Rick,
Well, it's not all formula. I'm lazy.
The first one is the basic power formula: Watts = voltage * amps. But to use that you need to know the current through the speaker load. If you factor that around with a little algebra and the above formula, you'll eventually come up with the one I use the most because of my laziness: Watts = (voltage squared) / impedance.
Working that backwards (i.e. solving for the unknown voltage) for the beta at 4 ohms, you'll get 100 watts at 4 ohms = 100 * 4 = 400 and taking the square root of that... you get 20 volts. So across a 4 ohm load the beta delivers a 20 volt signal for a 100 watt output.
Ok... now let's take that 20 volt signal, square it to get 400 and now divide by an 8 ohm load and we'll get... 50 watts? Yep absolutely correct.
What about that 65 watts? Well, that is a "rule of thumb" and takes into account things that the power formula doesn't consider. For instance:
With a 50% load (8 ohms is less of a load than 4 ohms), 50% less current is needed. Use that first equation I don't use. That equation also assumes the voltage remains the same and under that assumption, the current for 1/2 the power is 1/2 the current.
So... the power amp is required to deliver only 1/2 the current. With less of a current requirement, the supply voltage to the power amp will sag less, maintaing a higher voltage.
Also, with less current through the output transistors, the voltage drop across those transistors will also be less. Both of those aspects provide more voltage for the power amp to work with.
One extra thing. You know how some people claim that tubes "round" the output when overdriven but transistors clip cleanly when overdriven? Yeah, that one - totally false.
Look at any spec sheet for your favorite output transistor and find a graph typically called "collector current vs Beta". Beta is a term used for the current gain of a transistor, and... (typically) as more current is drawn through a transistor, the lower the Beta (or gain) of the device. Conversely, the lower the current, the higher the gain.
Factor these things in together (higher supply voltage and higher gain of the output transistors), and you wind up with the power amp being able to deliver approximately 15% higher output voltage into that 50% less load.
Going back to our 100 watt into 4 ohms being 20 volts and factoring in the above and doing the math we now get 23 volts (15% more) into 8 ohms... roughly 66 watts.
Since the math is easier, the 65% I gave is a rule of thumb.
The power to db... another rule of thumb. Less than 3 db is pretty much useless, which is good because the rule of thumb requires you to memorize two relationships.
A 3db increase is twice the power... a 10db increase is ten times the power. Add to that that 3db will be barely noticed and that 10db will be twice as loud.
So... a 6db increase becomes a 4 times increase in power, a 9db increase becomes an 8 times increase in power... and now we are at 10db. But if you wanted to continue the previous sequence, a 12 db increase would be a 16 times increase in power.
If necessity is the mother of invention, then surely laziness is the papa!!
