# Gas station without pumps

## 2013 February 26

### Better model for loudspeaker

Filed under: Circuits course — gasstationwithoutpumps @ 11:34
Tags: , , , , , ,

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 model we developed in class for the 10W loudspeakers that the students bought in their parts kits.

The improved model that matches the measured magnitude of the impedance better at high frequencies.

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 $Z=\sqrt{j\omega}K$ (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:

Three simple models of a loudspeaker: R+L+(R||L||C), R+L+(R||L||C) + (R||L),
and R+L(f)+(R||L||C), where the frequency dependency in the final model is simply a power law.
The numbers may be slightly different from previous fits, because I redid the fitting, and small changes in the initial estimates or the order  in which fits are done can affect the result slightly.

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:

Model for the loudspeaker with one frequency-dependent component (the inductor).

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:

Power to the loudspeaker for various LC low-pass filter designs.  Note that they probably won’t be using a ±3.3V power supply, and they’ll probably have a series resistor on the loudspeaker, for observing current waveforms on the oscilloscope.

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).

1. WOO HOO! Congratulations! Quadruple decimal digits is a milestone, even if you can still count them on your fingers!

Comment by Mark Guzdial — 2013 February 26 @ 11:37

2. […] Better model for loudspeaker […]

Pingback by Twenty-first day of circuits class | Gas station without pumps — 2013 February 27 @ 20:04

3. […] use in lab: a AIUR-06-221 power inductor with 220µH and 0.252Ω.  The script also includes the loudspeaker model I’d developed for […]

Pingback by Twenty-third day of circuits class | Gas station without pumps — 2013 March 6 @ 21:23

4. […] to measure the two inductors that were confusing me in Colpitts LC oscillator, using the same method I’ve used before to model loudspeakers, that is, applying a known frequency sine wave to the unknown inductor in series with a known […]

Pingback by Fitting L and R values | Gas station without pumps — 2013 June 24 @ 23:55

5. […] then mentioned one hack (that I have played with in the past) that is commonly used to model loudspeakers, of adding more “knees” to the curve to […]

Pingback by Loudspeaker analysis | Gas station without pumps — 2014 April 30 @ 21:39

6. […] so we had to do a lot of guessing reasonable values and seeing if we got convergence.  (See my post of 2 years ago for models that worked for measurements I made […]

Pingback by Very long couple of days | Gas station without pumps — 2015 April 23 @ 21:49

The Rubric Theme. Blog at WordPress.com.