A short scope ground prevents a nasty resonance.
Imagine that you have a self-contained, battery-powered, 3.3-V CMOS oscillator. This oscillator floats magically in space, touching nothing. The external case is plastic. No grounds are exposed. It has one output terminal.
If you connect the output terminal to your oscilloscope, what signal, if any, do you suppose you will see?
My colleague Bill Ritenour gave me the answer to this question several years ago. His analysis crystallized my thinking about probe grounding. Figure 1 schematically represents the major components of the measurement system. While looking at Figure 1, keep in mind that the current must always form a loop. It cannot exit the oscillator without having a path through which it may return. Even though it may initially seem as though the oscillator has no return path, because its internal ground node is not exposed through the plastic case, a return path nevertheless exists. The parasitic capacitance, C1, forms the path between the internal circuits of the oscillator and the nearby earth. Even if you conduct the experiment in space, there would still exist a parasitic capacitance directly between the oscillator and the scope.
In operation, current from the oscillator passes to the scope through the output wire. The same current then passes through the input impedance, Zin, of the vertical amplifier to the scope chassis. From this point, it passes through the scope's green-wire connection to the earth. From the earth, it passes in the form of displacement current, changing electric flux, through C1 and back to the oscillator. Along the way, the current produces some magnetic flux in the purple-shaded region of Figure 1 that acts like an inductor, L1, connected in series with C1 and Zin.
The impedances represented by C1 and L1 affect the magnitude of the current flowing through Zin, which affects what you see on the scope. Only when the impedances of L1 and C1 remain far less than Zin can you make accurate measurements.
You can replace the scope with an active FET probe and check to see over what range of frequencies, if any, impedances L1 and C1 remain below Zin. Assume you have a 1-pF probe near a digital circuit, and the probe has no ground connection to your system. The probe cable simply drapes over the circuit board. This arrangement creates a parasitic capacitive coupling between the probe shield and the pc-board ground of about C1=10 pF, in series with an effective loop inductance of L1=500 nH. The probe, Zin, equals 1 pF in parallel with 105 Ω. Over an operating range of 100 kHz to 100 MHz, which is a wide enough band to oftentimes see some kind of reasonable result, this crude arrangement initially appears to work. The impedances of C1 and L1 remain below the input impedance of the probe over the entire band but not by much, which means that the position of the probe cable will have an effect on the shape of the signals you see on the scope. Try it.
Another nasty artifact of a no-ground probe arrangement is the resonance associated with the combination of the rather large inductance, L1, and the 1-pF input capacitance of the probe. This resonance is called a probe resonance. It happens at frequencies high enough that C1 is essentially shorted out; the input impedance of the scope looks essentially capacitive, so the circuit reduces to a series combination of the oscillator, the scope capacitance, and the loop inductance. This series-resonant circuit exhibits a severe resonance at a frequency equal to 1/(2π(L1CPROBE)½). With the numbers assumed in this example, the probe resonance happens at 225 MHz. If your oscillator, or its harmonics, happen to hit that particular frequency, the response goes nuts.
A short, explicit ground connection made between the scope ground and the equipment under test shunts around both C1 and L1, eliminating their influence on the measured result and pushing the probe resonance up and out of the band of interest. All good probes come with short, tiny ground attachments to prevent such problems. For single-ended measurements, don't depend on mysterious ground connections. Always use a good, short ground connection.