Capacitors
September 2000 — Greenwood SC
Harmonics and Capacitors —
Dynamic Overvoltages
KILOVAR BRIEFS
Issue 3 of the Kilovar Briefs discussed steady state
sources of harmonics. These included arc furnaces,
solid state energy conversion equipment, rotating
machinery and transformers. During steady state operation of an AC power system, harmonics in the transformer magnetizing current are a small percentage of
the load current. Energizing a transformer results in
substantial amounts of harmonic current in the transformer inrush current. This is a transient event lasting
up to a few seconds. If the transformer is switched with
capacitors, it is possible that a resonant condition may
exist that results in dynamic overvoltage caused by an
interaction of the inrush current and capacitors.
Dynamic overvoltages are defined as long-term resonant overvoltages lasting many cycles and even
seconds that can result in damage to capacitors, transformers, surge arresters and/or solid state power
conversion equipment. In this issue of the Kilovar Briefs,
we will examine the mechanism of creating dynamic
overvoltages and the means to prevent them from
occurring when switching capacitors.
Consider the basic circuit shown in Figure 1, which
includes a supply transformer feeding an industrial load,
a switch used to energize the load and a load transformer. Often, these types of systems are energized
with no load applied to the transformer. Figure 2 shows
a typical inrush current wave form following transformer
energization. This is a non-sinusoidal wave that exponentially decays to a steady state magnetizing current
level.
Typically, inrush current is rich in second, third, fourth,
and fifth harmonics. A Fourier analysis of the harmonic
components of this wave form is tabulated in Table 1.
The exact shape and magnitude of the inrush current
depends on the transformer characteristics and the
initial condition of the residual transformer flux.
Substantial low order harmonic currents momentarily
exist in the transformer inrush current. Figure 3 shows
the bus voltage on the load side of the switch. Despite
the harmonic content of the current, the voltage is not
noticeably distorted.
6
Figure 1.
Typical Industrial Load Transformer.
A
M
P
S
0.00
TIME
400 MSEC
Figure 2.
Typical Inrush Current Wave Form Following
Transformer Energization.
2.00
PU
-2.00
0
TIME
400 MSEC
Figure 3.
Bus Voltage Following Switching in Figure 1.
TABLE 1
Harmonic Components of Inrush Current in Transformer
Energization
Harmonic
% of Fundamental
Amps
0
64.70
275.0
1
100.00
425.0
2
25.40
108.0
3
8.49
36.1
4
5.46
23.2
5
3.32
14.1
September 2000 • Supersedes 11/85
Printed in USA
1
Harmonics and Capacitors — Applying Capacitors in the Harmonic Environment
In Issue 5 of the Kilovar Briefs, a filter design was
discussed as a measure for preventing steady state
harmonic resonance caused by the fifth harmonic
content of the steady state load current. The equivalent
circuit and the impedance diagram is shown in Figure 6.
The presence of the filter reduces the system parallel
resonant frequency from 310 hertz to 211 hertz (3.5 x
the fundamental frequency). A filter always forces the
parallel resonant frequency to a level below that of the
filter tuning frequency. Since the transformer inrush
current is richer in the lower order harmonics, applying
a filter circuit can actually make the switching problem
more severe. Figure 7 shows the dynamic overvoltage
in this case is more onerous than without the filter.
We, therefore, have the situation where capacitors can
be applied successfully in a steady state harmonic environment, possibly requiring the use of a filter reactor to
detune the circuit. However, the design may not be
adequate for the transient harmonic environment;
caused by switching the capacitor and transformers
together.
Z (n)
S
I (n)
S
I (n)
I (n)
F
Z (n)
H
F
Figure 4a.
Equivalent Circuit for Power Factor Connector
Capacitor.
12
Parallel Resonant
Frequency
8
Impedance (Ohms)
In Issue 4 of the Kilovar Briefs, we discussed this same
circuit. A shunt capacitor was applied on the converter
bus to correct the system power factor from 75% to
98%. The equivalent circuit is shown in Figure 4 along
with its impedance spectrum. This circuit is resonant at
310.2 hertz. This means that any fifth harmonic current
present in the circuit will be substantially amplified,
resulting in a resonant overvoltage. The inrush current
shown in Figure 2, whose harmonic components are
shown in Table 1, is the current injected into the circuit
from the transient current source. The fifth harmonic
content of the inrush current results in a resonant
dynamic overvoltage. An example of the dynamic overvoltage is shown in Figure 5. This overvoltage condition
is a direct result of injecting a harmonic current into a
tuned tank circuit, which amplifies harmonic currents at
the timed frequency, resulting in the high overvoltage. If
the capacitor had tuned the circuit to the 4th harmonic,
the overvoltage would have been more severe.
ZS(n)
4
0
-4
-ZS(n)
ZF(n)
-8
-12
1
2
3
4
5
6
7
8
Harmonic Number
Figure 4b.
Impedance Spectrum.
1.50
PU
-1.50
0
TIME
400 MSEC
Figure 5.
Dynamic Overvoltage across Capacitors Following
Transformer Energization.
2
KILOVAR BRIEFS
Z (n)
6
A
S
I (n)
S
I (n)
I (n)
F
Z (n)
H
B
L
Figure 6a.
Equivalent Circuit for Tuned Filter.
Figure 8.
Typical Industrial Load with Capacitor Filter.
8
There is a simple solution to this problem — switch
the capacitor and/or the filter circuit separately from
switching the transformer. The network configuration for
this solution is shown in Figure 8. For energizing the
circuit, switch A should be closed to energize the load
transformer, followed by closing switch B some seconds
or minutes afterward to energize the capacitors and/or
the filter network. By following this simple switching
scheme, the capacitors are never in the circuit during
the time that the inrush current is present. As a result,
the circuit is not tuned to a frequency present in the
transformer inrush current and, therefore, there is no
chance of a dynamic overvoltage occurring!
Impedance (n’s)
4
ZS(n)
0
-ZS(n)
-4
ZF(n)
-8
Parallel
Resonant
Frequency
-12
1
2
3
4
5
Harmonic Number
6
The distribution engineer reading this discussion may
wonder if reclosing a distribution feeder with its capacitors and transformers could also result in dynamic overvoltages. The resonant frequency of the circuit is
analyzed in a manner similar to that already described.
With the capacitors in service, it is likely that the system
resonant frequency is 300–500 hertz. Fortunately, there
is normally adequate load on the distribution feeder and
this load acts as damping, preventing dynamic overvoltages from occurring. The combination of switching an
unloaded transformer and a capacitor bank is most
susceptible to dynamic overvoltages. The lower the
system natural frequency is, the more likely is the
chance of dynamic overvoltages occurring following
switching.
Figure 6b.
Impedance Spectrum.
2.00
PU
-2.00
0
TIME
400 MSEC
Figure 7.
Dynamic Overvoltage across the Capacitor Bank
following Switching of the Filter and Load
Transformer Together.
3
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Waukesha, WI 53187
http://www.cooperpower.com
©2000 Cooper Industries, Inc.
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