Phase Margin Criterion
Phase margin of 45 degrees is a somewhat common compromise between good step response and stability and a step response which overshoots and has ringing. Makers of OpAmps like to push gain bandwidth of their products and that reduces phase margin (PM). Here is a plot of closed loop amplifier peaking used to show the approximate amount of overshoot that can be expected in a unity gain OpAmp when exposed to a input step voltage. 



You can see that with PM of 60 degrees there is no overshoot, while with PM of 45 degrees overshoot is about 1.3 times the step amplitude. Things get rapidly worse from there as PM declines. 

Edit: For those who would like background about the above chart, and more information relating \$Q\$, Overshoot (os), Peaking (m) to phase margin. 



Cocktail Napkin (CN) Approach to Peaking with Unity Feedback 

To generate the above chart, a cocktail napkin style approach was used to derive an expression of peaking in a unity feedback closed loop amplifier. Start with the most simple form of feedback equation: 

\$\frac{V_o}{V_{\text{in}}}\$ = \$\frac{G(\text{j$\omega $})}{1+G(\text{j$\omega $})}\$ 

and for magnitude: 

\$\left|\frac{V_o}{V_{\text{in}}}\right|\$ = \$m\$ = \$\frac{|G(\text{j$\omega $})|}{|G(\text{j$\omega $})+1|}\$ 

where \$G(\text{j$\omega $})\$ is open loop amplifier gain. It's not necessary to know any particulars about the open loop gain, except that it crosses unity at some frequency \$\omega_c\$ with some phase margin (PM). 

Rewriting in phasor form and at the crossover frequency where G = 1: 

\$\text{mCN}\$ = \$\frac{1}{\left|1+e^{-j (180-\text{PM})}\right|}\$ 

for PM = \$45^{\circ }\$; \$\text{mCN}\$ = \$\frac{1}{|0.29-0.707 j|}\$ = \$1.3\$ 

It may seem silly to think that such a simple approach could calculate peaking response of a closed loop amplifier, but look: If PM = 0 then mCN = \$\infty \$ -- which matches the Barkhausen criteria, and if PM = 90 then mCN = 0.707 -- which is just what a unity gain amplifier with that open loop phase margin would do. 

Advantages here are: 


Really simple development --- Cocktail Napkin! 
Directly relates phase margin to peaking.
For peaking it's surprisingly accurate, as you'll see in the next section. 
For Overshoot in time space it's an OK approximation. Accuracy of +5% to -10% when pm is between \$45^{\circ }\$ and \$60^{\circ }\$, which is where most of the work gets done. 


Of course, such a simple model doesn't comprehend anything about the actual crossover frequency. While it's accurate for higher Q and PM less than \$50^{\circ }\$, becomes less accurate for lower Q. As Q drops, so does frequency of the peak, moving away from the crossover point. This method doesn't adjust and so misses the actual peak for lower Q. 

Doing better than this requires a better model of the open loop amplifier, meaning an estimate of open loop amplifier gain and pole locations. 

Doing Better 

The process starts in a similar fashion to the cocktail napkin approach, with the feedback equation. But an amplifier model that includes DC gain and low frequency and high frequency poles is used to obtain a second order system stated in Laplace form. The Laplace form can then be transformed into a time domain result for overshoot calculation. 

Any worthwhile control or stability text, like "Linear Control System Analysis and Design" by D'azzo and Houpis (DH) for example, will develop these equations to calculate peaking (frequency space) and overshoot (time space) for a two pole system. I will not derive them here, but will state them (from DH) and then add a piece that they are missing to make them useful. 

Peaking: 

\$\text{mDH}\$ = \$\frac{2 Q}{\sqrt{4-\frac{1}{Q^2}}}\$ 

Note this equation is singular for \$Q\$ of 0.5. The useful domain of the equation is \$Q &gt; 0.707 \$. 

Overshoot: 

\$\text{osDH}\$ = \$1 + e^{-\frac{\pi }{ \sqrt{4 Q^2-1}}}\$ 

Phase Margin and \$Q\$ 

To make these two equations for peaking and overshoot useful in common practice, an equation relating phase margin and \$Q\$ is needed. The most concise form of this that I've seen is from work by Christophe Basso -- "Transient Response Counts When Choosing Phase Margin", which culminates with Eq. 22: 

\$Q\$ = \$\sqrt{\cos (\text{PM})} \csc (\text{PM})\$ 

A chart of \$\text{mCN}\$, \$\text{mDH}\$, \$\text{osDH}\$, and \$Q\$ is shown below for comparison between the CN approach and the fully formal analytical approach of DH. 



mCN and mDH converge for PM values less than ~ \$55^{\circ }\$. As expected, mDH is more accurate above \$55^{\circ }\$ PM, but doesn't really exist above ~\$65.5^{\circ }\$. For very low values of PM, \$Q\$ mCN and mDH converge, because the peaking frequency converges with the resonant frequency.