Thiele - Small Analysis of Loudspeaker Enclosures

This section is based on the classic work of Neville Thiele, Richard Small, and the earlier work of Bart Locanthi. It is a bit technical, and unavoidably requires some familiarity with electrical circuit theory. The first section presents a derivation of the equivalent circuit of a speaker in sealed, and vented (bass-reflex), enclosures. Then the relationship between the equivalent circuit response and sound level is presented. The effect enclosure volume has on efficiency, frequently misunderstood, is discussed. Then infinite baffle, acoustic suspension, and bass-reflex enclosure designs are compared. Additional issues regarding bass-reflex designs are briefly reviewed. Sensitivity of results to misalignment is discussed, and finally the interaction between enclosures, crossovers, and cables is investigated, as well as the effect of the amplifier damping factor.

Thiele-Small analysis overview

The Thiele-Small approach is to first analyze the electro-mechanical behavior of a speaker voice coil, magnet, and cone, interacting with the cone suspension and the air in and outside the enclosure. The resulting equation is mathematically identical to the equation describing a purely electrical "equivalent circuit" consisting solely of resistors, capacitors and inductors. The sound produced by the loudspeaker can then be obtained via a relatively simple circuit analysis. The highly evolved theory of filter synthesis can be used to adjust circuit parameters to obtain a desired frequency response. The parameters can then be translated back into physical quantities, such as enclosure size, to build the loudspeaker. This procedure provides a scientific framework (veneer?) for the art of loudspeaker design. A very useful result is that after the speaker has been assembled, the electrical impedance at the speaker terminals can be measured and compared to the theory. If it differs, the design can be tuned based on this measurement, which is both simple and accurate. The location and height of the impedance spike at resonance are sensitive to any errors in the design, as will be shown. (However, as noted below, for a bass-reflex design some shift in the location of the peaks can be caused by mutual coupling, rather than by a design error per se). This analysis is particularly valuable for designing bass-reflex enclosures where a ducted port (also called a vent) allows air inside the enclosure to radiate in conjunction with the speaker cone.

The driver parameters, which determine several of the equivalent circuit elements, are known as "Thiele parameters," and are fairly standardized; a table given here[13.5kb] provides precise definitions and units, and also points out some potential pitfalls in using published values. A method for measuring the parameters is given on the Subwoofer DIY page.

The notation in this section generally follows Small's June 1973 paper (he alters his notation between analyzing drivers, sealed enclosures, and vented enclosures, but all three situations are covered by the circuit derived here).

Back to the top

The equivalent circuit

Small's papers and Dickason's book skip the derivation of the equivalent circuit. You don't need this to build a loudspeaker, but I like to feel as though I understand what is going on physically, so I'm going to go through it in detail. Besides, I enjoy starting from bedrock physics. There are three basic equations:

  1. the relationship between the voice coil motion and voltage induced by the driver magnet
  2. the equation for pressure inside the enclosure
  3. the relationship between forces and motion of the speaker cone assembly.

The cone is assumed to be rigidly connected to the voice coil. The analysis is performed for a sinusoidal frequency ω radians per second, and as usual can be generalized via a Fourier transform. All equations are in MKS units. The goal of the equivalent circuit analysis is to solve for the velocity of the cone, and velocity of the air in the vent. The far-field sound pressure produced by a known velocity is then obtained from equation P6 in the physics section on piston radiation. This equation is an exact solution for a flat piston mounted in an infinite baffle, and is generally a good approximation for real speakers.

Voltage induced by voice coil motion in the magnetic field

The voice coil of a real driver is ring-shaped, and rests within a concentric magnet structure (see illustration [8kb]). However the equations can be derived from a simplified rectangular structure shown here [3.5kb]. The voice coil is represented by the yellow wire loop in the simplified figure. The driver permanent magnet produces a field B, shown in red, pointing in the positive y direction. A coil movement in the +x direction is defined to be moving away from the enclosure interior. When the wire loop in the figure moves sinusoidally with a peak displacement xc, applying Maxwell's equations (Stratton's equation 1, page 2) shows that the motion induces a voltage across the loop terminals given by

The length l is the total length of wire within the field, in meters. For a real driver using the common "underhung" design, the magnetic field covers the entire voice coil, and l is the total length of wire in the voice coil. For an "overhung" design, as shown in the driver illustration, l is the length of wire within the magnetic field. When a current Ic passes through the wire, it produces an additional voltage that must be added to vx to obtain the total voice coil voltage vc. This term will be added later.

Pressure inside the enclosure

The cone motion produces a change in the pressure inside the enclosure. A vent in a bass-reflex design also affects the pressure. The enclosure walls vibrate as well, but this contribution to the pressure is assumed to be negligible. A side view cutaway shows the geometry [2.6kb]. A speaker cone with effective area SD moves sinusoidally with a peak excursion xc. There is a vent of cross-sectional area SV and length LV. It is assumed that the air within the vent moves sinusoidally as a rigid mass with peak excursion xv(discussion regarding this assumption later). Finally, it is assumed that the wavelength is large compared to the dimensions of the enclosure, so the pressure can be approximated as uniform throughout the interior. Then using equation 25, and equation 30 in the Physics section, applying the divergence theorem, and temporarily neglecting losses, it is found that the change from atmospheric pressure is

where the air density ρo and velocity of sound c are numerically given by

For an empty enclosure VB is equal to the physical volume; filling the enclosure with acoustic material can increase the effective size. The theoretical limit of the increase is 40%, and Small suggests that the practical limit is 25%. Small shows that for a bass-reflex design frictional losses due to the air flowing through material in the enclosure, through the vent, and leaks, can all be represented by a vent loss term -jωxvRV/SV. In other words, RV represents all enclosure losses. The value of RV is determined by a measurement of the enclosure Q as discussed below. This term must be appended to equation T2 for the bass-reflex case. For a sealed enclosure, these losses are lumped in with the suspension loss, and the vent loss term is omitted. The air mass in the vent is accelerated by the force produced by the interior pressure, resulting in a value of xv given by

Including the vent loss term in T2, and substituting T3 yields the final equation for pressure inside the enclosure in terms of the cone excursion

For a sealed enclosure, the two terms involving SV are omitted.

Forces and motion of the speaker cone assembly

The pressure in the enclosure produces a force on the cone given by SD times equation T4. The force driving the speaker is produced by the current Ic flowing through the voice coil. For the simplified figure [3.5kb], the current traverses the magnetic field in the positive z direction, and produces a negative x direction force (Stratton notes that this force equation essentially defines "magnetic field")

The next forces are due to the cone suspension via the surround and spider. There is a spring force, specified by a compliance CMS, and a frictional resistive force RMS. As noted above, for a sealed enclosure RMS is modified to include absorptive enclosure losses as well. Then the equation is

The resultant of the above forces acts to accelerate the cone assembly and air in front of the cone, defined as the moving mass MMS. The final force equation is

Equivalent circuit components

Now the procedure is to convert the physical parameters in equation T7 into equivalent inductors, resistors, and capacitors. Equation T1 is used to substitute for xc. Then the following definitions are made

The components LCES, RES, and CMES are determined by the "Thiele parameters", and are therefore selected by selecting an appropriate driver. The driver also determines SD and Bl. In practice REL is determined by the enclosure Q, denoted QL. Initially QL is typically assumed to have a value around 7, and is subsequently measured for the completed enclosure, and the design fine-tuned if required. The values of LCEB and CMEP are enclosure design parameters that may be selected to provide a desired response, and then the above equations are used to determine the corresponding values of VB, SV, and LV. In theory there are an infinite number of vent solutions, but there are rules of thumb relating the vent area and length, given for example by Dickason. For a sealed enclosure REL and CMEP are omitted from the equivalent circuit, and LCEB is the sole free design parameter, which then defines VB. Substituting these definitions into equation T7 yields the equation of the main part of the equivalent circuit of the driver and enclosure

Additional electrical components

as noted above Ic causes an additional voltage due to electrical resistance RE, and inductance LE of the voice coil. The full circuit also includes the amplifier, with its output impedance RA, speaker interconnect cable with resistance RC, and the crossover with resistance RXO. Therefore the full equivalent circuit of the system is as shown here [6.5kb]. For a bi-amped design minor modifications are made to the circuit, and for higher-order crossovers additional components are added. The components of equation T9 do not physically exist, but Ic is the real current through the voice coil, and vc is the real voltage across the driver terminals.

Back to the top

Sound output of the system

As stated in the beginning, the goal is to obtain the velocities of the cone and vent air. The amplifier is assumed to output a voltage with equal amplitude and phase at all frequencies. The equivalent circuit response is then analyzed, and the voltages vx and vv, as defined in the equivalent circuit schematic, are obtained. The physical peak cone velocity is then given by

The peak velocity of the air in the vent is

At this point, all references I have seen use the real part of the radiation impedance of a flat piston to compute the total output sound power. An exact solution of the radiation impedance of a piston, and a numerical solution for a cone, are given in the physics section. For a peak velocity that is constant with frequency, the real part of the impedance causes a power increase of 6 dB per octave in the "piston range," roughly where the speaker diameter is less than a wavelength. The power then levels out for higher frequencies. However the sound level on axis continues to rise at the 6 dB per octave rate because the sound becomes focused in a beam that gets narrower as frequency increases. In fact, for an ideal flat piston moving with a constant peak velocity, the sound level on axis grows 6 dB per octave for all frequencies, "DC to light" as engineers say. This is shown by equation P6 in the Physics section, the exact solution for sound produced by a piston in an infinite baffle, which is valid for any frequency. (It is surprising to me that this on-axis behavior is not mentioned in the references I have seen, or if it is I missed it). Although total output power is important, in my opinion on-axis sound level is more important. At a distance on 1 meter from the speaker, the sound pressure on axis for the driver cone and vent are

Neglecting mutual coupling, the two pressures are added coherently to obtain the total pressure, and the peak SPL level computed using

Subtract 3 dB to get the RMS value.

In the initial version of the crossover design section I stated, without really thinking it through, that a speaker cone would produce a square wave by jumping forward, sitting still, and then jumping back. Equation T12 shows that the sound pressure is actually proportional to the second derivative of the position, the cone acceleration. To produce a square wave of sound pressure, the cone velocity must be a triangle wave. Since the derivative of a sinusoid (meaning a sine or cosine) is another sinusoid, a sinusoidal cone velocity produces a sinusoidal pressure, but this is an exceptional case.

Consider a simple case of a sealed enclosure, no crossover, and neglect the voice coil inductance. For an input voltage that is constant in amplitude and phase, the cone velocity peaks at the resonant frequency, and drops by 6 dB per octave on either side of resonance. There is a -90-degree and +90-degree phase shift well above, and below, resonance respectively. If the cone velocity were constant with respect to frequency, the sound pressure would increase 6 dB per octave. For the cone velocity as described, the sound pressure is constant above resonance, and drops 12 dB per octave below resonance. The sound pressure phase varies linearly with respect to frequency and distance from the speaker, which simply represents a time delay. Neglecting this, there is an additional -90-degree phase shift between cone velocity and pressure. Therefore the sound phase is shifted 180-degrees with respect to the input voltage. But assigning a label of "positive" and "negative" is arbitrary in the analysis, and reversing the labels yields the result that the sound pressure has zero phase shift with respect to the input voltage, above resonance. The bottom line is that the wave shape of the input voltage is replicated by the sound pressure, for frequencies within the operating range of a single driver. In general, the wave shape is changed dramatically by the crossover, and by multiple drivers.

Back to the top

Efficiency

Efficiency is defined as the ratio of radiated sound power to electrical input power. The sound power is equal to the square of the sum of the velocities (equations T10 and T11) times the real part of the radiation impedance, which is given by equation P11 in the physics section. The input electrical power can be defined as the real part of the product of the voice coil current and voltage, or it can be defined as the real part of the product of the amplifier output current and voltage, which will give a slightly lower value. Efficiency is a function of frequency. Within the primary operating region of the driver the efficiency is close to the "reference efficiency" defined by

T14 is identical with equation 31 in Small's June 1972 paper. The efficiency is a function of the driver parameters, but is not a function of the enclosure. In other words, for a given driver the efficiency will be the same regardless of the enclosure it is put in. Dickason also points this out. Yet there is a frequently reproduced graph from a different paper by Small that appears to show that efficiency decreases as enclosure volume decreases. What's going on here? Equation T14 can be re-written in a form where efficiency is proportional to the 3rd power of the enclosed driver resonance times the enclosure volume. So what is actually changing as the volume decreases is that the resonant frequency is increasing. It is true that for a fixed low frequency response a larger volume can provide higher efficiency. But you have to buy a different driver to achieve the efficiency increase.

Back to the top

Design comparisons

Sealed enclosures are normally classified as "Infinite Baffle" or "Acoustic Suspension." If the compliance of the speaker suspension divided by the compliance of the air in the enclosure (equal to VAS divided by VB in the table below) is greater than 3, Dickason classifies the design as acoustic suspension. The driver QTS should be less than 0.3 or so for acoustic suspension designs, and greater than this for infinite baffle designs. Bass reflex designs should have driver QTS values of between 0.2 and 0.5. There are rafts of possible enclosure design "alignments," corresponding to Butterworth, Bessel and other filters. In my opinion there is not a huge difference between the alignments, neglecting plainly bad designs. For comparison purposes I have selected three designs given by Dickason: an infinite baffle and an acoustic suspension second-order Butterworth, and a SBB4 bass-reflex. In selecting the parameters given below an attempt was made to produce a good result for each design. I found real drivers in the Solen catalog that are fairly close to the selected parameter values, except that the compliance used in the acoustic suspension design is about twice as large as any I could find. A rather compliant driver is desirable for this design. A value of QL=7 was used for the bass-reflex design. For clarity, the voice coil inductance is zeroed out in the following comparisons.

Parameters common to all three designs:

Nominal driver diameter: 12-inches Voice coil DC resistance: RE = 6 Ohms

Amplifier, cable, and crossover resistance: 0.44 Ohms

Force factor Bl = 9.34 N-amp-1 Total system QTC = .707 Suspension loss RMS = 3 kg - s-1

Parameters that vary between designs

Design

QES

QMS

QTS

MMS

CMS

fS

fB

VAS

VB

Infinite Baffle

.51

2.48

.45

.040

.71

30.0

47.1

.275

.187

Acoustic Susp.

.30

1.43

.26

.040

2.1

17.3

47.1

.824

.129

Bass-Reflex

.35

1.72

.31

.033

1.2

25.0

25.0

.475

.187

Infinite baffle compared to acoustic suspension

The infinite baffle and acoustic suspension designs were selected to produce identical responses, including an identical efficiency of 1.4%, and sensitivity of 94.2 dB SPL. The physical configuration is not terribly different in this case. Scaling the infinite baffle cabinet dimensions by a factor of .88 provides the correct volume for the acoustic suspension. Pressure inside the enclosure is inversely proportional the cube of the scale factor (see equation T4). Stiffness of a clamped panel is roughly inversely proportional to the square of the scale factor. So my guess is that wall vibration is somewhat more problematic for the acoustic suspension design. There is also less room for filling to absorb the back wave. As shown below, the acoustic suspension design is slightly more sensitive to misalignment. I don't see a major difference here, but unless size is very important, I would choose the infinite baffle.

Sealed compared to bass-reflex

The bass reflex design was selected to have the same cabinet volume as the infinite baffle. The efficiency is 2.0%, sensitivity is 95.8 dB SPL, so there is only a 1.6 dB advantage here compared to the sealed enclosures. The frequency response curve [32kb] also shows slightly better relative bass in the region 20-50 Hz - about 1.2 dB better at 20 Hz. The transient response curve [29kb] is a plot of the sound produced by a step-function voltage input; that is, an abrupt rise from 0 volts to 1 volt DC. The response of the sealed enclosure is tighter, but not grossly different (More on transient response below). Probably the biggest difference is in the peak cone excursion [35kb]. The RMS input voltage for this graph is 2.83, which is about 1.3 Watts. The peak cone excursion for the sealed enclosure is 1.7 millimeters. For the bass reflex the peak is about 1.1 millimeters until the frequency drops below 20 Hz, when it begins to shoot up. A rumble filter is really essential for bass-reflex speakers to prevent excessive cone excursions at very low frequencies. Otherwise the frequently quoted advantage of smaller cone excursion for vented enclosures is simply not true. The vent diameter in this design is a sizable 11cm (4.3 inches). The peak air movement in the vent at 20 Hz is ±19 millimeters (note that it is scaled down by a factor of 10 in the figure). This is over 3/4 of an inch, for a relatively low power! Its not unusual for me to play music 10 dB higher than this, where the motion would be 2.5 inches. At 20 Hz the air motion in the vent is about 12 times larger than the cone motion for the sealed enclosure. Most of the difference, a factor of 5.5, is due simply to the ratio of cone area to vent area. There is a factor of 1.4 due to the higher sound output at 20 Hz for the vented design. Finally there is a factor of 1.5 due to the fact that the vent and cone are somewhat out of phase, so they have to work harder to produce the sound output. Consider the behavior of the sound in free space, 1 meter in front of the loudspeaker. At a very high SPL of 117 dB RMS the peak air motion is ± 0.4 millimeters at 20 Hz (this assumes far-field behavior and is obtained from equation 38 in the physics section). So the level of air movement in the vent is grossly higher than a "real" sound wave. The most serious problem is that the flow in a 4.3-inch diameter vent becomes turbulent for a peak excursion of more than 2.5 millimeters or so, at 20 Hz. So the vent flow is well into the turbulent region. The resistive force in the vent is proportional to velocity for laminar air flow, but proportional to the square of velocity for turbulent flow. Therefore turbulence in the vent creates distortion.

Back to the top

Additional effects for Bass-Reflex Enclosures

Mutual coupling between the driver and the vent alters the behavior somewhat, and is not included in Small's analysis. One effect is that the impedance resonant peaks are shifted from the values predicted by the theory. Locanthi reports that the upper peak typically moves down by 4-7 Hz and the lower peak moves up by 2-4 Hz. Jacobsen has studied the effect of mutual coupling on radiated sound pressure. Unfortunately the details are rather complex. The effect is typically largest near the vent resonance, where Jacobsen shows a difference of about 3 dB when mutual coupling is included. Even without turbulence the air in the vent doesn't move as a rigid mass. The effective length of the vent is actually larger than the physical length due to end effects. Dickason provides an equation for vent length that includes a correction. Dickason states that vent non-linearity is always a problem. Distortion due to turbulent airflow has already been discussed. Harwood indicates that for a 2000-ft2 room the vent in the design example will produce significant distortion for sound levels above 95 dB SPL, which is not all that loud. Vents also can produce wind noise - hardly surprising considering the substantial air motion. Pipe resonances within the vent, and transmission of enclosure resonances can also be troublesome. Bottom line for me is that a bass-reflex design adds complexity and problems, without that much payoff in return.

Back to the top

Misalignment

Enclosure designs are actually quite fault-tolerant. For the sealed enclosure designs, reducing the box volume by a factor of 2 produces the changes shown in the table below. The nominal case is the correct design. The lower two rows are the result of cutting the volume in half. The first column shows the resonant frequency of the driver, and the second the relative response at 30 Hz. A bump is created in the relative response near resonance, shown in the third column. Ringing in the step response is indicated by the peak negative excursion given in the fourth column. Considering the gross change in enclosure volume, the changes in response are modest. The input impedance spike doesn't change in height, but moves to the resonant frequency given in the table, so the design error is easily observed via an impedance measurement.

Sealed enclosure, effect of halving the nominal volume

Case

Resonance

30Hz Resp.

Resp. peak

Step ringing

Nominal

47.1 Hz

-8.4 dB

.0013 dB

-0.20

Infinite baffle

59.4 Hz

-11.2 dB

0.67 dB

-0.27

Acous. susp.

64.3 Hz

-12.4 dB

1.07 dB

-0.29

For the vented enclosure design, reducing the box volume by a factor of 2, or reducing the port air-mass by a factor of 2, produces the changes shown in the table below. The nominal case is the correct design. The middle row is for half of the design volume. The lower row for half the port air-mass (note: due to end effects the length decreases by more than a factor of two when the air-mass is halved).

Vented enclosure, effect of halving the nominal volume and port length

Case

Resonance

30Hz Resp.

Resp. peak

Step ringing

Nominal

24.9 Hz

-6.5

0 dB

-0.28

Half volume

35.3 Hz

-12.5

1.11 dB

-0.35

Half port air-mass

35.3 Hz

-7.4

0.79 dB

-0.37

Again the changes are relatively modest. A graph of the frequency response [31kb] shows the difference of port and volume misalignment. A graph of the step response [33kb] shows that the misalignments do cause a significant increase in the ringing. Finally the impedance plot [35kb] shows that the misalignment is readily apparent.

Back to the top

Crossover, cable and damping factor effects

Crossover interaction with the enclosure

For the next cases the voice coil inductance of 1.7 mH is included. The infinite baffle sealed enclosure and vented enclosure are used as examples. A 4th order Butterworth filter with a 300 Hz crossover and no impedance compensation produces a significant bump in the frequency response [29kb]. A Zobel did not help. Bi-amping with the same crossover eliminates the problem. This interaction problem is similar to the one covered in detail in the crossover section. In my opinion the most interesting effect is on the transient response, as shown here [30kb]. The solid blue curve shows the result without a crossover. The 4th order crossover delays the initial positive response spike by about 2 milliseconds. The enclosure acts like a filter with an effective frequency of about 50 Hz, so it reacts much slower and is more significant for the transient response than the crossover, creating the negative response ringing. In the bi-amped configuration the crossover causes a slight increase in the ringing. Without bi-amping, it causes a lot more ringing. For the bass-reflex design the result is particularly bad. Minimum ringing produces a tight-sounding bass. A large amount of ringing produces bass variously described as tubby or muddy.

Effect of speaker cable resistance and amplifier damping factor

The results presented in this section include the effect of .04 Ohms amplifier output resistance, corresponding to an amplifier damping factor of 200, 10 feet of 16 AWG zip cord speaker cable with .04 Ohms per side, and .32 Ohms crossover resistance. (see the equivalent circuit schematic [6.5kb]). For the bi-amped configuration the crossover resistance is absent, and the circuit is more sensitive to cable and amplifier resistance. For this case, the effect of increasing the amplifier damping factor to infinity, and increasing the cable size to infinity is simply to produce an imperceptible 0.2-dB increase in SPL level. Did the infinite damping factor give the amp "greater control" over the transient response? No, the difference is imperceptible. So the effect of increasing the damping factor beyond 200 is insignificant. And the effect of cables is insignificant, unless the cables are a lot longer, and/or a lot smaller. I know many audiophiles will not believe this result. I would only say that there are many things that engineers do not understand well, but circuit analysis is not one of them.

Back to the top

Back to Design of the Sound System