Transcript AC Power Concepts

Fundamental Electrical Power
Concepts
Instantaneous Power:
p t   v t  i t 
1
Average Power: P 
v  t  i  t  dt

TT
RMS (effective value):
1
2
I
i  t  dt

TT
AC Power Concepts
•Source voltage waveform is assumed to be an
undistorted sinusoid with zero phase angle.
•Current waveforms may contain harmonic
distortion components, which increases the RMS
value of the current waveform, and hence the
apparent power (but not real power).
Current Distortion
A distorted current waveform can be decomposed into a set of
orthogonal waveforms, (e.g. by Fourier analysis). The RMS
value of the composite waveform (I) may be computed as the
root-sum-squared of the RMS values of all of the orthogonal
components {Ih}.
2
1
2
2
2
2
I   i  t  dt  I 0  I1  I 2  I 3 
TT
2
o The DC component I0 is usually (but not always) equal to zero.
o The fundamental component, I1 is the only component that
contributes to real power.
o All the other components contribute to the RMS harmonic
distortion current, Id :
I I I 
2
d
2
2
2
3
Total Harmonic Distortion
Total Harmonic Distortion (THD) is defined as the ratio
of the RMS harmonic distortion current Id to the RMS
value of the fundamental component I1 :
Id
THD 
I1
thus…
I 2  I12  I d2
I  I1 1  THD
(assuming zero DC)
2
Apparent Power
Apparent power, S, is defined as the product of RMS
voltage V, and RMS current I :
S  VI  VI1 1  THD
I1 
S
1  THD
2

2
VI
1  THD
2
Real Power
The real power contribution of the fundamental
component of the current waveform is given by :
P  VI1 cos 
Where  is the phase angle
between the voltage and
fundamental current component.
Displacement Power Factor
cos  is defined as the “displacement power
factor” (DPF).
Power Factor
We now can express real power in terms of apparent power S,
DPF and THD :
P
VI cos 
1  THD
2
S
DPF
1  THD
2
Power Factor is defined as the ratio of real power to
apparent power:
P
DPF
cos 
PF  

2
2
S
1  THD
1  THD
Complex Power
(Assumes current and voltage waveforms are undistorted sinusoids)
S  VI *  P  jQ  S   
V : Voltage Phasor
I *: Conjugate of Current Phasor
S : Complex Power Phasor
S : Apparent Power
: Phase Angle of Load
P: Real (average) Power
Q: Quadrature Power
Magnetics Concepts
AmperesLaw:
B  H
H
dl  i
 =0 r 0  4  107 h/m
For a restricted magnetic path of length l and cross section A
having linked current ni :
ni
l

 =  B da 
R
B
R
A
A
A
d
Faraday's Law: v  n
dt
Inductors and Transformers
n2
n2
n2
L


(last term is leakage inductance)
RT Rcore  Rgap RFS
vs
ns i p


(ideal transformer)
v p n p is
2
 np 
Z p    Z s (impedance transformation)
 ns 