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Simulating impedance curve of a loudspeaker

Posted: 23 Dec 2015     Print Version  Bookmark and Share

Keywords:power amplifier  oscillation  resistor  speaker emulator  passive-component 

I finally got down to testing a prototype of a power amplifier I had been designing for a while. This amp has a bit of a unique output topology, and I was keen to see how it behaved when pumping 100 watts into a load – in particular, how well it did on stability when feeding into a reactive load, both capacitive as well as inductive. While it did not quite burst into oscillation, I could see snippets of 5MHz at various points, depending on the frequency, waveform, and load. Trouble is, since I was using a dummy resistor, I had to intentionally add a bit of capacitance or a bit of inductance at each stage, which left me wondering if I could not do this in a more efficient manner given I did not want to plug in a real target speaker this early in the game. Needless to say, I'm not the first or the only person to face this need.

So how does one go about addressing the problem? It's actually pretty easy to build a speaker emulator for a given driver – use a bunch of inductors, capacitors, and resistors to synthesise the impedance curve of the target driver. This is a passive-component approach. However if you want to make something more generic – a single circuit whose characteristics can be varied to emulate multiple drivers and therefore do a what-if analysis – you need a different approach, one in which you use active simulators for the inductors and capacitors, thereby making them variable. This is what was attempted by Niels Elk, Jaer Iversen, and Arnold Knott, and reported in their paper titled "Small Signal Loudspeaker Impedance Emulator," published in Journal of the Audio Engineering Society, Oct 2014. Unfortunately this paper is not available as a free download (as yet), but the idea is straightforward enough to understand without having to refer to the full paper.

The approach in brief
The idea of simulated inductors and simulated capacitors is not new – standard op-amp books will provide you circuits for these. But it's the application of these in the present case that is innovative.

Figure 1: The simplified model of the loudspeaker.

The simplified model of the speaker used by the authors is shown in figure 1 (taken from the authors' paper). This is of course a pretty drastic simplification, but it's a first approximation to a speaker in free air. What the authors do is to substitute the inductor and the capacitor with standard textbook op-amp-based simulators. The advantage of doing so is that you can simulate large-valued inductors and capacitors using small-valued capacitors and inductors respectively, and vary the simulated values by only varying resistors. Figure 2 shows a standard simulated inductor and figure 3 shows a standard simulated capacitor (figures 2 and 3 are taken from the authors' paper). These circuits are special forms of a gyrator, which essentially converts an impedance to its dual. The value of the simulated inductor can be adjusted by varying RL, and the value of the simulated capacitor can be adjusted by varying RC.

Figure 2: Simulated inductor.

Figure 3: Simulated capacitor.

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