Transcript EEE 302 Lecture 7 - Arizona State University

EEE 302
Electrical Networks II
Dr. Keith E. Holbert
Summer 2001
Lecture 7
1
The First of Many Definitions Today
• 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
Lecture 7
2
Power Factor (pf)
• Derivation of power factor (0  pf  1)
average power
P
pf =
=
apparentpower 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
Lecture 7
3
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
Lecture 7
4
Class Example
• Extension Exercise E9.11
Lecture 7
5
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 voltamperes reactive, or var
Lecture 7
6
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
Lecture 7
7
From a Load Perspective
• Recall phasor relationships between current,
voltage, and load impedance
VIZ
V
rms
VM  v  I M  i Z  Z



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 
Lecture 7
8
Real Power (P)
• Alternate expressions for the real or average
power (P)
P  ReS  Vrms I rms cos v  i 
 S cos Z   I rms Z  I rms
2
 I rms
ReZ 
Lecture 7
 ReZ 


 Z 
9
Reactive Power (Q)
• Alternate expressions for the reactive or
quadrature power (Q)
Q  ImS  Vrms I rms sin  v  i 
 S sin  Z   I rms Z  I rms
2
 I rms
ImZ 
Lecture 7
 ImZ 


 Z 
10
Power Triangle
• The power triangle relates pf angle to P and Q
Q reactive/quadraturepower
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) power
Lecture 7
Im
Q=I2rms Im(Z)
v- i
P=I2rms Re(Z)
Re
11
Summarizing Complex Power (S)
S  P  jQ  I
2
rms
ReZ   j I
2
rms
ImZ   I
2
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
Lecture 7
12
Class Examples
• Extension Exercise E9.12
• Extension Exercise E9.13
Lecture 7
13