Electro-Magnetic World: January 2020

In the early seventies, several technical papers were presented to the AES (Audio Engineering Society) that resulted in the development of what we know today as ‘Thiele-Small Parameters’. These papers were authored by A.N.Thiele and Richard H. Small. Thiele was the senior engineer of design and development for the Australian Broadcasting Commission and was responsible at the time for the Federal Engineering Laboratory, as well as for analyzing the design of equipment and systems for sound and vision broadcasting. Small was, at the time, a Commonwealth Post-graduate Research Student in the School of Electrical Engineering at the University of Sydney.

Fs (Hz)

This parameter is the free-air resonant frequency of a speaker. It is the frequency where the driver moves with minimal effort. In other words, it is the point at which the weight of the moving parts of the speaker becomes balanced with the force of the speaker suspension when in motion. If you’ve ever seen a piece of string start humming uncontrollably in the wind, you have seen the effect of reaching a resonant frequency. It is important to know this information so that you can prevent your enclosure from ‘ringing’. If you tap a speaker (or any object for that matter), it will make a sound, which has the same frequency as its resonant frequency. With a loudspeaker, the mass of the moving parts, and the stiffness of the suspension (surround and spider) are the key elements that affect the resonant frequency. When the driver reaches the resonance frequency, its response starts to roll off. As a general rule of thumb, a lower Fs indicates a woofer that would be better for low-frequency reproduction than a woofer with a higher Fs. This is not always the case though, because other parameters affect the ultimate performance as well. The lower the Fs , the better. A woofer with Fs of 50 Hz will not play well at 40 Hz, and a woofer with Fs of 30 Hz will play well at 40 Hz. Woofers can have resonant frequencies of 20 Hz or even lower. While you can’t hear those frequencies, you can feel them. Fs for midrange drivers and tweeters is irrelevant, as they will probably play above that frequency anyway.

Re (Ohms)

The Re parameter is the DC resistance of the driver measured with an ohm meter and it is often referred to as the ‘DCR’. This measurement will almost always be less than the driver’s nominal impedance. Consumers sometimes get concerned the Re is less than the published impedance and fear that amplifiers will be overloaded. Due to the fact that the inductance of a speaker rises with a rise in frequency, it is unlikely that the amplifier will often see the DC resistance as its load.

Le (mH)

This is the voice coil inductance measured in millihenries (mH). The industry standard is to measure inductance at 1 kHz. As frequencies get higher there will be a rise in impedance above Re. This is because the voice coil is acting as an inductor. Consequently, the impedance of a speaker is not a fixed resistance, but can be represented as a curve that changes as the input frequency changes. Maximum impedance (Zmax) occurs at Fs. When current is applied to the voice coil, at the same time, an additional current flow is created, in the opposite direction of the current flow, called back EMF, which is short for electromotive force. As current flows through the voice coil, it moves it into a certain direction, and back EMF tries to move it in the opposite direction. That is why the impedance spikes at resonance frequency. At that frequency the speaker easily reaches high excursions and back EMF is working hard to pull it back. Inductance causes the impedance to rise as the frequency goes up. Large Le values will translate into poor high frequency response. To improve the high frequency response, a technique called shorting ring or Faraday loop can be used, but that's another subject for discussion.

Impedance (Ohms)

The impedance is the AC resistance, which is not a fixed value because the speaker is moving and the impedance varies with frequency. Impedance will have a high value at resonance frequency. Usually the manufacturers quote one number, like 4, 6 or 8 ohms. A 8 ohm speaker can have impedance vary from 6 ohms to 25 ohms or more, but for the most frequencies it will be around 8 ohms.

Bl (Tm)

The Bl parameter is product of B, which is the flux density, and l, which is the length of the voice coil. In other words, this is a measurement of the motor strength of a speaker. Think of this as how good a weightlifter the transducer is. A measured mass is applied to the cone forcing it back while the current required for the motor to force the mass back is measured. The formula is mass in grams divided by the current in amperes. A higher Bl will translate in higher efficiency. Of course, the efficiency is determined by lots of factors, so a higher Bl doesn’t necessarily mean a higher SPL. To get the bigger picture, the bigger magnet and bigger coil equals bigger motor. Neodymium magnets are stronger than normal ferrite magnets and they don’t need to be as large. The strength of the motor is in direct correlation with the size and the weight of the cone, size of the coil, size of the magnet, size of the basket. If we modify Bl, that will change a lot of things. A high Bl speaker will be suitable for loaded horn applications. Also, a high Bl will translate into better transients (sudden sounds). The motor has enough power to move the cone with a fast reaction time. Bl is in accordance with the size of the speaker, so it’s hard to give an estimate of which is high and which is low. Bl of around 10 is pretty average.

Q Parameters

Qms, Qes and Qts are measurements related to the control of a transducer’s suspension when it reaches the resonant frequency (Fs). The suspension must prevent any lateral motion that might allow the voice coil and pole to touch (this would destroy the loudspeaker). The suspension must also act like a shock absorber.

Q (Unitless)

Also called quality factor or damping factor. The damping of the speaker, is a characteristic that helps the speaker to resume its rest state. Without adequate damping, a speaker would move uncontrollably at resonance frequency. Q actually stands for quality factor and is the inverse of damping. As damping goes up, Q goes down, but it is widely accepted that Q is a measurement of damping. There are 3 types of speaker damping: mechanical, electrical and pneumatic.

Qms

Qms is the mechanical Q or the damping made by the suspension of the driver: the surround and the spider of the speaker. Qms is a measurement of the control coming from the speaker’s mechanical suspension system (the surround and spider). View these components like springs.

Qes

Qes is the electrical Q or the damping made by the coil – magnet assembly. Qes is a measurement of the control coming from the speaker’s electrical suspension system (the voice coil and magnet). Opposing forces from the mechanical and electrical suspensions act to absorb the shock. When the coil moves through the magnetic field, it generates a current which opposes this motion (hence the electrical damping). Another factor which contributes to the electrical damping is the amplifier. This depends on the particular amplifier. The Qes provided by the speaker manufacturer does not include amplifier damping.

Qts

Also called total Q – The damping made by Qms and Qes combined. Qts is called the ‘Total Q’ of the driver and is derived from an equation where Qes is multiplied by Qms and the result is divided by the sum of the same: 1/Qts = 1/Qms + 1/Qes. This is the Q we should look for if we plan to use it in the free air.

Qtc

This is the pneumatic damping. This parameter exists only when there is a box in the equation. Depending on the size of the box, the air inside it will act like a spring and contribute to the dampening of the speaker. You can say that Qtc = Qts + Q of the box. This is the Q we should look for if we plan to use a sealed box.

Qts of values of 0.6 or higher, will demand a very large box. The predefined bass-reflex alignments can be used for Qts values lower than 0.7. Higher Qes values suggests that the woofer is more suitable for sealed enclosures, while lower values recommends the bass-reflex. As a general guideline, Qts of 0.4 or below indicates a transducer well suited to a vented enclosure. Qts between 0.4 and 0.7 indicates suitability for a sealed enclosure. Qts of 0.7 or above indicates suitability for free-air or infinite baffle applications. However, there are exceptions.

Cms (m/N)

Cms is the compliance of the speaker. Cms is measured in meters per Newton. Cms is the force exerted by the mechanical suspension of the speaker. It is simply a measurement of its stiffness. The suspension of the speaker (the surround and the spider) has a certain stiffness. If the suspension is stiff, the driver is not compliant. So, the easy it is to move the speaker, the more compliant it is. Considering stiffness (Cms), in conjunction with the Q parameters gives rise to the kind of subjective decisions made by car manufacturers when tuning cars between comfort to carry the president and precision to go racing. Think of the peaks and valleys of audio signals like a road surface then consider that the ideal speaker suspension is like car suspension that can traverse the rockiest terrain with race-car precision and sensitivity at the speed of a fighter plane. It’s quite a challenge because focusing on any one discipline tends to have a detrimental effect on the others.

Compliance affects the resonant frequency. A higher Cms will yield a lower Fs. If Cms goes up => Fs goes down. Like, for example, a ball on a spring. The stiffness of the spring determines the compliance. If the spring is stiff, then it is less compliant and the ball will bounce at a higher frequency (short and fast bounces). If the spring is not stiff, or more compliant, the ball will make long bounces - reduced frequency.

Vas (l)

Vas represents the volume of air that when compressed to one cubic meter exerts the same force as the compliance (Cms) of the suspension in a particular speaker. The air inside the cabinet has its own compliance. When you try to compress the air inside a box, you will encounter resistance. If the box is small, the air is harder to compress and therefore less compliant, and if the box is larger, the air is easier to compress, therefore more compliant. In conclusion, Vas describes the volume of the air inside the cabinet, where the compliance of the speaker matches the compliance of the air inside the box. Vas is one of the trickiest parameters to measure because air pressure changes relative to humidity and temperature — a precisely controlled lab environment is essential.

Xmax (mm)

Short for Maximum Linear Excursion. Speaker output becomes non-linear when the voice coil begins to leave the magnetic gap. Although suspensions can create non-linearity in output, the point at which the number of turns in the gap (see BL) begins to decrease is when distortion starts to increase. Xmax is the maximum distance a speaker can travel without distorting. The coil has a certain length and moves up and down inside the magnetic gap of the motor. If the coil travels too far and leaves the magnetic gap, the speaker will distort, as the magnet has a reduced control on the voice coil. Don’t confuse this Thiele/Small parameter with Xmech.

Xmax = ((height of the voice coil) – (height of the magnetic gap)) / 2

Xmech (mm)

Is the maximum distance a speaker can travel without damaging the driver. When a driver is exceeding the quoted Xmax, distortion is introduced into the sound. However, if the driver exceeds the quoted Xmech, the mechanical limits of the driver are reached and damage can occur to the driver. When the driver travels forward, it will stretch the surround until it can’t move forward. It looks and sounds disturbing. On the way back, the voice coil will hit the back plate of the magnet and will sound like loud bangs/knocks. The exceeded Xmech of the speaker can damage it.

Sd (m^2)

The effective area of the cone or the actual surface area of the cone, normally given in square cm. This is important, if you want to reach high pressure levels (Xmax also). You are probably wandering why 2 speakers of the same quoted dimensions have different Sd ? It is because only half of the surround is considered cone area, so larger surrounds will yield a smaller Sd.

Vd

This parameter is the Peak Diaphragm Displacement Volume — in other words the volume of air the cone will move. It is calculated by multipying Xmax (Voice Coil Overhang of the driver) by Sd (Surface area of the cone).

MMS

This parameter is the combination of the weight of the cone assembly plus the ‘driver radiation mass load’. The weight of the cone assembly is easy: it’s just the sum of the weight of the cone assembly components. The driver radiation mass load is the confusing part. In simple terminology, it is the weight of the air (the amount calculated in Vd) that the cone will have to push.

EBP

This measurement is calculated by dividing Fs by Qes. The EBP figure is used in many enclosure design formulas to determine if a speaker is more suitable for a closed or vented design. An EBP close to 100 usually indicates a speaker that is best suited for a vented enclosure. On the contrary, an EBP closer to 50 usually indicates a speaker best suited for a closed box design. This is merely a starting point. Many well-designed systems have violated this rule of thumb! Qts should also be considered.

Mms and Mmd (g)

This is the total moving mass. If you place on a scale the cone, the coil, half of the surround and half of the spider, you got yourself the value of Mmd. If you add to this equation the weight of the air in front of the speaker, then you will get the Mms value. When the speaker is moving, the pocket of air directly in front of it, will move with the cone. This air has its own mass and has to be accounted for, when calculating the total moving mass (Mms). If Mms goes up, the Fs goes down (imagine a ball hanging on a spring. If the ball is heavier, the ball will bounce at a lower frequency). If Mms goes up, the efficiency goes down (more amplifier power is needed to push the cone).

SPL (dB)

SPL stands for sound pressure level. The higher the number, the higher the efficiency. Good SPL rating is around 88 – 90 db, at 1 W / 1 m. This means that the manufacturer picks a certain frequency (depending on the type of the driver : woofer, midrange, tweeter), places a microphone at 1 meter from the speaker, and plays a 1 W tone at that frequency. How many decibels the microphone picks up is the actual SPL. The higher the efficiency, the better. This takes impedance into account. At 8 ohms there is no difference. So 90 db measured at 2.83 V / 1 m are the same as 90 db measured at 1 W / 1 m. But if the speaker is 4 ohms, 90 db measured at 2.83 V / 1 m is equal to 90 db measured at 2 W / 1 m, which is equivalent to 87 db 1 W / 1m. For a 2 ohm speaker 90 db 2.83 V / 1 m is equivalent to 84 db 1 W / 1 m and so on.

Usable frequency range

This is the frequency range for which Eminence feels the transducer will prove useful. Manufacturers use different techniques for determining ‘Usable Frequency Range’. Most methods are recognized as acceptable in the industry, but can arrive at different results. Technically, many loudspeakers are used to produce frequencies in ranges where they would theoretically be of little use. As frequencies increase, the off-axis coverage of a transducer decreases relative to its diameter. At a certain point, the coverage becomes ‘beamy’ or narrow like the beam of a flashlight. If you’ve ever stood in front of a guitar amplifier or speaker cabinet, then moved slightly to one side or the other and noticed a different sound, you have experienced this phenomenon and are now aware of why it occurs. Clearly, most two-way enclosures ignore the theory and still perform quite well. The same is true for many guitar amplifiers, but it is useful to know at what point you can expect a compromise in coverage.

Power handling

This specification is very important to transducer selection. Obviously, you need to choose a loudspeaker that is capable of handling the input power you are going to provide. By the same token, you can destroy a loudspeaker by using too little power. The ideal situation is to choose a loudspeaker that has the capability of handling more power than you can provide lending some headroom and insurance against thermal failure. To use an automobile as an analogy; you would not buy a car that could only go 55mph if that were the speed you always intended to drive. Generally speaking, the number one contributor to a transducer’s power rating is its ability to release thermal energy. This is affected by several design choices, but most notably voice coil size, magnet size, venting, and the adhesives used in voice coil construction. Larger coil and magnet sizes provide more area for heat to dissipate, while venting allows thermal energy to escape and cooler air to enter the motor structure. Equally important is the ability of the voice coil to handle thermal energy. Eminence is renowned for its use of proprietary adhesives and components that maximize the voice coil’s ability to handle extreme temperatures. Mechanical factors must also be considered when determining power handling. A transducer might be able to handle 1,000W from a thermal perspective, but would fail long before that level was reached from a mechanical issue such as the coil hitting the back plate, the coil coming out of the gap, the cone buckling from too much outward movement, or the spider bottoming on the top plate. The most common cause of such a failure would be asking the speaker to produce more low frequencies than it could mechanically produce at the rated power. Be sure to consider the suggested usable frequency range and the Xlim parameter in conjunction with the power rating to avoid such failures. The Eminence power rating is derived using an EIA 426A noise source and test standard. All tests are conducted for eight hours in a free-air, non-temperature controlled environment. Eminence tests samples from each of three different production runs and each sample must pass a test exceeding the rated power by 50 to 100W. The Eminence music program is double that of our standard Watts rating.

Sensitivity

This data represents one of the most useful specifications published for any transducer. It is a representation of the efficiency and volume you can expect from a device relative to the input power. Loudspeaker manufacturers follow different rules when obtaining this information — there is not an exact standard accepted by the industry. As a result, it is often the case that loudspeaker buyers are unable to compare ‘apples to apples’ when looking at the sensitivities of different manufacturers’ products. Eminence sensitivities are expressed as the average output across the usable frequency when applying 1W/1M into the nominal impedance. ie: 2.83V/8 ohms, 4V/16 ohms.

Formulas:

Fs – Resonant frequency

According to 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:

Fs = 50*Pi*sqrt(1/(Cms*Mms))

Units of measurement:

Fs in Hz;
Cms in mm/N (millimeters/newton);
Mms in g (grams).

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

>> Increasing Cms will decrease Fs. This can be done by making the suspension looser.
>> Increasing Mms will decrease Fs. This can be done by choosing a heavier material for the cone.

Vas – Equivalent compliance in liters

Vas 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, Vas is the compliance of the speaker expressed in liters.

Vas = 0.0014 * Sd^2 * Cms

Units of measurement:

Vas in l (liters);
Sd in cm^2;
Cms in mm/N.

Increasing the size of the speaker or the compliance (looser suspension) will increase Vas 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:

Res = Z0 – Re

Qms = Res / (BL^2 * Cms * 6.283 * Fs)

Qes = Re / (BL^2 * Cms * 6.283 * Fs)

Units of measurement:

Res is calculated by subtracting the voice coil resistance (Re) from the impedance peak measured at resonance (Z0) – all measured in ohms;
BL in Tm (Tesla * meters);
Cms in m/N (meters/newton);
Fs in Hz.

Qts = (Qes * Qms) / (Qes + Qms)

Calculate Qts by adding Qes and Qms like resistances in parallel.

Conclusions: Increasing BL, Cms or Fs will reduce Qes and Qms. Increasing Re will result in increasing Qes. 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 Qms.

Mms – 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:

Mmr – the air mass load – the air in front of the cone that follows the cone motion;
Mmd – 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:

Mmr = 0.000575 * Sd^1.5

Mms = Mmd + Mmr

Units of measurement: the masses are in grams and Sd 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.

Le – Inductance

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

Le = 1.592 * 10^-5 * (R10k – Re^2)^1/2

Where R10k is the resistance at 10 kHz measured in ohms, and Re is the DC resistance measured in ohms.

n0 and SPL rating – Efficiency

n0 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.

n0 = (9.7822 * 10^-10 * Vas * Fs^3) / Qes

n0 above is a ratio, not a percentage. To make it a percentage multiply by 100;
Vas is in liters;
Fs in Hz;

SPL @ 1W/1m = 112.2 + 10 * log(n0)

The SPL rating is in direct proportion to n0. Important to note is that the efficiency coefficient (n0) 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.

Electro-Magnetic World

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Measuring Thiele/Small parameters

Thiele-Small Parameters