Download Normalized calculation of impulse current circuits for given impulse

Normalized calculation of impulse current circuits
for given impulse currents
HAEFELY
Translated from Bulletin ASE. Bd. 67 (1976) 22
M. ModruSan
El-34
Normalized calculation of impulse current circuits
for given impulse currents
by M. ModruSan
With suitable normalization measures, the major parameters of the current impulses are illustrated commonly and without discontinuity
both in the dampingperiodic and in the aperiodic ranges. A survey of the correlations between analytical and technicalparameters of the
current impulse Cfront time and time to half-value) is given. Additionally, an illustration of the effect of circuit element changes on
amplitudes and impulse duration makes it possible to rapidly modtyy impulse circuits for changed impulse data.
1.
Introduction
Rapidly changing currents of high amplitudes are being used
more and more often for various testing and research purposes.
For practical reasons. the current impulse is usually predetermined
with technical parameters, i.e. with the front time and time to halfvalue as well as with the amplitude (fig. 1) [ 1: 2 I ‘).
A simple series oscillating circuit (fig. 2) with generally known
analytical equations for the equalizing current is usually used to
generate such current impulses. When designing an impulse
current circuit for a predetermined current impulse, it is first
necessary to determine the correlation between the technical and
analytical characteristics of the respective current impulse. This
correlation is relatively easy to find with numeric calculations. For
practical applications, however. the form in which this correlation
is presented can have major repercussions. The objective of this
paper is to provide a generally applicable method suitable for practical purposes as well as calculating fundamentals for the impulse
current circuit. Some indications are already given in 131.
2. Normalization of the analytical expressions
To permit the normalization of the impulse current equations
and of the expressions derived from these equations. reference
values were introduced for current, time. damping, etc. (appendix
1). These reference values are all characteristics of the oscillating
circuit.
In a circuit according to fig. 2, the form of the equalizing current
will be closely correlated to the degree of damping. For this reason,
the relative damping R, = R/Rap defined in equation (1 A) is very
important. The current pulse is damped periodic for 0~ R, <l and
aperiodic for R, 2 1, whereby the aperiodic borderline case occurs
with R, = 1. Accordingly. the circuit would be non-attenuated in
the even that R, = 0.
In all following considerations, the relative attenuation R,
introduced above will be regarded as an independent variable
which yields considerable advantages in the common representation of both ranges of current oscillation (periodic and aperiodic
ranges). All further characteristics of the current impulse can then
be represented as functions of the relative damping thus in
traduced.
When inserting normalized quantities into the generally known
current equations (appendix 2) they are also obtained in a
normalized form (appendix 3). Apart from the normalized time
tn, the shape of the normalized current i, is then only dependent on
the normalized damping Rr. The other characteristics ofthe current
impulse, such as the efficiency q
rl =im/Im
Fig. 1
Current pulse as a periodically damped oscillation (1).
Comparison with non-damped (2) and aperiodicaUy damped
oscillation (3)
Tf, T,
im
il
Front time, time to half-value
Current amplitude
Overshoot
(1)
as well as the peak time t,,, power P,, the voltage drop across the
resistor U&r, and current overshoot il, are pure functions of relative
attenuation RI. This fact makes it possible to graphically illustrate
these magnitudes simultaneously (figs. 3 and 4). The fact that all
these quantities continuously run from the periodic into the
aperiodic range is of special interest.
The course of power P, at the current peak exhibits an extreme
at a damping of RI = 0.553 which according to equation (27 A) is
P,, = 0.2992~0.30.
Relative overshoot il, decreases very rapidly as the damping
increases. Obviously, damping Rr = 1. i.e. the aperiodic critical
oscillation, is not an economical solution whatsoever because the
efficiency q is only 36.8%. If a negligible overshoot of approx.
1.5 % is permitted (R, = 0.8), a current amplitude which is approximately 15% higher can be obtained with otherwise unchanged
conditions.
With the known efficiency n, the peak current value im in a given
circuit can be calculated as
im= r) U/Z
(2)
according to fig. 2 with equations (1) and (5 A).
This expression is true for any damping included in q.
3
3. Correlation between technical and
analytical characteristics
As already mentioned, the design of an impulse current circuit
for a predetermined impulse also requires the correlation between
characteristics Tfand Tt (fig. 1) referred to as technical characteristics and the corresponding analytical characteristics of the current
impulse such as damping R, and the circuit time constant T = m
This correlation is shown here in a normalized form and times
II”, tzn and txn in accordance with lo%, 90% and 50% of the
current amplitude were obtained with a numerical approximation
method for current shape according to equations (21A) and (22A).
Thereby, the damping was varied in steps of AR, = 0.05 as a parameter. This yielded a normalized front time of
Tf” = Tf/T = 1.25 (tzn - tin)
(3)
and a normalized tail half-value time of
Tt = Tt/T =
t3n
as a function of
related to the
oscillations (R,
normalization is
+ 0.125 (tz, - 9 trn)
(4)
the corresponding damping RP If these times are
corresponding times for undamped current
= 0), a more easily surveyable form of time
obtained with
Tfn = Tfn/Tfn co,;
Tfn = TtnlTtn to)
(5)
as well as for the peak value time with
tdn = tmnltmn(01.
The three normalized times for undamped oscillation are
Tf,(o)= 1.275; Tt,(o)=2.645; tmn(q= 1.571.
Figure 4 shows the new relative times according to this
definition plotted against damping again. The front or peak value
times become shorter as damping increases while the tail time to
half value has a minimum for an damping R, = 0.42 approximately
4.2% below the reference value at R, = 0.
In practical calculations, times Tf and T will normally be given.
To simplify matters, figure 6 shows the main parameters plotted
against the time ratio TdTc Remarkably enough, the curve of
attenuation R, exhibits a relatively weak slope so that expression Tt/Trcould also be regarded as a measure of attenuation.
4. Calculation procedure
As already mentioned, the design of an impulse current circuit
will usually depart from the given data of a current imp‘ulse Tf,
Tt and im. A further variable which will usually be the charging
voltage Umust also be selected. Often, however, the inductance of
the test specimen will be given so that together with the normally
known inherent inductance of the impulse circuit, the entire inductance L will already be defined. Thus, two calculating methods are
distinguished.
4.1 Charging voltage U predetermined
For the predetermined impulse Tf, Tt and i,, figure 4 will yield
attenuation Rr, efficiency n and the normalized front time Tf,.
Before carrying out any further calculations, the relative overshoot
il, should be checked in figure 3. If required, damping Rr is increased to such an extent that overshoot will remain within a certain absolute range. For the new R, fig. 5 will show the new time
ratio Tt’/Tf’. If the change
Tt’ITf’
___ = 1 -1 2a,
Tt/Tf
0
Fig. 3
A
Efficiency Q, voltage drop URN, power P,and overshoot
il, in relation to damping R,= R/Rap
Fig. 4
03
I
130
Rr -
135
Influence of damping on front time T’f, front time f’,,,” and time to
half-value T’,,
411 values are related to the undamped current oscillation
290
the new times will be
Tt’ = Tt (1 + a) and Tf’ = Tf (1 - a)
(7)
I31.
With the data thus obtained, all circuit elements can be deter
mined. Thus, the time constant (7A) is obtained as T = Tf/Tf”
from equation (3) and the circuit impedance is found with equation
(2). With these two parameters, it is now possible to define C and
L: equations (3A) and (7A) provide
C = T/Z and
03)
L = T2/C.
(9)
t OS8
R,
036
Finally, equations (1A) and (2A) yield damping resistance
R = 2 Rr Z.
(10)
4.2 Inductance L predetermined
Fig. 5
If the inductance Lo of the test specimen is given, the inductance
of the entire impulse circuit will be L = Lo + L, for a known
inherent inductance L, of the test circuit. On the other hand, the
largest current slope for a current impulse of given form (i.e. damping, duration and current amplitude)
u
=-=jy
L
(11)
is also given as an impulse-related constant; the required charging
voltage is thus directly proportional to the inductance.
For K, equation (2) yields
This constant can be derived from the known data Tf, T, and
i,,, via Tf, and q from fig. 5, and again, a temporary correction of
times Tfand Tt should be considered for smaller overshoot values.
Figure 6 shows a new type of impulse current generator. The
capacitor battery consists of 8 individual capacitors which are
located in a semi-circular fashion around the spark gap. This
provides complete symmetry of the individual circuits as well as
minimum inherent inductance. Due to the substantial flexibility of
the individual circuit elements, this generator can be used to
produce more than 100 different current impulses and as an additional feature, the current amplitude can be adjusted from 5 to
100% of the rated value.
Efficiency q. normalized damping R,and normalized front time Tfn
in relation to time ratio T,/Tf
5. Matching the impulse circuit
to changing current impulse data
In the impulse current generators built, it is often necessary to
change the circuit elements C, L, and R in such a manner that a certain new current impulse with a desired amplitude or duration is
obtained.
To simplify this task, the change of impulse characteristics
(amplitude and front or time to half-value) were determined in
conjunction with the change of only one circuit element. Appendix
4 lists the equations for such changes. Additionally, they are
graphically illustrated in figures 7 to 10. Figure 7 shows the effect
of each of the three circuit elements on the current amplitude with
initial attenuation R,I as the parameter. With the specified
attenuations 0.4 < R,l < 1.2, it is possible to review practically
all possible current impulse cases. In figures 8 to 10, the relative
changes of the front times and time to half-value are also plotted
against the individual circuit elements. With these curves, it is
possible to rapidly and surveyably implement circuit corrections
for new impulse data.
4.3 Approximation equation
Most current impulses are selected in the damped periodic current
range due to the higher degree of efficiency. Based on relations
T’f, =fl (R3 and T;, = r2 (R,) shown in figure 4, the following
approximation equations can be established:
Tt
Ti” ~
Ttn IOI e 2 05 eo.6’ Rr
-=~.
Tf
7% Tfn (0) ’
T811 a l/(0,78 + 0,62 R,)
(13)
whereby the deviation against the accurate values of T,/Tf and
Tf” remains below 1% throughout the entire periodic range.
Together with the other analytical expressions, the impulse current
circuit can therefore also be designed on a purely mathematical
basis.
Fig.6 Complete lUOi;V. 8JI:J impuis~ CUXSIII genunioi
5
1,: 5--
I,( )--
OS: 5--
I
'b/R,-
12/h- Cz/C1-
0
1
2
4
45
OS2
-Fig. 7 Effect of changes of circuit elements on current amplitude. Initial
damping = parameter
I
02
Fig.8
05
'
Rz/%-
1
2
f
4
Effect of resistance changes on immdse duration. Initial damoing =
parameter
I
I
I
In
2,5
42
Fig.9
6
Oj
i
i
4
Effect of inductance changes on impulse duration. Initial damping =
parameter
Fig. IO
Effect of capacity changes on Impulse duration. Initial damping =
parameter
Appendix 1: Normalized quantities
Designation
Deliniton
a) Normalized damping
- Damping resistance for
aperiodic critical
oscillation
- Circuit impedance
R,
= RIRm
(1A)
RW
Z
=2z
G’N
=1/L/c
(3N
b) Normalized current
- Amplitude of the nondamped current oscillation
in
= i/In,
(4‘4)
I n,
= u/z
(5N
c) Normalized time
- Time constant
t”
= t/T
(W
d) Normalized maximum power
- Maximum power
- Characteristic power
e) Normalized voltage drop
across resistor
- Voltage drop across
resistor
T
= ]‘E
A
P”
= i/P,
j = i,,,zR
Pk
= (/2/z
URrn = Un/U
(7.4)
cm
VA)
(‘ON
(“A)
(17-4)
f) Relative current overshoot
U3N
Appendix 2: Analytical expressions
Quantity
Oscillation range
Damped period (0, R, _ 1)
-
aperiodic CR, ,I )
ii = R/2L (14A)
Damping factor
Damping criterion
6” < l/LC
62 > IILC
Current course
I = II e-St sm 01t
(15.4)
Current constant
II =$
(16‘4)
Circuit frequency
0, =
Peak time
1
tm - -aarctan~
01
6
1 - 6”
_
I/ L C
i :
21:e-%inh(t1/6?)
12-
(17A)
f&g
(18‘4)
(19A)
t”, _ In (VU3 - 1 + 6 VLC)
J
,ppendix
(16.4)
(2’3-4)
68- &
3: Normalized analytical expressions
Current course
in = v,e~~r2
Peak time
r 111” =
Efficiency
3 = exp
arctan 1/( 1 - Rr2)/Rr2
\!l - Rr”
i
Power
sin (tn 1’1 - Rr3)
Relative
overshoot
(23A)
R,
- -=
arc sin vF%)
vl - Rr2
mtn, = 2 R,.exp
Rr
l/l - Rr2
arc sin 1’1)
PA)
ilr =exp
-
e-R&
i,, = 7 sinh (tn i/R,?--)
1, R,” - 1
r.,n _ ln (RI + 1/R,” - 1)
j/R,” - 1
?/ = (Rr + I/&? - ’ )-Rrh’I;, =
2
R, (R, + 1/j.&’ _ , )--2dl’Rr2-I-
j
(27~)
-
t
(25A)
2 Rr
,,’ _ R,s arcsi” 1” - Rr”
in = 2 R, . exp
t
Voltage drop
across resistor
(2lA)
R Rr
1/l - Rr2
(31N
UR,n zz
2
R, (Rr + ~‘Rr?--I)-Rr/~Rr”-l
W’A)
(24A)
26A)
WA)
(3W
Appendix 4: Change of impulse characteristics when changing only one circuit element
Change of
circuit element
R
(C, L = const)
Changing
impulse
characteristics
Damping R,?
Amplitude &,2/i,,,{
R2
Rrl RI
6-z
-G
Front time Tn/Tfl
1
Time to half value Ttl/Ttt
T811.2
TS.,l
Trn.2
TXl,l
& L = const)
fk, C = const)
Literature
Address ofthe author
[l] Erzeugung und Messung van Hochspannungen. Teil 3: Besttmmungen fiir die
Erzeugung und Anwendung van StoDspanmmgen und StoBstrBmen fiir Priifzwecke. VDE-Vorschrift 0433 Teil 3/4.66.
[Z] Techniques des essais B haute tension. Zepartie: Mod&es d’essais. Publication
de la CEI No G--2(1973).
[3] M. ModrAm: Realisation of the prescribed exponential impuls currents for
different kinds oftest samples. In: Internationales Symposium Hochspannungstechnik, 9. . . . 13. September 1975, EidgenGsische Technische Hochschule,
Ziirich. Ziirich, SEV, 1975; Bd. 1, S. 155...160.
Dr. Martin Modmfan, Emil Haefely. Cie AC?. 4028 Basel/Switzerland
High
vontol~e
&i%
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systems
HIGH VOLTAGE TEST SYSTEMS
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Printed I” Switzerland
1 1977 2 0 0 0
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