Ok, let's recall what Peter wrote:


Let's discuss, whether a battery can add noise to the signal:

A battery's equivalent circuit consists of an EMF cell and a source impedance. This source impedance generates thermal noise according to Un = SQRT(4 x k x T x R x B), where k = 1.38 x 10^-23J/K is the Boltzmann constant, T the absolute temperature in Kelvin, R the source impedance and B the bandwidth.

Assuming a source impedance of 10Ohm yields a noise of Un = SQRT(4 x 1.38 x 10^-23 x 300 x 10 x 20000)V = 58nVeff.

An old battery shows a much higher source impedance. Assume it's 100Ohm. Then, the noise is 182nVeff.

Remember, this is only the noise on the supply voltage, not in the signal path. Good design practice assumed, at least 40dB PSRR (power supply rejection ratio) can be expected. So, damped by 40dB this noise results in a SNR (signal to noise ratio) refered to a 10mV signal of guitar of 145dB for the fresh battery and 135dB for an old battery.

A single coil pick-up shows a source impedance of about 6kOhm! This results in a noise of 1.4µVeff and a SNR of 77dB refering to a 10mV signal of guitar. So, we can see finally, that the additional noise of battery is so much less than the noise of guitar, that it's contribution to the noise floor is absolutely undetectable! Even if there was not any useable PSRR of the circuit, which is highly unprobable, then the noise contribution of battery was negligible.

On the other hand, there are indeed situations with guitar sound effect devices, where you can clearly hear a higher noise floor with an old battery! This can happen, when a circuit is faulty designed, means when the circuit totally runs out of the idle point, when the battery voltage falls under a certain level. This can be the case for instance, when an input stage sees a constant current source as load and this begins to turn-off due to the under voltage.

But in all these cases, this has nothing to do with the increase of thermal noise of old battery, but with effects resulting from a decrease of battery voltage!

A guitar player having experienced such a situation can indeed have the feeling that the one battery is noisier than the other. But this has nothing to do with the type of battery, but only that he more heavily discharged the "noisier" battery once in the past.

That the wirless set shows different noise with equally fresh batteries but from different manufacturers is nonsense, of course! Unless they show drastically different supply voltages and the wireless set is a faulty design...

To the hum:
As pick-ups of guitars are working magnetically, they are intrinsically a universe higher susceptible against hum than any loops formed by the wiring inside the guitar, even if the battery would form a special unproper loop, somehow. So, stating that the one battery causes more hum than the other would be entirely absurd!

Kai