One of the less widely understood subjects in audio, it would appear, is damping and transient response, particularly in relation to loudspeaker systems. The following is a general outline of the problem, and an attempt to clear up some of the more prevalent misconceptions.

Some of the material here presented, by virtue of the fact that it is contrary to many popularly accepted (and even published) ideas, may appear to be radical in approach. It is, however, entirely conservative. The subject has been well investigated in the literature; the main concepts in this article, for example, appear in a much more complete and mathematically rigorous form in Beranek's Acoustics, *1 and a motor engineer should easily recognize the lack of novelty of the basic ideas relative to electro-magnetic damping.

*1 Leo L. Beranek, "Acoustics", McGraw-Hill Book Co., 1954.
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Before proceeding further we must be clear about the meaning of our terms. Damping refers exclusively to the introduction of a resistive element into a vibratory oscillatory system. This resistive element may be electrical, mechanical, or acoustical.

If we introduce alternating energy into an electrical or mechanical system - we could apply a.c. to an electrical circuit, or vibratory force to a mechanical device - the system will respond, oscillating in the grip of the applied stimulus.

The extent to which the system will oscillate, fondly referred to as its "response" by audiofans, depends on its impedance. Impedance may be thought of as mechanical, acoustical or electrical intransigence - the unwillingness to be moved or to pass current under the particular conditions involved.

The reactive part of the impedance, associated with such characteristics as mass, elasticity, inductance, etc., allows the load to accept energy for storage only, not for absorption.

A frictionless system of a weight on a spring would go on bobbing forever once it was started. The real or resistive part of the impedance, associated with electrical resistance, friction, and viscosity, permits the load to accept energy permanently, i.e., to absorb it (or, as in the case of radiation resistance, to accept energy for one-way transmission).

When we concern ourselves with how the system acts, not during the time when it is working steadily, but at the very start, on the "attack," and also at the end, after the stimulating and presumably controlling force has been removed, (the "decay"), we are dealing with transient rather than steady-state response.

When the drumstick falls it produces a deformation of the stretched skin. The velocity of the initial, complex movement of the membrane over the distance travelled will not be in step with the natural frequency of the drum's mechanical-acoustical system. The strike sound, instead of having the same pitch as that to which the drum is tuned, will exhibit fundamental components of much higher frequency.

The amplitude of the steady-state sound that will ultimately appear due to the blow will depend on the impedance of the drum's primary moving system, relative to the applied force.

The amplitude and duration of the initial, higher-pitched attack sound will depend on the impedance of the drum to higher frequency stimuli, and the Q at these higher frequencies. The more amenable the drumhead is to moving at velocities and amplitudes corresponding to higher frequencies than its fundamental, the crisper will be the attack sound. The nature of the transient acoustical attack is therefore a function of the frequency response of the drum - the relative amount of sound it puts out when stimulated at different frequencies.

Once the drumstick bounces off, the drum is on its own. It can operate only on the energy that was supplied in the single stroke, as it will receive no more energy until it is hit again.

It would be useful to consider concrete examples of the transient response of mechanical systems. Let us consider two such examples:

1. the response of a kettle drum to the impact of the drumstick, and
2. the response of a loudspeaker to a signal representing the drum's recorded sound.

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We know, of course that the sound continues. If the drum were totally undamped it would continue to vibrate forever, but mechanical and acoustical resistance provide light damping to absorb the vibratory energy gradually, and the sound takes a relatively long time to die away unless it is checked by the player. This is the decay; the length of decay time depends on the amount of damping.

In the case of a loudspeaker reproducing the kettle drum's sound a similar analysis applies. The initial stimulus is provided, not by an external blow, but by a surge of signal current from the amplifier, and the attendant magnetic field built up around the voice coil.

And since the speaker should vibrate only when the signal so dictates, it must have a highly damped mechanical system. If the speaker cone, like the drumhead itself, continued to vibrate after the controlling stimulus had stopped there would be a hopeless confusion of sound.

The quality of the reproduced attack sound, as in the case of the drum, is a function of the speaker's response to higher-frequency sound components. Thus the attack sounds reproduced by multispeaker systems are controlled, not by the low-frequency performance of the woofer, but by the performance of whatever unit is assigned to reproducing the mid and higher frequencies, and may involve the woofer itself little or not at all, depending on how low the crossover frequency is.

By definition, a woofer which covers only the low-frequency range cannot and is not intended to respond to most transient attack sounds. Its contribution to a crisp drum beat is to move, however lumberingly, in accurate reproduction of the fundamental and lower harmonic frequencies only; the sharper attack components are reproduced and contributed by other speakers.

So much for the general background of the problem. We may now turn our attention to the more specific question of loudspeaker damping.

Speakers are damped, in their main resonance region, in three ways:
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1. mechanically, through friction in the suspensions,
2. acoustically, through various methods of applying acoustic resistance and
3. through the air load resistance, and magnetically.

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In bass-reflex and horn systems acoustic damping normally predominates; in direct-radiator systems most of the burden of damping falls on the electro-magnetic system. Damping of cone break-up modes of vibration, at higher frequencies, also takes place in the cone material and in its edge termination, but this is not the subject of the present article.

Magnetic damping results in an additional mechanical resistance being applied to the moving system. This mechanical resistance can be investigated directly in a very simple manner - if one shorts out the terminals of a loudspeaker containing a fairly heavy magnet, and then tries to work the cone back and forth manually, it will feel as though the voice-coil has been immersed in a viscous fluid. The apparent viscosity disappears as soon as the terminal short is removed. When the speaker is connected to an amplifier with a low source resistance the amplifier source resistance replaces our experimental short. If the source resistance is raised in value (lower damping factor) the mechanical damping resistance is correspondingly decreased.

The effects of speaker magnetic damping are twofold:

• 1. It prevents cone vibration from continuing after the signal has stopped (hangover).
• 2. It controls bass response in the frequency region of resonance, perhaps an octave on each side.

The first of these effects is generally known and widely commented upon, while the second is not so well known.

The mechanical resistance introduced by magnetic damping may become the major element in the speaker's mechanical impedance in the region of resonance, where mass and compliance reactances cancel each other out. Actually, the influence begins at some frequency above resonance, when the mass reactance becomes equal to the damping mechanical resistance.
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The extent of the influence of damping is a function of the value of moving mass in relation to the damping resistance, more precisely, on the mechanical Q of the system.

Where the resistance is small relative to the mass reactance at resonance (a high-Q system) the effect of damping on bass response is small; where the value of resistance is large in relation to this mass reactance the effect on bass response is great.

This is a simplified way of saying that which is described exactly by the well-known family of curves representing the frequency response of resonant systems for different values of Q. Figure 1 reproduces a set of such curves, specifically applied to the acoustic output of speakers. *2

*2 Ibid., p. 226. Also see D. E. L. Shorter, "Loudspeaker cabinet design,' p. 382, Wireless World, Vol. 56, No. 11, Nov., 1950.
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All dynamic loudspeakers, of course, are mass-elasticity resonant systems.

The crux of the matter is that for that value of Q which will bring the resonant peak down to a flat curve, the damping will also be such as to prevent any hangover. For lower values of Q the hangover will continue to be damped out in neither better nor worse fashion, but other things being equal, there will be an attenuation of bass response as indicated in Fig. 1.

The dynamical analogy to the mechanical system of a loudspeaker, including the mechanical resistance due to magnetic damping, is shown in Fig. 2. Since the amplifier source resistance determines the value of this magnetic damping mechanical resistance, a variable damping factor control can be used, particularly with a direct-radiator speaker, to control the Q over a fairly large range of values.

It can be seen in Fig. 2 that at frequencies well above resonance the equivalent electrical circuit is inductance controlled, that is, the net reactance is inductive, representing mass control in the speaker's mechanical system. As the frequency is lowered in the direction of resonance the net inductive reactance decreases, and current flow (velocity in the mechanical system) correspondingly increases. This is as it should be; the cone velocity of a direct-radiator speaker, for constant acoustical power, must double with each lower bass octave in order to offset the progressive decrease in airload resistance.

At some frequency, depending on the speaker used - perhaps an octave above resonance - the net inductive reactance will become equal to the total resistance.
B will thenceforth, as the frequency is lowered, act to reduce current progressively, compared to the rising value that would exist in a pure LC circuit. That value of B which produces a Q of about 1 gives an approximately flat curve, with neither resonant peak nor bass attenuation.
If BD is swamped by large values of other resistive elements due to the nature of the speaker system, its effect will obviously be minor.

Below resonance the net capacitive reactance of the circuit begins to mount, until it is greater in value than! the total B. An octave or so below resonance, then, B again loses its influence.

It should be clear at this point that the absolute value of the mass of the speaker's moving system has no relation whatever to damping or hangover.

It is the mass-resistance ratio that influences the Q. The only exception to the former statement is provided by the new electrostatic units, where, the mass of the very light diaphragm may be kept so low that the controlling resistive element is the actual air load resistance.3 In such a case all system constants become tied to a fixed reference of resistance.

Nor does the absolute value of the mass influence attack performance. What is needed for the proper reproduction of attack sounds is: (a) the same level of system response at the attack frequencies as at the fundamental, however this is achieved, and (b) uniform response in the region of attack frequencies (corresponding to proper damping in this range), so that the attack frequencies themselves don't ring.

So much has been said and written contrary to some of the above conclusions that it was felt that a set of actual field measurements, illustrating the main points of discussion, would prove both interesting and informative. Accordingly a direct-radiator speaker system of known characteristics was fed by an amplifier with controllable damping factor, and facilities for measuring the speaker frequency response and decay characteristics were provided, as illustrated in Fig. 3.

The test set-up in which the speaker is sunk into the ground in the middle of an open field, its face flush with the surface, have been described by the writer.4 The speaker sees a controlled solid angle of 180 deg., and test conditions conform to ASA and RETMA specifications. Validation of the frequency-response curves of the speaker used as representing essentially fundamental output was also described in the article referred to.

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• AR-1W Acoustic Research speaker system (woofer only)
• Fairchild 275 power amplifier, with variable damping factor
• Bruel and Kjaer beat frequency oscillator BL-1014, mechanically coupled to:
• Bruel and Kjaer level recorder (automatic) BL-2304
• Electro-Pulse pulse generator 1310A
• Bruel and Kjaer microphone amplifier BL-2601
• Altec 21-BR-150 capacitor microphone

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The acoustic output of the speaker over its frequency range was measured, using the automatic frequency-level recorder, at an input power of 20 watts. These curves were re-run on the same graph paper *5 with all conditions the same, except for a change of setting of the damping factor control (thereby changing the damping resistance and the speaker's mechanical Q). The results are reproduced in Fig. 4.

It will be seen that the curves conform closely to the theoretically plotted curves of Fig. 1, especially when they have been corrected for the calibrated errors in recorder and microphone {Fig. 5).

The effect of the increased amplifier source resistance on the upper frequency range can also be seen: this is due to the rising electrical inductive reactance of the voice-coil.

Figure 6 is a series of three oscilloscope photos of the wave forms of the acoustical output of the speaker, in response to the step front of a low-frequency square wave. They represent magnetic damping conditions associated with each of the three curves of Fig. 5. Note that there is no significant difference between the recorded hangover associated with a properly damped and an over-damped system. Ringing at the speaker's resonant frequency is clearly seen, however, in the under-damped condition, as is a large increase in the slight initial secondary ringing, at a higher frequency, which shows up as a disturbance halfway down the first decay slope.

It should be possible at this point to see the error in that misconception about damping which denies ability to the amplifier source resistance to damp the speaker mechanically because of the latter's low conversion efficiency.

Magnetic damping is, of course, powerless to control effects which take place in the course of energy transfer between the mechanical system of the speaker and the surrounding air, but it is of paramount importance in controlling the mechanical system itself. It should also be noted that the magnetic damping of the speaker is a function of the magnetic field strength and of the amount of copper in the gap.

Since these two factors do not uniquely determine electro-acoustic efficiency (the mass of the moving system, and the method of coupling the diaphragm to the air are at least as important), there is no direct relationship between electro-acoustic efficiency and damping.

The AR-1W, used in these tests, for example, a speaker with very low over-all efficiency, has unusually high magnetic damping due to its heavy magnet and to the large amount of copper in the gap. It is, as a matter of fact, in danger of being over-damped when improperly used, as under conditions similar to those of the lower curve of Fig. 3 (180 deg. solid angle of radiation, high damping factor), in which bass attenuation can be seen.

Figure 7a is a recorded graph of the frequency response of the speaker in a living room, with the two extremes of damping factor used in Fig. 4. Note that the over-all shape of the curve is affected by changing the damping factor, in the same way as it was in Fig. 4, but that the irregularities due to the acoustical environment of the room are completely uninfluenced.

This illustrates the independence of room ringing - associated with peaks and dips in the steady-state frequency-response curve - from damping in the speaker system itself. The only damping that can have any effect here is that connected with the room surfaces; neither magnetic, mechanical, nor acoustical damping of the speaker's moving system can affect a cure. The latter point is further illustrated in Fig. 7b, a frequency response record of the same speaker in a different part of the same room.

It may have been noted that the condition of high damping factor produced bass attenuation when the speaker radiated into 180 deg. in the open field, but that the same high damping factor is associated with essentially uniform response, down to 30cps in the indoor measurement (ignoring room-derived irregularities, and correcting for microphone and recorder).

In the room the lower damping factor produces a somewhat exaggerated bass. The primary reason for this lies in the fact that the speaker in the room was mounted so that it faced into a reduced solid angle (90 deg.) - in a corner, off the floor.

Figure 8, also borrowed from Beranek's Acoustics, shows the change in bass response produced by restricting the solid angle seen by a speaker. Higher-frequency components are concentrated in the area ahead of the cone, and if the environmental solid angle seen by the speaker does not similarly restrict the non-directional bass, it will be thinned out relative to the treble. As might be expected, below the frequency at which the speaker's signal becomes essentially non-directive each successive halving of the solid angle doubles the bass power, or raises the response curve by 3db.

It would seem to be. a good idea for someone to design an equalizer network to produce variable bass boost to compensate for this effect on performance due to change, in solid angle. In the meantime the closest approximation to such a circuit is a variable damping-control, which gives the user additional flexibility in tailoring the low bass response of his system to the conditions of speaker mounting. Lowering the damping factor may also affect the mid and high frequencies, and a circuit which only varied the damping factor (from a high value down to one-half or so) over the bass frequency range would be useful. *6

I would like to add some further comments to this article in an attempt to Jay to rest some of the old wives' tales about speaker damping. The insertion of a few numbers into the general relationships that we have discussed should help in this connection.

1) Let us study the case of a speaker whose nominal impedance is 8 ohms. The d.c. resistance of its voice-coil will be of the order of 6 ohms. Thus the total d.c. resistance that the speaker sees, looking back at the. amplifier, is equal to the sum of the amplifier source resistance, the d.c. resistance of any series choke from a crossover network, and its own d.c. resistance. (The. representation of internal resistance by an external resistor of equivalent value is standard practice for generator diagrams.)

The d.c. resistance of the series choke is likely to be about 0.5 ohm. The source resistance of the amplifier, with a damping factor as low as 4, will be 2 ohms. The total resistance seen by the speaker is then 8.5 ohms.

Eliminating the choke (a component which is sometimes severely frowned upon) gives us a reduction from 8.5 ohms to 8 ohms. Doubling the damping factor (halving the source resistance) gives us another sweeping reduction, to 7 ohms. In brief we must remember that, even with the speaker terminals shorted out by heavy copper wire of .001 ohm resistance, the smallest braking resistance we can ever achieve is 6.001 ohms. There is thus little to be gained by worrying about small resistive components in the speaker line, or by increasing the damping factor to astronomical values.

2) If we connect a second, identical speaker in series, the total internal d.c. resistance is increased to 12 ohms. But the ratio between resistance and reactance remains the same, as we now have a 16-ohm system, and the damping is unchanged. (Each 8-ohm voice-coil may be thought of as one-half of a 16-ohm voice-coil.) The series connection is perfectly good practice.

3) Another well quoted misconception relates to the fact that the coupling, at bass frequencies, between an infinitely baffled cone and the air into which it radiates decreases as the frequency is lowered, and that this decrease is compensated by progressively increasing speaker cone velocity, as discussed previously.

The belief has somehow gained ground that the loss in acoustical coupling referred to has to do with the low bass regions only, below one or two hundred cps, and that the compensating increase in cone velocity is related to speaker resonance: that is, that the resonant peak is used to "fill in" the acoustical losses.

Actually the air-load resistance presented to the cone decreases with frequency at an orderly rate (a factor of 4 per octave), below a frequency which is a function of the cone diameter - for a 12-inch speaker about 800 cps. No change in this progressive loss occurs in the extreme low bass. The theoretically ideal compensation for the decrease in air load resistance would be provided by a purely mass-controlled mechanical system, without resonance, which would dictate a doubling of cone velocity for each lower octave. (The electrical analogy is a purely inductive circuit - for the same applied voltage, current through the choke will double with each lower octave, due to the progressively decreasing inductive reactance.) Such a system is a non-existent entity, but if the speaker's resonant peak is properly damped the mechanical system acts as if it were purely mass-controlled at frequencies above resonance, and the proper compensation is provided.

4) As the frequency of the input signal to a loudspeaker is lowered in the direction of resonance, the electrical impedance of the speaker rises far above its nominal value, perhaps by 5 or 6 times. With a high-damping-factor amplifier the voltage across the speaker remains essentially constant, involving a severe drop in the electrical power drawn from the amplifier; with a lower damping factor the drop in electrical bass power is less severe; and with an even lower damping factor electrical power may remain constant, or may increase towards resonance. That value of damping factor which achieves the most uniform acoustical output and optimum performance is not tied to a condition of uniform electrical power, but is a function both of the particular speaker used, and of its conditions of mounting. While high damping factors are generally most suitable for horn or resonant-type systems, the same is not necessarily true for direct-radiators. No special virtue can be attached to that value of damping factor which produces constant voltage, constant power, or some intermediate type of relationship betwen amplifier output and frequency, if speaker system performance is un known.

I would like to express my appreciation to Dr. J. Anton Hofmann for his patient reading of the draft of this article and for his valuable suggestions.

geparkte Bilder und Texte


4 Edgar M. Villchur, l (Commercial acoustic suspension speaker," p. 18, Audio, July, 1955.

5 Unfortunately, 30 db per decade (American standard) graph paper was not available, and 20 db per decade paper had to be used.

6 Arthur A. Janszen, "An electrostatic loudspeaker development,'' p. 89, JAES, Vol. 3, No. 2, April, 1955.

* The writer has, since completing the draft of this article, learned of such an amplifier design available commercially - in the Mcintosh MC-30A and MC-60A. Tests on a sample MC-60A showed it to perform precisely according to expectation.


Bilder:

Fig. 1. Response of a direct-radio tor speaker system, in the region of resonance, at different values of Q Sf its mechanical system, (After Beranek)

Fig. 2. Simplified mechanical system ing the mechanical by magne
electrical analogy to of a speaker, includ-resistance introduced tic damping

Fig. 3. Test set-up for measuring speaker performance. The speaker sees a controlled solid angle of 180 deg.

Fig. 4. Recorded speaker response curves for different values of amplifier damping factor, open field conditions. The calibration curve for the recording equipment, not including microphone, appears at the top. (See Fig. 5 for corrections).

Fig. 5. Response curves of Fig 4, corrected for the calibrated errors of recorder and microphone (calibration curve for the latter appears in inset).

Fig. 6. Acoustic output of speaker, as monitored by microphone and oscilloscope, in response to step-front of low-frequency square wave: A (left), With amplifier damping factor of 6; B (center), with damping factor of 1; C (right), With damping factor of 0.1.

Fig. 7. A (top), Response curve of speaker mounted in two-sided corner of room. Lower curve is for damping factor of 6, upper curve is for damping factor of 0.1. B (bottom), Response curve of the same speaker in a different posi-t i o n in the same room at damping factors of 6 and 1.

Fig. 8. Effect of restricting the solid angle seen by a loudspeaker. The top curve (A) is for a solid angle of 45 deg.; each succeeding lower curve represents an in-crease of the solid angle by a factor of 2. (After Beranek)