Bandwidth Frequency

As we mentioned before, we can relate bandwidth frequency to closed-loop natural frequency and damping ratio. Thus, we can relate bandwidth frequency to damping ratio and peak or settling time with the following equations.

## Settling Time

In order to make our life easier, we usually plot these equations instead of plugging in the values:

## Peak Time

Again, we usually look at the graph instead of plugging in the values.

## MATLAB Code

However, if you want an exact number given a damping ratio and peak or setting time, you can copy the function wbw into an m-file:
```
function[] = wbw()
DR = input('Damping Ratio?    ');
n = 0;
n = input('Settling Time(0) or Peak Time(1)?   ');

if n == 0
ts = input('Settling Time?   ');
temp1 = sqrt((4*DR^4) - (4*DR^2) +2);
temp2 =  1- (2*DR^2);
temp3 = 4/(ts*DR);
Wbw = temp3*sqrt(temp1 + temp2)
end
if n ==1
ts = input('Peak Time?   ');
temp1 = sqrt((4*DR^4) - (4*DR^2) +2);
temp2 =  1- (2*DR^2);
temp3 = ts*sqrt( 1- DR^2);
temp4 = pi/temp3;
Wbw = temp4*sqrt(temp1 + temp2)
end
```

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