Measuring thiele small parameters Part 1

How to measure the TS parameters of a speaker ?

Measuring thiele small parameters is not something that only people in white coats get to do. With the appropriate tools and know-how, and assuming your IQ is not below 80, I feel confident that you won’t experience much difficulty in finding the TS values of your speaker. You might be asking yourself : why should I measure the TS parameters? The manufacturer quotes the values on the specification sheet of the speaker. Even though they are made through the same manufacturing process, every speaker has its own unique set of TS values.

Of course, they are very close to what the manufacturer quotes on the tech sheet, but only by measuring your particular speaker, you can get the exact values. Sometimes, a minor flaw in the manufacturing process, can make a good physical looking speaker have minor deviations in specs. Other times, you simply have an unknown speaker and don’t know the TS values, so measuring thiele small parameters is the only way to find them.

Should I break-in my speaker before I start measuring thiele small parameters ?

This is debatable. While certainly not a strict rule, it’s good practice to do so. Breaking-in the speaker will alter the TS parameters. If you take a freshly unboxed speaker and measure it, you will get a set of values. Measuring thiele small parameters after break-in, will result in a different set of values. After break-in, resonant frequency is lower, different values for Q_ts, V_as etc. Since the speaker will break-in by itself, after prolonged use, it is common sense that you should deliberately break-in the speaker, before you start measuring thiele small parameters.

To break-in the speaker, you should place it in an open baffle (not enclosure), play a tone close to its resonant frequency, sufficiently loud enough so the speaker reaches 70% of its X_max. Do this for about 12 hours and your speaker break-in should be complete. After this period, the suspension (spider and surround) will be all loosened up.

Before vs after break-in

If you got time, breaking-in the speaker is a good habit, but not mandatory. By looking at the TS parameters, before and after break-in, it would seem like they are 2 different drivers. In reality, if you compare F_s / Q_ts ratio of the before and after break-in, you will see that you will end up with the same number. Let’s say you make a sealed enclosure using the parameters from before break-in. Using this exact enclosure, you model the frequency response for a speaker with the parameters from before break-in, and a speaker with the parameters after break-in. You will get 2 curves which will almost overlap.

Notice from the graph below, that the 2 curves are ever so slightly offset. You could argue there is a difference, but it’s negligible, most certainly unnoticeable in the real world. However, if you measure the TS parameters before and after break-in, and you get different F_s / Q_ts numbers, you probably got a faulty driver. This is another reason why you should break-in a speaker. Starting out your project with a damaged speaker is a recipe for failure. Even if you don’t measure the parameters before break-in, making the speaker play at  high excursions for 12 hours will almost certainly reveal flaws in the suspension, or voice coil defects.

Now that we cleared that thing up, we can move on into describing how to measure each individual parameter.

How to calculate resonant frequency (f_s) ?

Equipment needed (Amazon affiliate links) :

• Sine wave generator.
• Frequency counter (optional).
• Multi-meter (or an AC voltmeter).
• One resistor (1000 ohm / 5 watt).

The frequency counter is optional, only if you have a high end sine wave generator. Otherwise, it is good to have a frequency counter, to give an exact reading on the frequency generated. This equipment is needed for a measuring technique called the voltage divider method. Here is graphical overview of the setup  :

Voltage divider method

Before you start doing any measurements, position the speaker correctly. Because the resonant frequency is measured in free air, placing the speaker onto a baffle is not permitted. Normally, you would suspend the speaker in the air, using a cable or chain. This creates some problems. As the cone moves, the speaker might start to swing. A good solution is to make a frame, to which you can clamp the speaker. The frame should be of minimal size, but sufficiently sturdy to keep the speaker stable. Place the speaker and frame in an open area, with at least 1 m from the nearest object in all directions.

Another thing you should do, before you start measuring, is to set  your equipment to the lowest nominal voltage level. Less than 1 V, if possible. All these parameters are small signal mathematical models, so measuring thiele small parameters should be done with “small” amount of voltage.

To measure the resonant frequency, you start by varying the frequency of the sine wave generator. Depending on the speaker you are measuring, you will start with different values. If you are measuring a woofer, you should sweep from 10 to 100 Hz. If you are measuring a midrange or tweeter, there is no point in going that low, and you should sweep the appropriate frequencies. As you sweep through the frequencies, you continuously read the voltmeter. As you start to read higher values, you are getting close to the resonance frequency. Fine tune the sine wave generator, until you get the maximum voltage. That is the resonant frequency in free air, which will be displayed on your frequency counter, or your sine wave generator (if it is a high quality one, and shows accurate and exact frequency numbers).

How to determine speaker impedance

Equipment needed :

• The same equipment needed for voltage divider method.

From the start, let’s clear up some usual mistakes. Don’t confuse R_e (which is DC resistance) with the impedance (which is AC resistance). Measuring R_e is easy. Just set your multi-meter to ohms, and take a measurement at the speaker terminals. All done ! Impedance, on the other hand, doesn’t have a fixed value, and is dependent on frequency. So, the impedance reading is actually a curve. Therefore you will need a graph paper to draw your curve. The graph should be semi-log, which means that one axis is logarithmic (frequency), and the other is linear (impedance).

We are using the same equipment like with the voltage divider method. Unlike the previous measurement, before we start to take any reads, we have to do some calibration.

Calibration :

Calibrating the equipment is particularly useful in making easy conversions (from volts to ohms). First, replace the speaker with 10 ohm resistor, and adjust the sine wave generator level, until you read 1 V on the voltmeter. Now, when you set different frequency waves, it is easy to convert the volts which read on the voltmeter to ohms.

So when the voltmeter reads 1 V, the impedance is 10 ohms, 0.5 V = 5 ohms , 10 V = 100 ohms, 2.5 V = 25 ohms , and so on.

Since the calibration is complete, replace the 10 ohm resistor with the speaker back again. Now you have to vary the sine wave generator to different frequencies between 10 Hz and 20 kHz, and manually record as many values as possible to make the graph more accurate. Each voltage read is easily converted into ohms (since we calibrated the setup) and marked on the graph. When you have taken an appropriate number of reads, then you can plot your impedance curve on the graph.

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The impedance curve generated using this technique is not true impedance. The problem with using the voltage divider method, is that the higher the impedance of the speaker, the less accurate the measurement will be. The difference is pretty small, but it is something to be aware of. This should not discourage you, as speakers usually have low impedance and this technique requires minimum equipment and works well.

This is how an impedance chart should look like. Impedance spikes at resonance frequency, and it slowly rises as frequency goes up.

How to measure speaker moving mass ?

Equipment needed :

• The same equipment needed for voltage divider method.

Let’s split moving mass into its components :

• M_md = The assembly mass (parts of the speaker that move = cone, coil, half of surround and half of spider)
• M_mr = Air-mass load (poket of air in front of the cone, that moves in sync with the cone)
• The total moving mass : M_ms = M_md + M_mr

To calculate M_ms, we first need to calculate M_md and M_mr.

M_mr is dependent on the size of the speaker. To be more precise, it is dependent on S_d, which is (the area of the cone) + (the area of half of the surround). Find out the M_mr value, using this formula :

M_mr = 0.575 * (S_d)^1.5

We can calculate M_md by placing the cone, the coil, half of the surround and half of the spider on a scale. To do this, you need to tear up your speaker, and that might be a deal-breaker. However, if the manufacturer quotes the Mmd, you should rely on that number and not try to measure it yourself. If you don’t have an M_md value in your specs sheet, there is a way to measure it without tearing apart your speaker (delta mass method).

Delta mass method

Take a small amount of clay and place it on the cone of the speaker. Spread it around evenly around the cone and dust cap junction. The mass of the clay must be known exactly (down to 0.1 grams), and it is noted as M_a. This clay is to modify the resonant frequency of the speaker. Depending on the size of the speaker, you will place an adequate amount of clay, so that you modify the resonant frequency by at least 25%. This would roughly mean 10 g of clay for 6″ speakers or smaller, and 40 grams or more for larger woofers.

After you have successfully applied the clay onto the cone of the speaker, you will measure the new resonant frequency, using the same method described at the beginning of the article. The new resonant frequency is f_sa.

To calculate the M_md, use the following formula :

M_md = M_a / [(f_s/f_sa)² – 1]

Since M_md and M_mr are now known, adding them up will give you the total moving mass (M_ms).

How to measure speaker compliance ?

You should be quite comfortable with measuring thiele small parameters by now. Compliance (C_ms) is easily calculated if you know, or measured, the parameters above.

To calculate the compliance of the speaker use this formula :

Cms = [(6.283 * f_s)² * M_ms]^-1

C_ms is measured in meters / newton. M_ms is given, and measured in kilograms.

End of part one

Because this article would have gotten too long, I decided to split it into two parts. Since there are a lot of Thiele Small parameters, learning how to measure them, will take some patience. Make sure to click on the link, for the second part on measuring Thiele Small parameters.

References

1. Loudspeaker Design Cookbook 7th Edition by Vance Dickason (Audio Amateur Pubns, 2005). (Amazon affiliate link)
2. Image source : link.