Fluke 6105A - Power (P) Accuracy Calculations; Power Example

Introduction
Electrical Specifications 1
1-25
1-60. Power (P) Accuracy Calculations
Real power is the sum of the products of volt/current/phase-angle at each harmonic frequency.
nnn
IVP Φ=
∑
cos
Watts
where n is the harmonic order of the components.
Calculation of power accuracy uses the same techniques shown previously. The uncorrelated uncertainty
components of voltage, current and phase are combined using root sum of squares for each frequency.
222
2
2
)]([]
)(
[]
)(
[
)(
f
f
f
f
f
f
f
phaseu
I
Iu
V
Vu
P
Pu
++=
where
)(xu
is the uncertainty of the component
x
and
phase
is the phase angle between the current and voltage
at frequency
f
. It is easiest to express each of these contributions as ppm.
The contribution of phase angle accuracy varies with the set phase angle as shown below.
Φ
+Φ
−=
cos
))(cos(
1)(
φ
u
phaseu
where
Φ is the set phase angle and
)(
φ
u
is the phase accuracy.
The power uncertainties for each frequency, modified by the appropriate sensitivity coefficient c
i
, are then linearly
summed to give the combined uncertainty u
c
(linearly summed because voltage components are correlated, as are
those of current and phase).
∑
=
=
N
i
iic
PucPu
1
)()(
1-61. Power Example
Voltage channel output is 109 V on the 180 V range at 60 Hz with 3
rd
harmonic at 15 V. The voltage 3
rd
harmonic has
0 ° phase angle relative to the voltage fundamental.
The current channel output is 7 A on the 10 A range at 60 Hz with 3
rd
and 5
th
harmonics at 0.7 A and 0.3 A
respectively. The current fundamental phase angle is 12 ° relative to the voltage fundamental. The current 3
rd
harmonic has a phase angle of +25 ° relative to the current fundamental, for example, the phase angle between the
3
rd
current harmonic and the 3
rd
voltage harmonic is 25 ° + (3 x 12 °) = 61 °. As the current 5
th
harmonic is not
matched by a voltage 5
th
harmonic, there is no 5
th
harmonic power contribution.
Voltage uncertainty values are given in “Voltage and Sine Amplitude Specifications” and “Voltage DC and Harmonic
Specifications”, current uncertainty values are given in “Current Sine Amplitude Specifications” and “Current DC and
Harmonic Amplitude Specifications”. Phase uncertainty values are given in “Current to Voltage Phase Specifications”.
The accuracy used is that of the 6100B.
Converting all values to ppm, accuracy contribution at the fundamental frequency
ppm
V
V
ppmVu 152
109
100044.0
112)(
6
1
=
×
+=
ppm
A
A
ppmIu 198
7
1000024.0
164)(
6
1
=
×
+=
ppmephaseu 1161
)12cos(
)003.012cos(
1)(
1
=×
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−=
Combined accuracy for the fundamental frequency components:
ppmPu 25011198152)(
222
1
=++=
Power in the fundamental frequency:
WattsIVP 3266.7469781476.07109cos
1111
=××=Φ= so: