Audio Related formulae for RPN/RPL calculators.
Audiophile are a strange breed, and we try to make our decisions based on what we hear. However, in order to set up our audio systems, some calculations are in order, if we are to have a system that works together. Failure to follow these formulas may end up in wasting a lot of money (although no disaster will happen if you stay within the sane boundaries of average audio gear - only the cutting edge may make you bleed - pun intended)
For example: usually audio listening is done with volumes between 75 and 100 dB. While 100dB can be considered very high volume, there are musical peaks that are higher than that. In general, it is accepted that a high quality audio system should be able to reach 105 db on peaks without distortion (cfr. Anthony Michaelson). You need to make sure your system is able to accommodate these peaks
Speaker Sensitivity and Amplifier power
In the past, when before the transistor came, all amplification was done with vacuum valves (called "tubes" on the other side of the pond), and amplifier watts were limited. Typical values were between 15 and 40 watts. Earlier on, the technology was based on single ended tubes, where power was less than 10 watts most of the time.
At the same time, the loudspeakers were developed with very high sensitivity, to make the most of that meagre wattage. Sensitivities in the region of 100 db/watt were normal. The designs were based on single drivers, occupying the whole audio bandwidth. This meant that the frequency extremes were not properly served. At the same time, the amplifiers were optimized for a given loudspeaker impedance, and the typical values of the time were from 8 to 16 ohms. Speakers were big at that time, and they used horns to increase sensitivity.
In the late fifties and the sixties the transistor took the audio world completely, and the power relation was reversed. Power was cheap, and the audio designers took ample power as a given. Gone were the single driver, bandwidth-limited loudspeakers, and the multi-driver loudspeaker arrived. In order to assign each driver the range of frequencies best suited to it, the incoming amplifier energy had to be diverted to each through high pass and low pass filters, in what was called the crossover. There was a lot of energy spent in the crossover, but it did not matter, since there was plenty. Strategies were used to extend bandwidth in a linear way; most of the times it had to do with dampening, ie. consuming more power. Sensitivity came down to 85 to 90 db/watt (from 10 to 30 times less sensitive than the above units)
During the last few years, there has been an increase of tubed amplification. It is felt by many audiophiles that tubes bring warmth and fullness to the sound, and give tridimensional qualities to it that seldom solid-state can attain. However, physical limitations remain, and there are few amplifiers over the 60 watt mark, while there are loudspeakers as low as 80 db/watt in sensitivity.
Let's picture a situation of a given audiophile, trying to decide on his system. Let's imagine that he wants a couple of Quad 2805 electrostatic loudspeakers. While they are excellent at showing the musical picture, they are very inefficient (at 82 dB/w) and require good, powerful amplification.
Let's see: Volume required: 100 dB
speaker efficiency: 82 dB
Amplifier must be able to supply 18 dB of power: 10^(18/10)= 63 w
If 105 dB peaks are required: 10^((105-82)/10) = 200 watt !! - loudness is expensive
On the other hand, if you take a Lowther-based horn loudspeaker, you will be confronted with an efficiency of 100+ dB/W, that is, with a single watt you can produce peaks of 100 dB. That's loud indeed!
To reach 105 dB you would then need an amp providing 5 dB: 10^(5/10) = 3,16 W. Before you laugh at it, these are not necessarily toy amps: there are two families of tubed amplifiers that will produce this kind of wattage, cost dearly (from 2000 to 20000 €) and make audiophiles salivate: Single Ended Triodes (SET) and Output TransformerLess (OTL) The firsts are based on using a triode (the basic valve) for both the upper and lower part of the audio waveform (as opposed to the push-pull topology, where two valve devices amplify positive and negative sides of the signal). The second can be push-pull, but are based in connecting directly the valves to the speakers, without a transformer to adapt for the very different impedances. Both families are claimed to be "magical· from their supporters (staunch in both cases, I must say)
Other indicators to check amplifier-speaker compatibility: current delivery, damping factor, loop feedback, amplifier gain
Having into account that this loudspeaker (Quad 2805) has an average impedance of 8 ohm (quite resistive in fact), then the amp for the Quad must be able to provide SQR(R * W) = 40 v RMS, i.e., 40 * sqr(2)*2 = 113,1 V of swing. If you are into DIY for your amplifier, you will need to provide a transformer able to give voltage enough to accommodate for that.
Bear in mind that the current required is significant as well: as W = I^2*R, then I = SQR(200/8) = 5 A. As our speaker has a benign impedance curve, current is kept at bay. Typical dynamic driver designs have a 4-ohm impedance, with dips of 3 or less ohms, mandating much higher currents and making them unsuitable to most tubed amplifiers.
Another issue to match the amp to the loudspeaker is the damping factor. This is the speaker impedance divided by the amplifier output impedance. With solid state designs, it is not difficult to find damping factors of 100 to 200; but the output impedance of many tubed amplifiers is around 1 ohm, giving damping factors of 4 to 8, depending on the speaker impedance. The effect of this low damping factor is on the amplifier's ability to control the woofers. The impedance swings of these will have an effect on the frequency response of the amplifier.
A tool a designer has in order to lower output impedance is feedback. It also helps to extend the bandwidth and reduce distortion. Solid state amplifiers use feedback with liberality - in fact, it is quite rare to see a zero-feedback solid state amplifier. In the tube world, as the valve is inherently a more linear device, there is the possibility of designing zero-feedback amplifiers. It is said that these keep all the nuances of the music and the ambient information present in the recording. The logic behind: the feedback signal is so very slightly delayed in time, so that you are introducing corrections on the signal that was there shortly ago, not on the current signal. You don't need golden ears to hear that amplifiers without feedback preserve better the cues of the recording site, have a better 3D effect and wider soundstage. SET amplifiers often are zero-feedback, but have output impedances above 1 ohm, and power below 10 watts in most cases.
The lack of feedback and concomitant higher output impedance can be heard as uncontrolled bass (due to the small damping factor), higher distortion, imprecise image specificity and limited bandwidth.
The calculations of feedback levels are circuit-dependent and we will not discuss it here.
In a lot of amplifier literature, amplifier sensitivity is specified, which is the signal voltage required to drive the amp to maximum power. This is a misleading measure that does not say a lot about the amp, or how it does interact with the rest of your system. It is better to state the amplifier gain: the factor times which the amp multiplies the incoming signal. It is expressed in decibels and it is likely between 25 and 30. Added to that of the preamp, you end up with a total system gain of around 30-35 dB. The real gain is less, since the preamplifier operates an attenuator (the potentiometer you twist to reach the volume you want - the above gain would be with the potentiometer wide open). It is worth noticing here that some of the best preamp are passive: either with resistor networks or multi tapped transformers, they just take whatever signal is thrown at them and attenuate its voltage, either by voltage dividers (resistive networks) or trading it off for current drive (in the transformer volume controls or TVCs). You need to take into account this fact when calculating your total gain.
By the way, a common misconception is to think that your amplifier is really very powerful, since you cannot turn the volume know more than 9 o'clock because it is too loud. This just means that your system gain is so high that you are using just a quarter of your volume knob: it will be difficult for you to adjust the volume, and the lower reaches of the volume know tend to have left-right imbalance. It may well be, as well, that your amplifier is clipping (reaching its maximum power in a painful way - a better description coming later) already by 11 o'clock !!)
If your system has a total gain of below 20, and the loudspeaker is not above 95 dB/W/m, you will be using all the time the higher reaches of your volume pot. Eventually, you will require more gain. For speakers between 85 and 90 dB/W/m, a gain of 30 dB should be enough. The gain we are discussing is from line level to loudspeaker. It is assumed that line level is 2 V RMS. In the past (seventies, before the advent of CD), line level was considered 0,707 V , i.e. 9 dB less than now.
After the above concepts, here are the formulas: