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8. Permanent magnet 9. Open web supporting

FIG. 7.1. Cross-sectional sketch of a direct-radiator loudspeaker to be mounted in an baffle.

of these traveling waves and of resonances in the cone itself are t o produce irregularities in the frequency-response curve a t the higher frequencies and to influence the relative amounts of sound radiated in different directions.

I n Fig. 7.1, the loudspeaker is shown mounted in a flat baffle assumed t o be of infinite extent. By definition, a is any means for acous- tically isolating the front side of the diaphragm from the rear side. For purposes of analysis, the loudspeaker diaphragm may be considered a t low frequencies to be a piston of radius a moving with uniform velocity over its entire This is a fair approximation a t for which the distance Fig. 7.1 is less than one-tenth wavelength.

Part D I R E C T - H A D I A T O R L O U D S P E A K E R S 185 7.3. Electro-mechano-acoustical Circuit. Before drawing a circuit diagram for a loudspeaker, we must identify the various elements involved. The voice coil has inductance and resistance, which we shall call L and respectively. The diaphragm and the wire on the voice coil have a total mass The diaphragm is mounted on flexible sus- pensions a t the center and a t the edge. The total effect of these suspen- sions may be represented by a mechanical compliance and a mechan- ical resistance = where the mechanical responsiveness.

FIG. 7.2. (a) Mechanical circuit of direct-radiator loudspeaker; electromechanical analogous circuit of the mobility type; (c) electrical circuit showing motional elec- trical impedance; (d) analogous circuit of the mobility type with electrical quantities referred to the mechanical side.

The air cavity and the holes a t the rear of the center portion of the dia- phragm form a n acoustic network which, in most loudspeakers, can be neglected in analysis because they have no appreciable influence on the performance of the loudspeaker. However, both the rear a n d the front. side of the main part of the diaphragm radiate sound into the open air.

A radiation impedance is assigned t o each side and is designated as

= where is the radiation mobility.

We observe that one side of each flexible suspension is a t zero velocity.

186 D I R E C T - R A D I A T O R L O U D S P E A K E R S [Chap.

in the suspensions. We already know from earlier chapters that of the mass and one side of the radiation mobility must be con- sidered as having zero velocity. Similarly, we note that the other sides of the masses, the compliance, the responsiveness, and the radiation mobilities all have the same velocity, that of the voice coil.

From inspection we are able to draw a mechanical circuit and the electromechanical analogous circuit using the mobility analogy. These are shown in Fig. and b, respectively. The symbols have the follow- ing meanings:

= open-circuit voltage of the generator (audio amplifier) in volts.

= generator resistance in electrical ohms

= inductance of voice coil in henrys, measured with the voice- coil movement blocked, &., for = 0.

= resistance of voice coil in electrical ohms, measured in the same manner a s L

B = steady air-gap flux density in webers per square meter. 1 = length of wire in meters on the voice-coil winding.

= electric current in amperes through the voice-coil winding.

f, = force in newtons generated by interaction between the alternating and steady mmfs, that is, = Bli.

u, = voice-coil velocity in meters per second, t h a t is, =

where e is the so-called counter emf.

= radius of diaphragm in meters.

= mass of the diaphragm and the voice coil in kilograms.

= total mechanical compliance of the suspensions in meters per newton.

= = mechanical responsiveness of the suspension in meters per newton-second (mks mechanical

= mechanical resistance of the suspensions in newton-seconds per meter (mks mechanical ohms).

= = = mechanical radiation mobility in mks mechanical mohms from one side of the diaphragm (see Fig. 5.4). The German indicates that varies with frequency.

=

+

= mechanical radiation impedance in newton- seconds per meter (mks mechanical ohms) from one side of a piston of radius a mounted in an infinite baffle (see Fig. 5.3). The German indicates that varies with frequency! The circuit of Fig. 7.26 with the mechanical side brought through the

to the electrical side is shown in Fig. The

nobility = zero if the diaphragm is blocked so t h a t no

is a mobility ohm. See Par. 3.3 for discussion.

Part XVII] L O U D S P E A K E R S 187

motion (u, = 0) but has a value different from zero whenever there is motion. For this reason the quantity called the motional electrical impedance. When the electrical side is brought over to the mechanical side, we have the circuit of Fig.

The circuit of Fig. will be easier to solve if its form is modified. First we recognize the equivalence of the two circuits shown in Fig.

FIG. 7.3. The electrical circuit (referred to the mechanical side) is shown here in equivalent forms. The circuits are of the mobility type.

Acoustic Electrical Mechanical

L

v

FIG. 7.4. ( a ) Low-frequency analogous circuit of the impedance type with electrical

quantities referred to side. given by Fig. 5.3. T h e quantity represents the total force acting in the equivalent circuit to produce the voice-coil velocity (b) Single-loop approximation to Fig. valid for

and b. Next we substitute Fig. for its equivalent in Fig. Then we take the dual of Fig. to obtain Fig.

The performance of a direct-radiator loudspeaker is directly related t o the diaphragm velocity. Having solved for it, we may compute the acoustic power radiated and the sound pressure produced a t any given distance from the loudspeaker i n far-field.

voice-coil Velocity at and Low Frequencies. T h e voice-coil

188 D I R E C T - R A D I A T O R Fig.

where

Voice-coil Velocity at Low Frequencies. At low frequencies, assuming n addition that

>>

we have from Fig. that

= = mass in kilograms contributed by the air load one side of the piston for the frequency range in which ka

<

0.5. The ka equals the ratio of the circumference of the diaphragm t o the wavelength.

The voice-coil velocity is found from Eq. using Eqs. (7.2) and for and respectively.

7.4. Power Output. The acoustic power radiated in watts from both ,he rear and the front sides of the loudspeaker is

assuming

<<

+

7.5. Sound Pressure Produced a t Distance Low Frequencies. In 4 we showed t h a t a piston whose diameter is less than one-third wavelength (ka

<

1.0) is essentially nondirectional a t low frequencies. we can approximate it by a hemisphere whose rms volume velocity

= where is the projected area of the loudspeaker cone. the projected area, we mean of Fig. 7.1.

From Eq. (4.3) we see t h a t the magnitude of the rms pressure a t a in free space a distance from either side of the loudspeaker in an nfinite baffle

is assumed in writing this equation that the distance is great enough that i t is situated in the "far-field." Hence, the pressure a t is

Equation (7.8) is also readily derived from Eq. (7.6) observing from 5.1 (page 124) that, a t low frequencies,

and

w

= =

where I is the intensity a t distance in watts per square meter.

Medium Frequencies. At medium frequencies, where t h e radiation from the diaphragm becomes directional but yet where the diaphragm vibrates as one unit, as a rigid piston, the pressure produced a t a dis- tance depends on the power radiated and the directivity factor Q.

The directivity Q was defined in Chap. 4 as the ratio of the intensity on a designated axis of a sound radiator t o the intensity that would be produced a t the same position by a point source radiating the same acoustic power.

From Eq. (7.10) we see for a point source radiating to both sides of an infinite baffle that,

For a directional source in an infinite baffle such a s we are considering here,

where = acoustic power in watts radiated from one side of the loudspeaker.

= directivity factor for one side of a piston in a n infinite plane Values of Q are found from Fig. 4.20. Note that equals and, a t low frequencies where there is no Q = 2, so that Eq. reduces to Eq. (7.11 ) at, frequencies.

sound pressure is by substituting (7.6) divided by 2 into

7.6. Frequency-response Curves. A frequency-response curve of a loudspeaker is defined as the variation in sound pressure or acoustic power as a function of frequency, with some quantity such a s voltage or electrical held constant. Inspection of and

(7.13) that the

+

in the in

A 7

In document LA NUEVA REFORMA DE LA FINANCIACIÓN AUTONÓMICA: ANÁLISIS Y PROPUESTAS (página 163-169)