Is this a linear regulator? Or switching regulator?
For a linear regulator:
1. the regulator can probably handle it just fine. It's the electronics on the output side of the regulator or the input side of the regulator that don't like the noise being generated by the other side. (Read part 2)
2. Regulators of all kinds (linear and switching regulators and things like op-amps too) have trouble filtering out high frequencies (it has to do with their gain-bandwidth product which is commonly seen to specify op-amps). So any high frequency noise or transients (due to source voltage dips/changes or current load changes) generated on either side of the regulator (output or input) can basically travel through the regulator with relatively little supression. THis means that noisy devices on either input or output side of the regulator can contaminate each other (it's a two way street). Additional filtering like inductors is needed if this lack of high frequency noise isolation between the two sides of the regulator becomes a problem.
Like if you have a motor on the input side of the regulator, it might send high frequency noise through the regulator and disturb your electronics. Or if you have two regulators in parallel powering analog and digital circuitry, the digital noise might pass through both regulators (output->input of the digital regulator, and then input->output of the other regulator) and contaminate the analog power supply. This happens with both linear and switching regulators, though switching regulators tend to be worse since they themselves introduce switching transients as a result of their operation.
FOr switching regulators:
1. A switching regulator's output is not constant DC at all and you need a regulator to smooth the current (and capacitor to smooth the voltage) to produce an output with lower noise, otherwise your load will misbehave due to a noisy power supply. Without it, you get a squarish waveform for current which is very noisy.
2. For a step-down switching regulator the inductor is needed to slow down the rate of current rise to a speed where the regulator's control loop can react fast enough to keep it under control. WIthout it, the current will basically short-circuit every time the converter switch closes producing excess current and overheating, as well as causing the output voltage to be equal to the input voltage (not a good thing for a step-down regulator since it will probably fry the load due to overvoltage).
Personally, if this was a linear regulator I would put the inductor on the input rather than the output. Inductors have lots of resistance, and placing it between the regulator and load introduces a resistive voltage drop that the regulator cannot see and therefore cannot compensate for. Putting it on the input eliminates this problem- you just have to make sure that this same current-iduced voltage drop doesn't cause the regulator input voltage to go into dropout. WIth switching regulators, you don't have a choice where the inductors go (though you can add more inductors to input and output to increase filtering).
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