Any formula about settling time and loop-filter bandwidth?

As we can see, the wider the bandwidth,the shorter the settling time.
Now I want to know the exact formula revealing this relative!
Pls help me,thank you!

BW * T = 1

BW... bandwidth
T ... Tau

As I can remember it is called the "law of Kupfmüller"

ciao

eda4you: Thank you for your reply!

I'm puzzled by this formula.

If comparision frequency is 25MHz,and suppose the loop bandwidth is 500KHz,according to the equation BW*Tau = 1,
Tau should be 1/500,000=2uS.
But in PLL circuit, it is impossible to achieve so fast settling time.

How to explain it?

Long ago I was busy with this topic. I also didn't believe this law, but as I can remember: for a LTI (linear time independent system) it is true as I verified myself.
You may not mix up settling time and tau! In (almost?) every case tau is shorter than the settling time, depending on the order of the system and its parameters.

Further maybe the PLL can't be seen as a LTI system, due to nonlinearities.

However I am very interested about this topic. Let me hear if you get something new.

for CP PLL type 2 second order

for any second order if damping factor is .2
the 3db BW=2*Wn
Wn natural frequncy

and the sttling time
ts=1/(damping factor *Wn) *ln(frequencystep/(settling error*(1-damping^2)^.5)

this equation i used them to calculte the ts of my PLL

more details are here
www-mtl.mit.edu/~perrott/Distribution/ PllDesign/pll_manual.pdf

Najfi, this link cannot work!

khouly,
damping factor is .2 ? I think the PLL with this value does not have good performance.
The recommended value is 0.707.

eda4you,
What's the meaning of Tau?Pls explain it!