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## Homework Statement

In my notes it is stated that an integrator adds a phase lag of -Pi/2 and thus can cause instability. I want to understand what this really means and am deviating from the notes somewhat so do not know if I am barking up the wrong tree.

## Homework Equations

Given a system with a transfer function G(s) input U(s), the output X(s)=G(s)U(s)

## The Attempt at a Solution

To understand this I have been investigating the general response of a system to a harmonic input. I performed calculations based on G(s) = 1/s. I first chose a cosine input u(t) = cos(wt) and then repeated the calculations with a second input u(t) = sin(wt) and looked at the steady state response.

For the cosine input, the output at steady state was a cosine output with altered magnitude and phase, giving me the phase lag of 90 I was looking for. From thinking about it, I understand that if the feedback path to a summation for an error is 180 out of phase, it will effectively switch the sign of the summation and therefor create a runaway condition, so adding phase lags of -90 could be dangerous ... if that makes sense!?

However, for the sine input, the output at steady state was a sine output with altered magnitude and phase, again giving me the phase lag of 90 I was looking for except this time there was an added omega, which I interpret as a dc offset, which contradicts something later in my notes.

I have been through it many times, and keep getting the DC offset, is this correct?

With both of these, I came to the general conclusion that for any given G(s) the steady state response to a harmonic input will be a waveform of equal frequency yet altered magnitude and phase + sX(s) evaluated at s=0. Is this also correct or do I need to look into more inputs?

Thanks!