# Thiele Small parameters equations – How each one affects the others

## Thiele small parameters list

The Thiele Small parameters equations reveal how they are calculated and how they interact with one another. If you modify the value of one, most likely you will affect a few more as a result. However, altering the value of a parameter is not like moving a slider up or down. It is more like a physical modification, for example : longer voice coil wire, stiffer spider, heavier cone, adding a series **resistor**, etc. **If you are not familiar with the Thiele Small parameters, I invite you to read this first.**

**Check the units of measurement for each equation. Sometimes the formula is a bit different from what you might know. This is because more convenient units have been chosen. For example : grams for the moving mass (instead of kilograms). However, sometimes the international units have been used, so please check for each one.**

### F_{s} – Resonant frequency

We all know the ball hooked up to a spring analogy. The ball will bounce differently (frequency) depending on how heavy the ball is or how stiff the spring is. If the ball is heavy, it will take long bounces. Therefore, reduced frequency, as it takes longer to complete a cycle. However, if the spring is stiffer, it pulls the ball back faster. In conclusion, higher frequency. Since we’re talking about thiele small parameters equations, here is the equation for the resonant frequency :

Units of measurement :

- F
_{s}-> Hz. - C
_{ms}-> mm/N (millimeters/newton) . - M
_{ms}-> g (grams).

Altering the compliance or the moving mass will directly affect the resonant frequency :

- Increasing C
_{ms}will decrease F_{s}. This can be done by making the suspension looser. - Increasing M
_{ms}will decrease F_{s}. This can be done by choosing a heavier material for the cone.

### V_{as} – Equivalent compliance in liters

V_{as }expresses the compliance of the speaker in terms of volume. Imagine a syringe without the needle. Close up the nozzle with your finger. If you try to push the plunger, you will encounter resistance from the air trapped inside the tube. This amount of air has a certain compliance. If the syringe is bigger (higher volume of air), the air is easier to compress, therefore, higher compliance. Having said this analogy, V_{as} is the compliance of the speaker expressed in liters.

**V _{as} = 0.0014 * S_{d}^{2} * C_{ms}**

Units of measurement :

- V
_{as}-> l (liters). - S
_{d}-> cm^{2}. - C
_{ms}-> mm/N.

Increasing the size of the speaker or the compliance (looser suspension) will increase V_{as} as a result.

### Qes, Qms and Qts

These 3 thiele small parameters equations have more to do with the interaction, rather than the calculation. To calculate them, you would use the impedance curve, rather than the following equations :

**R _{es} = Z_{0} – R_{e}**

**Q _{ms} = R_{es} / (BL^{2} * C_{ms} * 6.283 * F_{s})**

**Q _{es} = R_{e} / (BL^{2} * C_{ms} * 6.283 * F_{s})**

Units of measurement :

- R
_{es}is calculated by subtracting the voice coil resistance (R_{e}) from the impedance peak measured at resonance (Z_{0}) – all measured in ohms. - BL -> Tm (Tesla * meters).
- C
_{ms}-> m/N (meters/newton). - F
_{s}-> Hz.

**Q _{ts} = (Q_{es} * Q_{ms}) / (Q_{es} + Q_{ms})**

Calculate Q_{ts} by adding Q_{es} and Q_{ms} like resistances in parallel.

Conclusions :

- Increasing BL, C
_{ms}or F_{s}will reduce Q_{es}and Q_{ms}. - Increasing R
_{e}will result in increasing Q_{es}. This can be done by adding a series resistor, but this will also affect the efficiency of the driver in a negative way. - A higher impedance peak at resonance will translate in a higher Q
_{ms}.

### M_{ms} – moving mass

This is one of the most obvious thiele small parameters equations. First of all, let’s talk about the components of the moving mass :

- M
_{mr}– the air mass load – the air in front of the cone that follows the cone motion. - M
_{md}– the assembly mass – the mass of all the components that move (cone, voice coil, half of the surround, half of the spider).

Here are the equations for calculating the moving mass :

**M _{mr} = 0.000575 * S_{d}^{1.5}**

**M _{ms} = M_{md} + M_{mr}**

Units of measurement : the masses are in grams and S_{d} is in cm^{2}.

Clearly the air mass load is highly dependent on the size of the speaker. The moving mass is the sum of the assembly mass and the air mass load.

### L_{e} – Inductance

L_{e} is the voice coil inductance and it’s measured in millihenries (mH). The equation for the voice coil inductance is :

**L _{e} = 1.592 * 10^{-5} * (R_{10k} – R_{e}^{2})^{1/2}**

- R
_{10k}is the resistance at 10 kHz measured in ohms. - R
_{e}is the DC resistance measured in ohms.

### n_{0} and SPL rating – Efficiency

n_{o} is a percentage, showing how efficient the driver is at converting an electrical signal to an acoustical one. As a result, the bigger the number, the greater the reference sound pressure level.

**n _{0} = (9.7822 * 10^{-10} * V_{as} * F_{s}^{3}) / Q_{es}**

- n
_{0}above is a ratio, not a percentage. To make it a percentage multiply by 100. - V
_{as}is in liters. - F
_{s}in Hz.

**SPL @ 1W/1m = 112.2 + 10 * log(n _{0})**

The SPL rating is in direct proportion to n_{0}. Important to note is that the efficiency coefficient (n_{0}) is highly dependent to the resonant frequency, because it’s at the power of 3. In conclusion, tweeters and mid-range drivers will be more efficient versus subwoofers.

### Conclusion on Thiele Small parameters equations

You can find the value of the thiele small parameters in different ways, either through direct measurement, or by using formulas. However, by knowing the formulas, you can see how they interact with one another, and draw conclusion just by inspecting the driver. For example, if you look at a speaker, and see its got a big magnet, you can deduce the Bl is high. As a result Q will be lower (better damping) and sensitivity will be higher.

*References*

- Loudspeaker Design Cookbook 7th Edition by Vance Dickason (Audio Amateur Pubns, 2005).
- Speaker Building 201: A Comprehensive Course in Speaker Design by Ray Alden (Audio Amateur Pubns, 2004).
- Image source : link.