# Bad Circuit Modelling Episode 2: Component Aging

In this article, we’ll be discussing another way that circuits in the real world can behave in a non-ideal way: component aging. Real circuit components always change with age, and usually for the worse. In this article, we’ll be focusing specifically on resistors and capacitors, but in theory the same approach can be used for any circuit component.

For the programmers in the room, as always, this article can also be seen as a Jupyter Notebook.

# Circuit Analysis

As with our previous article, we’ll be examining a Sallen-Key lowpass filter.

As before, we’ll be assuming that the op-amp is ideal, and examining what happens to the circuit for non-ideal resistors and capacitors. Once again, the filter cutoff frequency and Q value are as follows:

And again, we’ll design our filter for fc=1 kHz, and Q=2; C=4.7 nF, R=33.8 kOhms, R2=1.5 kOhms, and R1=1 kOhms.

# Resistor Aging

As a resistor grows old, the resistance tends to increase. For a typical thin-film resistor, the age dependence is described by [1]:

where t is the length of time the resistor has been used (measured in hours), and T is the operating temperature of the resistor, measured in Kelvins.

The plot above shows what happens to a 1 kOhms resistor at an operating temperature of 400 K, over a period of 100 years. While these changes may seem relatively small, they compound over all the resistors in the circuit. Below we show the frequency response of the Sallen-Key lowpass filter at different ages, again with an operating temperature of 400 K.

# Capacitor Aging

For audio, it is typical to use class II X7R capacitor, since this type of capacitor has a minimal amount of voltage dependence, meaning the resulting circuit will have a minimal Total Harmonic Distortion (THD). For this class of resistor, the capacitance decreases by ~2.5% per decade hour (one decade hour means the number of hours has been multiplied by 10) [2]. Below we show how a capacitor ages over a period of 100 years.

And again, we show how the frequency response of the Sallen-Key lowpass filter changes as the capacitors age.

## Capacitor Failure

While Capacitor aging makes a pretty small difference, a bigger issue in old electronic circuits tends to be capacitor failure. For the class of capacitors considered here, the expected lifetime is approximately:

where TT is again the operating temperature in Kelvin. While there is some fluctuation in actual failure times, capacitor lifetime tends to follow a normal distribution [3]. When we include failure in our capacitor model, we see:

In this case the capacitor failed after 100 years, causing the large jump in the cutoff frequency of the filter.

# Putting It All Together

Ultimately, when we combine the results of resistor and capacitor aging and failure, we see a dependence on age that looks like this:

Once again, to give an example of the aging effect in a function digital audio effect, I’ve implemented the above circuit model as an audio plugin (VST/AU). Source code is currently available on GitHub. The pure aging effects of the components make a subtle, but noticeable difference in the output sound of the effect, but I’m most impressed with the change that happens when a capacitor fails, compared to my own experience working with older circuits, I think it sounds very accurate.

# Finally…

So that’s a brief look at how component aging can affect the performance of an audio circuit. I hope you’ve been inspired to think about how some of these ideas could be incorporated into your own work, I’d love to see some plugins out there that “age” the longer you run them. For next time, I’m planning to take a look at how operating temperature can affect all components, even new ones. Thanks for reading!

# Sources

[1] Brady, et al., *Thermal Oxidation and Resistivity of Tantalum Nitride Films*, Thin Solid Films 66 (1980), pp. 287–302

[2] England, C., *Ceramic Capacitor Aging Made Simple*, https://johansondielectrics.com/downloads/ceramic-capacitor-aging-made-simple.pdf

[3] Parler, S., *Reliability of CDE Aluminum Electrolytic Capacitors*, https://www.cde.com/resources/technical-papers/reliability.pdf