The purpose of a power supply in an amplifying device - and let's be clear, by amplifier I am referring to anything that amplifies, even sources - is to provide a steady and uninterrupted source of DC voltage.
On the surface there can be nothing better than the 'ultimate' source of DC voltage, a battery. Choose the voltage you require, a battery will give you that without any hint of AC on it. Should be perfect. Right? Wrong. So let's see why this is not a great idea.
You know how some things seem like a perfect solution until you attempt to implement them? Batteries are like that. A battery CAN be a perfect source of power for an amplifier but its practical application is typically so problematic a good power supply it does not make.
I remember years ago when a friend of mine and I thought we had the perfect replacement for gasoline. Water. Water? Sure, water consists of oxygen and hydrogen - one a terrific sources of energy, the other necessary for releasing that energy: two perfect ingredients to replace gasoline. I mean, think about it! Oops. I did think about it. Sigh. Another one of my brilliant inventions down the drain. But you get the point. Not all obvious ideas work.
Batteries are not practical for a perfect power supply. Remember our discussion of the amplifier's valve turning up and down to let more or less of our power supply out to feed a loudspeaker? The valve moves with the music at its input. And if we have a perfect power supply and valve, then our amplifier's perfect. Unfortunately 'perfect' is not part of the engineer's lexicon.
Batteries have impedance. This means you can model a battery with a resistor in series with it. The resistor causes the voltage of the battery to fluctuate with the valve's actions, adding a funny type of distortion. Worse, the 'resistor' of a battery is odd; it is not fixed. It varies with many parameters including temperature, battery type, chemical properties, amount of charge, age, etc. In fact, we could say it depends on the battery's mood and not be too far off the mark.
Can we solve this problem? Sure! Like anything in engineering the problem's easy enough to solve. To make a battery have less of this funny distortion, you just go bigger. A lot bigger. Like the size of a car battery. That get's you pretty close to good if the battery remains charged, the load isn't great, it doesn't age and you don't mind a lead-acid-gas-leaking beast in your home.
There are better, more practical power supplies than batteries.
