Hi

I would like to know can anyone explain to me how to obtain the loop bandwidth for all digital PLL(ADPLL). I can define this for the second transfer function to be 3dB bandwidth but when I transform everything to the Z domain and using ADPLL how I am going to obtain the loop bandwidth. How even loop bandwidth is defined for the all digital PLL ?

Your confusion confuses me. If you really know how to do this in the Laplace domain then doing in the z domain is a simple matter of variable substitution.

Make a Bode plot of the response from the z domain transfer function of your choice. Wherever a Laplace-domain plot would indicate the loop bandwidth for that transfer function, so will a z-domain plot.

E.g., for an open-loop gain of \(\frac{a}{s}\) in the Laplace domain, the loop closes at \(a\), where \(a\) is in radians/second. Similarly, for an open-loop gain of \(\frac{b}{z-1}\) in the z domain, with \(b \ll 1\), the loop closes at pretty close to \(b\), where \(b\) is in radians/sample.

I need a analytical expression for loop bandwidth vs other parameters like damping ratio and natural frequency like the 3dB bandwidth of continuous time system. Can I replace z with DTFT version and find frequency in which power is 1/2 of its maximum value. I need mathematical expression for loop bandwidth in digital domain

Z-transform variable z = exp(j*w), where j*j = -1.

w = 2*pi*f*Ts where f = analog-frequency and Ts = time-period of uniform-sampling.

So f = Fs corresponds to w = 2*pi and f = Fs/2 (Nyquist) corresponds to w = pi.

If you have expression for analog-PLL loop-BW, you can either numerically or analytically (if the expression is simple) solve for loop-BW in digital-PLL.

Thanks gmsk1 if I have loop bandwidth in terms of natural frequency can I represent Wn=2*pi*fn/Fs, that is the only place I am using frequency component. Also is there any other changes beside Wn in discrete and continuous?

Yes. You should be able to simulate continuous-time-PLL using Simulink and discrete-time-PLL using C-code and compare the two thus checking if your conversion is correct.