**1. Introduction**

The measurement of high-energy current pulses is performed for lightning current measurements, measurement of partial discharges, measurements of the parameters of high voltage and high current generators. These pulses are generated for electromagnetic compatibility verification and testing in order to evaluate the system or equipment shielding effectiveness and its resistance to a high electromagnetic field [1,2]. These pulses are also generated and measured in magnetic flux compression generator tests where very high current values, up to 1 MA, are used to generate electromagnetic fields capable of damaging electronic devices [3,4]. The same is for Marx generators where very fast voltage pulses ranging up to 1 MV are generated, and the energy achieved allows for electronic equipment malfunction [5–7]. These high value pulses can be measured with the use of transducers utilizing the optical Faraday effect [8], Ampere's law by means of Rogowski coil and electromagnetic field probes [9].

In most cases mentioned above, both high voltage and high current measurements are performed indirectly by means of E and H field probes. This approach increases the safety of the measurement as there is a galvanic isolation of the measuring circuit and the tested circuit. On the other hand, it becomes necessary to perform additional conditioning or processing of the measurement signal. Among other popular methods used, there is the integration of measurement signals. It is performed when inductive or capacitive sensors are used to measure the field. The voltage induced at the coil terminals is directly proportional to the derivative of the magnetic field. For the capacitive sensors on the other hand, the current induced in the capacitor circuit is directly proportional to the derivative

**Citation:** Jó´sko, A.; Dziadak, B.; Starzy ´nski, J.; Sroka, J. Derivative Probes Signal Integration Techniques for High Energy Pulses Measurements. *Energies* **2022**, *15*, 2244. https:// doi.org/10.3390/en15062244

Academic Editors: Marcin Kami ´nski and Angel A. Juan

Received: 27 January 2022 Accepted: 14 March 2022 Published: 18 March 2022

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of the electric field. They require integration in both cases. The problems are presented in [10,11]. Moreover, in [12], Marconato et al. discuss the possibility of using different types of analogue and digital integrations in a circuit for the magnetic field measurement in a plasma machine. The proper selection of the integration interval plays a very important role, which was described in [13–15] and illustrated with an example of the Analog to Information (A2I) converter pre-integrator. On the other hand, an easy to implement algorithm using a second-order generalized integrator to control an induction motor is presented in [16].

Numerical integration brings the risk of accumulation of mean value present in the processed signal, which is manifested by the occurrence of a significant drift in the integrated output signal. The proposed methods described in the measurement instruments documentation work well in practice for periodic, stationary signals. For single pulse and in particular, floating signals, numerical integration with these methods often does not give good results.

The main goal of the work is to develop a method of numerical integration of signals which gives results comparable to analogue (hardware) integration (figures in the article). An additional goal is to develop a method not demanding computing power, so that it can be efficiently carried out directly on the oscilloscope (not always equipped with dedicated software and high computing power). For this reason, the starting point is the fundamental method of determining the mean value for the entire signal, the method commonly given in the documentation. For this method it is assumed that the mean value is constant all over the acquired signal. The Authors' proposition is to split the signal into segments (dependent on the signal form) and independently compute local mean values applied in the following signal processing. The digital filtering tool is also not excluded from the research field. An additional reason is the fact that digital filtering can also be used to remove slow varying signal frequency components closely related to the local mean values.

Thus, in the area of electromagnetic field measurement, the need to integrate a derivative signal is quite common, but implementing the appropriate integration method for a particular measurement case is not straightforward. Our paper focuses on discussing the basic methods of integrating the signal from E and H field probes with special emphasis on regions of averaging introducing significant differences in the case of numerical integration. The paper is composed as follows:

