A Brief Study of Nonlinearities Using Klippel’s SIM Module: Harmonic Distortion


This is part of an ongoing study focusing on the various electro-mechanical design impacts on nonlinear distortion propagated in the sound field. This page will be updated to reflect progress with the study so please check back from time to time to view further updates. I will be focusing this page on the study of large signal performance (high output) on Harmonic Distortion. Once I complete this study I will move on to the effect of driver design on Intermodulated Distortion and will be posting it on a separate page for the sake of keeping the level of confusion down as much as I can.



Large Signal Identification (LSI) is the study of electro-mechanical performance of a driver in the large signal domain measured by Klippel’s LSI Module. In essence, it can tell you the engineering design behind a driver in a rather simple set of measurements to give better insight and ability to tweak the transducer design to better meet goals; whether that be rooted in performance or cost or both. This type of non-linear analysis leads us to the following discussion on driver simulation of non-linear performance using Klippel’s SIM Module.

Much of what I will discuss here can be found in the white paper Loudspeaker Nonlinearities – Causes, Parameters, Symptoms. I plan to try to make this a bit more of a visual learning approach as it’s the way I understand easier. I’m a hands on learner and I think others may benefit from this tactic as well.

Moving ahead…

Klippel offers a Simulation (SIM) Module to be used with their Distortion Analyzer R&D product which allows the end user to take known LSI parameters/curves – or create new ones from scratch – and modify them to see the effect of electro-mechanical nonlinear parameters on nonlinear distortion measured in the sound field. This can be broken up in to individual LSI components such as the big three: Bl(X), Kms(X), and Le(X). Force factor vs displacement, stiffness vs displacement, and inductance vs displacement, respectively. The following additional parameters can also be incorporated in to this SIM analysis:

  • L2(X)
  • R2(X)
  • Fm(X)
  • Doppler Radiation


The obvious benefit with SIM is the ability to model and predict nonlinearity in the sound pressure field without having to build engineering sample drivers or modify existing ones. It has been my experience that measured performance in both Harmonic Distortion (HD) and Intermodulated Distortion (IMD) vs that of the SIM predictions are often very close and sometimes spot on. Clearly this is a very powerful module that helps manufacturers explore potential tradeoffs and benefits of designs vs cost and allows testers such as myself to gain further knowledge in how measurements likely translate to in real world listening performance with relative ease.

As I have touched on in some earlier tests, nonlinear distortion can be summarily comprised of harmonic and intermodulated distortion components. Each have their place in transducer evaluation, however, a good data set should incorporate both sets of measurements to better understand driver performance namely because the two data sets differ in exercise. Harmonic distortion is simply a sweep of frequencies with measured distortion components relative to the fundamental. This means that a sine sweep from 20hz to 20khz provides d2, d3, dn, dn+1… as functions of a single tone at a time. This simply means that one tone is played and measured at a time. Rather than drawing out the process and playing one tone at a time with time between tones, HD measurements are quickly done via a sine sweep. While the results are a good indicator of performance, they are not wholistic simply because one does not listen to a single tone at a time. A loudspeaker transducer plays multiple frequencies at a time. As such, there is a need to measure a driver’s performance while playing multiple tones; at least 2 at a time. This leads to intermodulation performance analysis and is the basis for IMD measurements. Klippel offers a way to measure both forms on nonlinear sound pressure performance via their DIS Module and TRF Module. Also, as noted above, Klippel offers a way to simulate these parameters via the SIM Module.

I thought this would be a good time to explore some basic SIM performance models as a means to kick off both my own and the audio community’s understanding of nonlinear transducer performance and how one can potentially locate the root cause of particular performance down to something as simple as an asymmetrical suspension linearity or voice coil shift.

The Baseline

Let’s first look at some basic examples using only Bl(X), Kms(X), and Le(X) and evaluate only the Harmonic Distortion.

The simulated driver has the following specifications:

Re 3.5 Ohm DC resistance of voice coil
Bl 5 N/A force factor
Kms 1 N/mm siffness of suspension
Le 1 mH voice coil inductance
R2 0.5 Ohm electrical resistance due to eddy current losses
L2 1 mH para inductance of voice coil
Mms 14.3 g mechanical moving mass including air load
Rms 1.06 Ns/m mechanical resistance of driver suspension losses
fs 42.09 Hz driver resonance frequency
Qms 3.56 mechanical loss factor of driver in free air considering Rms only
Type of enclosure Driver in baffle type of enclosure used in model
Piston, 2-pi, anechoic
Distance 1 m distance between diaphragm and listening position
X (t=0) 0 mm initial displacement


Next we have the three primary LSI parameters for this study: Bl(X), Kms(X), and Le(X). Here are the baseline LSI simulation results:

Both drivers have linearly-symmetrical curves with no shift or non-linearity (ie: each graph is a mirror image of itself in both coil directions and in shape). Both have ±6mm mechanical excursion. The Bl curve given above was chosen as it mimics a standard overhung coil configuration design. The Kms(x) curve was somewhat arbitrary regarding values. The Le(X) indicates the use of a shorting ring used to keep inductance in and out the same.

With the given Thiele-Small specs, this driver is essentially a standard midwoofer. I took a default SIM model and changed the cone mass and Bl shape. Otherwise, that’s it.

If one were to use the 10% distortion parameters to define Xmax (linear) based on the above, this simulated driver would be limited by Bl at approximately 3mm (82% Bl) and limited by suspension at 6mm (75% Kms(x); inverse of Cms).

Running a sine sweep from 20hz to 10khz with applied voltage range of 1v to 3v in 4 increments, I obtained the following HD results for my baseline:

What you see is this baseline driver’s harmonic distortion (relative to the fundamental) is comprised almost entirely of 3rd order distortion. You can see the clear increase in 3rd order distortion as input voltage is applied. At 50hz, the Fs of the driver, the 2nd order distortion is at approximately 0.12% while 3rd order at 50hz is about 4.3% (relative to the fundamental).

Above 100hz, the component distortion from both orders input is negligible.*

*Le(X) effects will be discussed later and will help explain what you see above 1khz.


Modifying the Curves: Kmx(X)

Offsetting the Suspension

In this example, I have shifted the suspension by approximately +1mm (to the coil out position). This makes the curve offset. The shape is still symmetrical, though.

In red is the nominal performance with the driver’s suspension symmetrical and linear in respect to xmax+ (coil out) and xmax- (coil in).

In blue you can see the shifted response overlaid.

Here are the results of the SIM:

In the baseline Kms(X) graph, it was determined the linear Xmax was ±6mm. In this case, however, the Kms Xmax would be 4.8mm, since 0.75 is achieved at -4.8mm due to the forward offset. While 0.75 on the positive X axis is obviously going to be further out, the linear Xmax value is based on the lower of the two points, thanks to the offset.

While 3rd order distortion has pretty much gone unchanged, 2nd order distortion has increased by approximately 1.6% at 50hz, impacting the total harmonic distortion (THD) and increasing it by about 0.5% from the baseline study.


De-Linearizing the Kms(X) Curve

What happens when I take the linearly-symmetrical curve (baseline) and make it both asymmetrical and nonlinear? Let’s see….

As with the above, red is the baseline and blue is the new SIM curve:

Here you can see the suspension is again shifted 1mm forward. This time, however, I also tilted the Kms(X) curve 0.0168mm to give it a nonlinear shape. In this case, the Kms Xmax would be 3.98mm, since 0.75 is achieved at (-)3.98mm.


I ran the same sweep of 1v to 3v on the simulated model and obtained the following results:

Let’s again look at 50hz for the analysis point compared to the baseline

  • 2nd order distortion has increased from practically zero to 2.5%.
  • 3rd order distortion has decreased by 0.05%.
  • Overall THD has increased by only 0.7%.

Comparing these asymmetrically nonlinear results to the shifted-only results taken before using 50hz as the analysis point:

  • 2nd order distortion has increased 0.8%.
  • 3rd order distortion has decreased by 0.01%.
  • Overall THD has increased by 0.5%.

At 50hz analysis, it seems the biggest difference maker is simply the shifting of the suspension by +1mm while the tilting of 0.0168mm had little effect.

The real change to note is below Fs, where 2nd order distortion increases quite drastically as each change is made. In the case of the asymmetrical nonlinearity, the 2nd order distortion at 30hz increased nearly 8% driving the THD to increase by 2.3% from the baseline while 3rd order at 30hz actually decreased 0.5% between the two. More on this in a bit.


Let’s get back to the tilting of the Kms(X) curve, though. I’m not sure I changed it enough to really see what happens so let’s tilt the an additional 0.07mm for a total of 0.090mm tilt, reflected in Green below. Now it’s much more nonlinear. In fact, in this case, the Kms Xmax would only be 2.5mm, since 0.75 is achieved at (-)2.5mm.



Now let’s see how that changes the baseline SIM measurements…

As before, let’s look at Fs just as a data point to analyze compared to the baseline

  • 2nd order distortion has increased by approximately 5.5%
  • 3rd order distortion has decreased by 0.2%
  • THD increased by about 2.6%


So, 2nd order distortion increased quite a bit. Compared even to the shifted only curve, it increased 5.4% in 2nd order.

What’s really interesting, however, is the shape of the 3rd order distortion curve. While previous simulations resulted in fairly linear distortion increases in voltage steps, this particular SIM showed a clear drop below 30hz in 3rd order distortion approaching 3v as seen below.

At this point, I’m unsure what could be causing this. It may be due to a DC displacement but I’m not sure just yet. However, I did want to note it as I find it very interesting.


In general, what you can see is that assymetry affects 2nd order distortion while nonlinear curves affect 3rd order distortion. With Kms(x) the area impacted is around Fs in these studies and should give us a better idea of where to look for suspension related issues in further test data.

That does it for the Kms(X) evaluation. If you have any questions, feel free to ask.

I will pick up with Bl(X) next.

To be continued …


Dated: 08/05/2012, 0227 CST.