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I'm using an oscilloscope and observing the voltage output of an amplifier.

First case: Osc. is not grounded circuit is not grounded and I see a very distorted signal.

Second case: I ground the negative output of the circuit(which directly goes to the osc. as well) to the earth and the output signal is neat.

My question is, even though it is a well known phenomena, why does grounding to the earth only eliminates the noise and does not affect the original signal? Why does 'only' noise currents flow to the earth?

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    \$\begingroup\$ So to be clear, you are looking the the noise on your 'scope? Did you know that your ground on your probe is referenced to earth (AKA safety). To measure a unreferenced signal you use a differential probe or use two inputs and use the A-B function of the 'scope. \$\endgroup\$ – placeholder Dec 18 '12 at 20:08
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Its not the fact that its the grounding to earth that causes the clean signal... its the fact that your references are now the same.

Voltage is just a reference between two points, what you are experiancing in the first case is called a floating ground. My connecting the ground of the device you are measuring to the ground of the probe you are in essence telling the two devices (your OSC and the unit your testing) that this is what you define as "0V".

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It is not simply the case that the main line in the oscilloscope probe carries a signal, and that the ground line is just something to catch noise.

Both lines carry the signal.

We need two wires out of a device, because inside that device, there are two points between which a voltage is developed which is understood as some signal.

To bring that signal to another device, we have to convey both points, and so we need two conductors.

We call one of these lines "ground" or "return" or "common". But electrons do not know that; they do not read schematics. Electrons do not know that this line is intended to carry signal, and that other one catches noise. It doesn't work that way.

If you sever the ground, the whole arrangement of the oscilloscope and the DUT will still exhibit some kind of "signal". The scope continues to measure the potential difference between the probe tip and the ground clip. However the ground clip is no longer the correct voltage point against which the signal is properly defined.

Any two points in space have some potential difference between them (possibly zero), regardless of whether current can easily flow between them.

The oscilloscope scope is a high impedance input which requires very little current to flow. It measures pure voltage, or almost. Without a ground in place, the scope then still measures the potential between two points in space: the disconnected ground and the point in the DUT being probed. The potential between those two points includes stuff other than the intended signal, like noise induced by stray electric fields.

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