Hardware and Functional Development of Protection Devices and Systems - Digital Signal Processing in Power System Protection and Control - page 21

One may observe that the signal model consists of several terms, including the
fundamental frequency, decaying DC, higher frequency and transient oscillatory
decaying components, where:
I 1m is a magnitude of the fundamental frequency component, I a , T a are initial
value and time constant of the decaying DC component, I km are magnitudes of
higher frequency components, I hm , T h are magnitudes and time constants of
transient oscillatory decaying components.
The useful information is usually contained in the fundamental frequency
component, and sometimes in the selected higher frequency components (2nd, 3rd
or 5th). Therefore, all the other components are to be rejected. Particularly dan-
gerous are components with very high frequencies (close to the sampling fre-
quency) since they may cause irreversible corruption of the digital signal.
Therefore, selection of the sampling frequency f S is a compromise. It must not
be too low, to enable reproduction of the components that are vital for the relaying
decisions. On the other hand, it must not be too high, to avoid unnecessary burden
for the digital processing.
Basically, the minimum sampling frequency results from the Shannon-Ko-
tielnikov theorem that defines conditions for possibility of signal reconstruction
after sampling. According to that, there should be at least two samples of the signal
taken within the period of the signal component that should be represented in
digital form without loss of information about frequency [ 10 , 15 ].
If the component that should be reproduced correctly has the frequency f k then
the sampling frequency ought to be:

f S 2f k : ð 3 : 6 Þ

In real installations the sampling frequency is seldom lower than 800 Hz (16
samples per one period of the fundamental frequency 50 Hz component, 4 samples
per one period of the 4th harmonic, etc.). Contemporary digital protection relays
offer sampling rates up to several kHz. Special solutions, where higher frequency
components are used for generating the trip decision, may have sampling rates in
the range of many hundreds of kHz [ 12 , 13 ].
The sampling process leading to extraction of the sampled values in time
domain has also significant consequences in frequency domain. One can prove that
the spectrum of an analog signal becomes duplicated after signal sampling. The
spectrum of sampled signal is a sum of copies of original spectrum shifted left and
right by a multiple of sampling frequency (or angular frequency), according to:

8
<

9
=

Z

1

X X

1 1

X ð jx Þ¼ 1
T S

¼ 1
T S

x ð t Þ exp ½ j ð x kx S Þ t dt X ½ j ð x kx S Þ :

: ;

k ¼1 k ¼1

1

ð 3 : 7 Þ

This feature of digital spectrum is illustrated in Fig. 3.9 . One can conclude that
the analog signal can be reconstructed from its samples (by using an appropriate