Q: "What's the difference between a balanced and unbalanced signal?"
A: An unbalanced signal is one that is traveling through an audio path (such as a cable) that contains two conductors; a hot wire to transfer the actual signal and a shield to act as a ground. Instrument cables fall into this category which is why they are often referred to as unbalanced cables or TS (Tip, Sleeve) cables. An unbalanced signal is a perfectly serviceable method of connecting your guitar and amp, but you do have the possibility of running into noise or hum issues if you happen to be using a long cable, with the problem getting worse the longer the cable.
A balanced audio path on the other hand contains three conductors: a hot wire, a cold wire and shield. The extra cold wire is used to transfer an identical signal as the hot wire, only in reverse-polarity. Having two identical signals traveling next to each other in reverse polarity cancels out the frequencies where electromagnetic interference can reside. This also prevents ground issues in the signal and allows you to run much longer cables without problems. Noise reduction isn't the only benefit: compared to an unbalanced signal, a balanced signal also adds about 6 to 10 dB of extra headroom, meaning your audio equipment can sound louder with less gain. An XLR cable is an example of common balanced audio cable.
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Have you ever used a balanced cable with an unbalanced input? What, if anything, did you notice?
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