On a Method for Measuring Pulse Energy
Abstract Generated abstract
The paper proposes a method for determining the energy of short electrical pulses using the relationship between a Laplace transform and the reading of a simple electrodynamic meter circuit. It shows that, for exponentially varying pulse voltages and for double exponential impulse waveforms common in high current practice, pulse energy can be expressed through the operational image of the current evaluated at the relevant damping coefficients. The method uses calculated or oscillographically determined current functions, with a worked procedure for currents represented as a sum of two exponentials whose coefficients are obtained from an oscillogram. Its applicability is limited to linear or conditionally linearized pulse circuits, and the author notes that accurate determination of damping coefficients is essential to avoid large errors.
Full Text
On One Method for Measuring Pulse Energy
T. Yu. Mogilevskaya
(Presented by Professor, Doctor A. A. Vorob’ev.)
The Laplace transform \(F(p)=\int_0^\infty e^{-pt} f(t)\, widely used for determining the operational images of functions in the calculation of transient processes, can be represented visually by means of a very simple circuit [1], shown in Fig. 1. The operational image of the voltage \(u(t)\), which is a function of time:
\[ U(p)=\int_0^\infty e^{-pt}u(t)\,dt. \tag{1} \]
If the voltage winding of the meter is connected to the voltage under investigation, and the current winding—to the discharge circuit of the capacitor \(C\), then the meter reading will be equal to:
\[ W=\int_0^\infty i_1 u(t)\,dt. \tag{2} \]
For \(U_1\), numerically equal to \(R\), the capacitor discharge current will be:
\[ i=\frac{U}{R}e^{-\frac{t}{RC}}=e^{-\frac{t}{RC}}=e^{-pt}, \tag{3} \]
where \(p=\frac{1}{RC}\) is the damping coefficient.
The meter reading in this case will be:
\[ W=\int_0^\infty e^{-pt}u(t)\,dt. \tag{4} \]
On the basis of equalities (1) and (4),
\[ W=\int_0^\infty e^{-pt}u(t)\,dt=U(p), \tag{5} \]
which makes it possible to check the correctness of the transform found for any form of the original function by substituting the quantity \(\frac{1}{RC}\) into the found transform in place of the operator \(p\). The numerical value obtained by such substitution must prove equal to the reading of the meter connected according to the circuit of Fig. 1.
Using the indicated relationship between the energy and the operational image of a function, the inverse problem can also be solved—the determination of the energy
of the pulse by means of the Laplace integral, since the energy of a single short pulse often proves to be very small, and it is difficult to measure it not only with an ordinary meter, but even by more refined methods (for example, by means of thermistors). Determining the pulse energy from oscillograms of voltage and current [2] requires either a two-beam oscillograph or repetition of the pulses, which is not always convenient.
The proposed method is as follows:
If the voltage of the pulse under investigation varies according to the law
$u(t)=U_0 e^{-at}$ (the voltage between the electrodes after breakdown of the spark gap varies almost in this way), then the energy of this pulse is
\[ W=\int_0^\infty u(t)i(t)\,dt =U_0\int_0^\infty e^{-at}i(t)\,dt =U_0 I(p)_{p=a}. \tag{6} \]
In high-current practice (for insulation testing, discharge investigation, etc.) a pulse with exponential fronts of the type
\[ u(t)=U_0\left(e^{p_1t}-e^{p_2t}\right). \tag{7} \]
is usually used, obtained from an impulse-voltage generator.
In this case, considering the circuit parameters to be linear, one may use the superposition principle and apply the Laplace transform to each of the component functions in turn. The energy of such a pulse is determined by the expression:
\[ W=U_0\left[I(p)_{p=-p_1}-I(p)_{p=-p_2}\right]. \tag{8} \]
Fig. 1. Diagram of an electric circuit for a visual representation of the Laplace transform.
The current curve $i(t)$ may be obtained by calculation or taken from an oscillogram and expressed by an analytic function, which presents some difficulty, but is almost always possible.
If the form of the mathematical expression for the dependence $i(t)$ is unknown, it is first necessary to choose the form of a formula satisfying the experimental data [3], and then determine the coefficients entering into it. In this case the formula obtained is purely empirical. If, however, as a result of previous calculations or investigations, the character of the mathematical relation between current and time has been determined beforehand, then it remains only to find experimentally the coefficients of the equation relating them (a semi-empirical formula is obtained).
In a number of cases (in particular, with linear parameters of the impulse generator and of the object to which the pulse is applied, and in the absence of inductance), the curve $i(t)$ proves to be the result of superposition of exponentials with the same damping coefficients as the voltage components. In a particular case the mathematical expression for $i(t)$ may have the form:
\[ i(t)=Ae^{p_1t}+Be^{p_2t}. \tag{9} \]
The coefficients $A$ and $B$ may be determined from the oscillogram, the form of which is shown in Fig. 2.
The initial value of the current
\[ i(0)=m_I N, \tag{10} \]
where $N$ is the maximum ordinate of the curve,
$m_I$ is the current scale.
The time $t_1$, when the current $i=0$, can be determined by recording on this same oscillogram the voltage from a calibrated high-frequency generator.
The last curve will not only give the time scale, but will also make it possible to draw the zero line (the time axis) with sufficient accuracy.
To determine \(A\) and \(B\), two equations are sufficient:
\[ A+B=i(0)=m_lN; \tag{11} \]
\[ Ae^{p_1t_1}+Be^{p_2t_1}=0. \tag{12} \]
From equation (12),
\[ B=-Ae^{(p_1-p_2)t_1}. \tag{13} \]
Then
\[ i(0)=m_lN=A[1-e^{(p_1-p_2)t_1}]. \tag{14} \]
From (13) and (14) we obtain the expressions for the coefficients:
\[ \left. \begin{aligned} A&=\frac{m_lN}{1-e^{(p_1-p_2)t_1}},\\[6pt] B&=-\frac{m_lN}{1-e^{(p_1-p_2)t}}\,e^{(p_1-p_2)t_1}. \end{aligned} \right\} \tag{15} \]
As is evident from (15), to determine \(A\) and \(B\) it is sufficient to know only the difference of the damping coefficients of the component exponentials.
Fig. 2. Approximate form of the current oscillogram.
The damping coefficients \(p_1\) and \(p_2\) may be determined by calculation from the parameters of the pulse generator, as the roots of the characteristic equation, or from a previously recorded oscillogram \(u(t)\). With an unchanged pulse shape, these coefficients do not depend on the voltage on the discharge capacitance of the pulse generator.
It should be noted that the proposed method requires an accurate determination of \(p_1\) and \(p_2\), since otherwise it gives a large error. In addition, if the roots \(p_1\) and \(p_2\) differ very greatly from one another (by a factor of 100 or more), then it is better to carry out the calculation not with a logarithmic slide rule, but with an arithmometer.
The proposed method for measuring pulse energy is applicable in linear or conditionally linearized pulse circuits. For a not complicated dependence \(i(t)\) of type (9), this method gives a simpler solution of the problem than exact calculation or multiplication of separately recorded oscillograms \(u(t)\), \(i(t)\).
References
- Gauster V. F., On the Visualization of the Laplace Transformation, Elektrotechnik und Maschinenbau, 71, No. 9, 236, 1954.
- Stekolnikov I. S., Pulse Oscillography and Its Application, Academy of Sciences of the USSR, 1949.
- Blanter M. E., Methods for Investigating Metals and Processing Experimental Data, Metallurgizdat, 1952.