Transcript Slide 1
Lab 13 – Complex Power
EE 188L
Instantaneous Power
Time domain
Instantaneous Power :
pt it vt
SinusoidalVoltage :
vt VM sint v
SinusoidalCurrent :
it I M sint i
Instantaneous Power
pt it vt
1
1
VM I M cos v i VM I M cos2t 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