Oscilloscopes are the workhorse instruments for time domain measurements. Most current digital oscilloscopes include about twenty-five built in measurement parameters as a standard complement. Adding application specific options, the parameter count increase to over a hundred. Even with this plethora of measurement capability there are some measurements that must be derived using the existing measurement tools. One of these is the measurement of the time constant of an exponential signal.

Many physical phenomena related to charging and discharging energy storage devices like capacitors and inductors result in waveforms with exponential rising or falling edges where the exponential time constant reveals information about the underlying process and component values. It is useful to be able to measure the exponential time constant using an oscilloscope to better understand the circuit operation. There is no measurement parameter to directly read out exponential time constant. This article will show how the exponential time constant can be measured both with manual cursor measurements or using the oscilloscope’s signal processing and built-in measurements capabilities to read the time constant directly. Let’s start with a review of exponential signals.

A typical exponential process can be defined by either of these equations dependent on the slope of the exponential:

Rising exponential: V(t) = 1 – a*e ^{-t/}^{τ} + b

Decaying exponential: V(t) = a*e ^{-t/}^{τ} + b

where:

V(t) is the voltage as a function of time in volts,

a and b are arbitrary constants,

τ is the exponential time constant in seconds,

and t is the time in seconds.

**Figure 1** is an example of an exponential pulse showing both rising and decaying edges acquired on an oscilloscope. The constants for the exponentials in this example are a=1 and b=0. The waveform is averaged to improve its signal to noise ratio and improve measurement accuracy.

**Figure 1** Using the oscilloscope’s cursors to measure the time constant of the decaying or falling edge of an exponential pulse.

Considering the decaying exponential equation, when the time t is equal to the time constant τ the value of the voltage is equal to 1/e or 0.368 for the constant a=1 and the constant b=0. This is the key to measuring the time constant on the oscilloscope. By setting the cursors so that they measure a change in amplitude of 0.368 times the constant a the time difference between the cursors is the time constant. In the example the left-hand cursor reads an amplitude value of 860.4 mV. The right cursor is adjusted until its amplitude readout is as close to 36.8% as possible of that value, in this case 317.6 mV. The indicated time difference between the cursors is 100 ns, this is the time constant, τ, of the falling edge.

The time constant of the rising edge can also be determined as shown in **Figure 2**.

**Figure 2** Measuring the time constant of the rising edge of the exponential pulse.

Looking at the equation for the rising edge, at one time constant from the start the voltage value is 1- 0.368 or 0.632 of the maximum value. Again, setting the cursors so that the amplitude difference is 0.632 volts from zero for this 1 V peak signal the time constant is 100 ns. This method uses a traditional technique and can be done on any oscilloscope, it provides reasonable results, but it does require a fair amount of setup. Accuracy is dependent on the user’s ability to set the cursors correctly. It is better to use the oscilloscope’s measurement parameters, if possible, to get the most accurate results.

If the oscilloscope’s available math operations include a natural logarithm function and its measurement parameters include a slope or slew rate measurement it is possible to read the time constant directly.

Taking the natural logarithm of an exponential function results in a linear function whose slope is equal to the time constant of the exponential as seen in **Figure 3**.

**Figure 3** The natural logarithm of an exponential function is a straight line with a slope proportional to the exponential time constant.

The natural logarithm of the acquired signal yields a straight line. This is a good test to assure that the acquired waveform is indeed exponential. If taking the natural logarithm of a signal is not a straight line, then waveform is not an exponential. The slope of the linear natural logarithm can be calculated by measuring the signals slew rate which is the change in amplitude per unit time (Δv/Δt). The result is shown in measurement parameter 1 in the figure. The result is 9.9965 Megavolts/second. Note that the slew rate measurement requires the user to select the slope of the signal being measured, in this case the signal has a negative slope. The time constant is the slope of the line and is the reciprocal of the slew rate or (Δt/Δv). This oscilloscope supports calculations using parameters including sum, difference, product, ratio, reciprocal, and rescale of parameters. The reciprocal of P1 is calculated in parameter P2 which, when applied to the parameter P1, returns a negative slope of 100 ns per volt. This is the time constant of the exponential waveform.

Exponential signals commonly appear as the modulation on a high frequency carrier, they occur naturally as the RF carrier is keyed on or off. Measuring the time constant of this type of signal requires extracting the modulation envelope as seen in **Figure 4**.

**Figure 4** Measuring the time constant of the exponential amplitude modulation of a carrier requires demodulating the modulated signal to extract the modulation envelope.

In this example the decaying exponential amplitude modulates a 100 MHz carrier. The math function F1 uses an optional demodulate function to extract the exponential modulation envelope which is displayed over the modulated signal. From this point the natural logarithm function is applies to the exponential envelope and parameters read the slope of the natural logarithm trace as before. The result is a time constant of 100ns.

An alternative demodulation technique, if the oscilloscope doesn’t have a demodulation function, is to perform an RMS detection on the modulated carrier. This involves squaring the modulated carrier, filtering the square function, and then taking the square root of the filtered function as shown in **Figure 5**.

**Figure 5** Measuring the time constant of an exponentially modulated carrier using the square, filter, and square root functions to perform an RMS detection.

RMS demodulation is a classic technique. The demodulated waveform is truncated by the filtering operation, but this does not hinder the measurement of the exponential time constant. Using the slew rate measurement and parameter math to take its reciprocal the time constant is determined.

**Figure 6** provides a practical example in measuring the time constant of a RF burst in a remote keyless entry fob signal. The fob generates encoded signals using RF pulse bursts with a carrier frequency of 390MHz.

**Figure 6** Measuring the time constant of the exponential decay of an RF signal burst from a remote keyless entry fob.

The fob produces 21 RF bursts of varying widths shown in the top trace. EMI considerations generally require the RF keying to be controlled with finite attack and decay times to minimize spectral ‘splatter’ caused by fast on/off keying. The fifth pulse burst is horizontally expanded using a zoom trace in the next lower trace. The leading and trailing edges of the burst are exponential. This trace is further expanded to show the entire decaying amplitude in the third trace down. Applying the demodulate function to extract the exponential envelope is shown in the bottom trace. Just above that trace is the natural log of the modulation envelope. The slew rate parameter readout shows a slew rate of 165.5kV per second and its reciprocal, the time constant is 6 µs. The signal amplitude will decay to zero after about five or six time constants.

Digital oscilloscopes have a great deal of flexibility built in so derived measurements, like time constant, can be made with the existing measurement tools using a little creativity.

*Arthur Pini is a technical support specialist and electrical engineer with over 50 years experience in electronics test and measurement.*

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