Phase locked loops (PLLs) have been used for years to stabilize signal sources such as oscillators. In the past, loop bandwidths tended to be small compared to the sampling frequency, but with modern communications systems, requirements for faster switching times mean that this is no longer the case. Narrow-bandwidth PLLs can be effectively modeled and simulated by means of linear analysis, but these same approaches fall short for wide-bandwidth-sampled PLLs. In a sampled PLL, when the sampling frequency is large compared to the loop bandwidth, a linear simulation provides a fairly close approximation of the PLL’s behavior. But when the loop bandwidth is a considerable percentage of the sampling frequency, as in fast-switching frequency synthesizers, linear analysis may not provide accurate predictions. This opening installment of a three-part article will explore a nonlinear approach to the analysis of the effects of sampling on PLL performance.

Figure 1 shows the essential components of a typical PLL in block-diagram form, with the reference phase, fr(s), given by Eq. 1:

There is an argument that the output phase, f_?(s)/? and reference phase, f_r(s), should already be sampled by the action of the PLL’s dividers, However, this makes little difference since both sources of phase noise are synchronously resampled by the sampler that leaves the result unchanged, as shown by the equality in Eq. 2:

The sampled version of fe is described by Eqs. 3-5,

with the additional descriptions given in Eqs. 6, 7, and 8:

With a little rearrangement, it is possible to solve for the output phase as a function of the input noise

(see equation 9)
(see equation 10)
(see equation 11)
(see equation 12)
(see equation 13)

Putting f’_o(s) into the expression for f_o(s) results in Eq. 14. Note that throughout this article series, the “prime” symbol, such as in f’_o(s), is used to denote a sampled signal.

At this point, this analysis will proceed to separate out the noise at the output due to reference noise and VCO noise. The first step is to find the noise due to the reference by means of Eq. 15:

(see equation 15)

where

f_or(s) = the output phase due to the reference, and can be found by Eqs. 16 and 17.

This is effectively the sampled reference noise modified by the transfer response of the loop.

Now, consider the contribution to output phase noise, f_o, due to VCO noise, N_o(s)

It is now necessary to develop the mathematical functions for performing the phase-noise analysis. Let the impulse sampled f(t), f ‘(t) be defined as in Eqs. 20 and

where T = 1/F_s.

This may be represented by the Fourier series

where

Now, call F’(s) the Laplace transform of the sampled time function, f’(t), where

F'(s) = L(f'(t))

Then, apply this to the reference noise:

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