THE CONCEPT OF ‘Q’ OF LOUDSPEAKERS
This discussion relates to only low frequence performance of loudspeakers.

Resonance:

A parallel electrical resonant circuit consisting of an inductance and a capacitance shows a peak at the resonant frequency in its impedance v/s frequency graph. An ideal lossless circuit will show a very sharp peak;  i.e., a high amplitude of peak on a very narrow spread over frequency band. This sharpness property is called ‘Quality’ or just ‘q’ of the resonant circuit. The losses in the components cause a reduction in the height of the peak resulting in a broader peak, consequently having a smaller ‘q’. These losses are resistive as they are independent of the frequency and are known as ‘damping’.

In a loudspeaker, there are two elements that form a similar resonant system:

1. The mass of the moving components – inertia-equivalent of inductance in an electrical circuit.

2. The restoring spring action of the spider and suspension-compliance- equivalent to capacitance in electrical circuits. Compliance is the inverse of stiffness. Stiffer the system is, lesser is the compliance.

Like the electrical equivalents, both the above store energy.

A resistance is also present in the mechanism. This is equivalent to damping in an electrical circuit. This damping restricts the ‘q’ of the speaker system which is actually useful.

If there is no damping the system will move excessively at the resonant  frequency  - a property not good in a loudspeaker. Hence, we say that the ‘q’ of the loudspeaker (or speaker system) should not be too high.

It must be remembered that a driver ‘q’ is controlled by the mass of the moving parts, complance and damping.

Effect of cabinet:

In practice, a speaker is not used in isolation. It is enclosed in a cabinet to avoid back and front cancellation of air compression as the cone moves back and forth. ‘Air’ itself is a compressible medium and as the cone moves, the air inside the enclosure also contributes in storing energy. Thus when the speaker is mounted in a cabinet there is an additional ‘compliance’ of the air in the cabinet. Technically, the driver and cabinet compliances appear as a series circuit in electrical equivalence. Thus the total compliance is like the resultant value of capacitors in series:

C= (c1 x c2) / (c1 + c2)

(Note that adding capacitors in series reduces the final value.)

Hence the lower value of compliance dominates and the final value of compliance in a driver-cabinet combination is always lower than the lower of the compliance between the two.

If the volume of the cabinet is very large, there will be very small effect on the cone movement but if the volume is small then the air will get compressed (high stiffness) and affect the speaker behavior. A large air volume has a large compliance while small air volume has a small compliance. A Lower compliance also results in a higher ‘q’. So, when a driver is fitted in a cabinet, the combined compliance is reduced and overall ‘q’ is increased.

To keep the overall ‘q’ low, we need a driver of low q or a large  compliance of the air in the cabinet. That is to say, we need a cabinet of large volume (or by filling glasswool in the cabinet which has a similar effect).

In short, the driver mounted in a ‘sealed’ cabinet will tend to raise the ‘q’ of the overall system and increase the resonant frequency.

Selecting q:

Note that ‘q’ is a pure number with no units. From network principals we know that the value of ‘q’ around 0.7 gives a maximally flat frequency response. Hence we have to choose the driver and cabinet combination to achieve the combined ‘q’ of around 0.7. For a given driver we can achieve the required q only by properly choosing the cobinet compliance i.e. cabinet volume.

Therefore, if we have a driver of high compliance then the compliance of the cabinet can be small to effectively keep low overall compliance and hence lower q. This means a relatively smaller cabinet. On the other hand if we have a driver with a low compliance then we need to use a cabinet of higher compliance or a bigger cabinet. In other words,  a soft/loose cone assembly needs a small cabinet while a stiff cone assembly needs a large cabinet.

If the cabinet is too large, it will have a very high compliance and dominate the resulting system ‘q’. A very low ‘q’ makes the system inefficient though the frequency response may be smooth to a lower frequency. It also reduces the power handling capability of the driver. Conversely, too small a cabinet and a higher q of driver will have a very high system ‘q’ resulting in a ringing of the driver at resonance, creating a confused sound quality and loss of distinction of bass notes.

Note that if the ‘q’ of the cabinet with driver is the same as ‘q’ of the driver alone, i.e. the air volume in the cabinet has no effect on the final q of the system,  then the resultant resonance frequency of the system will be the same as that of the driver in free air.

(graphic from  http://www.ht-audio.com/pages/SpeakerBasics.html)

Red curve shows too low Q of the system; Green curve shows optimum Q of the system; Blue curve shows a too high Q of the system

Note:

1.     Overall lower output in low frequency region in red curve.

2.     Peaky response in LF response in blue curve.

3.      Green curve shows optimum response.

Value of overall q between 0.5 to 0.7 is often used for  deep bass reproduction but it comes with a rduced overall bass loudness. Lower values give a better transient response.

A higher value of q from 0.7 to about 1.1 is used to produce a warm bass but at the higher value it tends to be boomy.

Effect of damping:

Higher the damping lower will be the sharpness of resonance peaks and the ‘q’ will be lower. Hence a good damping is required in a speaker system. (glasswool)  The damping is not only provided by the speaker system but also by the source impedance of the amplifier that drives the speaker as an electrical damping. Therefore it also important to keep the resistance of cable connecting the speaker to the amplifier as low as possible.

How electrical damping works: A speaker consist of a magnet and a coil forming a dynamo like structure. If the cone moves it produces a voltage at the speaker terminals. If this voltage is short circuited by a low resistance- like the source resistance of the driving amplifier- the cone movement is resisted by a back e.m.f. This dampens the cone movement. This damping controls the movement of the cone at resonance. Therefore the cone will quickly stop moving as soon as the amplifier signal stops.

Calculations:

Terminology-

‘Air equivalent’ of the speaker compliance is termed as Vas. This is in (cubic) volume units.( It can be viewed as an air volume having the same stifness as the driver suspension); ‘ fs’ is the free air resonant frequency of the driver. Qts is the driver q.

In practice, to design a sealed cabinet, we take the value of Vas and Qts from the manufacturer’s data (or better found from tests) and  find the volume of the cabinet Vb for the desired total q of the system (Qtc) as follows:

Formula to be used are-

Vb = Vas / (Qtc²/Qts²    - 1)

fc =  fs x Qtc/Qts

Note that Qtc is always greater than Qts. Hence we must choose a driver having a Q smaller than the desired final Q of the system.

For example, if the driver fs is 45 Hz, Qts is 0.5; Vas is 50 lit. ; desired Q of the system Qtc is 0.7, then the cabinet volume Vb is -

Vb= 50/ {(0.7/0.5)2  - 1}

=50/ 1.96  -1

=50/0.96

=52 Lit.

The final resonant frequency fc of the system will be found as:

Fc= 45 x 0.7/0.5

=63 Hz.

Thus the resultant resonant frequency of the system will be raised from 45 Hz to 63 Hz

It is worth noting that the smaller the Vas, smaller will be the resulting cabinet volume for the same final q. Hence it is good to choose a driver of low q and low Vas.

If we have a diaphragm fitted to a closed box, and if require the compliance of the enclosed volume of air, it can be readily calculated as

C= V/ ƍ0 c2d2

Where

C is the required compliance, in m/ N

V is the volume of enclosed air in m3

ƍ0 is the density of air which is 1.2 Kg/m3

c is the velocity of sound in air which is 343 m/sec.

Sd is area of the diaphragm used in m2

While constructing a cabinet the net volume should be taken as Vb. Actual volume will be bigger considering the volume takenup by the driver itself and the bracings. If glasswool is included then the effective volume increases by about 10%. Hence proper allownces must be made while working out final dimensions. Remember that these are all inside dimensions of the cabinet.

A sealed cabinet shows a 12 db/octave roll off in the frequency response below resonance. Therefore the bass reproduction is smooth.

Different considerations apply while designing a vented cabinet because the roll- off below the resonant frequency in a vented cabinet is very rapid (24 db/octave) and cosiderable low frequency phase shift occurs.

Sarang Lonkar