The effect of the differential delay can be found by derivation of the time dependent equations governing the system operation. Assuming the conditions appropriate for Equation 3.35 the IPLL system can be redrawn as Fig. 3.13 in which the delay paths are combined in a single delay = (r^ + - r^). The control voltage to the oscillator frequency tuning port is given by:
Vg(t) - Kisin($osc(C)) + [V^f(t)]*[sin($pp(t))] 3.36
where * denotes the convolution integral. The oscillator phase can be found by substitution into Equation 3.3.
lm[û(s)]
Re[G(s)]
diff “
Fig. 3.14 Nyquist diagram showing the failure of the injection phase locked loop to meet the stability criterion when the differential phase is equal to w.
G(s) - — — n + 7- cos(^diff^
CHAPTERS 53
(a) First order IPLL systems
Assuming lock, and for simplicity zero offset frequency from the free running frequency, then the first order time dependent equation derived from 3.36 and 3.3 reduces to:
sin($Q5ç(t)) - ^ sin(fl>pjj(t)) 3.37
The size of the phase errors at the two points of measurement is dependent on the relative magnitude of the two loop gains.
The dynamic response under these conditions can be derived by representing the phase errors as a time invariant part and a varying phase of small magnitude. Writing ^Qsc(t) = <>osc ^^(0 and $p^(t) - $p^ + A$(t) where A$(t) is small, it can be shown from 3.4:
- A$(t)[KiCOs($osc) + KpCOs($po)]
+ [Kisln($osc) + KpSln($pD)] + Wf - Wq
dt ... . . . . . . 3_33
The equation shows that the loop amplification is given by K^cos(*q5^) +
KpCOs(^pjj).
(b) Second order IPLL systems
In the case of the second order type II loop, the integration of the PLL reduces $p^ to zero and the differential phase resides entirely across the injection locked oscillator. The IPLL loop gain therefore is:
G(s) -
*
KjCosCddiff) 3 35As the differential phase increases, so the effective magnitude of the injection locking gain within the IPLL system decreases. For differential phases greater than +n/2 or less than -*/2 the injection locking gain subtracts from the PLL component gain. The loop is unstable if > K^K^F(s) at any frequency for which |G(s) I > 1. Instability in second order IPLL systems due to the differential phase is illustrated by Fig. 3.14.
33.4 Limits of PLL model for ii\jection locking
The assumptions used to derive the PLL model for an injection locking effect restrict the application of the IPLL analysis. The principal suppositions are in the basic assumptions of the Adler equation - small signal injection and linear reactance. Kurokawa [68] considers the injection locking of microwave oscillators. He uses, firstly, a quasi-static analysis to describe the principal injection locking features including stability and large signal injection effects. Secondly he uses a dynamic analysis to derive an Adler-type injection locking equation in a simple form, which also describes the effects of non-linear reactance of the active device negative impedance, in and out of lock operation and noise effects.
This section discusses the effects on the injection locking effect if the assumptions are relaxed. However the extension of the discussion to IPLL has not been attempted at this stage and requires further study.
(a) Non-linear reactance effects.
Adler’s equation neglects the variation of the RF signal amplitude and therefore the effect of non-linear device reactance which can be significant in microwave oscillators. Kurokawa [68] shows the effect of an amplitude dependent active device reactance, assuming small signal injection will modify the time dependent injection locking equation such that the injection locking range becomes:
îii + 1»] 3 40
W_ cos(^)
where
and where Z(A) = R(A) + jX(A) is the amplitude dependence of the active device in the circuit and -*/2 <\f> < n/2. Qualitatively, i/> is the slope of the device impedance locus. If ^ = 0 then the injection locking effect follows an Adler-type behaviour. As ^ increases so the injection locking range increases and the limits for the static phase error change from their + /- n/2 values. The stability of the locked condition can be derived from transient analysis for a small perturbation. For general small signal injection the condition for stability is cos(6^(t) + ^) > 0.
CHAPTER 3 55
(b) Large signal.
Under large signal injection, the oscillating current amplitude is perturbed from its free-running value, invalidating the dynamic analysis and therefore equation 3.17. Locking stability is no longer assured.
The condition for stability under large signal injection can be found by graphical methods based on a quasi-static analysis [68] Qualitatively large signal effects can be observed from the changes to the spectrum immediately following loss of lock. Under small signal injection, loss of lock is characterised by a break down from the locked spectrum to a comb of spectral lines crowded near the free running oscillation frequency and spaced at the difference frequency Aw. Under large signal injection the spectrum changes, on loss of lock, to groups of spectral lines of similar appearance to the small signal case, but repeated at set frequency intervals.
The locking range dependency on the locking gain deviates from the square root relationship and the phase error limits can exceed the ir range found in small signal conditions. Experimental results [68] show the onset of large signal operation at locking gains of greater than 18 dB.
3.3.5 Noise considerations
Since there are two input terminals for the signal there are two means with which noise can enter the loop. Under circumstances in which the noise accompanying the signal is the largest noise source in the system, there is correlation between the noise at the two inputs and this can lead to either constructive or destructive interference. The magnitude of the effect is dependent on the injection angle and the loop damping coefficient. Runge [70] shows phase noise suppression when the differential phase is within a 20° range centred at 18°. A maximum suppression of 4 dB is found when the IPLL damping coefficient equals 4. The interference effect equals zero for in-phase injection and the phase noise level is enhanced at differential phase angles outside this range.
Of more interest to the systems considered in Chapters 6 and 7 is the case in which the two input noise sources are uncorrelated. This will be the more likely condition in microwave IPLL systems in which the injection signal AM noise is negligible compared to the noise of the microwave oscillator. Therefore, the PM sidebands, due to the AM noise at the PLL input, will be the only noise source that requires consideration.
The effects of noise in a PLL are discussed in Appendix A. The input AM noise accompanying the signal is represented as an equivalent phase noise source at the phase detector output. A schematic for the IPLL with noise at the input to the PLL is shown in Fig. 3.15. In this case it is easy to show that the output phase is given by:
3. 42