Advanced Frequency Synthesis by Phase Lock

APPENDIX L

LOOP RESPONSE

L.1   PRIMARY LOOP

Unity gain is at 58 kHz and there is a filter zero at 16.8 kHz and two filter poles at about 310 kHz plus a steep, low-pass, elliptic filter with 1 MHz bandwidth.

L.1.1   Open-Loop Transfer Function

The forward gain is

where EF(5, 1, 120, 1 MHz) is the transfer function of a 1 MHz bandwidth fifth-order elliptic filter with 1 dB in-band ripple and 120 dB out-of-band attenuation. Figure L.1 shows the open-loop gain and phase of G_F/N, where we will be using N = 10 for the divide number.

FIGURE L.1   Open-loop gain and potential phase margin (i.e., 180° + phase). From MATLAB script opent58.m.^L1

We encounter a delay equal to half a cycle at 10 MHz in some of our circuits. This decreases the phase margin 1.0° at 58 kHz (180° × 58 × 10³/10⁷).

This loop has only 37° phase margin and 9 dB gain margin. While this has not been a problem for our experiments, it will cause overshoot in the time and frequency responses and could cause it to be subject to instability due to other parameter variations if it were used in a practical application. Without the elliptic filter, the phase margin would be a much more generous 53° and it would be 45° if the filter cutoff frequency were just doubled to 2 MHz. The existing, lower, corner frequency and the resulting ...