Bad Circuit Modelling Episode 3: Temperature
Today, we’ll give a brief discussion of another method by which real-world analog circuits can exhbit non-ideal behavior. Circuit components tend to behave differently at different operating temperatures. So a guitar pedal, for instance, might sound a little bit different at a chilly winter concert in Montreal, compared to a sweltering summer gig in Austin, Texas. As with the previous two articles in this series, we’ll be focusing on resistors and capacitors, but a similar thoertical approach can be applied to any circuit component.
As always, this article can be seen as Jupyter Notebook, available here.
Circuit Analysis
Once again, we’ll be using a Sallen-Key lowpass filter as an example circuit. And again, we’ll assume that the circuit op-amp is ideal, and only consider the non-ideal behavior of the circuit resistors and capacitors.
The filter cutoff frequency and Q values can be calculated as follows:
Let’s design our filter for fc=1 kHz, and Q=2. Thus, our component values should be C=4.7 nF, R=33.8 kOhms, R2=1.5 kOhms, and R1=1 kOhms.
Capacitor Temperature Dependence
While the resistance of the circuit resistors as well as the wires in the circuit are affected somewhat by the circuit operating temperature, these effects are mostly negligible. The most noticeable effect that temperature fluctutations will have on the circuit is that the capcitors will be noisier at higher temperatures. The standard deviation of the capacitor noise is defined by:
where kB is Boltzmann’s constant, T is the temperature in Kelvin, and C is the capacitance value (for more information on capcitor noise of this type, see here). We can then determine the voltage that the noise from each capacitor will add to the output.
For the capacitor in the upper feedback loop, the noise voltage can be added directly to the output. For the capacitor connected to the positive terminal of the op-amp, the noise voltage will be added through the voltage divider made up of R1 and R2.
Implementation
Visualizing the effects of this capacitor noise is a little bit difficult, so to demonstrate these effects I have developed a simple audio plugin using the JUCE framework that implements the capacitor noise into a model of the Sallen-Key LPF circuit. Source code is available on GitHub, and a video demo is available on YouTube.
Finally…
Thanks for tuning in for another edition of Bad Circuit Modelling! Up to this point, we’ve been assuming the op-amps used in our circuits are ideal. For next time, I’m hoping to remove that assumption, and take a look at some of the imperfections that can arise from op-amps that don’t behave quite how they’re suppose to.
