power is the average of the instantaneous power
over one period.-------real power
1
P
T

T
0
p(t )dt  Vrsm I rsm cos 
Note : There are other methods to calculate P.
1)
1)
V  Z I , P  I 2 Z cos 
 Z cos   Re Z ,
 P  I 2 Re Z (or P  U 2 Re Y )
2)
P   Pk
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Instantaneous power, real power
Instantaneous power waveforms for a voltage of 2V peak and a current of 1.5A peak
Flowing separately in a resistor, a capacitor and an inductor
3
v t n
2
4
v t n
2
i t n
i t n
P  t n
2
2
1
0
P  t n
0
1
2
2
0
0
1
2
3
tn
2
4
2
2
0
1
0
tn
4
Resistor case
3
4
4
Inductor case
v t n
Average power
Pav=0.5Vm*Im
Pav=vrms*irms
2
i t n
1
Pav = 0
0
P  t n
1
2
2
0
0
1
2
tn
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4
4
Capacitor case Pav = 0
9
Apparent power
S=VrmsIrms
(VA)
 P  Vrms I rms cos 
 P  S cos 
Power factor
P
  cos  
S
 (lead ) or  (leg )
current leads voltage or current lags voltage
<0
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or
 >0
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Reactive power
p(t )  VI cos( 2t   )  VI cos 
 VI cos(1  cos 2t )  VI sin  sin 2t
Q  Vrms I rms sin 
(VAR)
Resistor: Q=0
Inductor: Q=VrmsIrms
QL  2WLav
Capacitor: Q=-VrmsIrms
QC  2WCav
To any passive
single port network
Q  2 (WL  WC )
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The power triangle
S  P Q
2
2
2
S
Q
tg 
P

P
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Q
12
EXAMPLE
Find the average power delivered to the load to the right of the
interface in Figure 8-64.
SOLUTION:
Fig. 8-64
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Complex power
Complex power is the complex sum of real power and
reactive power
So
~ =P+jQ
S
~ =VI*
S
Where V is the voltage phsor across the
system and I* is the complex conjugate
of the current phasor.
The magnitude of complex power is just apparent power
~
2
2
S  S  P Q
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Are these equations right?
P   Pk
Q   Qk
S   Sk
~
~
S   Sk
( )
( )
( )
( )
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Maximum power transfer
Fig. 8-66: A source-load interface in
the sinusoidal steady state.
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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
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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)
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(b)
Question:
If the load must be a resistor,
how get the maximum power
on it?
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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 RT2  ( X L  X T ) 2 as possible
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
Z L  ZT
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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?
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