# What is the reason for exponential decay of ceramic capacitor leakage current?

I came across two Tektronix documents describing how to measure leakage current. Both of them say that current decays exponentially, which I don't understand. I expect a constant leakage current once the capacitor is fully charged.

The first one is from Tektronix's Low-Level Measurements Handbook 7th ed.

I simulated the transient response of a ceramic capacitor using LTspice and the manufacturer's equivalent circuit like this:

The diode was suggested in the Tektronix handbook but result is the same without the diode.

The result looks like this:

Basically, a current peak while charging the capacitor and then a constant leakage current given in this case by R1. Eitger my simulation model is not capturing something of a real ceramic capacitor or there's a measurement issue.

Can someone explain why there would be an exponential decay of ceramic capacitor leakage current?

The equivalent circuit values come from TDK's equivalent circuit model library:

• 4 megohms seems like an excessively low leakage resistance. Why did you choose that value? And why 22 pF? Apr 15, 2023 at 7:55
• Values come straight from TDK's equivalent model library, I added the info in OP Apr 15, 2023 at 13:19
• SPICE models are an approximation; to what degree, is up to the manufacturer. Evidently they aren't interested in modeling dielectric absorption or the like. Evidently they didn't care much about modeling leakage current either. Apr 15, 2023 at 14:18
• This model TDK describe as "simple" so it apparently lack something. If you want something better then they also ofer "precision" model, maybe you will find that that model describe leakage better. Apr 15, 2023 at 14:31
• Some of that exponential decay is just due to the capacitor charging, a process which in the ideal case never ends. Apr 15, 2023 at 15:27

Ok, so you've started with the full model and that's perhaps causing you problems, let's start with just the two elements that we need to think about self-discharge, capacitance, and leakage resistance.

In this simulation, I've just put a capacitor, value of 22u (looking at your data that's the correct value of 22e6 pf), and 4M Ohm leakage resistance. I've set the inital voltage accrosss the capacitor to be 120V, and you can see in red the voltage, and in green the current out of the capacitor, both decay exponentially as expected.
So why the exponential. Well at the start, the capacitor has 120V, and it's discharging this into a 4M resistor, so as you'd expect
I= V/R I = 120/4e6 = 30uA
and that's where the curve starts. However, as the capacitor discharges, its voltage drops, therefore the current drops, and therefore the rate of discharge also drops, and this creates an exponential relationship. Because the rate of discharge is decreasing over time, the relationship is no longer linear, but exponential.

You might notice that above the time is very long, 500s on the graph. Roughly speaking, a capacitor takes 5RC seconds to discharge, and in this case that would be 439 seconds. If I now add the rest of the lumped elements in the curves look like this

At first glace it looks like the current is zero, but actually this is becuase of the scale. The initial current in is so high (>100A) that our discharge at 30uA is impossible to see. I have to do a bit of fussing to be able to see it.

In this case, I have to start the simulation at 5us, after the charging pulse, and constrain the axis a bit to get it in veiw. Without this delayed start, my simulator gives me a warning and shows some weird stuff.

Swapping back to your settings, 22pF for the capacitor and 30uS simulation time, I get these results.

Which is what I'd expect, some spikes around the switching event, but then a nice exponential discharge. You might think those discharges were linear, untill you look at then on a longer timebase, hence why calculating 5RC was important.

• Thanks! You're right the capacitance value should be 22 uF Apr 15, 2023 at 19:35
• The capacitance-charging curve has a time constant of ESR x C. The dielectric absorption effects discussed in the question have time constants if several seconds at least, sometimes minutes. Apr 16, 2023 at 14:05

The leakage resistance is in general not constant over time but is history dependent.

https://en.m.wikipedia.org/wiki/Dielectric_absorption

When you suddenly change the voltage across a capacitor dielectric, one can consider this effect to mean: Not all of the capacitance is immediately "connected". A small part only starts to take part only delayed.

The result is the curves you describe, but also effects like spontaneous voltage build-up upon sudden discharge.

A related but not immediately obvious result is that the ESR of capacitors rises at low frequency because there is more time for energy to dissipate in the "slower corners" of the capacitor.

That is why capacitor datasheets should specify how long they applied a voltage before taking the leakage measurement.

• Thanks but I'm not sure it's the same physical mechanism: dielectric absorption, which (if I understand correctly) is related to an increase in voltage in open circuit whereas the issue I wrote is a decrease in current when a source is applied Apr 15, 2023 at 19:34
• @KenGrimes Yes it is. If you check the Theory section in the link, you'll find an equivalent circuit. Try to model it. e.g. 99 nF in parallel with 1 nF, where the 1 nF has a series resistance of 1 GOhm. ( as the most simple conceivable case) Apr 16, 2023 at 4:59