A. Stolp
3/31/09,
2/20/10
11/23/13
corrected
p4 '14
ECE 2210
Introduction to AC Power, RMS
RMS
AC Power
I
DC Power
P
= ?
P
=
.
V
I
=
V
2
R
=
.
I
2
R
v
(
)
t
=
.
V
p
cos
(
)
.
ϖ
t
R
V
R
v
(
)
t
V
p
t
p
(
)
t
V
p
2
R
Couldn't we define an "effective" voltage that
would allow us to use the same relationships
for AC power as used for DC power?
average or
"effective"
power =
V
p
2
R
2
t
P
ave
=
=
=
V
p
2
R
2
V
p
2
2
R
V
p
2
2
R
v
(
)
t
2
V
p
2
S
quare
/
V
eff
=
V
p
2
2
=
V
p
2
=
V
rms
=
.
1
T
d
0
T
t
(
)
v
(
)
t
2
average =
V
p
2
2

\
M
ean (average)
R
oot
t
RMS
R
oot of the
M
ean of the
S
quare
Use RMS in power calculations
Sinusoids
V
rms
=
.
1
T
d
0
T
t
(
)
v
(
)
t
2
=
.
1
T
d
0
T
t
.
V
p
cos
(
)
.
ϖ
t
2
=
.
1
T
d
0
T
t
.
V
p
2
1
2
.
1
2
cos
(
)
.
.
2
ϖ
t
=
.
V
p
2
.
1
T
d
0
T
t
(
)
1
.
1
T
d
0
T
t
cos
(
)
.
.
2
ϖ
t
=
.
V
p
2
1
0
ECE 2210
AC
Power
p1
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Common household power
ECE 2210
AC
Power
p2
f
=
.
60
Hz
Neutral, N
Line, L
black, 120V
V
rms
.
120
V
white
ϖ
=
.
377
rad
sec
(also ground)
V
p
=
=
.
V
rms
2
170
V
T
=
.
16.67
ms
Ground, G, green
What about other wave shapes??
v
(
)
t
Triangular
v
(
)
t
Square
V
p
V
p
t
t
v
(
)
t
2
V
p
2
v
(
)
t
2
V
p
2
average =
V
p
2
average =
V
p
2
3
t
t
V
rms
=
V
p
3
V
rms
=
V
p
Works for all types of triangular and sawtooth waveforms
Same for DC
v
(
)
t
How about AC + DC ?
.
/\
Vp
\/
V
rms
=
.
1
T
d
0
T
t
(
)
v
(
)
t
2
V
DC
t
=
.
1
T
d
0
T
t
.
V
p
cos
(
)
.
ϖ
t
V
DC
2
=
.
1
T
d
0
T
t
.
V
p
cos
(
)
.
ϖ
t
2
.
.
2
.
V
p
cos
(
)
.
ϖ
t
V
DC
V
DC
2
=
.
1
T
d
0
T
t
.
V
p
cos
(
)
.
ϖ
t
2
.
1
T
d
0
T
t
.
.
2
.
V
p
cos
(
)
.
ϖ
t
V
DC
.
1
T
d
0
T
t
V
DC
2
   zero over one period   
=
V
rmsAC
2
0
V
DC
2
=
V
rmsAC
2
V
DC
2
ECE 2210
AC
Power
p2
ECE 2210
AC
Power
p3
rectified average
V
ra
=
.
1
T
d
0
T
t
v
(
)
t
sinusoid:
V
rms
=
V
p
2
I
rms
=
I
p
2
V
ra
=
.
2
π
V
p
I
ra
=
.
2
π
I
p
triangular:
V
rms
=
V
p
3
I
rms
=
I
p
3
V
ra
=
.
1
2
V
p
I
ra
=
.
1
2
I
p
square:
V
rms
=
V
p
I
rms
=
I
p
V
ra
=
V
rms
=
V
p
I
ra
=
I
rms
=
I
p
Most AC meters don't measure true RMS.
Instead, they measure
V
ra
, display
1.11V
ra
,
and call it RMS.
That works for sine waves
but not for any other waveform.
waveform + DC
V
rms
=
V
rmsAC
2
V
DC
2
Some waveforms don't fall into these forms, then you have to perform the math from scratch
0
1
2
3
4
5
6
7
8
9
10
11
12
6
4
2
2
4
For waveform shown
v(t)
(volts)
The average DC (
V
DC
) value
=
.
.
2
V
(
)
.
4
ms
.
(
)
.
5
V
(
)
.
2
ms
.
6
ms
0.333
V
time
(ms)
The RMS (effective) value
0
1
2
3
4
5
6
7
8
9
10
11
12
5
5
10
15
20
25
v(t)
2
(volts
2
)
Graphical way
=
.
.
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 Staff