Single Pole Amplifier

My text book says, "Single Pole amplifiers are always stable,hardly surprising, because in the worst case it can never go beyond 90 degrees."

But it did surprise me and I cannot figure out why a single pole amplifier cannot go beyond 90 degrees. What is the reason?

Answer

I will address your direct question of why a single pole amplifier never goes beyond 90°, and not why this fact makes an amplifier stable, which has already been covered in another answer.

A single pole H(jw) generally means something like: $$ \frac{1}{jw+p}$$ A bode plot for magnitude and phase can be made and tells you what the filter would do to each of the frequency components of the input.

The above equation simply evaluates to a complex number for each frequency value w. Plotted in the complex plane, H(jw) will have a real component (x-axis) and an imaginary component (y-axis). The angle of this phasor with respect to the real axis determines the phase shift of the input signal at the frequency w at which this complex number was calculated.

If you choose any p>0, and evaluate the single pole H(jw) equation above for w going from zero to infinity, you'll get:

H(w=0)    = 1/p      (angle is   0°) 
H(w=p)    = 1/(jp+p) (angle is -45°)
H(w=inf.) = 1/j*inf. (angle is -90°)

So a single pole H(jw) simply never evaluates to a complex number that shifts any input frequency by more than 90 degrees.