To send an electrical signal, you need to have a complete circuit so that the electricity makes a full 'round trip'.
So imagine the output signal from an electric guitar. This connects inside the guitar to the tip of the jack plug. The signal goes all the way down the center conductor of the cable to the input of the amplifier. The amplifier takes what it needs from the signal and passes it to the screen of the cable going all the way back to the guitar. The circuit is completed in the guitar's pickup.
This is an unbalanced signal. In general, the screen of the cable is connected to earth. If there is no connection to earth, as in battery-operated equipment, then the metal case of the equipment takes on that role. If there is no metal case or chassis, then one point inside the equipment will take the role of earth and everything that needs earthing will be connected to that.
In an unbalanced signal, the earthed screen of the cable is there to protect the signal from interference. Any interference that gets into the screen is shorted out to earth, keeping the signal clean. However some interference may still get through and you will hear it, as it is inextricably bound into the signal.
Now suppose you have a cable with an extra conductor, exactly parallel to the signal conductor. Better still, have them twisted around each other so they are cuddling as close together as possible. This conductor will pick up interference in exactly the same way as the signal conductor. Now all you have to do is invert the interference picked up by this new conductor, add it to the original signal+interference, and the interference will magically cancel. That's what happens in a balanced connection.
It's worth noting that this only works properly when the output impedance of the sending equipment and the input impedance of the receiving equipment are equal for both conductors. It's the job of the equipment manufacturer to get that right.
It is also worth noting that the second conductor doesn't need to have a signal on it for this to work. It's handy however to put an inverted version of the signal on the second conductor as when this is inverted it will add to the signal thus reinforcing it. This indeed is what is normally done.
Balancing is a brilliantly simply way of guarding against interference.
To gain the advantages of balancing, then you must use equipment that has balanced outputs and inputs. You must use cables that have two conductors and a screen. You can balance an unbalanced signal by connecting it to a DI (direct injection) box.
In general you can get away with unbalanced connections in the studio where conditions are controlled and signal paths are short. Balanced connections however are a distinct advantage in live sound and outside broadcast where cable runs can be very much longer, and interference-producing lighting equipment is used.
In short, if you are not experiencing problems with interference, you don't need to worry about balanced connections. If you do have problems with interference, then balancing will be a sensible step towards a solution.
P.S. This description refers to electronically balanced connections. Balanced connections can also be made using transformers.
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