ExamplesEdit
To fix the concepts introduced in the previous section, and to better understand them, let’s see a few examples.
Amongst them, just three will be further developed in the remainder of the book, because they represent the three really useful architecture models of CDRs.
They are:
The other examples are useful just as theory exercises and not so much as models of how practical CDRs should be made.
Notes:
 A CDR shall be called "a slave CDR" if its recovered clock is also used to output the recovered data.
 A slave CDR may simply deliver the recovered data to a buffer memory (end point CDR) or may extend the clock domain mastered by the remote transmit clock by sending forward the recovered data using the recovered clock (regenerator CDR).
 In some cases the characteristic of certain individual blocks in the block diagram (like the gain of the phase comparator, or the oscillator gain, or other characteristics) are not explicitly considered. More abstract quantities, like the natural frequency _{n}, or the damping ratio , are used instead for the formulas and for the time or frequency diagrams. This is done for two reasons:


 the characteristics of the blocks that operate at the (high) frequency of the data stream (phase comparator and local oscillator ) are frequently imposed by technology choices and are not for the designers to play with;
 the general behavior of the CDR is more generally represented and more easily understood.

1^{st} order, type 0Edit
and 0^{th} order, type 0 as a special case.
(For definitions of order and type, see: Order and type of a CDR )
This list could have started with the simplest example possible, where the output of the phase comparator drives directly the (phase of the) local oscillator, without any low pass filter in between.
That would have been a loop of 0^{th} order (with just constants in its closedloop transfer function and no dependence on the complex variable s) and of type 0 (because there are of course as it is of 0^{th} order no poles in the origin, i.e. no factors 1/s in its openloop transfer function).
As there is always some sort of filtering of the output of the phase comparator, the 0^{th} order type 0 loop can be seen as a limit case of the 1^{st} order, type 0 loop described here below. The model of the 0^{th} order type 0 loop can be useful when a high loop gain makes the (closed loop cutoff) frequency (of the of 1^{st} order, type 0 loop) so high to be negligible. In such case it is enough to put τ = 0 in following equations to obtain its model.
The only practical block diagram that corresponds to a PLL of 1^{st} order and of type 0 shows (see figure below) a local oscillator controlled in phase by the low pass version of the phase comparator output.
The PLL in this CDR is of unity feedback, of the first order and of type 0.
The relation of the output to the input (= the closed loop transfer function), is:
where G, the "open loop gain", is G_{φ} G_{f} G_{VCO}. The unit step response can be obtained from the (jitter) transfer function multiplying the closed loop trasfer function by 1/s , and reverse transforming the result:
Some points can be remarked:
 that there remains a steady state error, which is minimised if G is >> 1. In fact a type 0 loop like this one needs a small phase error Φ_{x(j0)}/G to track a phase difference Φ_{x(j0)} between input and the phase of the free running oscillator. In other words, if the local oscillator is synchronous with the received signal (i.e. both are within a single clock domain), this circuit provides a strong reduction of the phase difference between the two, that is reduced from Φ_{x(j0)} to Φ_{x(j0)}/G.
 that the cutoff frequency of the loop, ω_{n}, is 1+G times higher than the cutoff frequency of the filter block ω_{f} = 1/τ_{f} [rad/sec]; as G is large to render negligible the steady state phase error, this loop reacts very fast to any input phase change.
 that the loop tracks the jitter frequencies below ω_{n} = , and is unable to follow higher jitter frequencies (in all examples of PLLs there exists a natural frequency ω_{n} that depends closely on the open loop gain, and the jitter frequencies below ω_{n} are tracked, while those above ω_{n} elude the tracking ability of the loop); in this case of a type 0 loop, ω_{n} is pushed very high if the gain G is high, to the point that this loop cannot be used in CDRs.
A circuit of this architecture may find application, but never inside a CDR. It can be used to implement a DLL, (although DLLs can also be made with a 1^{st} order, type 1 architecture). As long as the residual phase error is kept small enough by a large loop gain, it can fit a DLL application. The unconditional stability and the fast response are useful, while the lack of input jitter filtering at high frequencies (mandatory for CDRs) is not important when the reference clock is sufficiently immune from high frequency noise.
1^{st} order, type 1Edit
This fundamental architecture is often used and also is, because it contains all the fundamental features of the PLLs used in practice, useful to fix concepts and to better understand the more sophisticated architectures of higher order.
It may present itself under different guises (three examples are given in the figure below), but the mathematical model remains the same.
It is maybe easier to derive the model equations from the first example in the following figure (the "slave"), although the third example (the "phase aligner") is the one most frequently found in actuality.
Oscillator controlled in frequency ("first order slave CDR")Edit
This example (its architecture is shown on the top left in the Figure above) is important because it introduces this type of architecture and allows an easy derivation of the model equations.
For slave CDR applications though, PLLs of the second order are preferred and this one can only be found in the simplest, undemanding, applications.
The second architecture of PLL in the figure above is a slight variant of the first (the one treated in this SubSection), and is not important for CDR loops. It can be important instead as model of the DLL.
The relation of the output to the input (= the closed loop transfer function), is:
where G, the "open loop gain", is defined as G_{φ} G_{f} G_{VCO}.
The unit step response can be obtained from the (jitter) transfer function multiplying the closed loop trasfer function by 1/s , and reverse transforming the result:
The model is described in detail in a page dedicated to the 1^{st} order, type 1 architectures later on in this book, owing to the practical importance of its phasealigner variant.
To be able and contrast this architecture with the others in this page of examples, the magnitude of the jitter transfer function, the response to a unit (= 1 radian) step variation of the input signal phase and the magnitude of the jitter tolerance function ( The jitter tolerance function ) are shown in the figure below.
The relevant equations are:
 Magnitude of the transfer function:
 Unit step response:
 Magnitude of the jitter tolerance: = , ( where Φ_{leo} is the lateral eye opening expressed in radian) .
This architecture clearly shows:

 a medium (20 dB/decade) jitter filtering towards high frequencies;
 a medium jitter tolerance, with the 20 dB/decade asymptote towards zero frequency. It comes from the pole 1/s present in the forward path of the control loop. This is a minimum characteristic when the CDR must operate as a slave and track frequencies that can wander around their nominal value;
 only one degree of liberty for the designer (that is G in the figures above, i.e. the open loop d.c. gain).
All these characteristics together make this architecture not preferred in actual designs of slave CDRs. For these, the second order, type 1 or type 2, architectures are chosen instead, depending on the requirements and on the technology available for the specific application.
Addition of the complementary delay ("phase aligner")Edit
Adding the complementary delay it becomes: the phase aligner.
Another example of unity feedback, 1^{st} order and type 1.
The local oscillator is fixed and an adder compensates the phase deviations of the incoming signal before its regeneration with the local clock.
This example (its architecture is shown on the bottom left in the Figure above) has value in that it it is widely used in practice wherever this function is required in a network.
This example is of particular interest because:
  a clock independent from the PLL operation (i.e. in the Figure above) is used for the regeneration, while it is y(s) instead that represents the output of the PLL system.
  as a consequence of the point above, the CDR incorporates an "elastic buffer" function, located in the processing of the input signal through the adder block and through the subsequent resampling by the local clock ().
This example in fact describes the abstract model of the phase aligner. The CDRs that match this model are found in network points where a signal (jittered and possibly regenerated by slave CDRs one or more times along a transmission path) is brought back in phase with the clock that was used to generate it^{[1]}
For sake of simplicity in the following calculations  and without loss of generality in the results the local clock is used as reference for the phase of all the signals.
Therefore its jitter is identically zero. Additionally it is assumed that the initial phase value .
 The only function (the only one that is not a transfer function) which does not belong to the linear model is obviously the one that depends on nonlinearities for its definition.
 The jitter tolerance function depends in the case of the phase aligner on two hard nonlinearities (phase comparator and phase adder), instead of on just one (the phase comparator) as in the case above.
 Jitter transfer and Unit Step Response are identical functions to those derived in the previus example, and not repeated here.
 The jitter transfer function in this case would describe a characteristic of the PLL as usual, but becomes meaningless for the whole CDR, because of the phase alignement resulting from the elastic buffer of this CDR.
 The Unit step response, in this particular case, as already pointed out for the transfer function, only describes the PLL operation, while it is meaningless for the whole CDR.
 It can be seen as the waveform at the output of the phase adder ( xy signal in the Figure above ) resulting from a phase step of the input ( x(s) in the Figure above ).

 Jitter tolerance function ( The jitter tolerance function )
The phase jitter tolerance of this circuit is limited both by the range of linearity of the phase adder and by the range of linearity of the phase comparator.
Let's see each one separately. Then, the tolerance function at any frequency will simply be the lower of the two, because the signal in either of the two circuit blocks does not influence the characteristic of the other block.


 Adder tolerance function ( Tolerance of phase adders )
 The adder is normally made by a delay line, that adds a delay to the input signal ( x(s) in the Figure above ) equal to the signal present on the control input ( y(s) in the Figure above ).

  Defining as D (measured in radian  if D is in seconds, then use Dω), the total phase delay that the delay line can add, and
  defining as zero the phase delay added when the control signal is set at the mid point of its control range,
 then the tolerance of the adder inserted in the CDR circuit of the above Figure is:
 Adder tolerance function ( Tolerance of phase adders )



 using the relation (see the relevant subsection on the jitter transfer function of this example)



 then



 Comparator tolerance function ( Tolerance of phase comparators )
 The tolerance limit is reached when the phase difference between and xy (again on the right part of the Figure above) exceeds an error :
 Comparator tolerance function ( Tolerance of phase comparators )



 using the relation (see the relevant subsection on the jitter transfer function of this example)



 then

The relevant diagram of the jitter tolerance function is shown in the Figure here below.
The figure above contrasts the case of the 1  1 loop (in blue) with the case of the other two important loops (21 and 22) only to better understand the former. The other two loops in fact are not used for phasealigners, as explained in the following subsections.
As stated above, the unit step response and the jitter transfer function of this CDR do not differ from those of other 1^{st} order, type 1 CDRs.
The jitter tolerance of this CDR shows a new example. It is limited at low frequencies by the jitter tolerance of the adder and, at medium and high frequencies, by the jitter tolerance of the phase comparator. In fact the actual design choices always assure that
In conclusion, this example shows a CDR that:

 is not able to tolerate a wander of the input signal frequency. There is no increasing jitter tolerance with a 6 dB/oct asymptote towards zero frequency, but just the flat tolerance limited by the adder range. This CDR can only phase align a signal (jittering but) synchronous with its local clock;
 can lock in quickly into a signal burst after a period of loss of signal;
 offers the inherent elastic buffer feature typical of phase aligners.
2^{nd} order, type 1Edit
Oscillator controlled in frequency ("regenerator CDR") 2  1Edit
(Another example of a tracking system based on a second order, type 1 loop can be found in Wikibooks, at: Control Systems/Examples/Second Order Systems).
Some of the characteristics of the first architecture in the figure above make it ideal for the regenerator application.
= the CDR with good dejittering characteristics.
This is the second architecture of three that must be remembered.
It is presented as the first example in the figure here above.

 The oscillator is controlled in frequency. It converts an input signal into a frequency, that is into the integral of a phase: its transfer function in the s domain is in fact V_{0}/s.
 Between the output of the phase comparator and the input of the oscillator, a single pole low pass circuit filters the higher frequencies of the error signal. (Combined with the oscillator, this low pass in the forward path makes the loop a second order, unity feedback, system).
 The presence of an independent block to perform the low pass with a precise cutoff frequency (represented by in the figure above) gives the designer one more degree of freedom to find the best performance compromise.
The figures that follow, as in the previous examples, plot the unit step response, the jitter transfer function and the jitter tolerance function of this architecture.
In order to appreciate some of the peculiarities of this new example (that is further developed in in a dedicated page further down in this book), also the corresponding curves for the fist order ( 1  1 ) slave have been plotted as well.

 Unit step response
The increased complexity of this architecture versus the first order one is responsible for the delayed reaction to an input phase step.
In fact this architecture is inferior to the first order alternative (no matter how smartly the design parameters are adjusted) when the lockin phase of the system is of prime importance.

 Jitter transfer function
The ability to filter out the high frequency components of the incoming jitter is the strong point of this architecture.
When a series of regenerations takes place along a chain of telecom links, the jitter accumulation must be avoided, and the low pass action must have a (relatively) abrupt cut off.
This second order type 1 model (see the relevant figure) has a jitter transfer with a 40 dB/decade rolloff, while the first order type 1 model only offers a 20 dB/decade performance.

 Jitter tolerance function
The advantages offered for the jitter filtering are partially offset by some weakness in presence of strong jitter components concentrated around the corner frequency of the PLL (fortunately not very likely in practice).
While the first order loop presents a tolerance larger than 1 Unit Interval at all frequencies, this second order loop always has a frequency range where its tolerance is lower than 1 UI.
This dip of the tolerance is more evident for lower values of the damping ratio ζ, and the above figure uses the two curves to give evidence of that.
The importance of this architecture is such that an entire chapter is devoted to it further down in the book, where all time and frequency functions of the previous diagrams are described in detail and further developed.
At this stage it is sufficient to have understood in which ways this architecture is different from the other architectures presented in this series of examples.
A modified representation of the same architectureEdit
The second example in the figure above is essentially the same architecture as the first.
It helps introduce the concept that the equations and the behavior of a VCO can be separated into an integrator and a phase controlled element, that is typical of phase aligners.
Linear modeling is the same as the case above, while tolerance obviously depends on the linear range of the new blocks.
It is not an important example for practical applications.
Addition of the complementary delay ("2^{nd} order type 1 phase aligner")Edit
This architecture, the third in the figure of the 2  1 loops above, is another interesting example for didactic purposes, but has no value for a real CDR application.
The fundamental blocks are very similar to those of the previous example.
 The last integrator block (an up/down counter in a digital implementation) corresponds to the oscillator controlled in frequency. It converts an input signal into a frequency, that is into the integral of a phase: its transfer function in the s domain is in fact G_{f}/s. (As there is no other 1/s factor in the open loop transfer function, this is a type 1 system).
 Between the output of the phase comparator and the input of the oscillator, a single pole low pass filters the higher frequencies of the error signal. (Combined with the oscillator, this low pass in the forward path makes the loop a second order, unity feedback, system).
The functions that belong to the linear model (like the USR and the jitter transfer function) are exactly the same as in previous example of the slave 2  1 just above.

 Jitter transfer function
For instance, the jitter transfer function can be derived with the same steps used for the first order phase aligner:
and it coincides with the jitter transfer function of the 2  1 slave of the previous example.
The ability to filter out the high frequency components of the incoming jitter is the strong point of the 2  1 architectures.
This type of performance is valuable as long as the CDR is a regenerator and sends the regenerated clock with the data stream to another CDR downstream. In this example of a phase aligner however, this performance is not useful at all, because the phase aligner uses the PLL output only for its internal operation, and does not use it for its output data stream.

 Unit Step Response
The USR and its time diagram figure are identical to the ones of the previous 2  1 example.

 Jitter tolerance function
The jitter tolerance function of this architecture is different because it does not depend on the nonlinearity of the phase comparator only, but also on the non linearity of the phase adder.
Whenever nonlinearities are considered for architectures of the same order and type, a linear model that is in fact the same for all architectures of the same order and type cannot be used. The functions that describe the operation of these architectures (like jitter tolerance, ..) are different if different nonlinearities are considered.
Following the approach of the first order phase aligner, the tolerance due to the adder must be combined with the tolerance due to the phase comparator.
The circuit tolerance, at any frequency, is the lower of the two.


 Tolerance of the phase adder ( Tolerance of phase adders )
 The adder is normally made by a delay line, that adds a delay to the input signal (X(s) in the figure above ) equal to the signal present on the control input ( Y(s) in the figure above ).

  Defining as D (measured in radian  if D is in seconds, then use D ω) the total phase delay that the delay line can add, and
  defining as zero the phase delay added when the control signal is set at the mid point of its control range,
 then the tolerance of the adder inserted in the PLL circuit is:



 using the relation found for the jitter transfer function of this example)



 then



 Tolerance of the phase comparator ( Tolerance of phase comparators )
 The tolerance limit is reached when the phase difference between and xy (again on the the Figure above) exceeds an error :



 using the relation (see the relevant subsection on the jitter transfer function of this example):



 then:

It is easy to see that the tolerance is equal (for > 0.707) to the tolerance of the first order phase aligner, or inferior for lower values of .
The inherent weakness of this type of architecture (undershoot below 1 U.I. of the tolerance at frequencies around _{n} ) remains, and it can not be offset by the corresponding jitter transfer advantages that are meaningless for a phase aligner system.
As a result this architecture is inferior to the first order alternative (no matter how smartly the design parameters are adjusted) during the lockin phase for a phase aligner application.
(On the other hand, it has the disadvantage of limited tolerance at low jitter frequencies, when compared to the classic 2nd order circuit for slave CDR applications, as it will be shown in the following paragraphs.)
In conclusion, the importance of this architecture is limited to its use as example for didactic purposes. Its first order alternative, simpler and more efficient, is preferred in practice for phase aligners.
2^{nd} order type 2Edit
If a 2nd order CDR PLL incorporates a single zero lowpass filter (instead of a single pole lowpass as in the examples above of the 2 – 1 PLL), then
the pole at frequency zero (that is implicit in the filter) makes the PLL change from type 1 to type 2.
.
The first example, in the figure of 2  2 PLLs just above, is the important one.
The other two are good exercises, but are not important for practical applications.
The model is described in detail in a page dedicated to the 2^{nd} order, type 2 architectures later on in this book, owing to the practical importance of this first case (see figure above).
Oscillator controlled in frequency ("monolithic slave CDR") 2  2Edit
The first of the three examples in the figure above is the third common PLL architecture for a (slave) CDR.
Important nonlinearities are inevitably associated with a monolythic implementation, and
the characteristic of robustness against them the have made this architecture the preferred choice for monolythic slave CDRs.
= the CDR with good rejection of generated noise.
This is the third architecture of three that must be remembered.
It is presented as the first example in the figure here above.
In the linear model it can be remarked that the loop filter provides a very high gain at very low frequencies.
This corrects any steady state error (squeezes the steady state error to zero) that may otherwise result from a difference between the frequency of the input signal and the freerunning frequency of the local oscillator.
This also strongly attenuates the low frequency noise generated inside the PLL itself.
The loop filter does not attenuate the high frequencies, though. This makes this 2  2 loop less efficient (than the 2  1 loop) in rejecting the high frequency input noise (i.e. the part of the input jitter spectrum that does not carry any useful information about the sending clock and its phase). On the other hand, this makes this loop inherently faster in acquisition than the 2  1 loop.
This loop always overshoots in the lockin phase, as the unit step response shows (see figure below).
The jitter transfer functionEdit
The transfer function can be derived, like for all other CDR PLLs that are unity feedback loops, using the two equations:
 E(s) = X(s)  Y(s)
 Y(s) = E(s) (gain of the loop forward chain)
and eliminating E(s) by substitution. In this case the result is:
and the relation between the circuit parameters and the loop parameters ω_{n2} and ζ with respect to G and τ_{z} can be obtained from:
Its magnitude is:
The unit step responseEdit
The unit step response can be obtained from the (jitter) transfer function multiplying the closed loop trasfer function by 1/s , and reverse transforming the result:
The jitter toleranceEdit
The normalized jitter tolerance can be obtained,
as X(jω)_{Ε(jω) = 1} that is:
The asintote towards low frequencies (in the loglog plot) has a slope of 40 dB/ decade (as opposite to the slope of 20 dB/ decade of the type 1 systems).
This is typical of all type 2 systems, that are more tolerant ( 20 dB/ decade more) of low frequency jitter than type 1 systems.
In the 2  2 loop, bandwidth and damping ratio increase when the gain increasesEdit
ζ_{22}^{2} = G τ_{z} / 4
This is important for a monolithic implementation with a bangbang phase detector (and sometimes also a bangbang VCO) because the gain of those nonlinear elements is very high when their input signal deviates very little from its center point.
Instability may occur at high gain conditions if the damping ratio decreased with increasing gain (like in a 21 loop), but luckily a 22 loop becomes more damped at high loop gain. Its bandwidth widens at the same time, and this is associated with more rejection of internally generated noise but also with less rejection of incoming line noise.
A modified representation of the same architectureEdit
The second example in the figure above is essentially the same architecture as the first.
It helps introduce the concept that the equations and the behavior of a VCO can be separated into an integrator and a phase controlled element, that is typical of phase aligners.
Linear modeling is the same as the case above, while tolerance obviously depends on the linear range of the new blocks.
It is not an important example for practical applications.
The example of the 2 2 phase alignerEdit
The figure presented earlier in this page for the case of the 2 1 phase aligner shows also the magnitude of the jitter tolerance for this example (Jitter transfer and unit step response remain in this case the same as for the 2  2 slave, because they do not depend on circuit nonlinearities).
In this figure, the green curve represents the jitter tolerance function for the 2  2 phase aligner.
Note that the jitter tolerance function is the same as the one derived for the 2  2 slave until the half depth of the delay line becomes a horizontal limit at low jitter frequencies.
This 2 2 phase aligner has no practical use, for reasons similar to the ones that make also the 2 1 phase aligner a non preferred option. The 1  1 phase aligner is the preferred choice.
The 2 2 phase aligner has a slightly better tolerance in a small region of frequencies, but this is a negligible advantage in practice and is offset by its risky behavior close to the natural undamped frequency. The loop ability to reduce strongly the low frequency noise of the local oscillator and its ability to reduce to zero the steady state frequency error make no sense for a phase aligner.
ReferencesEdit
 ↑ A robust phase detector for 1.25 Gbit/s burst mode data recovery (from the INTEC team of Ghent University) IEICE Electronic Express, Vol. 1, No. 18, 562567 http://www.jstage.jst.go.jp/article/elex/1/18/1_562/_article