jrWl1t
E N
I1t
2fr-~N
2
f(r
- 1
)fll
WEN
1
~
r.t
~
M, r ;t 1
(1.40)
(1.41)
(1.42)
f(r
+
t)l1tW
E N
l~r,t~M
In order to confirm that the derived conditions (1.40), (1.41) and (1.42) are appropriate for conditions
(1.30), (1.36) and (1.39), it is sufficient to look at relations (1.40), (1.41) and (1.42): here, the indexes
r
and t can be replaced with 2r-1 and 2t-1, for the case in which only odd harmonics (in a complex periodic
voltage) signal are present; these are processed according to the suggested measurement method. With
our forms, the derived conditions are very similar to those provided in the theory
of
synchronous
sampling, the difference being that in the latter it is necessary to fulfil the additional demands related to
the selection
of
parameters LItand
W,
with regard to the delay in processing and the necessary number
of
samples. During this, we must not fail to satisfy the Nyquist criterion, as a basic postulate in a digital
processing
of
analogue values.
Depending on the harmonic content, it will subordinate the error or, more precisely, the form
of
the
delay that will cause the accumulation
of
error.
1.3Mathematical Proofof Correctness in the ActivePowerProcessing
If calculating the average power by the proposed method, let us assume that the system voltage
Vinput(t),
and current
iinput(t)
signals can be represented as a sum
of
their Fourier components as follows:
M
Vinput = VI + .J2VR • L k.sin
(rmt
+ f//
r)
= Vinput
(t),
r=!
M
iinput = I I +
.J2
I R • L 1s sin
(s
OJt + ¢s ) =
(nput
(t)
s=!
(1.43)
(1.44)
where m.=21ifrepresents the angular frequency, krV
R
is the RMS voltage value
of
the rth harmonic, lsI
R
is
the RMS current value
of
the sth harmonic, f//r,
f/Js
are the phase angles
of
the rth and sth harmonic
of
voltage and current, V/, II are the average input voltage and current respectively. M is the number
of
the
highest harmonic, while, for the reasons
of
simplicity, we assume that the voltage and current signals
possess the same number
of
harmonic components. At the same time, we maintain general character in
our approach and conclusions. In real environment, the current signal usually possesses a "richer"
harmonic content. The delay
LItmust satisfy the same demands as in the processing
of
the RMS value
of
the net voltage (LltSf/(2M+1)), the difference being that
if
the current and voltage signals have a different
harmonic content, the conditions for the delay
LIt will determine a signal with a "richer" harmonic
content (which is usually the current signal because
of
its greater dynamics) In other words, we must
satisfy the demand that
LltSf/(2max(M, K)+1), where M is the number
of
the voltage harmonic
components, and K the number
of
current harmonics.
Average power
p is calculated using the following (definition) expression, while the processed power
is defined as
p", in accordance with the suggested measuring method:
T
p =t f
iinput
(I).
Vinput
(I
)d1,T
=]-
o
w
p * = t L
Vinput
(j(NT
+
fll
)).
iinput
(j(NT
+
~/))
j=l
Llt[
s] is the delay in processing presented in seconds, W is the number
of
samples needed for accurate
processing
of
the observed values, based on the suggested algorithm for digital processing.
14