Download Power in AC

AC Power
January 22, 2009
AC Circuit
SEE1023
1
Instantaneous Power: p(t)
For AC circuits, the voltage and current are
v(t) = VM cos(t+v)
i(t) = IM cos(t+i)
The instantaneous power is simply their product
p(t) = v(t) i(t) = VM IM cos(t+v) cos(t+i)
= ½VM IM [cos(v- i) + cos(2t+v +i)]
Constant
Term
AC Circuit
SEE1023
Wave of Twice
Original Frequency
2
Average Power (P)
• Calculate average power (integrate power over
one cycle and divide by period)
1
P=
T
t 0 T

t0
1
p(t) dt =
T
t 0 T
 V
M
cost   v  I M cost +  i  dt
t0
1
= VM I M cos v -  i 
2
• Recall that passive sign convention says:
P > 0, power is being absorbed
P < 0, power is being supplied
AC Circuit
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Average Power: Special Cases
• Purely resistive circuit: P = ½ VM IM
The power dissipated in a resistor is
VM2 1 2
1
P = VM I M =
= IM R
2
2 R 2
• Purely reactive circuit: P = 0
– Capacitors and inductors are lossless elements and
absorb no average power
– A purely reactive network operates in a mode in which
it stores energy over one part of the period and
releases it over another part
AC Circuit
SEE1023
4
Average Power Summary
Circuit Element
Average Power
V or I source
Resistor
P = ½ VM IM cos(v- i)
P = ½ VM IM = ½ IM2 R
Capacitor or
Inductor
P=0
Does the expression for the resistor power look
identical to that for DC circuits?
AC Circuit
SEE1023
5
Effective or RMS Values
• Root-mean-square value (formula reads
like the name: rms)
I rms
1

T
t 0 T

2
i (t ) dt
and
t0
• For a sinusoid:
Vrms
1

T
t 0 T

v 2 (t ) dt
t0
Irms = IM/2
– For example, AC household outlets are
around 120 Volts-rms
AC Circuit
SEE1023
6
Why RMS Values?
• The effective/rms current allows us to write
average power expressions like those used in dc
circuits (i.e., P=I²R), and that relation is really the
basis for defining the rms value
• The average power (P) is
1
Psource  VM I M cos v   i   Vrms I rms cos v   i 
2
2
Vrms
1
2
Presistor  VM I M  Vrms I rms 
 I rms R
2
R
AC Circuit
SEE1023
7
RMS in Everyday Life
• When we buy consumer electronics, the
faceplate specifications provide the rms voltage
and current values
• For example, what is the rms current for a 1200
Watt hairdryer (although there is a small fan in a
hairdryer, most of the power goes to a resistive
heating element)?
• What happens when two hairdryers are turned
on at the same time in the bathroom?
• How can I determine which uses more
electricity---a plasma or an LCD HDTV?
AC Circuit
SEE1023
8
Maximum Average Power Transfer
• To obtain the maximum average power transfer
to a load, the load impedance (ZL) should be
chosen equal to the complex conjugate of the
Thevenin equivalent impedance representing
the remainder of the network
ZL = RL + j XL = RTh - j XTh = ZTh*
AC Circuit
SEE1023
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Maximum Average Power Transfer
ZTh
Voc
+
–
ZL
ZL = ZTh*
• Note that ONLY the resistive component of
the load dissipates power
AC Circuit
SEE1023
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Max Power Xfer: Cases
Load
Load Equivalent
Characteristic
Complex
ZL = ZTh* = RTh - j XTh
Purely Resistive Z  R = R 2 + X 2
L
L
Th
Th
(i.e., XL=0)
Further reduces to ZL= RL=RTh
for XTh=0 (old DC way)
Purely Reactive No Average power transfer to
load; Not really a case
(i.e., RL=0)
AC Circuit
SEE1023
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Power Factor (pf)
• Derivation of power factor (0  pf  1)
average power
P
pf =
=
apparent power V rms I rms
V rms I rms cos v   i 
=
= cos v   i = cos  Z L
V rms I rms
 
• A low power factor requires more rms current for the
same load power which results in greater utility
transmission losses in the power lines, therefore utilities
penalize customers with a low pf
AC Circuit
SEE1023
12
Power Factor Angle (ZL)
• power factor angle is v- i = ZL (the phase
angle of the load impedance)
• power factor (pf) special cases
– purely resistive load: ZL = 0°  pf=1
– purely reactive load: ZL = ±90°  pf=0
Power Factor Angle
I/V Lag/Lead
Load Equivalent
-90 < θZL < 0
Leading
Equivalent RC
0 < θZL < 90
Lagging
Equivalent RL
AC Circuit
SEE1023
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From a Load Perspective
• Recall phasor relationships between current,
voltage, and load impedance
VIZ
V
VM  v  I M  i Z  Z
rms



2  v  I rms 2  i Z  Z
Vrms  I rms Z
• The load impedance also has several alternate
expressions
Z  Re( Z)  j Im( Z)  Z cos Z   j sin  Z 
AC Circuit
SEE1023
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Power Triangle
• The power triangle relates pf angle to P and Q
Q reactive/quadrature power
tan  v   i   
P
real/average power
– the phasor current that is in
phase with the phasor
voltage produces the real
(average) power
– the phasor current that is
out of phase with the
phasor voltage produces
the reactive (quadrature)
AC Circuit
SEE1023
power
Im
Q=I2rms Im(Z)
v- i
P=I2rms Re(Z)
Re
15
Summarizing Complex Power (S)
S  P  jQ  I
2
rms
Re Z  
jI
2
rms
2


Im Z  I rms Z
Complex power (like energy) is conserved, that is,
the total complex power supplied equals the total
complex power absorbed, Si=0
Reactive Power
Load
Power Factor
Complex Power
Q is positive
Inductive
Lagging
First quadrant
Q is zero
Resistive
pf = 1
Real valued
Q is negative
Capacitive
Leading
Fourth quadrant
AC Circuit
SEE1023
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More Power Terminology
• average power, P = Vrms Irms cos(v- i)
• apparent power = Vrms Irms
– apparent power is expressed in volt-amperes
(VA) or kilovolt-amperes (kVA) to distinguish it
from average power
AC Circuit
SEE1023
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Complex Power (S)
• Definition of complex power, S
S  Vrms I *rms  Vrms  v I rms    i
 Vrms I rms  v   i
 Vrms I rms cos v   i   j Vrms I rms sin  v   i 
 P  jQ
– P is the real or average power
– Q is the reactive or quadrature power, which indicates
temporary energy storage rather than any real power
loss in the element; and Q is measured in units of
volt-amperes reactive, or var
AC Circuit
SEE1023
18
Complex Power (S)
• This is really a return to phasor use of voltage and
current rather than just the recent use of magnitude and
rms values
• Complex power is expressed in units of volt-amperes like
apparent power
• Complex power has no physical significance; it is a
purely mathematical concept
• Note relationships to apparent power and power factor of
last section
|S| = Vrms Irms = apparent power
S = (v- i) = ZL = power factor angle
AC Circuit
SEE1023
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Real Power (P)
• Alternate expressions for the real or
average power (P)
P  Re S   Vrms I rms cos v  i 
 Re Z 
 S cos Z   I rms Z  I rms 

 Z 
2
 I rms
Re Z 
AC Circuit
SEE1023
20
Reactive Power (Q)
• Alternate expressions for the reactive or
quadrature power (Q)
Q  ImS   Vrms I rms sin  v  i 
 ImZ 
 S sin  Z   I rms Z  I rms 

 Z 
2
 I rms
ImZ 
AC Circuit
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