Posted by Al Sekela on November 23, 2002 at 10:51:52:
In Reply to: CLARIFICATIONS posted by Al Sekela on November 21, 2002 at 12:54:34:
After further thought I've concluded there is some amplifier effect on electrostatic membrane damping, but it is slight. Electric damping requires higher, rather than lower, amplifier output impedance, and should only be effective for amplifier output impedance on the order of 100 ohms or very high membrane resonant frequencies. Thus, amplifiers with lots of negative voltage feedback and with output impedances much less than 1 ohm (with high "damping factors" as calculated for cone speakers) will have negligible damping effect on electrostatic speakers. However, OTL amplifiers, with output resistances on the order of one ohm, may be able to damp higher-order membrane resonant modes, and this may explain the preference some people have for OTL amplifiers with electrostatic speakers.
Here is my reasoning for what its worth in entertainment value.
Start with a loss-free electrostatic speaker. The speaker is located in vacuum, the membrane material is perfectly elastic, the stators, transformer wires, and cables are made of superconductors, and the amplifier has zero output impedance.
The amplifier applies a voltage to the stators long enough that the membrane reaches some displacement at equilibrium and then suddenly the input signal goes to zero.
What will happen?
The membrane, with mass and under tension, will vibrate like a drum head. The charge stored on the stators, that pulled the membrane away from rest at the beginning, will flow back and forth between the stators and through the transformer in response to the motion of the membrane. Charge flow is current, so the transformer will cause alternating current to flow through the amplifier.
As long as there is no air resistance or electrical resistance in the system, the vibration will continue undiminished. The energy stored in the system will slosh back and forth between mechanical energy in the membrane and electrical energy in the charge on the capacitance of the stator/membrane structure. In circuit terms, the membrane looks like an inductor and this circuit is like a parallel L-C with infinite Q.
Damping would require energy dissipation. Let us add a little output resistance to the amplifier and see what happens. Now there will be some voltage developed across the amplifier as the current flows due to its finite output resistance. This voltage is a loss which reduces the amount of charge that can be transferred to the opposite stator and generates a retarding force on the membrane. The vibrations will diminish and eventually cease.
For the output resistance of the amplifier to apply significant damping (ie, close to critical), it seems reasonable that the peak voltage developed across the amplifier with zero input signal (while the membrane is vibrating) should have to be comparable in magnitude to the voltage applied to the stators at the beginning to create the initial displacement. If it is much less, then the vibration will continue for many cycles and the situation will be similar to an underdamped drum head vibrating in air.
Assuming a sinusoidal motion with drum head frequency fd and using the convenient properties of the sine and cosine functions, it is easy to show that the ratio of peak voltage developed across the amplifer to initial voltage, Va/Vo, is 2xPixfd times RC, where R is the amplifier output resistance and C is the speaker capacitance translated back to the amplifier through the step-up transformer. For this ratio to be on the order of unity, the RC time constant has to be close to the reciprocal of the drum head angular frequency.
To get an order-of-magnitude estimate, assume a square meter speaker with one millimeter spacing and 1000:1 impedance ratio for the step-up transformer. The drum head frequency of the membrane is 159 Hz. The speaker capacitance is about 0.01 microfarad, or 10 microfarads to the amplifier as translated by the transformer.
The amplifier output resistance would have to be 100 ohms to make the time constant comparable to the drum head frequency. Typical amplifier output resistances are much less than one ohm, so the amplifier damping effect is negligible and cannot be used to achieve critical damping of the membrane.
Why is the drum head frequency important in this situation? The current that flows through the amplifier is the time rate of change of the charge stored on the speaker stators (translated by the transformer). We assumed this to be an underdamped system to begin with, with the motion largely controlled by the mechanical properties of the membrane (mass and effective tension). This is a good assumption for typical amplifier output resistances, as shown above. The time derivative of the membrane displacement contains the angular frequency (2xPixfd) as a factor, and the higher this frequency, the larger the peak current. Thus, amplifier damping could be significant for higher membrane drum head frequencies.
This suggests OTL amplifiers may be capable of cleaning up higher-order resonant modes on some membrane designs, that conventional amplifiers (with high "damping factors" as calculated for cone speakers) would ignore. This might help explain why some people find OTL amplifiers a good match with electrostatic speakers.
This also suggests critical damping could be achieved if the amplifier were designed with negative current feedback instead of negative voltage feedback, so that it looked like a current source instead of a voltage source and had very high output impedance. Such an amplifier would not work well with cone speakers (Carver's cynical use of the term 'current source' to describe an amplifier with a one ohm resistor in series with the output notwithstanding), but could be adjusted to mate well with an electrostatic speaker membrane.