Transcript Slide 1

Lab 13 – Complex Power
EE 188L
Instantaneous Power
Time domain
Instantaneous Power :
pt   it vt 
SinusoidalVoltage :
vt   VM sint   v 
SinusoidalCurrent :
it   I M sint   i 
Instantaneous Power
pt   it vt 
1
1
 VM I M cos v  i   VM I M cos2t  v  i 
2
2
constant with
time = P (real or
average power)
time varying
frequency = 2
Phase Shift
f = 624 Hz
T = 1 / 624 Hz
= 1.60 ms
1
0.5
voltage leads current by
0.22 ms
v( t )
i( t )
0
(v - i) / 360°
= 0.22 ms / 1.60 ms
v - i = 50° (lagging)
0.5
1
0
5 10
4
0.001
0.0015
t
0.002
0.0025
For an impedance with v(t)
and i(t), θz = θv - θi = 50°
(resistive and inductive)
Complex Power
Complex power :
Frequency domain
using phasors
1 *
S  VI = P + j·Q
2
where
S is the complex power (VA)
V  VM  v
I  I M  i
P is the average power (watts)
I*  I M    i
Q is the reactive power (VAR’s
or volts amps reactive
1
S  VM I M  v   i
2
S  Vrms I rms  v   i
where
Vrms 
VM
2
I
I rms  M
2
Apparent Power
Apparent Power :
S  S  Vrms I rms
units: volt-amperes, VA
Power Factor, pf :
pf  cos( v   i )
Q
tan( v   i ) 
P
Load :
V VM
Z 
 v   i
I IM
Ideal pf = 1
lagging if (θv – θi) > 0º
leading if (θv – θi) < 0º
Inductive Load
S  S  Vrms I rms  P 2  Q 2
Neglecting Rs
R2
L = 100 mH
Vm = 1 V
f = 624 Hz
Real Power, Watts, W :
1 VM2 R1  R2 
P
2  2 L2  R1  R2 2
R1 = 100 W
Delivered to
R1 and R2
RS = 10 W
Reactive Power, volts- ampere - reactive, VAR :
VM2 L
1
Q
2  2 L2  R1  R2 2
Delivered
to L
Compensated Load
To increase power factor to 1, add compensating capacitor:
Again neglecting Rs
Real Power :
1 VM2 R1  R2 
(same as before)
P
2  2 L2  R1  R2 2
R2
C
L = 100 mH
Vm = 1 V
f = 624 Hz
Reactive Power :


1 VM2  L  C  2 L2  R1  R2 2
Q
2
 2 L2  R1  R2 2
[
Which is zero if pf = 1.

]
R1 = 100 W
RS = 10 W
Current
P  Vrms  I rms cos( v   i )
or
I rms 
P
Vrms  pf
Power Triangle
S
Q
v - i
P