Download Basic Concepts - Oakland University

AC Power
Discussion D9.3
Chapter 5
Steady-State Power
•
•
•
•
•
•
Instantaneous Power
Average Power
Effective or RMS Values
Power Factor
Complex Power
Residential AC Power Circuits
Instantaneous Power
v(t )  VM cos t  v 
i(t )  I M cos t  i 
p(t )  v(t )i(t )  VM I M cos t  v  cos t  i 
VM I M
cos  v  i   cos  2t  v  i  
p(t ) 
2
Note twice the frequency
Average Power
T  2 
1
P
T

t0  T
t0
1
P
T

1
p(t )dt 
T
t0 T
t0

t0 T
t0
VM I M cos t   v  cos t  i  dt
VM I M
cos v  i   cos  2t   v  i   dt
2
P  1 VM I M cos v  i 
2
Purely resistive circuit
Purely reactive circuit
 v  i  0
v  i  90
P  1 VM I M
2
P  1 VM I M cos 90  0
2
 
Effective or RMS Values
We define the effective or rms value of a periodic current
(voltage) source to be the dc current (voltage) that delivers the
same average power to a resistor.
1
PI R
T
2
eff
I eff

t0  T
t0
i 2 (t )Rdt
1 t0 T 2

i (t )dt

T t0
I eff  I rms
Veff2
1
P

R T

t0 T
t0
v 2 (t )
dt
R
1 t0 T 2
Veff 
v (t )dt

T t0
root-mean-square
Veff  Vrms
Effective or RMS Values
Vrms
Using
Vrms
Vrms
1 t0 T 2

v (t )dt

T t0
v(t )  VM cos t  v 
cos   1  1 cos 2
2
2
2

 VM 
 2

 VM 
 2


2 
0
2 
0
T  2 
and
 1  1 cos  2t  2v dt 
2
 2
 
1 
dt 
2 
1
2

 VM 
 2
2 
t
 
20



1
2
1
2
VM

2
Power Factor
Recall Average Power
Vrms
VM

2
P  1 VM I M cos v  i 
2
IM
I rms 
2
P  Vrms I rms cos v  i 
Power factor
P
PF 
 cos  v  i   cos  ZL
Vrms I rms
Power factor angle
Note that I rms
 ZL  v  i
P

Vrms PF
Thus, an industrial load that consumes
P watts with a high PF from a Vrms volt
2
R line losses.
line will have lower I rms
Complex Power
Complex Power
S = Vrms Irms

I rms
 complex conjugate of I rms
 rms  I r  jIi  I rms i
rms  I r  jI i  I rms   i
S = Vrms v I rms   i  Vrms I rms  v  i 
S = P  jQ
P  Re(S)  Vrms I rms cos v  i 
Real, average power
Q  Im(S)  Vrms I rms sin v  i 
Imaginary, quadrature power
Residential Power Systems
Read Section 5-8, pages 216 - 219