One of the key parameters that features in the specifications for loudspeakers is the cone resonant frequency.

The cone resonance frequency is a key parameter for loudspeaker units as it governs the bass performance and where the bass response falls away.

As a very broad indicator, larger loudspeaker units have lower cone resonant frequencies and hence their bass performance is better.

## Speaker cone resonance basics

The loudspeaker cone resonance is of particular importance because the output from the speaker falls at a rate of 12 dB / octave below this frequency. As a result, the lower the speaker cone resonance frequency the better the bass response.

To take the simplest case of a loudspeaker in free air, it is found that the resonant frequency is proportional to the square root of the reciprocal of the mass of the cone and the compliance of the suspension scheme for the cone.

**Where:**

f_{r} = speaker cone resonant frequency

M = cone mass in grams

C is the compliance in m/N

It can be seen that for the lowest resonant frequency the compliance should be as large as possible, but able to provide a stable degree of suspension. Also the mass of the cone should be large, but a larger cone mass also equates to a reduced level of sensitivity and a poorer high frequency response.

To gain good bass response a separate bass speaker is used and the high frequencies are often covered by loudspeakers tailored more to the high frequency response and two or even three way speaker systems used, each speaker covering a band of frequencies.

One point to consider for a bass speaker is that increasing the cone mass not only reduces the high frequency response but it also reduces the sensitivity. Reducing the sensitivity too far can result in requiring a high power bass drive amplifier and it will also be found that greater heat losses are incurred in the speaker coil and this needs to be taken into account.

## Loudspeaker compliance

The loudspeaker compliance is an integral element of determining the resonant frequency of the loudspeaker.

The compliance for the loudspeaker can be calculated from the formula below which has been rearranged from the relationship shown above:$C=\frac{1}{{(2\pi {f}_{r})}^{2}M}$

## Loudspeaker resonance in an enclosure

The resonance of a loudspeaker is altered when it is placed within an enclosure. A sealed enclosure has the effect of adding further stiffness to the cone as it has to compress or expand the air within the sealed box.

The additional stiffness has the effect of reducing the compliance.

The overall effect of placing a loudspeaker in a sealed box or enclosure is that it increases the speaker resonant frequency.

Smaller enclosures add greater levels of stiffness because there is a smaller amount of air to be compressed - a larger cabinet has more air enclosed and this can be compressed more easily. Accordingly large speaker enclosures tend to provide better bass response.

## Loudspeaker damping

A peak in the response can be seen around the resonance frequency. This type of peak can be seen in many resonant circuits.

At the peak, the cone movement and hence the back EMF is greater along with the counteracting magnetic field - this helps with self damping.

The damping of the counteracting field depends on a variety of factors including the EMF generation (flux density x coil length) as well as being inversely proportional to the coil resistance, cone mass and resonant frequency.

The value of Q for the speaker can be calculated:

$Q=\frac{2\pi {f}_{r}MR}{{(Bl)}^{2}}$

**Where:**

M is the cone mass

f_{r} is the resonant frequency

B is the flux density

B is the flux density

l is the coil length

A consistent unit scheme must be used across all units.

Choice of the level of Q for the loudspeaker has an effect on the performance. It is found that a Q of 1 gives a lift in the response of 3 dB just above the resonant frequency, but the optimum speaker damping at the resonant frequency. A Q of 0.7 (√ 2) gives a flat response but the response rolls off occurs slightly higher in frequency. The smoother roll off this provides is often preferred for high quality equipment, those requiring bass boost may opt for the Q of 1 or more.

If the loudspeaker is within an enclosure, this will also have a significant effect on the damping. There is a critical enclosed volume for any particular loudspeaker that will provide the required level of damping.