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 2^{nd} order and type 2 has become the preferred architecture for high performance CDR (regenerators) as soon as the ever advancing IC technology has allowed to integrate all CDR blocks inside one silicon chip.
 All blocks can be integrated ( high gain and low noise amplifiers, A/Ds, D/As and DSPs ) with very low manufacturing costs.
 Even the VCO is a block that is integrated, but it can not reach the accuracies possible with a crystal oscillator. This is compensated by the more sophisticated architecture and some additional circuitry (PFD).

This architecture uses the option of a zero (the gain increases at lower frequencies) in the filter stage between phase comparator and VCO.

 The zero is at angular frequency ω_{z} =1/τ_{z}, ω_{w} = 1/τ_{w} is the frequency of gain = 1 = 0 dB . τ_{z} = G_{f} τ_{w} .
 Lower frequencies are amplified more (20 dB / dec) and the lowest frequency the d.c. jitter component gets a theoretical infinite amplification.
 The decrease of the amplification continues up to ω_{z} and becomes flat (G_{f}) at higher frequencies.
 In practical applications of type 2 CDRs: G_{f} << 1 (often referred to as "the filter attenuation" and called β) and ω_{w} << ω_{z} .
The maximum output of the comparator corresponds to much less than the maximum frequency deviation of the VCO, deviation that can be reached gradually thanks to the integration of low frequencies made the loop filter.
The response to an input variation always ends with catching up completely (zero steadystate error in this type 2 loop), even if (if the loop gain is much less than "infinite" or the "latency" is relevant ^{[1]} ) every CDR design is tailored to specific applications to avoid that the response overshoots significantly.

 It is like using either the accelerator or the brake, increasing pressure indefinitely on the pedal until the response of the system becomes sufficient.
 It is easy to understand how this architecture reacts immediately but only to a limited extent, and then increases the correction slowly but progressively if the immediate response has not proven sufficient.This is especially evident during the acquisition phase.
The low gain at high frequencies attenuates more the high frequency input jitter (in many cases it also attenuates the bangbang tracking jitter) while the integration of the input phase error (assisted by a PFD) provides for acquisition in a much wider range of input frequencies.
Where used and how madeEdit
The 22 architecture is well suited for monolithic implementations, which explains its widespread use.
This architecture finds its practical applications where a bangbang phase detector must be used and where the E_{s} must be reduced to a minimum (possibly zero).

 The strong reduction of the lowfrequency noise from the VCO is a further bonus.
A bangbang phase detector is inevitable in monolithic applications at very high line frequencies ^{[1]}, and monolithic implies also an onchip oscillator, that is relatively noisy and relatively inaccurate in ω_{fr}.
On the other hand, the high volume applications of today are correspondingly very cost sensitive, and require that a single silicon chip accommodate the whole CDR, and also much more.
In practice a bangbang phase detector is always used, and the VCO is a linear monolithic one, either a ring oscillator or a lowQ LC.
Consequently all the monolithic:
 line regenerators,
 slave clocks in Telecom networks
 CDRs of portable electronics
are made with this 2  2 architecture.
A long runlength can affect the tolerance margin in sampling the received signal, but this is normally mitigated by the use of a ternary phase detector.

 This loop type is the most “powerful” of the three, because it incorporates a loop filter that offers a very high gain at low frequencies.
 This is the characteristic feature of this architecture, that generates its strong and weak points.
 The high gain at low frequencies allows the compression to very low value of any steady state error, unlike the other two loops ( 1  1 and 2  1 ). The filter gain is maximum at very low frequencies, and decreases up to ω_{z}, then it flattens to the asymptotic value and stays constant in the bandwidth of interest (until parasitic poles, always present, make it drop). G_{f} is very low, in all cases below 1, so that it is often referred to as "the filter attenuation".
 The flat gain of the loop filter at high frequencies allows a good tracking of medium frequency jitter, unlike the other second order loop ( 2  1 ).
 The very low gain of the loop filter at mid and high frequencies keeps the tracking jitter (i.e. the jitter generated by the bangbang frequency jumps) adequately low.
 The high gain at low frequencies allows the frequency and phase acquisitions even when the VCO freerunning frequency is shifted much more than ω_{z} from the frequency of the signal to recover.
This 2  2 loop tracks very well, as long as the loop filter and the VCO operate within their range of normal operation.
Care must be paid to the very high (closed loop) gain at low jitter frequencies when either:
 the lack of data transitions temporarily opens the feedback path (this is mitigated by the use of a ternary PD, but presents anyway the risk of a phase error that may increases out of control as long as the loop is open);
 the (sinusoidal) input jitter has a significant amplitude. The filter gain at high frequencies is kept low to limit the jitter generation resulting from the bangbang, but this limits the ability of the loop to track a fast and large swing of the input phase. The jitter tolerance depends on the corner frequency of the loop filter and on the length of runs without transitions.

 Closed loop gain and corner frequency of the tolerance curve are concepts easily identified if the loop is linear.
 When the loop incorporates a bangbang detector, and the data transitions are random, it is necessary to restrict the study only to conditions of practical interest and to use simulations.
Introductory exampleEdit
This simulation diagram helps understand the operation of the 2 – 2 architecture with bangbang phase and frequency detector.

 The two waveforms of the two phase detectors of the PFD are shifted higher in the diagram, for easier interpretation. The other waveforms are not shifted.
 The outputs of the PFD and of the loop filter are scaled differently, with the filter output more amplified in the representation. (The filter output does not reach the clamping level(s), and the VCO drive signal is not different from it, i.e. would be exactly overlapped to it in the diagram. The high frequency parasitic pole of the charge pump is at 200 Mrad/sec, and its effect is barely visible in the shape of the filter bang bang spikes).

 To the left the waveforms show that there is no incoming signal. LOS is asserted and the loop is open. The VCO (=the CDR) output simply drifts away with a slope proportional to the difference between the signal frequency and the VCO freerunning frequency.
 After 1.28 μs (= 150 simulation steps) the signal appears with a phase mismatch equal to 4.0 radian, which happens to reduce the cumulated phase drift that had reached 10.06 rad.
 LOS is disasserted at that moment and the loop starts catching up.
 Initially the output of the PFD is a constant positive level (meaning the VCO is slower than the incoming signal. The filter, that integrates the low frequencies of the comparator output, adds a further positive ramp.
 The output phase undershoots the input phase before lock, with even a small overshoot when the filter makes the first negative bang and starts a negative ramp to finally reach a stable continuous bangbang condition.
 During the frequency acquisition three slips have createdted a gap of exactly 3 π between input and output. The gap remains constant from then on, apart from a small additional phase error.
 As soon as the loop has caught up with the input, the typical pattern of bangbang starts: the loop is in lock. The comparator bangs rapidly between is two output states and the filter output maintains a d.c. bias that compensate the ω_{p}  ω_{fr} distance.
 After 8.53 μs (= 1000 simulation steps) the input signal phase starts a sinusoidal jittering with a large amplitude (3.20 rad) and a frequency of 0.9 rad/μs.
 This jitter brings the loop close to its tolerance limit, which is shown by the detector and by the filter outputs that is alternatively forced out of bang bang, as well as by the error signal that shows deviations from its average of 3π4 rad in correspondence with those periods of difficult tracking. The error signal deviations are not large in this case, and remain within +0.25 and 0.26 rad.
In this example the transition density is 100%, but the random nature of the input signal must also be taken into account.
Single zero filterEdit
This loop type is the most “powerful” of the three, because it incorporates a loop filter that offers a very high gain at low frequencies, that is the key feature of the 2  2 architecture and that generates its strong and weak points.

 The filter gain is maximum at very low frequencies, and decreases up to ω_{z}, then it flattens to the value G_{f} and stays constant in the bandwidth of interest (until parasitic poles, always present, make it drop).
 The high gain at low frequencies allows the frequency and phase acquisitions (a ternary PFD is used in these applications) even when the VCO freerunning frequency ω_{fr} is shifted much more than ω_{z} away from the frequency ω_{p} of the signal to recover.
 The high gain at low frequencies allows the compression to very low value of any steady state error, unlike the other two loops ( 1  1 and 2  1 ).
The (almost) infinite gain at low frequencies gives this architecture some very useful properties, but it can also be the origin of unexpected troubles.
As shown in the figure above, G_{f} is very low, in all cases below 1, so that it is often referred to as "the filter attenuation" and called β. As a consequence, ω_{z} is higher than ω_{w}, the zerogain frequency.
 In the 1  1 architecture the output of a bangbang binary PD always makes the VCO jump from one end of its control range to the opposite end. This leaves a strong residual "tracking" jitter in the output phase of the PLL, because only the 1/s slope of the VCO characteristic does filter the sharp and large swings.
 The 2  2 architecture in principle behaves the same for high frequency jitter as its loop filter does not filter out the high frequency components (higher than ω_{z}) coming from the comparator output. The filter passes the high frequency components of the jitter to the VCO with a flat transfer function. But the value of the filter gain above ω_{z} ( although flat up to where parasitic poles at even higher frequencies make themselves felt ) is much smaller in a 2  2 architecture than the equivalent gain in a 1  1 architecture and that makes the tracking jitter proportionally smaller.
 The use of a ternary phase detector further reduces the peak tracking jitter ^{[2]}.
This 2  2 loop tracks very well, as long as the loop filter and the VCO operate within their range of normal operation.
Loosetracking conditionsEdit

 When the phase detector outputs a constant request for higher, of for lower, frequency for a significant number of clock cycles ,

 a temporary lack of bangbang around the locking condition occurs, while the VCO lags behind a rising or falling input phase .
As this interval grows, the tracking error increases (see the linear ramp of the step response of the filter), and this might result into a phase error beyond the tolerance limit.

 This 2  2 loop may drift away from lock more than the other type 1 loops, if the phase information is not refreshed, as indicated also by its linear model.
This may occur for two different causes:
 the input signal has too few transitions (runlength problem, that can be approximately described using the stability factor ξ) or
 the phase of the input signal varies too fast for the loop to track (slewrate problem, that is investigated by simulations with sinusoidal input jitter close to the tolerance limit).
Both causes can reduce the tolerance of the CDR, and the effects of each can be reduced at the expenses of other CDR performances.
Runlength problemEdit
The actual statistic of the transitions in the input signal is so difficult to predict that only a simple worstcase approach can be of use.
The very extreme case when transitions come periodically and separated by a constant number of linepulse periods may give some insight and explain why large design margins are found in real monolithic CDRs.
In a condition of very rare but periodic transitions, the fundamental parameter is the time that the loop waits before the next update of the input phase, update that comes from the next transition of the input signal.

 When a transition comes as soon as possible, the update time is: t_{update} = 1/f_{p}.
 When a bit of the same sign follows in the input signal, t_{update} at least doubles. More precisely:
It is convenient to define a parameter ξ:
The longer the run length, the more critical the loop response may become . In fact, ξ is called the stability factor^{[3]}.
Let's focus now on what happens in the condition of bangbang tracking.

 The last bang corrects the filter output, higher or lower, by a jump of G_{φ}G_{f} [v], and a ramp follows, in the same direction, with slope G_{φ}G_{f} ω_{z} [volt/sec]. (See the figure above).

 The VCO frequency jumps G_{φ}G_{f}G_{VCO} = G [rad/sec] and then ramps with a slope of G_{φ}G_{f}G_{VCO}ω_{z} = Gω_{z} [rad/sec^{2}]

 (G_{φ}G_{f}G_{VCO} = G, where the phase detector outputs can be Gφ, +Gφ if the detector is binary and Gφ, 0, +Gφ if it is ternary)
 The quantity G is very closely related to the quantity f_{bb}^{[4]}. Both measure the frequency jump during bangbang tracking, and are related by the formula G = 2π f_{bb}.
 While f_{bb} is easy to interpret as the frequency jump and is fixed by the circuitry of the CDR, G is used as a nominal value for the open loop gain, in the nominal conditions of D_{T} = 1 and maximum phase error (i.e. minimum G_{φ}). In fact, G is always found multiplied by D_{T} in the formulae that describe the loop behaviour.
The VCO phase, as a function of the time t, is the sum of a linear ramp Gt plus a parabolic ramp Gω_{z}t^{2} .

 After t_{update}, the VCO phase has increased (or decreased) by a linear part G t_{update} [v] plus a parabolic part G ω_{z} t_{update}^{2} [rad].



 the ratio of the linear part of the phase increase to the parabolic part of the phase increase is exactly the stability factor ξ:



 When the update takes place, the new bang makes the filter output jump in the opposite direction by a step of G_{f} [v], followed by another ramp, now in this new direction.
 In order to stay in tracking and not to sidetrack out of lock, the phase drift during t_{update} must not make the VCO phase drift outside the lateral eye opening:
 When the update takes place, the new bang makes the filter output jump in the opposite direction by a step of G_{f} [v], followed by another ramp, now in this new direction.

 The equation yields always one positive real root^{[5]}corresponding to ξ ≥ 2.
The value of G however decreases from its nominal value proportionally to the reduction of D_{T} from its maximum of 100 %.
The ability of the CDR to tolerate with minimal phase drift some very long runlengths, and/or periodic repetitions of them, can be increased increasing the value of τ_{z} at the design stage (ξ = 2τ_{z} / t_{update}).
This reduces the bandwidth of the loop filter ω_{z} and has the adverse effect of reducing the frequency lockin range and of increasing the lockin time.

 In practice, values of ξ in excess of 1000, even with low transition densities as found in SONET transmissions, are not used.
Slewrate problemEdit
The 2  2 CDR is able to vary rapidly its frequency. If the input signal offers the maximum transition density, the loop can respond to an input phase step with a frequency step equal to G [rad/sec], i.e. with a phase ramp of slope G.

 The value G (calculated with D_{T} = 100%) represents the bang bang frequency step but also the reference loop gain ( G_{φ} is calculated with the largest phase input error, typically ±π , multiplied by D_{T}, G_{f} is the value of the filter gain in the flat region, G_{VCO} is a linear approximation around the f_{p} working point ).
But if the phase of the input signal varies more rapidly than G rad/sec^{2}, then a phase error appears and it may grow and possibly affect the CDR tolerance.
This problem may be investigated using a sinusoidal input phase jitter. This is also useful as the tolerance curve of the CDR is measured as a function of a sinusoidal input jitter.
A sinusoidal function (of time in the case of jitter) has its maximum rate of variation at the zero crossings, and its maximum rate of variation is the product of its amplitude by its angular frequency, Aω in the figure here below:
Maximum phase slewrate = maximum frequency deviation.
The value of ω_{z} is kept low in order to have a stability factor ξ large enough.
But ω_{z} can not be too low, or else the slew rate (and the jitter tolerance limit!) at medium/low frequencies become too low as a consequence. (develop this point with comments and a figure)
There always exists the possibility that the VCO is not able to follow the rapidly changing phase of the input, because the rate of change of the VCO phase is insufficient. The VCO is "slewrate" limited.
This is not normally due to late response of the VCO driven by the signal from the filter output.

 The frequency deviation limits of the CDR are often set by the characteristics of the loop filter, rather than by the extremes of the frequency range of the VCO itself.
 "The VCO is designed to respond fully in one update time. This is usually very easy to achieve in ringoscillators and possible with some care using lowQ VCOs." ^{[6]}).
The slewrate is originated by the loop filter that has a very limited gain (= an attenuation) at frequencies higher than ω_{z}.

 In fact, the G_{f} at high frequencies is kept as small as possible to reduce the "tracking" jitter due to the bangbang jumps, but can not be reduced too much or else the "slewrate" becomes excessively small.
Jitter Bandwidths in 2^{nd} order type 2 bangbang CDRsEdit
The bangbang CDRs are made up by most of the 11 (PD) and practically all of the 22 (PFD).
These loops are intrinsically nonlinear, primarily because of the bangbang nature of the phase detector: this is the first nonlinearity.

 The very large (≡ infinite) gain in the detector needs a compensation by a signal level limitation is another loop point.
 (The level limitation keeps signal levels within the physical capability of the circuit elements).
 The level limitation takes place either:
 1. in the limitation of the VCO drive signal made by the circuits driving the VCO, or
 2. in the extremes of the VCO voltagetofrequency characteristic.
It is a frequency limitation. This is the second nonlinearity. In both cases of frequency limitation, when the limit is reached, the output phase waveform of the CDR enters a slewrate condition.

 The VCO intrinsic characteristic, at both ends, depends very much (and often unpredictably) on manufacturing and environmental variations.
 Therefore, the frequency range is always deliberately limited by a clamp of the output range of the VCO drive stage, +/ V_{dr} .
Overall, two nonlinearities (PD or PFD, and VCO frequency constraints) combine making transfer (e.g. jitter transfer) functions become families of functions.
Each function in a family is associated with a specific input waveform.
In the (transfer functions') case of sinusoidal inputs, each function of ω_{j} is associated with the input amplitude only.
The output is also periodic, and the transfer function is defined as the ratio of the output peak amplitude to the input peak amplitude A_{j} (that is constant at all frequencies each function in the family).
V_{bb} = the absolute value of the voltage step that takes place in these CDRs, at the input of the VCO, when the PD bangs up or down from its intermediate level.
 It coincides with the value of G_{f}(ω) for ω >> ω_{z}ω. In fact, the PD output is either +1 or 1 [volt] (or 0 V in the case of a ternary PD).
 In 1 1 loops +/ V_{bb} is the total drive range of the VCO and coincides with +/ V_{dr} .
 E_{d} is generated by a deviation of the duty cycle away from 50% in the VCO drive waveform.
 In 2  2 loops +/ V_{bb} is much smaller than the drive range of the VCO +/V_{dr} (e.g. V_{bb} / V_{dr} = 10^{3}).
 The VCO drive waveform shows a tiny bangbang threelevel ripple added to a slower and much larger waveform, that is made by the filter amplification of low jitter frequency components.
 E_{d} is obtained by a variation of the mean level of the bangbang ripple in the drive waveform.
Jitter transfer functions in bangbang CDRs of 2^{nd} order and type 2 ^{[7]}Edit
This is a type 2 system, which means that during tracking the mean values of the input and of the output phase waveforms do not differ .

 The difference of mean values is the steady state sampling error Es, and Es = 0 in these systems.
 In other words, the average level of the the difference between the input and output waveforms is zero (= the average phase error is zero ).
In 2^{nd} order systems the signal processing between PD and VCO consists of two regions along the frequency axis:

 1. a flat attenuation above ω_{z} (that makes the drive signal so small that , in tracking, the VCO frequency bangbangs within a small range around f_{p}).
 2. a region of increasing amplification as jitter frequency decreases from ω_{z}, that slopes at 20 dB/dec.
The two regions of the loop filter define the loop behaviour unless either of the two VCO frequency limits is reached (with an input jitter large enough), where the resulting slewrate corresponds to either of the VCO limits (+/ V_{dr} or +/_V_{bb}).
The two frequency regions of the loop filter, with jitter levels that make V_{bb} slewing evident in both of them, can be identified in the figure already shown in the jitter tolerance page.
In the left hand part of the figure, input and output phases have the same peak value: flat region of the jitter transfer, with ratio of peak amplitudes = 1 i.e. 0 dB. In the right hand part of the figure, the tracking is poor.

 The type 2 nature of the loop makes the mean value of the output waveform equal to the mean value of the input, but the output jitter remains in V_{bb} slew rate during every entire half period of the input jitter.
 The peak value is clearly lower.
For any given value of the input amplitude A_{j} (of the sinusoidal jitter), there is a region of high ω_{j} values where the output is triangular like in the right hand side of the figure above.

 There the triangular output has fixed slopes (+/ V_{bb}G_{VCO}) and It takes one quarter of the jitter period, 1/4 * 2π/ω_{j} for the output to vary from 0 to its peak value.

 Its peak value is: . As the peak varies inversely to ω_{j}, in this region it rolls off at 20 dB/dec.

 The ratio of the output peak value to the input peak value (A_{j}) in the V_{bb} slewrate region is therefore :
The overall jitter transfer curve has one horizontal asymptote at 0 dB towards low frequencies, and a  20 dB/dec asymptote towards high frequencies.
The transition between the two asymptotes can be approximated using the model of a first order linear system, whose corner frequency ω_{jc} is where the sloping asymptote intersects the 0 dB horizontal asymptote. (The approximation given by this model is more than adequate for most engineering purposes) :
The jitter transfer can therefore be modelled as:
Jitter tolerance curves of bangbang CDRs of 2^{nd} order and type 2Edit
External ReferencesEdit
 ↑ ^{a} ^{b} C. Response to Phase Step
 ↑ Richard C.Walker article, A. Runlength and Latency, pg 10.
 ↑ Richard C.Walker article, A. Stability Factor, pg 4.
 ↑ Richard C.Walker article, p. 3.
 ↑ Roots of G ω_{z} (2τ_{z}/ξ)^{2} + G (2τ_{z}/ξ)  LEO = 0 are: .
 ↑ Richard C.Walker article, VII. C. VCO Tuning Bandwidth, pg 10.
 ↑ Jri Lee, Kenneth S. Kundert, Behzad Razavi (SEPTEMBER 2004). "Analysis and Modeling of BangBang Clock and Data Recovery Circuits". IEEE JOURNAL OF SOLIDSTATE CIRCUITS, VOL. 39, NO. 9 pages 1571 .. 1579, III., JITTER ANALYSIS, A. Jitter Transfer. http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=1FD71150D9F00B8D9F8A32252D812190?doi=10.1.1.134.3922&rep=rep1&type=pdf. Retrieved 2015125.
 ↑ Lee, Kundert, Razavi paper of sep. 2004, JITTER ANALYSIS, A. Jitter Transfer ω_{3 dB} = (12)