Damping factor effects on loudspeakers
Premise
Like any other parameter in choosing an audio power amplifier, the damping factor is often underestimated and of very high relevance in audio quality.
The damping factor affects the excursion of the speaker, generating distortion in a way that cannot be measured with a simple constant signal or a frequency sweep. This paper will explain how it works in detail and how the speaker reacts to its variation.
Loudspeaker basics
An audio amplifier is ideally a constant voltage generator, and constant voltage generators have very precise ideal specifications: infinite input impedance, unlimited speed, unlimited bandwidth and zero output impedance.
The parameter to take into consideration is the output impedance: the damping factor is directly correlated (and linked) to the output impedance of an amplifier and the impedance of its load.
In the real world, it is clearly not possible to obtain an impedance equal to zero, but it is possible to obtain a very low value: so that the signal quality is acceptable and the distortion is inaudible, or is negligible compared to the basic distortion of the speaker.
To understand how the damping factor works, one must first understand how a speaker works.
First of all, a classic loudspeaker consists of a permanent magnet, a coil, a cone, and its suspensions. The suspensions keep the coil in the center of the air gap, which would be the top of the magnet. The cone and the suspension are glued to the coil.
When an electrical signal passes through the coil, it generates a magnetic field, and depending on the direction of the current, the coil aligns or moves away with the magnet as it produces opposite or equal magnetic fields.
Since the cone is attached to the coil, the movement of the coil moves the cone, thus the air: creating pressure, and therefore the sound.
But a loudspeaker is also a reversible machine. By moving the cone, the coil is displaced relative to the magnet, and this causes the electrons in the coil to move thanks to the magnetic field of the magnet, generating current at its terminals.
The effect generated is the same as that of a solenoid: a magnet passing inside an inductor generates voltage across the inductor itself.
Discovering the problem
If a loudspeaker was an ideal component, at the end of an input signal, the movement of the cone should stop at the same time.
But this does not happen because a speaker is not only composed of its magnetic core and its voice coil, but also of the cone, which is connected to the basket by a spider and a membrane.
These have the task of returning the cone and the voice coil to the starting position once the input signal has ended. However, since these components are elastic, they will act like springs before the various frictions completely stop the cone.
The graph below will exactly describe the behavior described above, once the signal stops, the cone will oscillate until the frictions stops it. Obviously, the amplitude of the oscillation and its duration depend on many factors, including: the amplitude of the initial signal, how long it takes to stop, the hardness of the suspensions, the mass of the cone, air density, and the compression of the box where the speaker is placed.
This behavior of the cone is in all respects a generated distortion: the cone should have stopped when the signal stopped, if the cone continues to move it means that it also moves air and therefore continues to generate sound pressure (sound). And if the generated sound is not identical to the initial one, it means that it is distorted.
Clearly the case of the graph is extreme, but in the reality of the musical signal it happens continuously: the cone must follow a signal that has multiple intermodulated frequencies, for this reason the direction of the cone often must change suddenly.
This is where the damping factor comes into play, a mechanism of the amplifier to control the inertia of the speaker and force it to follow the original signal.
To understand how damping factor forces the speaker to maintain the same trend of the signal, is important to understand also how the amplifier feedback network works and how output impedance is calculated.
The damping factor is calculated by dividing the load impedance by the amplifier output impedance.
The output impedance of an amplifier depends on various things, these include the specific frequency at which it is measured, the impedance of the load, but most importantly, the strength of the amplifier’s feedback.
An amplifier, when it does its job, being neither it, nor the load, ideal components, it will always introduce distortions of some kind. This is where feedback comes into play, which deals with comparing the input with the output. The feedback system compares the input and output signals, negatively adding a part of the output signal to the input signal to obtain the most undistorted output signal possible.
There are various types of feedback in electronics, the most common are local feedbacks, generated by degeneration resistors or error correction systems, and global feedbacks, which enclose the entire amplifier in a loop circuit, connecting the output to a differential system in the input stage.
That’s how the damping factor works: when oscillating, the speaker generates current at its terminals. This current will ends in the amplifier feedback system, that will negative sum to the input, creating a contrasting current to corrects the speaker oscillation.
The output impedance is determined, in the first place, by the strength of the feedback, and then by everything between the feedback point of the amplifier and the speaker terminals, which are not the connection terminals of the enclosure, but the terminals of the speaker itself.
Meaning that also amplifier output networks, terminals, connection cables and especially passive crossovers reduces the damping factor by increasing the output impedance.
Always to remember, that the impedance of speaker cable is calculated summing the impedance of both cables, not only the hot (+) one. Summing up: everything between will always decrease the influence of the damping factor in the control of the speaker.
Also, the output impedance is not always equal on all frequencies of the bandwidth: it tends to increase with the frequency.
The damping factor, however, becomes more important at low frequencies, not because it is not important at high frequencies, but because in a musical signal, normally the power density is greater at low frequencies, consequently the excursion at low frequencies is greater and therefore the effects of damping are far more audible.
To understand how much the damping affects the load, an ideal circuit is built inside a circuit simulator program.
The amplifier and load will be considered ideal, so the load will not be as complex as a real speaker, but a simple resistive load: this means that any effect demonstrated will certainly be significantly worse in the presence of a real load. Furthermore, output networks, cable impedance and crossover impedance will not be considered.
The only thing that will vary will be the output impedance of the amplifier, which since it is ideally zero, will be highlighted by the variation of a Zout resistor placed after the feedback. In the simulation, a constant current generator is placed in the direction of the amplifier, configured to obtain the exact reverse current generated by the opposite displacement of the cone and simulating its inertia.
In the simulation demonstration, you can not only see how the ringing of the speaker is stopped less as the output impedance rises, but that the original signal itself is modified because the power transferred to the speaker decreases and the amplifier does not manages to push the cone sufficiently because its inertia at t=0 tends to keep it stationary.
A lower output impedance, therefore a higher damping factor manages to keep the signal more similar to the original one. Below is a table of the measured data.
| Output impedance | Resulting damping factor | |
|---|---|---|
| 8 Ohm load | 4 Ohm load | |
| 1 mOhm | 8000 | 4000 |
| 5 mOhm | 1600 | 800 |
| 10 mOhm | 800 | 400 |
| 50 mOhm | 160 | 80 |
| 0.1 Ohm | 80 | 40 |
| 0.2 Ohm | 40 | 20 |
| 0.5 Ohm | 16 | 8 (almost no effect on load threshold) |
| 1 Ohm | 8 (almost no effect on load threshold) | 4 |
| 2 Ohm | 4 | 2 |
| 4 Ohm | 2 | 1 |
| 8 Ohm | 1 | 0.5 |
Here is a measured data from table, values are referred to 1 Vpk half sine wave with 0 seconds of decay time.
Overshoot voltage refers to the first, lowest, peak of the wave after the signal stops, the THD is the percentual of the signal distorted taking in consideration the overshoot voltage compared to the output signal (1 Vpk). The voltage attenuation is how much of the original signal is attenuated before arriving to the load.
| Load impedance | 8 Ohm | 4 Ohm | ||||
|---|---|---|---|---|---|---|
| Output impedance | Overshoot voltage | THD | Voltage attenuation | Overshoot voltage | THD | Voltage attenuation |
| 1 mOhm | 60 uV | 0.006 % | - 30 uV | 120 uV | 0.012 % | - 60 uV |
| 5 mOhm | 300 uV | 0.03 % | - 140 uV | 600 uV | 0.06 % | - 280 uV |
| 10 mOhm | 600 uV | 0.06 % | - 270 uV | 1.2 mV | 0.12 % | - 540 uV |
| 50 mOhm | 3 mV | 0.29 % | - 1.4 mV | 6 mV | 0.58 % | - 2.8 mV |
| 0.1 Ohm | 6 mV | 0.59 % | - 2.7 mV | 12 mV | 1.18 % | - 5.4 mV |
| 0.2 Ohm | 12 mV | 1.16 % | - 5.4 mV | 24 mV | 2.32 % | - 10.8 mV |
| 0.5 Ohm | 28 mV | 2.81 % | - 13 mV | 56 mV | 5.62 % | -26 mV |
| 1 Ohm | 53 mV | 5.30 % | - 26 mV | 106 mV | 10.6 % | -52 mV |
| 2 Ohm | 96 mV | 9.66 % | - 44 mV | 192 mV | 19.32 % | - 88 mV |
| 4 Ohm | 160 mV | 16 % | - 72 mV | 320 mV | 32 % | - 144 mV |
| 8 Ohm | 240 mV | 24 % | - 100 mV | 480 mV | 48 % | - 200 mV |
Level of distortion can be enormous when damping does not intervene in controlling the speaker excursion.
The damping factor must be as high as possible, because when the impedance of the load drops, the distortion will rise clearly.
But problems, don’t ends here. Another effect of reducing the damping factor is also the reduction of the speaker’s bandwidth linearity. The damping factor is basically the amount of control the amplifier has over the speaker: if the control becomes too low, the effective bandwidth becomes a servant of the impedance module of the speaker itself.
Taking for example a circuit where the speaker is now a real-world one, the impedance module now is non-linear.
The frequency response becomes subject to the speaker impedance the higher the amplifier output impedance.
Here is another real-word computation: effects of amplifier damping factor on the frequency response when connected to a simulated impedance load typical of a two-way closed box loudspeaker system.
Unfortunately, the problem is not solved even with a Zobel network. A Zobel network is normally used to linearize the impedance modulus of a loudspeaker. However, the effects of band variation are still present, with two major problems: the bandwidth is attenuated and have a strong bending in the Fs region of the speaker. Something that does not happen when the output impedance is really low.
Using a passive crossover will make the situation much worse, making the frequency cut that had made the band linear by design, now with a completely different response.
Even with the amplifier having a very low output impedance, using passive components as a frequency cutter adds its own amount of impedance to the circuit. And the greater the amount of components, the greater the impedance added. Even a very simple low pass made by a simple inductor can be extremely deteriorative: standard inductors on the market do not have impedances below 0.5 Ohm, which is an enormous value.
In the simulation, a Zobel network is used to linearize the impedance module of the speaker and is used an ideal inductor with zero impedance, only variating the Zout.
This shows that even with a perfect crossover calculation, the bandwidth changes significantly depending on the amplifier used.
Conclusions
From the study, it is clear that it is not possible to first determine the damping factor distortion with a simple frequency sweep or a constant signal, since it would only result in attenuation.
Using a pulsed signal instead, it is possible to study the behavior of the speaker and its excursion, and the effects of the damping on it.
It also highlights how the lack of damping factor has really significant results on the distortion introduced and how this can also modify the bandwidth of the speaker if low.
Furthermore, it is very important to keep the impedance of the area between the feedback point and the speaker terminals as low as possible, paying attention to the use of output networks or cables that are too thin. If possible, it is recommended to eliminate crossovers and switch to multi-amplification with active cut.