*2.2. Signal Integration Methods*

The following methods can be used to integrate the measurement signal:


Hardware integrators are built with RC circuits. The advantage of these integrators is the operational simplicity. This approach gives a signal directly at the output of the integrator that is proportional to the measured quantity, in this case the magnetic and electric fields. The output signal is not enormously lagged behind the raw signal before integration and can be directly recorded with an oscilloscope. This allows the operator for quick reading of the fundamental parameters of the measured field such as: its amplitude, frequency, rise and fall times or pulse length.

The condition for the correct integration of the signal is the preservation of time parameters of the signal and integrator. It should be remembered that the integration interval (0-t) should be shorter than the time constant τ = RC. This is a disadvantage of hardware methods, because the time parameters of the signal (field) being measured, must be known before the measurement. This makes it necessary to have a set of multiple integrators with different time constants ready when starting an experiment for which the frequency of the signal (field) is not known at all. The application of passive analogue integrators is relatively easy. Nevertheless, one should remember several practical aspects, which are already pointed out by the probe manufacturers themselves [17]. First, the integrator should be connected directly to the oscilloscope's input without any additional connecting cables. This is shown in Figure 1c (recommended connection). Laying the lead wires, one should remember to avoid, as far as possible, any bends in the cable. Manufacturers of probes do not recommend using e.g., angled BNC connections. It is also probably dictated by the fact that in the case of bent connections it is difficult to maintain constant geometry of the circuit and consequently impedance parameters. It is especially important at high frequencies. It should also be kept in mind that correct integration results are easier to be obtained when the entire measurement path is at a common ground reference potential. Therefore, a semi-rigid cable is commonly used in this type of measurements. However, it is not recommended to use passive analogue integrators in a measurement system with the optolink connections. This is mainly due to the low voltage range of the optoelectronic circuit and noticeable noise deteriorating the measurement signal. In this case, one should use numerical integration directly in the oscilloscope or perform that stage in the post-processing activity.

**Figure 1.** Probe manufacturer recommendations for the passive integrator connection: (**a**,**b**) not recommended, (**c**) recommended connection.

Numerical integration is the approximate calculation of definite integrals [18]. The methods approximate the integral by using the sum of the values of the function being integrated at several points. To obtain a more accurate approximation, the integration interval is divided into small fragments. The final result is the sum of the estimates of the integrals in each subinterval.

For the numerical integration of signals, the rectangle or trapezoid method can be used, but the latter is more accurate and popular.

In the trapezoid method, the approximation improves because one approximates each of the sub compartments linearly, which can be written as follows (3):

$$\int\_{t0}^{\text{tn}} \mathbf{u}(\mathbf{t})d\mathbf{t} \approx \frac{\text{h}}{2} \sum\_{i=0}^{n-1} (\mathbf{u}(\mathbf{t}\_{i}) + \mathbf{u}(\mathbf{t}\_{i+1})),\tag{3}$$

where n is the number of subintervals of length h.

The use of simple numerical methods is caused by the fact that in case of high sampling frequency of the signal (this is usually the case), the error of integration is relatively small (very narrow subintervals of integration). This gives the possibility of carrying out the integration process directly on oscilloscopes recording the waveforms from measurement equipment. The trapezoid method, though it can be considered basic (as compared for example to Simpson method), is commonly used as an integration formula in oscilloscopes.

Numerical post-processing provides the most tools, methods and opportunities that can be applied (if only necessary) into the integration process. In this case, the signal recorded on the oscilloscope is processed using signal processing dedicated software and powerful computers, after the completion of measurement experiments. A relatively long delay in obtaining results is a disadvantage of this procedure, but the certainty of the process correctness as well as the possibility of performing additional analysis, e.g., spectral or wavelet analysis and additional filtering often compensates for the mentioned disadvantage.

#### *2.3. Practical Problems of the Signal Acquisition*

By definition, the output signals of B-dot and D-dot probes do not contain a constant component. However, this component may appear in the conditioned measurement signal immediately before the functional integrator block. There may be several reasons for this condition. First of all, it can happen as a result of imbalance of the measurement path. Another reason may be the accumulation of electrostatic charges, especially in high-impedance lines (leads of the Programmable Gain Instrumentation Amplifier). In practice, however, a lot depends also on the selection of the oscilloscope recording time (period). Documentation dedicated to measurement instruments commonly gives the integration of a periodic, rectangular signal generated by oscilloscope built-in test generator as an example. In this case, as the signal occupies the entire recording period (window), and especially when the screen presents the whole number of signal periods (with significant amplitude), removing the DC component is an uncomplicated task (Figure 2). In manual mode (convenient for demonstration but impractical especially for a series of measurements), the average value is found using the oscilloscope knobs.

**Figure 2.** Rectangular waveform integration process, (**a**) signal non-zero mean value (purple), integral signal (yellow), (**b**) signal with mean value extracted (purple), integral signal (yellow).

To automate the measurement, it is more convenient to determine the mean value using a dedicated function and subtract it from the samples of the recorded signal before integration. This technique works well for periodic and single shot signals under certain conditions.

For the single pulse signals, it is important to select the recording length as to record the entire phenomenon, possibly without an excessive "zero" space following the occurrence of the phenomenon. In each case, the trigger point is commonly set between the first and second oscilloscope display division of the time base (typical setting). For such a setting, the mean value before the trigger point (pre-trigger zone) for the background signal (zero would be ideal) can be determined. Depending on the approach, the mean value in the post-trigger region where the raw (not yet integrated) measurement signal (recorded phenomenon) is located, can be determined separately. This variant is especially helpful in cases where it is difficult to match the length of the registration window. Moreover, in the case when, apart from the constant component, there are slowly varying components in the signal, high-pass filtering, preceded by a spectral analysis of the signal, it turns out to be useful or even demanded.
