Today we’re going to look at how to rigorously measure the time constant with an oscilloscope,
as well as some of the time constant’s implications for your circuits.
Measuring the time constant with an oscilloscope is surprisingly easy. All it takes is a quick
calculation and a cursor measurement. But before we get to that, why do we care
about the time constant of a circuit? The time constant is useful because it gives
us information about how our 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. The TL;DR on this is that 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, tells us 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.
The time constant gets its name from its two key characteristics. Time, because it’s
a measure of time – seconds. And constant, because the period of time it takes to settle
doesn’t depend on the magnitude of the voltage step up or step down – it settles at a constant
rate. More specifically, it contributes to the charge
equation ∆ 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
Which gives us this: 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.
For this simple RC circuit with a square wave input, we see an exponential curve. For more
on step response, check out the “Parasitic Inductance and AC Step Responses” video
which I’ll link to in about a minute thirty – but you’ll probably want to see this
first To measure the time constant with an oscilloscope,
you simply need to pick two reference points on the decaying portion of your signal and
see how long it takes to grow 63.2%. Here’s a simple RC circuit being supplied
with a square wave going between 0 and 10 volts. For measuring the time constant, let’s pick 2 V as a starting point,
which means that, after one time constant, we should end up right at 7.056 V.
We’re probing the voltage across our capacitor, so it should be easy to measure the time constant
So lets turn on cursors, and to get a cleaner acquisition, let's go into "acquire" and turn on high resolution mode.
And let's bump the waveform intensity up to 100%.
Lets put our Y1 cursor at our starting voltage, which is 2 volts and
our Y2 cursor at 7.056 volts.
Now, let’s move the X cursors to the intersection the waveform and the Y cursors.
Let's zoom in a little bit here to get some better horizontal accuracy for our measurements.
The distance between the X cursors is our time constant. That value that we're measuring is 104 ms..
So we can say that for this circuit our time constant is 104 ms.
To prove the time constant is actually constant,
let’s do it again, but starting from 4 V.
So you can see again, we're measuring right around 104 ms.
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.
So now that we have a good idea of what our time constant is, we can compare it to our
calculated value. Before you can do that, we should probably
learn 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 the elegantly simple: τ = R *C for an RC circuit, and τ = R/L
for an RL circuit. For this circuit, we are using a 1 kOhm resistor
and a 100 µF capacitor, so our calculated value is 100 ms, which his pretty darn close
to our measured value. Just for kicks, let’s go a little deeper.
After one time constant, we know that the charge on the capacitor will be 638 microcoulombs
using Q=CV, and the energy stored in the capacitor will be roughly 2 milliJoules, based on W
= ½ If at some point this resistor load were to
be removed after one time constant - or there's a switch openeing up - 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.
It’s not all doom and gloom, though. This charge storage can also be quite useful. This
is actually 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.
The capacitor generally acts as a filter capacitor to help remove inherent ripple in this type
of design. This is useful because, unlike a resistive
voltage divider, you don’t lose half of your energy in that other resistor.
You could also go old-school and use RC and RL circuits in conjunction 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. That’s all for today, I’m Daniel Bogdanoff,
and thanks for watching! Make sure you subscribe to the Keysight Labs YouTube channel and the
Keysight Podcasts YouTube channel! If you want to dig a little deeper into inductance
and capacitance, check out the “AC step responses” video here, or watch our latest
video, which is right over my face.