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?
This is nearly a mathematical tautology, like saying that a positive number cannot go below zero. A one pole filter can introduce a phase shift which is at most ninety degrees.
Instability requires positive feedback at some multiple of 360 degrees with at least unity gain. But amplifiers use negative feedback. A two-pole amplifier is under threat, because it poles can potentially create up to 180 degrees of phase shift, which will combine with the 180 degree of phase inversion from negative feedback to create in-phase positive feedback.
In fact, instability under negative feedback begins at less than 180 degree of phase shift. As a rule of thumb, a safety margin (called phase margin) of at least 45 degrees is required, so 135 degrees of shift or less. A commonly used stability criterion is that at any frequency where there is unity gain or greater, the phase shift should be 135 degrees or less. If there is any frequency where this is not true, the amplifier may either self-oscillate, or at least exhibit overshoot and ringing: damped oscillations which are stimulated by input, and take several swings to die down.
One way in which amplifiers with multiple poles are stabilized is with the help of capacitors which create a "dominant pole" whose frequency roll-off is so great that the poles at higher frequencies basically do not matter (the gain is squashed at those frequencies). The amplifier basically "looks" like a single pole one.
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.
