Advanced Frequency Synthesis by Phase Lock

CHAPTER 8

OPERATION AT EXTREME BANDWIDTHS

The methods that we have been discussing offer the possibility of obtaining fine step sizes while maintaining wide loop bandwidth at the cost of some complexity. Before deciding to use them, we may try to achieve our design goals with a standard integer-N single-loop synthesizer. This could force us to widen the bandwidth to a point where the sampling process significantly affects our representation of the synthesizer loop. Here, we will extend the discussion of operation at these extreme bandwidths that we began in Chapter F.7.

8.1   DETERMINING THE EFFECTS OF SAMPLING

Control systems using sampled data are traditionally analyzed using z-transforms [Franklin et al., 1990], but we can also employ Laplace transforms, if we include additional terms. The open-loop transfer function then becomes

where f_s is the sampling frequency and G₀(f_m) is the transfer function in the absence of sampling.

In some ways, the use of Laplace transforms can be more convenient and permit a smoother transition from the usual analysis of continuous (not sampled) loops.⁴⁷ Whether we use z-transforms or Laplace transforms with added terms, the theory is based on the assumption of a fixed sampling rate. To that degree, the improved analysis may still involve an approximation, unless we have taken measures to make the sampling rate constant. Accurate simulation can ...