I'm going to be connecting a low-voltage 8 ohm amp module to a 4 ohm loudspeaker. Using the same amp and a different speaker, I've done this before without issues. I'm not expecting high-fidelity, just having fun with a project. A vendor suggested putting a 4 ohm resistor between the amp and the speaker which I haven't done in the past.
I have a question regarding the pure mathematics of the setups with and without the resistor. Will I get more or less power out of the amp and more or less volume out of the speaker with the resistor in series? The amp will be powered by a 9-volt battery. Will a 4 ohm resistor in series drain the battery faster?
Here's what it looks like with various loads at various output voltages.
The original load 8 Ohms, with original 2v output:
2^2/8=1/2 watt (0.5 watts)
The new load 4 Ohms, with 1.4142v output:
sqrt(2)^2/4=1/2 watt (0.5 watts)
The new load with extra 4 Ohm resistor, with original 2v output:
4+4 Ohms:
2^2/8=1/2 watt, (1/2)/2=1/4 watt (0.25 watts)
The new load with extra 4 Ohm resistor, with 1.4142 times the original voltage of 2v output:
4+4 Ohms:
(sqrt(2)*2)^2/8=1 watt, (1)/2=1/2 watt (0.5 watts)
This shows that it looks like you have to turn up the output voltage (with the volume knob) to 1.4 times the original output voltage in order to get the same output power. That means that you use twice the power to get the same volume output when using an extra 4 Ohm passive resistor in series with the new 4 Ohm speaker.
If you can get away with just a 4 Ohm speaker and no extra resistor, you do not have to turn the voltage (volume) up you can actually turn it down, but you get the same output power.
With 2v output and 8 Ohms the current would be 1/4 amp.
With 1.4142v output and 4 Ohms the current would be 0.3536 amps.
With 1.4142 times 2v output and 8 Ohms the current would be 0.3436 amps.
With these results and the results farther above, the output power with a 4 Ohm speaker would be the same as with a passive series resistor or not. The input power with the extra passive resistor would be twice of that without the extra resistor. However, since the current would be the same and batteries tend to run down based on current flow, the run time may be the same or nearly the same for either setup. If the amplifier could handle the 4 Ohm load though, I'd probably go with that except I would not turn up the volume all the way.
Of course amplifier distortion is dependent on load values so we would expect a change in the distortion level. There's a good chance it would go up, but since we can turn the volume up less than with the 8 Ohm load it depends on how the distortion changes with volume setting also. Some amplifiers are worse at low volume and some worse at higher volume.
We could do some simulations to verify these results, I did not do that yet due to time constraints.