What Are Thiele-Small Parameters?

Thiele-Small parameters, commonly abbreviated as T/S parameters, describe the low-frequency behaviour of a loudspeaker driver using a set of measurable parameters.

They allow a driver to be represented by an equivalent electrical and mechanical model.

Using these parameters, a designer can estimate how a driver will behave when installed in a particular enclosure.

The parameters are particularly important for:

• Sealed speaker box design
• Bass-reflex enclosure design
• Passive-radiator systems
• Transmission-line development
• Subwoofer design
• Driver comparison
• Loudspeaker simulation

Why Thiele-Small Parameters Matter

A loudspeaker cannot be selected for an enclosure simply by looking at its diameter and power rating.

Two 12-inch woofers can have completely different low-frequency characteristics.

One may be well suited to a small sealed enclosure while another may require a much larger bass-reflex cabinet.

The T/S parameters provide the information required to make this decision.

Important Thiele-Small Parameters

Fs — Free-Air Resonance Frequency

Fs is the natural resonant frequency of the driver when it is operating in free air without the additional air-spring loading of an enclosure.

It is normally expressed in hertz.

Fs = resonance frequency

For example:

 Fs = 35 Hz 

means that the driver's free-air mechanical resonance occurs at approximately 35 Hz.

Why Fs Is Important

Fs provides an important indication of the driver's low-frequency capability.

A lower Fs generally indicates that the driver can operate toward lower frequencies, although Fs alone does not determine the final frequency response.

The enclosure, Q parameters and other driver characteristics must also be considered.

How Fs Is Measured

Fs can be determined from the driver's impedance curve.

A free-air driver normally shows a prominent impedance peak near its resonant frequency.

 Impedance │ │ /\ │ / \ │ / \ │__________/ \________ │ └──────────────────────────► Frequency ↑ Fs 

The frequency at the impedance peak is approximately the driver's Fs.

See also: Speaker Impedance Testing .

Re — Voice-Coil DC Resistance

Re is the DC resistance of the voice coil.

It is measured in ohms.

Re is normally lower than the speaker's nominal impedance.

For example, an 8 Ω nominal driver may have a Re of approximately several ohms.

Why Re Matters

Re affects the electrical damping of the driver and is one of the parameters used in calculating the driver's electrical Q.

It also determines part of the electrical load presented to the amplifier under DC conditions.

Q — What Does It Mean?

The letter Q describes the damping or sharpness of a resonant system.

A high Q generally corresponds to a more strongly peaked resonance, while a lower Q indicates greater damping and a broader resonance.

Three important Q parameters are:

• Qms
• Qes
• Qts

Qms — Mechanical Q

Qms describes the mechanical damping of the driver.

It is associated with losses in the mechanical suspension and moving system.

A high Qms indicates relatively low mechanical losses.

Qes — Electrical Q

Qes describes the electrical damping associated with the voice coil and motor system.

The magnetic motor converts mechanical motion into an electrical interaction with the amplifier.

The electrical damping is therefore strongly influenced by the motor strength and voice-coil resistance.

Qts — Total Q

Qts represents the total damping of the driver at resonance, combining the mechanical and electrical contributions.

It is related to Qms and Qes by:

 1 / Qts = 1 / Qms + 1 / Qes 

or:

 Qts = (Qms × Qes) / (Qms + Qes) 

Qts is one of the most useful parameters when evaluating enclosure suitability.

Why Qts Is Important for Enclosure Design

Qts gives the designer an indication of the driver's total damping and helps determine which enclosure alignments may be appropriate.

It should not be used alone, but it is an important starting point.

The complete design also requires Fs, Vas, Sd and other parameters.

Vas — Equivalent Compliance Volume

Vas represents the volume of air that has the same acoustic compliance as the driver's suspension.

It is normally expressed in litres.

For example:

 Vas = 80 litres 

does not mean that the driver physically contains 80 litres of air. It represents an equivalent acoustic compliance.

Why Vas Matters

Vas is important when calculating enclosure volume.

A driver with a large Vas generally has a softer suspension and can require a larger enclosure for certain alignments.

A driver with a smaller Vas may be suitable for a smaller enclosure, depending on its other parameters.

Sd — Effective Diaphragm Area

Sd is the effective radiating area of the diaphragm.

It is normally expressed in square metres or square centimetres.

Sd is related to how much air the driver can move for a given cone excursion.

Why Sd Matters

A larger diaphragm area can move more air for the same excursion.

For a given displacement, the approximate volume of air displaced is related to:

 Vd = Sd × Xmax 

where Vd is the linear displacement volume.

Vd — Displacement Volume

Vd is the approximate volume displacement produced by the driver at its specified linear excursion.

 Vd = Sd × Xmax 

For example, if:

 Sd = 0.053 m² Xmax = 0.005 m 

then:

 Vd = 0.053 × 0.005 Vd = 0.000265 m³ ``` 
 which is approximately: 
 
 0.265 litres 
 

Mms — Moving Mass

Mms is the total moving mass of the loudspeaker system.

It includes the effective mass of components such as:

• Diaphragm
• Voice coil
• Dust cap
• Suspension contributions
• Other moving components

Mms is normally expressed in grams or kilograms.

How Mms Affects the Speaker

Moving mass influences the driver's resonance and high-frequency behaviour.

For a given suspension compliance, increasing the moving mass generally lowers the resonant frequency.

A simplified relationship is:

 Fs = 1 / (2π√(Mms × Cms)) 

Cms — Mechanical Compliance

Cms describes the compliance, or flexibility, of the driver's mechanical suspension.

It is associated primarily with the surround and spider.

A higher compliance means the suspension is mechanically softer.

Cms and Fs

The driver's resonance depends on both moving mass and suspension compliance.

The relationship can be expressed as:

 Fs = 1 / (2π√(Mms × Cms)) 

A softer suspension generally lowers Fs, while a stiffer suspension generally raises it, assuming the moving mass remains unchanged.

Rms — Mechanical Resistance

Rms represents mechanical losses in the moving system.

These losses come from mechanisms such as suspension damping and other mechanical frictional effects.

Rms contributes to the driver's mechanical Q.

BL — Motor Force Factor

BL describes the strength of the interaction between the magnetic field and the voice coil.

It is normally expressed in tesla-metres, equivalent to newtons per ampere.

A higher BL generally indicates a stronger motor force for a given current, although driver behaviour depends on the complete motor system.

Why BL Matters

The motor force acting on the voice coil can be represented approximately by:

 F = BL × I 

where:

• F = force
• B = magnetic flux density
• L = effective conductor length
• I = voice-coil current

The combined BL parameter is therefore important in determining how strongly the motor drives the moving system.

Xmax — Linear Excursion

Xmax represents the approximate maximum linear excursion of the driver.

It is normally expressed in millimetres.

For example:

 Xmax = ±5 mm 

means that the driver is specified to maintain approximately linear operation over a peak excursion of 5 mm in each direction, according to the manufacturer's definition.

Why Xmax Matters

Low-frequency output requires the driver to move substantial amounts of air.

When a driver reaches its excursion limit, distortion increases and mechanical damage may occur if the driver is driven beyond its safe limits.

Xmax is therefore particularly important for subwoofer design.

Le — Voice-Coil Inductance

Le represents the voice-coil inductance.

It is normally expressed in millihenries.

Because the voice coil is an inductor, its reactance increases with frequency.

 Xl = 2πfL 

Le therefore influences the driver's high-frequency impedance and electrical response.

Equivalent Circuit of a Loudspeaker

A loudspeaker can be represented by an equivalent electrical and mechanical model.

 Electrical domain │ ▼ Re + Le │ ▼ Motor BL │ ▼ Mechanical system │ ┌─────┴─────┐ │ │ Mms Cms │ │ └────Rms───┘ 

The electrical and mechanical systems are coupled through the loudspeaker motor.

Electrical, Mechanical and Acoustic Parameters

Relationship Between the Parameters

The T/S parameters are not independent numbers.

Many of them are mathematically related.

For example:

 Fs = 1 / (2π√(Mms × Cms)) 

and:

 Qts = (Qms × Qes) / (Qms + Qes) 

The parameters together describe the driver's low-frequency electromechanical behaviour.

How T/S Parameters Are Measured

T/S parameters are normally determined from electrical and mechanical measurements of the driver.

A typical measurement system can measure the driver's impedance curve and derive several of the parameters mathematically.

Additional information such as diaphragm area and known test masses may be used depending on the measurement method.

Basic T/S Measurement Setup

 Computer │ ▼ Audio interface │ ▼ Reference resistor │ ▼ Speaker driver │ ▼ Measurement software │ ▼ Impedance curve │ ▼ T/S parameters 

The test system normally uses a low-level frequency sweep to measure the driver's impedance.

Impedance Peak and Fs

The driver's free-air impedance curve normally contains a prominent resonance peak.

The frequency at this peak provides the basic measurement of Fs.

Additional characteristics of the peak are used to determine damping and Q parameters.

Qms and the Resonance Peak

Qms describes the mechanical losses associated with the resonance.

A highly damped mechanical system tends to have a broader resonance, while a lightly damped system can exhibit a sharper resonance.

The exact value is determined from the measured impedance behaviour.

Qes and Electrical Damping

The driver's electrical damping is strongly influenced by the motor strength and voice-coil resistance.

The amplifier effectively provides electrical damping through the interaction between the voice coil and motor.

This contributes to Qes.

Qts and Driver Selection

Qts is often used as a first indicator when evaluating a woofer for a particular enclosure type.

However, it should never be used as the only selection criterion.

Fs, Vas, Sd, Xmax, power handling and the intended frequency response must also be considered.

Thiele-Small Parameters and Sealed Enclosures

The T/S parameters allow the designer to predict the behaviour of a driver installed in a sealed enclosure.

Important parameters include:

• Fs
• Vas
• Qts

The enclosure volume changes the acoustic compliance seen by the driver.

Sealed Box Volume

A simplified relationship for a sealed enclosure is:

 Vb = Vas / ((Qtc / Qts)² - 1) 

where:

• Vb = enclosure volume
• Vas = equivalent compliance volume
• Qtc = desired system Q
• Qts = driver's total Q

This relationship is useful for basic sealed-box calculations.

System Resonance in a Sealed Box

The system resonance of a sealed enclosure can be estimated using:

 Fc = Fs × √(1 + Vas / Vb) 

where Fc is the approximate system resonance.

The actual acoustic response also depends on damping and the complete driver characteristics.

Qtc — System Q

Qtc describes the total Q of the driver and sealed enclosure system.

The enclosure changes the mechanical loading of the driver, so Qtc is different from the driver's free-air Qts.

Common design targets can include approximately:

 Qtc = 0.707 

but the appropriate target depends on the desired response.

Thiele-Small Parameters and Bass Reflex Enclosures

Bass reflex enclosure design depends strongly on the driver's T/S parameters.

Important parameters include:

• Fs
• Vas
• Qts
• Sd
• Re

The enclosure volume and port tuning are then selected to achieve the desired alignment.

Bass Reflex Tuning Frequency

The bass reflex tuning frequency is commonly represented by:

 Fb 

The desired Fb depends on the driver and enclosure alignment.

It is not simply equal to the driver's Fs.

Why Fs and Fb Are Different

Fs is the free-air resonance of the driver.

Fb is the tuning frequency of the complete bass reflex enclosure.

The enclosure volume and port form an acoustic resonant system that changes the overall behaviour.

T/S Parameters and Port Design

Once an enclosure volume and tuning frequency have been selected, the port dimensions can be calculated.

Port diameter, port area and port length determine the final tuning frequency.

The driver parameters are used to select an appropriate overall alignment.

T/S Parameters and Subwoofer Design

T/S parameters are especially important when designing subwoofers.

A subwoofer must move substantial amounts of air at low frequencies.

The combination of Sd and Xmax gives an indication of the driver's linear displacement capability.

 Vd = Sd × Xmax 

Large Sd vs Large Xmax

A driver can achieve high displacement through a large diaphragm area, large excursion, or a combination of both.

For example, a large woofer with moderate Xmax can move a substantial amount of air because its Sd is large.

A smaller driver may require much greater excursion to achieve the same displacement.

T/S Parameters and Power Handling

T/S parameters do not completely describe power handling.

A driver can have excellent T/S parameters but still have limited thermal or mechanical power handling.

Power rating, voice-coil temperature, excursion limits and other manufacturer specifications must also be considered.

T/S Parameters and Efficiency

The T/S parameters are also related to the driver's low-frequency efficiency.

A driver with a strong motor and suitable moving mass can achieve useful electromechanical efficiency, although final speaker sensitivity also depends on the complete acoustic system.

EBP — Efficiency Bandwidth Product

A commonly used rough indicator is:

 EBP = Fs / Qes 

EBP has historically been used as a rough guide when considering whether a driver may be more appropriate for sealed or vented applications.

It should not be treated as a definitive enclosure-selection rule. Modern design should consider the complete T/S data and simulate the actual alignment.

Example Driver Data

These values are only an example to demonstrate how T/S data is presented. They should not be interpreted as specifications for a particular commercial driver.

How to Read a Driver Datasheet

When evaluating a woofer for an enclosure, first identify the complete set of available T/S parameters.

At minimum, look for:

• Fs
• Re
• Qes
• Qms
• Qts
• Vas
• Sd
• Xmax

Additional information such as BL, Mms, Cms and Le can provide deeper insight into the driver.

Incomplete T/S Specifications

Some inexpensive drivers may provide only a small number of specifications.

For example:

 Power: 300 W Impedance: 8 Ω Sensitivity: 95 dB Fs: 40 Hz 

This information is not enough to accurately design many enclosure alignments.

Vas, Qts and other parameters may be required.

Measured vs Manufacturer T/S Parameters

T/S parameters can vary between individual drivers.

Manufacturers may publish nominal or typical values, while a measured driver may produce somewhat different results.

For critical enclosure development, measuring the actual driver can be valuable.

Why New Drivers Should Be Measured Carefully

The suspension of a new loudspeaker can change slightly during initial use.

This process is commonly called break-in.

Because suspension compliance can affect Fs and related parameters, measurements made at different stages can produce somewhat different results.

T/S Parameters and Speaker Repair

T/S testing can also be useful when repairing loudspeakers.

If a reconed driver behaves very differently from the original driver, its impedance and resonance measurements can help identify a problem.

Possible causes include:

• Incorrect voice coil
• Incorrect spider
• Incorrect surround
• Incorrect moving mass
• Improper centering
• Mechanical rubbing

Impedance Testing After Reconing

After reconing a woofer, measuring its DC resistance is a basic electrical check.

A low-level impedance sweep provides much more information.

The measured resonance can be compared with the expected behaviour of the repaired driver.

This can be especially useful for professional speaker repair.

T/S Parameters and Speaker Enclosures

The relationship between driver and enclosure can be summarized as:

 Driver T/S parameters ↓ Enclosure type ↓ Box dimensions ↓ Acoustic tuning ↓ Frequency response 

The enclosure should therefore be designed around the actual driver rather than chosen independently.

Computer Speaker Simulation

Speaker simulation software can use T/S parameters to predict the response of different enclosure alignments.

A designer can change:

• Enclosure volume
• Port tuning
• Driver parameters
• System Q
• Electrical filters
• Power input

The predicted response can then be used as a starting point for physical construction and measurement.

Important Limitation of T/S Models

Thiele-Small parameters primarily describe low-frequency electromechanical behaviour.

They do not completely describe the behaviour of a loudspeaker across the entire audio spectrum.

They do not fully capture:

• Cone breakup
• High-frequency resonances
• Detailed directivity
• Cabinet diffraction
• Nonlinear distortion
• Thermal compression
• Complex suspension nonlinearities

For complete speaker design, additional measurements are required.

T/S Parameters vs Frequency Response

T/S Parameters vs Impedance Testing

Impedance testing and T/S parameters are closely related.

A measured impedance curve provides much of the information needed to derive several T/S parameters.

This is why an impedance measurement system is a useful tool for speaker designers and repair technicians.

Common T/S Parameter Mistakes

• Choosing an enclosure using speaker diameter alone.
• Assuming nominal impedance is Re.
• Using Fs alone to predict bass extension.
• Using Qts alone to select an enclosure.
• Ignoring Vas.
• Ignoring Xmax in subwoofer design.
• Ignoring Sd.
• Assuming manufacturer parameters are exact for every driver.
• Using T/S parameters to predict high-frequency response.
• Ignoring the actual measured impedance and frequency response.
• Confusing driver Fs with bass reflex tuning frequency Fb.

Practical T/S Parameter Workflow

1. Obtain the manufacturer's complete T/S specifications.
2. Check whether the parameters are measured or typical values.
3. Verify the driver's nominal impedance and Re.
4. Check Fs.
5. Check Qms, Qes and Qts.
6. Check Vas.
7. Check Sd.
8. Check Xmax.
9. Determine the intended enclosure type.
10. Simulate the enclosure.
11. Build a prototype.
12. Measure the finished speaker.
13. Compare measured results with the simulation.
14. Adjust the design if necessary.

T/S Parameter Design Checklist

• Fs known
• Re known
• Qms known
• Qes known
• Qts known
• Vas known
• Sd known
• Mms known
• Cms known where available
• BL known where available
• Xmax known
• Le known where available
• Enclosure type selected
• Enclosure simulated
• Port tuning calculated if applicable
• Prototype constructed
• Impedance measured
• Frequency response measured
• Final design verified

Key Takeaways

• Thiele-Small parameters describe important low-frequency characteristics of a loudspeaker driver.
• Fs is the driver's free-air resonance frequency.
• Re is the voice-coil DC resistance.
• Qms describes mechanical damping.
• Qes describes electrical damping.
• Qts represents the combined total Q.
• Vas represents the equivalent acoustic compliance volume.
• Sd is the effective diaphragm area.
• Mms is the total moving mass.
• Cms represents mechanical compliance.
• Rms represents mechanical resistance.
• BL represents the motor force factor.
• Xmax represents the driver's specified linear excursion limit.
• Le represents voice-coil inductance.
• Vd can be estimated from Sd × Xmax.
• T/S parameters are particularly useful for enclosure design.
• Sealed and bass-reflex enclosures depend strongly on T/S parameters.
• Fs is not the same as the tuning frequency of a bass-reflex enclosure.
• T/S parameters do not completely describe high-frequency response, directivity or distortion.
• Measured impedance and frequency response should be used to verify the finished loudspeaker.