# Clock and Data Recovery/Structures and types of CDRs/The (slave) CDR based on a second order PLL
### Limits of the first order PLL

The PLL of the first order, type 1, in spite of its simplicity, has several good features and represents the best choice in many applications.
For instance, it is preferred to implement a phase aligner (Definition of phase aligner) because the increases of performance offered by loops of 2nd order cannot give any advantage in this particular application;
it may also be a good choice for a simple slave CDR when the incoming bit stream has a high transition density.
For other (slave) CDR applications, the flat gain block (evident in the figure of the slave loop of 1st order) can be modified and made evolve into a 1st order filter (making the loop become 2nd order).
It should be noted that the two other blocks of the first order loop (the phase comparator and the local clock) are necessarily working at the line pulse speed (that is the highest speed in a CDR).
This makes them more complex technologically and more expensive to modify when an improvement of the loop performances is required.
The added filter block works instead at lower frequencies, and its design is correspondingly more flexible.
The filter block is used to implement the desired values of ω$_n$ and ζ in the second order loops.
It is always a good design approach to implement a filter that be linear (at least in its practical behavior). The loop modeling and simulation remain possible in all conditions and are simpler than the modeling and simulation of a non-linear loop, whose models and simulations do not always exist.
The added filtering action allows to better fit specific requirements of other applications and to add some performances that the 1st order loop lacks:
* the input jitter high-frequency cut-off can be made sharper (from 6 dB/octave to 12 dB/octave), a performance important in regenerators (using the 2 - 1 loop)
* the steady state error resulting from a frequency difference f$_p$ - f$_{fr}$ can be compressed to zero, a performance important when G is widely variable and the VCO has poor centering (using the 2 - 2 loop)
* the jitter tolerance due to the end-of-range of the phase comparator (that in a 1st order type 1 slave PLL increases towards the lower frequencies with a slope of 6 dB/octave ) can be increased and present a steeper slope of 12 dB/octave (using a 2 - 2 loop )
* the low frequency noise generated by the VCO can be better rejected with a sharper low-frequency cut-off (from 6 dB/octave to 12 dB/octave), a performance important to use low cost, noisy VCOs (using the 2 - 2 loop).
It should be remarked however that the four points of possible improvement listed above are not relevant in the case of a phase aligner! This is in fact where the 1 - 1 loop finds its best fit.
Point by point in the list above, for a phase aligner:
* De-jittering the incoming clock is no use, because the recovered clock is not used after the data regeneration;
* f$_p$ exhibits some jitter with respect to f$_{fr}$, but cannot wander away from it. They both belong to the same clock domain;
* the jitter tolerance is limited horizontally by the half-depth of the phase adder, that cancels any improvement possible at low frequencies to the boundary of tolerance set by the phase comparator;
* the local clock is the reference, and is conceptually impossible to use f$_p$ as a reference to identify its low frequency noise!

### From 1st order to 2nd order
#### Addition of a filter between comparator and VCO

The first order, type 1 PLL is the simplest and corresponds to the simplest (and least expensive!) implementations of CDRs.
In many other applications (of slave CDRs) it is desirable to improve certain performances of the CDR itself, at the expenses of a little increase of complexity.
The "flat gain block" between phase comparator and VCO (clearly identified in the model of the slave 1st order type 1) acquires a first order structure, becomes "the loop filter" and makes the PLL loop a 2nd order loop.
One pole at frequency ω$_f$ and one zero at frequency ω$_z$ are added to this central block, whose transfer function becomes:
${\tfrac {G_{f}({\tfrac {s}{\omega _{z}}}+1)}{({\tfrac {s}{\omega _{f}}}+1)}}$
But not all possible choices of locations for the zero and the pole yield a good PLL for CDR. Some shall be excluded so that just two are left:
* The block can neither be a simple integrator 1/s nor a simple differentiator s. The PLL would oscillate!
* The pole shall always be at a lower frequency than the zero. If not, the rejection of the input jitter at high frequencies would be reduced to a finite flat attenuation. Worse, the filter block would amplify the frequencies higher than ω$_z$ with a 6 dB/octave slope: the PLL would become and behave like a type 0 loop. This is not acceptable for slave CDRs: the steady state error corresponding to a finite f$_p$ - f$_{fr}$ difference would be infinite!
* Therefore the pole at a lower frequency than the zero, and at least one of them at a finite (neither 0 nor ∞) frequency.
** They cannot be close to each other. The loop would be a 2nd order in name and cost but it would behave in practice not different and not better than a 1st order loop: a wasted effort!
** Either the pole at ω$_f$ and the zero at frequency ∞ (the loop in this case is a 2 - 1 loop), or the pole at frequency 0 and the zero at ω$_z$ (the loop in this case is a 2- 2 loop). Were they both at finite, though distant, frequencies, just one would be determinant for the loop behavior (the one closer to ω$_{n1}$ = G ).
The natural frequency ω$_{n1}$ of a first order loop is the same quantity as the open loop gain G. The natural frequency ω$_{nx}$ is a useful concept also in higher order PLLs. It represents in fact the corner frequency that the closed loop transfer function (= the jitter transfer function) would have if all frequency shaping was removed from the loop filter (that would be reduced, in such hypothesis, to just a flat gain G$_f$).
       In a 2 - 1 loop, the natural frequency ω$_{n21}$ cannot be made much lower than ω$_f$, because, as the next page shows, the loop would be underdamped:
       In a 2 - 2 loop, the natural frequency ω$_{n22}$ cannot be made much higher than ω$_z$, because, as the relevant page shows, the loop would be underdamped:
       If, for instance, the open loop gain G varies (because of manufacturing variability, or of sensitivity to temperature, supply voltage, .., or because of the non linearity of a circuit block), the "non filtered" frequency ω$_{n1}$ ( = G) may come close to the cut-off frequency of the filter ( = ω$_f$ or ω$_z$, depending on the loop type 1 or 2) that, in practical circuits, is much more stable than G.
          In a 2 - 1 loop this would happen if the gain G increased (or if ω$_f$ is pushed too much down to tighten the closed loop bandwidth more than ω$_{n1}$);
          In a 2 - 2 loop this would happen if the gain G decreased (or if ω$_z$ is pushed too much up to widen the closed loop bandwidth more than ω$_{n1}$);
       In both cases (increase of G in 2 - 1, decrease of G in 2 - 2), looking at the same dependence from another point of view, the damping ratio decreases:
       For both 2nd order loops, ζ should not go below 0.7 (although the type 2 is a little more robust, as the following pages will show).

#### Targeted to different applications, 1st and 2nd are very different in practice

When 1st and 2nd are contrasted (in conditions of same cut off frequency) their differences may not seem so large.
Those differences become more evident if the values of the ratio ω$_n$/ω$_p$, requested for different CDR applications, are considered.
1st becomes clearly preferred when a high ω$_n$/ω$_p$ (=a short acquisition) is specified, and 2nd when a small ω$_{n2}$/ω$_p$ (a continuous mode application) is specified. More precisely, in practice:
* 1st order loops are used when ω$_n$ needs to be just one or two decades lower than ω$_p$ (typical case of burst mode receivers), while
* 2nd order loops are used when ω$_{n2}$ must be three or more decades smaller than ω$_p$ (typical case of the continuous mode receivers).
Another important consideration exists, that addresses the noise generated inside the CDR and its deterioration of the output clock of the CDR.
The noise generated inside the PLL is rejected and does not affect the CDR output in the range of frequencies at which the PLL accepts the input signal as useful, but it propagates to the output if generated in the frequency range at which the PLL rejects the input noise ( i.e. the input jitter ) !
How the PLL reshapes the spectrum of the noise generated inside its own blocks is shown in a dedicated page: Clock and Data Recovery/Noise is shaped by the PLL structure. 2 - 2 is shown to perform better than 2 - 1 in this respect.
The 2nd order type 1 is mostly used in cases where ω$_{n2}$ << ω$_p$ and, at the same time, it is important that a "jitter clean" clock is regenerated.

#### Relative strengths and weaknesses of 2nd type 1 and 2nd type 2

Succintly, the 2 - 1 is best where the input/output requirements are tight and the circuit blocks have good characteristics.
It filters better than the other two loops the noise present in the input signal.
It requires a VCO with reduced noise at low frequencies and a phase comparator with linear characteristics. It is in fact less able to reject the low frequency noise of the VCO, and its damping ratio decreases when the loop gain increases.
It best fits the continuous-mode, regenerator applications in professional Telecom equipment.
The 2 - 2 instead is a best fit where some compromises can be accepted on jitter transfer and generation, but low cost and high integration correspond to circuit blocks with poor characteristics (= non-linear, noisy, ..).
It filters more sharply the low frequency noise of the (integrated ≈ noisy ) VCO ; its damping ratio decreases when the loop gain decreases.
Despite the intrinsic trade-offs of its model with respect to the 2 - 1 model, the 2-2 structure is a very frequent choice for monolythic ICs, especially when the line pulse frequency challenges the highest speed of the elementary components of the technology.
The phase detector is often a bang-bang PFD followed by a charge pump, and the VCO is often a LC monolithic oscillator.
It best fits the continuous-mode, end point applications in communication equipment.

#### In both cases, the 2nd order model is useful to describe the behaviour of complex CDRs

A linear model of the 2nd order, type 1 or type 2, is also useful because it "mimics" an existing complex CDR in a continuous mode application.
       In some cases, a generic linear model of the 2nd order is referred to. For instance:
       “a SEC will generally mimic the behavior of a 2-nd order linear analogue phase locked loop. This allows the use of the terms (equivalent) 3 dB bandwidth and (equivalent) damping factor, as they are used in analog PLL theory, irrespective of the fact that in the implementation of a SEC, digital and/or non-linear techniques may be used."
       ( SEC: a SDH equipment slave clock)
       In other cases, a complex function of clock recovery inside an equipment is studied using as a reference model a 2nd order type 2 linear CDR .

### External References

* ↑ Richard C. Walker (2003). "Designing Bang-Bang PLLs for Clock and Data Recovery in Serial Data Transmission Systems" (PDF). pp. 34-45, a chapter appearing in "Phase-Locking in High-Performance Systems - From Devices to Architectures", edited by Behzad Razavi, IEEE Press, 2003, ISBN 0-471-44727-7.
* ↑ ITU-T G.813 (03/2003) Timing characteristics of SDH equipment slave clocks (SEC), Appendix II: Considerations on bandwidth requirements, noise accumulation and payload wander accumulation, II.1 Introduction.
* ↑ ITU_T G.8251-2010 09 The control of jitter and wander within the optical transport network (OTN); Appendix IV 3R regenerator jitter accumulation analyses; IV.2 Model 1 "The regenerator is modelled as a second-order phase-locked loop (PLL) with first-order, proportional-plus-integral (PI) loop filter. ..." page 34; and: Appendix V Additional background on demapper (ODCp) phase error and demapper wideband jitter generation requirements; V.2 Demapper phase error "The demapper (i.e., desynchronizer) is modelled as a second order phase-locked loop (PLL) with 20 dB/decade roll-off. The model is the same as the 3R regenerator Model 1 described in clause IV.2 and illustrated in Figure IV.2-1; only the numerical values of the parameters are different. ...
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