OK, let me present in a much more simplified form about gain flatness, amp DF using JUST NON INDUCTIVE RESISTORS only. This will eliminate the complication of all the reactance, complex numbers.
I called speaker impedance as RL and amp output impedance + cable resistance as RS. I set RS=1ohm constant. I set output of the ideal amp VOUT=1V in all the following cases. Fig.2 show the circuit where the amp drive the speaker through the RS. A voltage divider forms between RS and RL and the formula is shown in the attachments.
In example (A) in the first drawing RL= 8ohm constant throughout the audio frequency. In Fig.3, you can see the VIN=0.8888V flat across the frequency. So you can see, even if RS is so high at 1ohm, as long as RL is non inductive resistor, the voltage VIN at the input of the speaker is flat.
In example (B) in the second drawing, RS is still constant at 1ohm. BUT I make RL dip to 4ohm at two points and peak to 16ohm at two points. Now we observe how this affect the voltage at the input of the speaker VIN. VIN calculation is exactly the same, just change the RL only.
I show the calculations when
i) RL=8ohm, VIN=0.888V as before.
ii) RL=4ohm and VIN =0.8V.
iii) RL=16ohm and VIN=0.94V.
Look at Fig.4. I just plot the gain flatness by plotting VIN to the speaker. You can see how much the gain change because of RL changing.
You can see, the math is very simple, people must have learn resistor voltage divider calculation in junior high. It's just that simple to see how the speaker impedance has everything to do with the gain flatness measured with real amp and real speaker cable that has impedance as RS.
I hate to use Damping Factor, but people relate better with DF than if I said output impedance of the amp. But they pretty much talking about the same thing.
As you can see, output resistance is just as important at high frequency as in low frequency in gain flatness. In fact, it's much easier to get higher DF at low frequency than in high frequency. Cheap amps use NEGATIVE FEEDBACK to raise the DF( or lower the output impedance). BUT the N feedback runs out of steam at higher frequency, meaning DF goes down at higher frequency. High end amps usually use 4 or more pairs of output transistors, Krell use 8+ pairs of output transistors. The DF does NOT depend on Negative Feedback. Output impedance remain very low because of that many output transistor pairs. Running higher bias current decrease output impedance of the amp also. That's the reason I judge the quality of the amp literally by looking at how big the heatsink, how many pairs of output transistors........How heavy the amp is.
These are all scientific, I hope when I remove the reactance part and just use resistor divider, this will make it easy for people to follow to see the effect of speaker impedance on the measurement. It is just this simple, nothing esoteric.
Own designed power amp, own designed preamp, JM LAB Focal Alto Utopia and Spectral 913.1 speakers, Rythmik F12SE sub.
Not hooked up: Nakamichi Stasis PA-7 power amp, Velodyne VA1210 sub, Kef Reference Series center, Kef Bookshelf speaker, Monitor Audio bookshelf speaker, Infinity rear speakers. Acurus 3X200W amp.
Last edited by alan0354; 05-22-2020 at 01:25 PM.