the jordan loudspeaker manual chapter 4

7/29/2019 The Jordan Loudspeaker Manual Chapter 4

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Chapter 4: THE MOVING COIL LOUDSPEAKER

The basic moving coil loudspeaker is such a neat application of the physicallaws that it is virtually impossible to make one that does not work. Even apiece of cardboard stuck on a coil immersed in a magnetic field will producedistinctly recognizable sounds. The basic construct is shown in Fig: 6

Fig:6

Nevertheless, this simple assemblage consisting of only four key parts beliesthe complexity of its analysis.

The conversion of an electrical signal into sound takes place in three stageswhere the electrical output from the amplifier is fed to the voice coil situatedwithin an annular magnetic field. The current reacts with this field to causemotion of the coil, which drives the cone. This, in turn, creates pressurewaves in the air at its surface to generate sound.

These acoustical electrical, and mechanical stages are now described indetail.

RADIATION IMPEDANCE

This is the impedance to the motion of the cone presented by the air at itssurface. It comprises a resistive term, Rma, in which the mechanical energyis converted into sound and a mass component, Xma In practice; both of these are very small compared to the mechanical impedance of a practicalcone assembly. To analyze the characteristics of Rma, an idealized cone isrepresented by a flat, circular, ridged piston having no mechanical mass,friction or restoring force. Radiation from one side only is considered and thepiston is assumed to be mounted in an infinitely large flat baffle to avoidcancellation from the rear radiation. For all practical purposes the effects of

the mass reactance can be neglected. The plots are shown in Fig: 7.

Jordan Manual 2011 Chapter 4 1



Fig: 7

Unlike electrical resistive components Rma varies with frequency.

The transition frequency, Ft occurs where the piston circumference is equalto half the corresponding wavelength

Below Ft, Rma increases in direct proportion to the square of the frequency.

Above Ft, after a few minor wobbles, Rma settles down to a steady stateindependent of frequency.

For those who like knowing the derivation, this is summarized below

Rma = ρckr2[{(2kr)2/2x4} – {(2kr)4/2x4x6} – {(2kr)6/2x42x62x8}…… etc].

Where ρ= Density of air, c = Velocity of sound in air, r = Radius of piston, k = 2πf/c, f = Frequency. c = velocity of sound.

At this stage, in order to simplify the understanding of the text and allrelationships, these will, where appropriate, be reduced to terms of the

proportionality of the key terms only. The symbol for proportionality is ‘α’ .

Applying this to the above gives;

Below Ft, Rma α f 2……………………………..….……...Eqn: 1.

Above Ft, Rma is independent of frequency….…....Eqn: 2.

Ft is inversely proportional to cone diameter. As an example, for a cone of 10cm diameter Ft will be approximately 2.2 kHz.

RADIATED POWER FROM PRACTICAL CONES

In practice, the loudspeaker cone is not infinitely rigid and has a significantfinite mass, ‘Lm’ . It. has to be supported by a suspension system, whichconventionally comprises a synthetic roll surround at the cone periphery andusually, a corrugated cloth suspension at the rear. These provide therestoring force determined by their compliance ‘Cm’ . In addition there will be




resistive components, due to internal friction within the suspension system ‘Rm’ and electro-magnetic damping, ‘Rme’

The reactance of Lm and Cm are given respectively by:

XLm = 2πf.Lm. and XCm = 1/2πf.Cm

The total mechanical resistance is given by:

Rmt = Rm + Rme

Total mechanical impedance is:

Zt = [(XLm – XCm)2 + Rmt2]

Assuming a constant driving force, ’F’, the cone velocity is given by

v = F/Zt

The resonant frequency, ‘Fs’ of the loudspeaker is given by:

Fs = 1/2π(Lm.Cm)1/2………………………….………..Eqn: 3.

The radiated sound power is then:

Pr = v2Rma…………..…………………………….…..……..Eqn: 4.

The power response of a practical loudspeaker can be divided into threefrequency bands: (0 to Fs), (Fs to Ft) and (Ft + )

Band ‘0 to Fs’, is dominated mainly by the reactance ‘Xcm’ of thesuspension compliance:

Xcm = Cm. 1/(2π.f.Cm)

Then from ‘Eqn: 1’ Pr α (f 2.2πf.Cm)2. f 2………………………… Eqn: 5.

Therefore, below ‘Fs’, Pr varies as the fourth power of frequency

At ‘Fs’, the reactive terms sum to zero and the cone velocity is determinedonly by the resistive components.

Pr= (F/Rmt}2. Rma……………………..…..Eqn: 6.

Over a limited bandwidth either side of Fs, ratio of the reactive to theresistive components increases. This defines the ‘Q’ of the speaker the valueof which can be adjusted by design to maintain a level power response. Thiswill be fully discussed later.

Band Fs – Ft With frequency increasing above Fs, the radiated power

becomes progressively controlled by the increased mass reactance.

Then from Eqn:1 Pr α F/(2.π.f.Lm)2.f 2…………………………….Eqn: 7.




Therefore, for a specific value of ‘Q’, Pr is independent of frequency

between Fs and Ft, . This is often referred to as the piston range.

Band Ft+ Assuming a ridged cone this is also dominated by the massreactance but the radiation impedance ‘Rma’ becomes independent of frequency. Then from ‘Eqn: 2’:

Pr α F/(2.π.f 2.Lm)2.f ……………………………….Eqn: 8.

Therefore Pr is inversely proportional to frequency.

Although this shows the radiated power to be falling at 6dB/Octave, theactual sound pressure level over an area in front of the loudspeaker mayactually rise with increasing frequency due, firstly, to the directivity effectconcentrating the available power forwards, and, secondly, at somefrequency determined by the concentric wave velocity within the cone, it willstart to flex thereby reducing the effective cone area and therefore its mass.

This is progressive with rising frequency and can further extend thefrequency response.

Directivity

The following illustrate the theoretical directivity patterns for an idealizedpiston of 10cm diameter. Due to cone flexure, these will not berepresentative of a practical loudspeaker.

0.5 Ft (1.1). 2Ft, (4.3). 3.5Ft, (7.7). 5Ft, (10).




Cone Flexure

Flexure, in conventional cones, takes two forms: radial or ‘Bell modes’ Fig:8a, and Concentric modes, Fig: 8b. These are shown below. Both modes canco-exist

Fig: 8a. Radial modes Fig: 8b. Concentric modes

The Radial or bell modes occur only at frequencies where the circumferenceis sub-multiples of one half wavelength within the cone. These tend to occurat lower frequencies but are minimized in cones with curved profiles. Theyare called ‘free modes’ since they do not follow the driving frequency

Concentric flexure occurs at all frequencies above which the cone side iscomparable to one quarter of a wavelength. Since these follow the voice coilfrequency they are called forced modes. At frequencies corresponding to aquarter wavelength within the cone material reflected waves would return

from the cone periphery and interact with the outgoing incident wave tocreate standing wave resonant modes.

Fig: 9. Shows uncontrolled radial and concentric modes in a 15cm straight-sided metal cone at ‘a’ 700hz, ‘b; 2.7 kHz and ‘c’, 6kHz. (These pictureswere produced around 60 years ago by sprinkling the cone with licopodiumpowder). a b c




THE CONE SUSPENSION SYSTEM

This normally comprises a synthetic roll surround at the cone periphery and acorrugated cloth suspension at the rear. In good design the rear suspensionprovides the principle restoring force keeping the cone at zero displacementin the absence of a signal. Together with the roll surround it also contributesto the centring of the cone and the coil whilst permitting limited axial

displacement. The roll surround is a multi-function component which isrequired to provide:

a) Mechanically stable centring of the cone in the chassis.

b) An acoustically opaque seal between front and rear radiation

c) Maximum compliance to the axial displacement of the cone. This willalso ensure the lowest resonant frequency.

d) An essential damping medium to minimise ‘free mode resonances.

These requirements are conflicting and it is almost impossible to evolve aperfect material and form to satisfy all of them. There will inevitably beinconsistencies in production and performance.


the jordan loudspeaker manual chapter 4

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