In *Seventeenth day of circuits class: inductors and gnuplot tutorial*, I introduced two linear circuit models for the JAMO 30466/30462 loudspeakers that we’re using in the class:

The second model fits better and is one of the standard ways of modeling loudspeakers, though I did not realize that at the time I developed the model. I looked into some other ways people have modeled the slower-than-f^1 rise of impedance for loudspeakers. There seem to be three popular approaches:

- Adding more (R||L) sections in series. This approach is handy when dealing with circuit simulators, since it remains a fully linear circuit. It has a lot of parameters, though, and tends to produce fits that waver around the observed data. The phase tends to be affected quite a bit by the extra sections.
- Adding a “semi-inductor” that has impedance (or a semi-inductor in parallel with an inductor).
- Using a frequency-dependent inductor with Lhi being proportional to a power of frequency, rather than constant.

I tried both semi-inductors and frequency-dependent inductors, and I was not very impressed with the semi-inductor approach. It took more parameters to get a worse fit. The frequency-dependent inductor fit is excellent, though, and really has about as few parameters as a model of this data can:

To really distinguish between the different models of the loudspeaker, I should have recorded data that would allow me to try to figure out the phase as well as the magnitude of the impedance. I don’t have the tools to measure phase directly (though I’ve been thinking about making a circuit to do that), but I could have measured three RMS voltages instead of just two in the 47Ω + loudspeaker test circuit. I measured the voltage across the resistor and across the loudspeaker, but not across the two in series. The extra measurement would have allowed me to estimate the phase difference between current and voltage (though not the sign of the phase difference). Maybe I’ll do that when I have a couple hours to spare to redo all the measurements.

In the lab handout for next week’s lab, I’ll explain Zobel networks for compensating the loudspeaker to get resistive behavior at high frequencies (but we won’t use a Zobel network, because it wastes a lot of power) and designing LC filters to suppress the high frequencies. The designs will be done with the frequency-dependent inductor model:

I’ll have to provide a lot of scaffolding for the LC design, as the students don’t have the programming ability (even for simple scripting in gnuplot) to do the optimization. I’ll probably give them a script that produces graphs like the following, so that they can choose their components without having to do messy calculations, though they’ll have to modify the script a little bit:

Incidentally, this post is the 1000th post on this blog. I was going to write something celebratory for the 1000th, but my son convinced me I should save the celebration for the real round number in post 1024 (which is where I run out of fingers for counting the posts).