How to Measure the Time Constant with an Oscilloscope

And more fun with RC circuits and RL circuits

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Measuring the time constant with an oscilloscope is easy. Simply make a quick calculation and a cursor measurement.

To measure the time constant of an RC circuit or an RL circuit with an oscilloscope, pick two reference points on an edge of your signal and see how long it takes to grow/decay 63.2%.

The time constant is useful because it gives information about how first-order circuits react to stimulus. First order circuits have only one energy storage component – an inductor or a capacitor – and can be described using a first-order differential equation.

TL;DR
When a first order circuit experiences a voltage step up or step down, the circuit will settle to a constant voltage. The time constant, τ [tau], tells how long that settling will take.

Also, if you know the settling characteristics, then you can also determine the charge of a capacitor or inductor at a specific point in time.

Charge = (∆ Source) (1 – (1/e^(t/tau)))
∆ Source is the change in voltage or current applied to the RC or RL circuit.
t is the time at which we want to know the charge on your inductor or capacitor (how long after the step up/step down)
and τ is the time constant.

If we set t = τ, our formula becomes
Charge = (∆ Source) * .632

What this means is that a capacitor will charge up to 63.2% of the source delta after one time constant.

From a settling perspective, if we wait a period of one time constant, we move 63.2% closer to our final value. After a second time constant, we move another 63.2%. Essentially it drops to 36.8% of its starting value.

Now think about this. For the period of the second time constant, we’re basically dealing with a new ∆ source value. Instead of moving from 10V to 0V, you’re now moving from 3.68V to 0V. So after the second time constant you’ll end up at 3.68 V * 36.8%, roughly 1.35V

After five time constants, you’ll be 99% of the way to your final voltage – After 5 time constants, people generally agree that, for all practical purposes, the signal has settled and the inductor or capacitor is fully charged or discharged.
So, if you were to plot this out, you’re signal will look like an exponential curve. That’s where the oscilloscope comes in.

In this video, we’re probing the voltage across our capacitor, so it should be easy to measure the time constant.

For a more robust time constant measurement, do this a few times with a few different captures and a few different start/stop voltages and take the average. I’d also recommend staying towards the middle of the decay, as it’s possible to get some non-linear effects right at the beginning. An example of that is the parasitic inductance video, linked above and in the description.

How to calculate the time constant?
Without going through the math, you can do some fancy substitutions using the formula for charge, Q = CV, and Kirchoff’s law and you end up with:
τ = R *C for an RC circuit, and τ = R/L for an RL circuit.

For this circuit, after one time constant, the charge on the capacitor will be 638 microcoulombs using Q=CV, and the energy stored in the capacitor will be roughly 2 mJ, based on W = ½QV

If at some point this resistor load were to be removed after one time constant, the capacitor or inductor will throw all of that stored energy back at your source. If you aren’t careful, that can cause serious damage.

Remember, inductors resist a change in current, so if you open a switch that is providing current to an inductor, the inductor is not going to allow that instantaneous current change

To protect against this, you can put a diode in parallel with the inductor to allow it to discharge. This is called a freewheeling or flyback diode. You could also build an RC snubber if you need a faster current decay.

This is also the fundamental theory behind switch mode power supplies. Take a buck converter, for example, which is used to efficiently step down voltage.

When the switching transistor is closed, the “on state,” the inductor is charging up. When the switch is open, the off state, the inductor is powering the load.

You could also go old-school and use RC circuits and RL circuits with a comparator to form an analog timer. Because the circuit’s decay profile is known, you can set a threshold level for the comparator that will cause the comparator to flip after a very specific wait period.

Graphics credit:
CyrilB~commonswiki assumed. Own work assumed (based on copyright claims). CC BY-SA 3.0 commons.wikimedia.org/w/index.php?curid=650955