Basic Of Power Measurement; Arithmetic Mean Value; Rectified Mean Value; Root-Mean-Square Value - Hameg HM8015 Manual

B a s i c s o f P o w e r M e a s u r e m e n t
Basics of Power Measurement
Abbreviations and symbols used:
W active, true power
VA apparent power
var reactiv power
u(t) voltage as a variable of time
u²(t) voltage squared as a variable of time
IÛI rectified voltage
V rms value of voltage
rms
û peak value of voltage
I rms value of current
rms
î peak value of current
ϕ
phase angle between voltage and
current
cos ϕ
power factor, valid only for sine waveform
PF power factor in general for arbitrary
waveforms
Arithmetic mean value (average)
T
∫
1
x = –– x
|· dt
(t) (t)
T
0
The arithmetic mean value of a periodic signal
is the average calculated for a full period T, it is
identical to its DC content.
– If the average = 0 it is a pure AC signal
– If all instantaneous values are equal to the
average it is pure DC
– Otherwise the average will constitute the DC
content of the signal

Rectified mean value

T
∫
1
x = x
| | –– | ||dt
(t)
T
0
The rectified mean is the average of the absolute
values. The absolute values are derived by
û
0
IuI
0
24
Subject to change without notice
P
S
Q
t
t
rectifying the signal. In general the rectified mean
is calculated by integrating the absolute values
for a period T.
In case of a sine wave u(t) = û sin
mean will amount to 2/π = 0.637 of the peak value
according to:
T
∫
1
û sin ωt
IuI =
–– |
T
0
Root-Mean-Square value (RMS)
The quadratic mean value of a signal is equal to
the mean of the signal squared integrated for a
full period
T
1
∫
2 2
x = x dt
––
(t) (t)
T
0
The rms value is derived by calculating the square
root
1
∫
T
2
x = x dt
––
rms (t)
T
0
The purpose of the rms value was to create a
value which allows the use of the same formulas
as with DC for resistance, power etc. The rms
value of an AC signal generates the same effect
as a DC signal of the same numerical value.
Example:
If an AC rms signal of 230 V is applied to an
incandescent lamp (purely resistive at 50/60 Hz)
the lamp will be as bright as powered by 230 V
DC.
For a sine wave u(t) = û sin ωt the rms value will
be 1/√2 = 0.707 of the peak value:
1
T
∫
U = û sinωt
–– ( )
T
0
U
eff
0
ω
t the rectified
2
û = 0,637û
| dt = ––
π
û
2 = 0,707û
dt = ––
2
2
u (t)
u(t)
t