CORRESPONSABILIDAD FISCAL (*)
IV. LOS MECANISMOS DE AJUSTE/RESPUESTA
a. Find t h e values of a n d SD from Par. 8.7 (pp. 228 t o
232). Approximate values may be obtained from Fig. 8.5, Table 8.1, a n d t h e sen- tence preceding Table 8.1.
b. Determine = = and =
c. Determine CA B, a n d from to
d. If the flow resistance and volume of t h e acoustical lining are known, deter-
mine from Fig. 8.8. Otherwise, neglect t o a first approximation.
e. Determine the actual (not the rated) o u t p u t resistance of the power amplifier,
All t h e constants for solving the circuit of Fig. 8.4 are now available. f. Calculate the total resistance t o t a l mass MA, and total compliance
from Eqs. (8.19) to (8.21). Determine a n d from Eqs. (8.22) and (8.23).
g. Determine the reference sound pressure a t distance from the loudspeaker b y Eq. (8.27).
h. Determine t h e ratios of t h e driving frequencies at which the response in
desired to the resonance frequency t h a t is, Determine the ratio
i. Obtain the frequency response in decibels relative to the reference sound pressure directly from Figs. 8.12 a n d 8.13.
provided in the box so t h a t changes in atmospheric pressure d o not dis- place t h e neutral position of t h e diaphragm.
Analogous Circuit. A closed box reacts on the back side of t h e loud- speaker diaphragm. This reaction m a y be represented by an acoustic impedance which a t low frequencies is a compliance operating t o stiffen the motion of the diaphragm and t o raise t h e resonance frequency. A t high frequencies, the reaction of the box, if unlined, is that of a
circuit. This is equivalent to an impedance that varies cyclically with frequency from zero t o infinity t o zero t o infinity, and so on. This vary- ing impedance causes the frequency-response curve t o have correspond- ing peaks and dips.
If is these resonances arc:
high the side of diaphragm
with an impedance to t h a t for the diaphragm in an radiating into free
At low frequencies, where diaphragm vibrates as one unit so t h a t i t be as a rigid piston, a complete
circuit can be drawn t h a t describes the behavior of box-enclosed loudspeaker. This circuit is shown in Fig. 8.4 and developed by
given in P a r t
Some interesting facts about loudspeakers are apparent from this circuit. First,, electrical generator (power amplifier) resistance
Front side of
Mechanical part diaphragm
Electrical of loudspeaker radiation
FIG. 8.4. Circuit diagram for a direct-radiator loudspeaker mounted in a closed-box baffle. This circuit is valid for about cps. The volume velocity of the diaphragm = IT
, ; = open-circuit voltage of generator; = generator resistance; = voice-coil B = air-gap flux density; = length of on voice-coil winding; = of diaphragm; = acoustic mass
of and voice roil; = total of t h e suspensions;
= acoustic rcsistancc in = acoustic-radiation imped-
ance from t h e front of diaphragm; = acoustic-loading impedance of the on the rear of diaphragm.
and t h e voice-coil resistance appear in the denominator of one of t h e resistances shown. This means t h a t if one desires a highly damped or an overdamped system, i t is possible t o achieve this by using a power amplifier with very low output impedance. Second, the circuit is of the simple resonant type so that we can solve for the voice-coil volume velocity (equal t o the linear velocity times the effective area of the dia- phragm) by the use of universal resonance curves. Our problem becomes, therefore, one of evaluating the circuit elements and then determining the performance by using standard theory for electrical series LRC
Values of Electrzcal-circuit Elements. All the elements shown in Fig. 8.4 are in units t h a t yield acoustic impedances in mks acoustic ohms (newton-seconds meter5), which means that all elements are trans-
formed t o the acoustical of the circuit. This accounts for the effec-
tive area of the the part of the
circuit. The shown are
flux density in the air gap in per square meter m2 = gauss)
= length of the wire wound on the voice coil in meters ( I m =
39.37 in.)
= output electrical impedance (assumed resistive) in ohms of t h e audio amplifier
= electrical resistance of t h e wire on the voice coil ohms a = effective radius in meters of t h e diaphragm
= = effective area in square meters of the diaphragm Values of the Mechanical-circuit Elements. The elements for t h e mechanical part of t h e circuit differ here from those of Part XVII in t h a t they are transformed over t o the acoustical part of the circuit so t h a t they yield acoustic impedances i n mks acoustic ohms.
MAD = = acoustic mass of the diaphragm and voice coil in kilograms per meter4
= mass of t h e diaphragm a n d voice coil in kilograms
= = acoustic compliance of the diaphragm suspen- sions in meters5 per newton (1 newton = dynes)
= mechanical compliance in meters per newton
= = acoustic resistance of the suspensions in mks acoustic ohms
= mechanical resistance of t h e suspensions in mks mechanical ohms
As we shall demonstrate in an example shortly, these quantities may readily be measured with a simple setup in the laboratory. I t is helpful, however, t o have typical values of loudspeaker constants available for rough computations, and these are shown in Fig. 8.5 and in Table 8.1. The magnitude of t h e air-gap flux density B varies from 0.6 1.4 webers/m2 depending on the cost and size of the loudspeaker.
T A B L E 8.1. Typical Values of 1, a n d for Various Advertised Diameters
of Loudspeakers Advertised diam, in. 4- 5 6-8 10-12 12 15--16 Impedance, 3 2 3 2 3 2 8 0 16 0 RE, mass of
ohms voice coil, g
Values of Acoustical elements
t o the some difficulty they
Part
Advertised diameter in inches Advertised diameter in inches
Advertised diameter in inches
FIG. Relation between effective diameter of a loudspeaker and its advertised
diameter.
FIG. Average resonance frequencies of direct-radiator loudspeakers when mounted in infinite baffles vs. the advertised diameters.
FIG. Average mass of voice coils and diaphragms of loudspeakers as a function
of advertised diameters. is the mass of the diaphragm including the mass of the voice-coil wire, and is the mass of the diaphragm excluding the mass of the voice-coil wire.
FIG. Average compliances of suspensions of loudspeakers as a function of adver- tised diameters. Note, for example, that 3 on the ordinate means 3 X
not well behaved. T h a t is t o say, the resistances vary with frequency, and, when the wavelengths are short, so do the masses.
The radiation impedance for the radiation from the front side of the diaphragm is simply a way of indicating schematically that the air has mass, t h a t its inertia must be overcome by the movement of the dia- phragm, and t h a t i t is able t o accept power from the loudspeaker. T h e magnitude of the front-side radiation impedance depends on whether the is very large so t h a t it approaches being an infinite baffle or whether
has of by m (7.6 in case the behavior is
VERY- LARGE- SIZED (APPROXIMATE I N F I N I T E BAFFLE)
radiation resistance for a piston in an infinite baffle in mks acoustic ohms. This resistance is determined from the ordinate of Fig. 5.3 multiplied by If the frequency is low so that the effective circumference of the diaphragm is less than X, t h a t is,
<
1 (where k = may be computed from,
-
C
= radiation reactance for a piston in an infinite baffle. Determine from ordinate of Fig. 5.3, multiplied by For
ka
<
given by,
0.318 ---a a
MEDIUM-SIZED BOX (LESS T H A N 8
= approximately the radiation impedance for a piston in the end
of a long tube. This resistance is determined from the ordinate of Fig. 5.7 multiplied by If the frequency is low so t h a t the effective circumference of the diaphragm is less than may be computed from
= approximately the radiation reactance for a piston in the end of a long tube. Determine from the ordinate of Fig. 5.7 multiplied
by For
<
given byand
(Rear-side) T h e acoustic impedance of a closed box in the loudspeaker is mounted is a reactance in series with a shall see below, neither nor
is well behaved for wavelengths than 8 times the smallest
217