Modern digital oscilloscopes offer a host of analysis capabilities since they digitize and store input waveforms for analysis. Most oscilloscopes offer basic math operations such as addition, subtraction, multiplication, division, ratio, and the fast Fourier transform (FFT). Mid- and high-end oscilloscopes offer advanced math functions such as differentiation and integration. These tools let you solve differential equations that you probably hated in your days as an engineering student. They are used the same way today in your oscilloscope measurements. Here are a few examples of oscilloscope measurements that require differentiation and integration.
Measuring current through a capacitor based on the voltage across it
The current through a capacitor can be calculated from the voltage across it using this equation:
The current through a capacitor is proportional to the rate of change, or derivative, of the voltage across it. The constant of proportionality is the capacitance. A simple circuit can be used to show how this works (Figure 1).
Figure 1 A signal generator supplies a sine wave as Vin(t). The oscilloscope measures the voltage across the capacitor. Source: Art Pini
In this simple series circuit, the current can be measured by dividing the voltage across the resistor by its value. The oscilloscope monitors the voltage across the capacitor, Vc(t), and the voltage Vin(t). Taking the difference of these voltages yields the voltage across the resistor. The current through the resistor is calculated by rescaling the difference by multiplying by the reciprocal of the resistance. The voltage across the capacitor is acquired and differentiated. The rescale function multiplies the derivative by the capacitance to obtain the current through the capacitor (Figure 2).
Figure 2 Computing the current in the series circuit using two different measurements. Source: Art Pini
Vin(t) is the top trace in the figure; it is measured as 477.8 mV RMS by measurement parameter P1, and it has a frequency of 1 MHz. Below it is Vc(t), the voltage across the capacitor, with a value of 380.2 mV RMS, as read in parameter P2. The third trace from the top, math trace F1, is the current based on the voltage drop across the resistor, which is measured as 5.718 mA RMS in parameter P3. The bottom trace, F2, shows the capacitor current, Ic(t), at 5.762 mA.
Parameter P6 reads the phase difference between the capacitor current and voltage traces F2 and M2, respectively. The phase is 89.79°, which is very close to the theoretically expected 90°.
Parameters P7 through P9 use parameter math to calculate the percentage difference between the currents measured by the two different measurements. It is 0.7%, which is respectable for the component tolerances used. Comparing the two current waveforms, we can see the differences (Figure 3).
Figure 3 Comparing the current waveforms from the two different measurement processes. Source: Art Pini
The two current measurement processes are very similar. Differentiating the capacitor voltage is somewhat noisier. This is commonly observed when using the derivative math function. The derivative is calculated by dividing the difference between adjacent sample values by the sample time interval. The difference operation tends to emphasize noise, especially when the rate of change of the signal is low, as on the peaks of the sine wave. The noise spikes at the peaks of the derivative signal are obvious. Maximizing the signal-to-noise ratio of differentiated waveforms is good practice. This can be done by filtering the signal before the math operation using the noise filters in the input channel.
Measuring current through an inductor based on the voltage across it.
A related mathematical operation, integration, can be used to determine the current through an inductor from the integral of the inductor’s voltage.
Another series circuit, this time with an inductor, illustrates the mathematical operations performed on the oscilloscope (Figure 4).
Figure 4 A signal generator supplies a sine wave as Vin(t). The oscilloscope measures the voltage across the inductor, IL(t). Source: Art Pini
The oscilloscope is configured to integrate the voltage across the inductor, VL(t), and rescale the integral by the reciprocal of the inductance. Changing the units to Amperes completes the process (Figure 5).
Figure 5 Calculating the current in the series circuit using Ohm’s law with the resistor and integrating the inductor voltage. Source: Art Pini
This process also produces similar results. The series current calculated from the resistor voltage drop is 6.625 mA, while the current calculated by integrating the inductor voltage is 6.682 mA, a difference of 0.057 mA. The phase difference between the inductor current and voltage is -89.69°.
The integration setup requires adding a constant of integration, thereby imposing an initial condition on the current. Since integration is a cumulative process, any offset will generate a ramp function. The constant in the integration setup must be adjusted to produce a level response if the integration produces a waveform that slopes up or down.
Magnetic measurements hysteresis plots
The magnetic properties of inductors and transformers can be calculated from the voltage across and the current through the inductor. The circuit in Figure 4, with appropriate input and resistance settings, can be used. Based on these inputs, the inductor’s magnetic field strength, usually represented by the symbol H, can be calculated from the measured current.
Where: H is the magnetic field strength in Amperes per meter (A/m)
IL is the current through the inductor in Amperes
n is the number of turns of wire about the inductor core
l is the magnetic path length in meters
The oscilloscope calculates the magnetic field strength by rescaling the measured capacitor current.
The magnetic flux density, denoted B, is computed from the voltage across the inductor.
Where B is the magnetic flux density in Teslas
VL is the voltage across the inductor
n is the number of turns of wire about the inductor core
A is the cross-sectional area of the magnetic core in meter2
The flux density is proportional to the integral of the inductor’s voltage. The constant or proportionality is the reciprocal of the product of the number of turns and the magnetic cross-sectional area. These calculations are built into most oscilloscope power analysis software packages, which use them to display the magnetic hysteresis plot of an inductor (Figure 6).
Figure 6 A power analysis software package calculates B and H from the inductor voltage and current and the geometry of the inductor. Source: Art Pini
The analysis software prompts the user for the inductor geometry, including n, A, and l. It integrates the inductor voltage (top trace) and scales the integral using the constants to obtain the flux density B (second trace from the top). The current (third trace from the top) is rescaled to obtain the magnetic field strength (Bottom trace. The flux density (B) is plotted against the field strength (H) to obtain the hysteresis diagram.
Area within and X-Y plot
Many applications involving cyclic phenomena result in the need to determine the area enclosed by an X-Y plot. The magnetic hysteresis plot is an example. The area inside a hysteresis plot represents the energy loss per cycle per unit volume in a magnetic core. The area within an X-Y plot can be calculated based on the X and Y signals. The oscilloscope acquires both traces as a function of time, t. The variables can be changed in the integral to calculate the area based on the acquired traces:
Note that both integration and differentiation are involved in this calculation. To implement this on an oscilloscope, we need to differentiate one trace, multiply it by the other, and integrate the result. The integral, evaluated over one cycle of the periodic waveform, equals the area contained within the X-Y plot. Here is an example using an XY plot that is easy to check (Figure 7).
Figure 7 Using a triangular voltage waveform and a square wave current waveform, the X-Y plot is a rectangle. Source: Art Pini
The area enclosed by a rectangular X-Y plot is easy to calculate based on the cursor readouts, which measure the X and Y ranges. The relative cursors are positioned at diagonally opposed corners, and the amplitude readouts for the cursors for each signal appear in the respective dialog boxes. The X displacement, the rectangle’s base, is 320.31 mV, and the Y displacement, the rectangle’s height, is 297.63 mA. The area enclosed within the rectangle is the product of the base times the height, or 95.33 mW.
Taking the derivative of the voltage signal on channel 1 yields a square wave. Multiplying it by the current waveform in channel 2 and integrating the product yields a decaying ramp (Figure 8).
Figure 8 The integrated product is measured over one input waveform cycle to obtain the area within the X-Y plot. Source: Art Pini
The area of the X-Y plot is read as the difference in the amplitudes at the cursor locations. This is displayed in the dialog box for the math trace F2, where the integral was calculated. The difference is 95.28 mW, which is almost identical to the product of the base and height. The advantage of this method is that it works regardless of the shape of the X-Y plot.
Practical examples
These are just a few practical examples of applying an oscilloscope’s integration and differentiation math to common electrical measurements that yield insights into a circuit’s behavior that are not directly measurable.
Arthur Pini is a technical support specialist and electrical engineer with over 50 years of experience in electronics test and measurement.
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