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Computation of Period Sensitivity Functions for the Simulation of Phase Noise in Oscillators Angelo Brambilla, Paolo Maffezzoni, and Giancarlo Storti Gajani
Abstract—Accurate phase noise simulation of circuits for radio frequency applications is of great importance during the design and development of wireless communication systems. In this paper, we present an approach based on the Floquet theory for the analysis and numerical computation of phase noise that solves some drawbacks implicitly present in previously proposed algorithms. In particular, we present an approach that computes the perturbation projection vector directly from the Jacobian matrix of the shooting method adopted to compute the steady-state solution. Further, we address some problems that arise when dealing with circuits whose modeling equations do not satisfy the Lipschitz condition at least from the numerical point of view. Frequency-domain aspects of phase noise analysis are also considered and, finally, simulation results for some benchmark circuits are presented. Index Terms—Floquet theory, oscillators, phase noise.
I. INTRODUCTION
D
URING these last years, wireless communication systems have gained large interest due to a continuous increase of the number of applications and, correspondingly, of the volume of commercialization. Their design requirements have become more stringent, mainly for what concerns separation between communication channels and input sensitivity of receivers. The study and simulation of phase noise, that represents one of the main aspects limiting channel compaction, is thus of great importance and is the main topic of this paper. Phase noise spreads the bandwidth of the fundamental frequency of oscillators causing, in the worst case, interference with neighboring channels. Literature reports a large number of papers on phase noise simulation [1]–[7]; most of them are based on Floquet theory [9], [13] and do a modal decomposition of the variational model associated to the autonomous circuit (oscillator) operating in the steady-state condition. The theory underlying the Floquet methods shows that the function describing period sensitivity with respect to state perturbation is a rescaled version of a specific eigenfunction that can be obtained from the variational system. In fact, each mode used to decompose the dynamics of the variational model is described by a scalar called the Floquet multiplier and an associated eigenfunction. Autonomous systems operating on a stable periodic trajectory must have a Floquet multiplier equal to 1; computation of the associated Manuscript received April 15, 2003; revised June 22, 2004. This work was supported by the Italian MIUR under Grant FIRB RBNE012NSW_004. This paper was recommended by Associate Editor M. Nakhla. The authors are with the Dipartimento di Elettronica e Informazione, Politecnico di Milano, I-20133 Milan, Italy (e-mail:
[email protected]). Digital Object Identifier 10.1109/TCSI.2004.842884
, for the adjoint variational eigenfunction, here denoted as problem is the first and most delicate step in phase noise analysis. Here it is shown that the perturbation projection vector introduced in [2], [3] is a scaled and sampled version of the eigenfunction. Even though Floquet theory was fully developed and has been known for more than a century, its application to numerical simulation is anything but easy and straightforward. As other authors showed (see, e.g., [1], [7]), problems arise when more than one multiplier is close to 1: in this case, the among all multipliers and eigenfunctions choice of and is very difficult or even impossible. This is true, in particular, in very high-Q oscillators. Furthermore, if back integration in time of the adjoint variational model is required, as would seem from a straightforward application of the theory, instability problems of the employed numerical method can arise. Finally, the differential equation modeling the autonomous circuit must satisfy the Lipschitz condition to ensure the existence of a solution for the variational model. Non-Lipschitz oscillator models are quite common in practical applications, but computation of perturbation effects is, in this case, very difficult, since the unit multiplier is in general not well defined. Note that the Lipschitz condition has different practical meanings if mathematical or numerical points of view are considered; even if it is sufficient from a mathematical point of view, in general it is not from a numerical one. Indeed the Lipschitz constant can be so high that it can not be “handled” by the finite relative precision of the floating point unit of the computer. Recent papers [5], [7] have dealt with some of the above problems, in particular those related to back integration, by adopting a method based on time sampling of the system monodromy matrix. The approach is very promising, but still suffers of a few drawbacks. In this paper, we address the above aspects and propose an approach that solves them. In the simulation result section we eigenfunction for some benchmark aucompute only the eigenfunction has been tonomous circuits, since, once the accurately computed, phase noise analysis can be carried out through one of the approaches reported in literature (e.g., [2], [6]). II. MATHEMATICAL FORMULATION In order to characterize phase noise in oscillators, we first analyze some general properties of autonomous systems that have a known periodic trajectory in state space. Consider the state equation
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(1)
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where : and : , being the state dimension of the system. Assume that (1) admits the asymptotiso that . cally stable -periodic solution Suppose that an input perturbation , with, in general , is applied to the oscillator,1 the state equation can be formally written as (2) The perturbed solution of the system can be expressed as , where represents the phase deviation relative to the periodic orbit of the unperturbed oscillator, reis an orbital deviation. The ferred to as phase noise, while term represents amplitude modulation due to perturbations and occurs along directions that are in the basin of attraction of the stable limit cycle representing the oscillation [9]; on the occurs along the limit cycle of the other hand, phase noise unperturbed circuit that always represents a neither stable nor unstable manifold with respect to state. This means that phase error may become arbitrarily large if the applied perturbation is does not pose in some way “polarized.” On the other hand, any problem since in practice it can be eliminated for example by means of a zero-crossing detector and a hard-limiting circuit followed by a bandpass filter with a suitable quality factor. The zero-crossing detector and hard-limiting circuit square the periodic waveform generated by the autonomous circuit and supply a constant amplitude output waveform which is filtered to keep only the fundamental component of the spectrum. For and asthis reason, in what follows we limit our attention to sume that the approximation holds, where is the vector of state variable variations. To describe the response of the perturbed system, we first , by linearise (2) in a neighborhood of the periodic solution adopting a Taylor expansion of truncated at the first-order terms
where
(4) Let us now consider the associated homogeneous problem (5) From the Floquet theory, we have that the state transition maof (5), defined as the matrix such that trix , , can be written as (6) and are T-periodic matrices, nonsingular , where and ; entries of matrix are the Floare the characteristic or Floquet exponents and be the columns of and quet multipliers. Let the rows of , it is known that they satisfy the biorthogonality condition , where is the Kronecker index [9]. The more general case of differential algegraic equations (DAEs) was considered in [4]. By differentiating both sides of (1) with respect to time, we have (7) , we see that is and, by choosing a nontrivial T-periodic solution of (5). This means that, being a function characterized by a zero mean value, the characteristic multipliers are bounded by with equal to 1 [13]. With no loss of generality, we at least one and . reorder multipliers so that Consider the adjoint homogeneous variational problem (8) where represents the transposition operator, its state transition matrix is given by
from which we have
(9)
Since is a solution of (1), we obtain the linear time varying system modeling the variational problem (3)
is assumed that the structure of the f (1) function is not altered by the addition of noise sources. 1It
where represents the conjugate transposition operator. The of the matrix, corresponding to , first row for plays for the adjoint system the same role played by the direct problem. It was shown that the eigensolution is crucial to determine the phase noise component of the oscillator [1], [5], [6]. A. Problems in the Numerical Computation of Numerical computation of the vector, i.e. of a time sam, poses some problems already reported in pled version of literature. One of them is evidenced by oscillators that, due to
BRAMBILLA et al.: COMPUTATION OF PERIOD SENSITIVITY FUNCTIONS
design requirements or, more simply, to numerical approximations, have more than one multiplier close to 1. To have some insight, we first introduce the approach proposed in [4]. , possibly using a technique such 1) Compute as the shooting method. . 2) Compute the monodromy state transition matrix , recall that . 3) Compute is an eigenvector of corresponding to 4) Floquet multiplier (it is immediate to verify the that, since all matrices involved are -periodic, we have ). To compute , first compute corresponding to the the eigenvector of Floquet multiplier, then scale this eigenvector so that . , , by nu5) Compute the periodic vector merically solving the adjoint system with as the initial condition. A comment found in [4] and [6] about step 4 is: “For high-Q oscillators, the iterative methods can run may have several other into problems, because eigenvalues that are close to 1 ” A similar approach was previously introduced in [1], [2] and again there is a quite interesting comment by the author: “As it is well known, the shooting methods get into numerical trouble when a second Floquet exponent of the system approaches zero (Floquet multiplier equal to 1); this has an eigenmeans that the fundamental matrix vector which is not the tangent vector at the limit cycle, . This may cause nuwith corresponding eigenvalue merical problems in analysing oscillators with high-quality resonators or if the oscillator operates near a bifurcation point where very small Floquet exponent may occur ” The above algorithm shows that the user has to choose by hand the Floquet multiplier equal to 1; numerical errors and the characteristics of the oscillator can make troublesome its choice, since, as it was evidenced, there can be several Floquet multipliers very close to 1. A solution to this problem, based on the geometrical properties of the eigenspace associated to the unit multiplier, is described in the next section. Similar approaches were previously described in [5], [7]. We show that a sampled can be computed in a straightforward way by version of considering the Jacobian matrix of the shooting method adopted to compute the steady-state behavior of the autonomous circuit. is obtained without bothFurthermore we evidence that ering about the Floquet multiplier equal to 1 since we do not as initial guess use the corresponding eigenvector of to numerically back integrate the adjoint system. in (1) does not satisfy A second problem arises when the Lipschitz condition. This problem is described with more details in Section IV, where we discuss a solution to it.
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phase noise. In the Section III-A a definition of a sampled verthat we denote as is first given; in the Secsion of tion III-B a robust numerical method for the computation of is then presented. A. Definition of Consider the numerical solution of the unperturbed system (1) when the T-periodic limit cycle is reached through, for instance, the adoption of the backward Euler (BE) integration method. In this paper we often limit our attention to BE but the same reasoning can be extended to other integration methods such as the linear multistep (LMS) ones. At the time instant, we have (10) where , is the integration time step, supposed fixed and chosen so that for some integer number of is the null -vector.2 We solve (10) by using the samples and Newton algorithm
where is the Newton iteration index and is the Jacofor . Once the Newton algorithm bian matrix of has reached convergence, the matrix (11) is obtained, where is defined in (4). Similarly, the numerical solution of the variational problem (5) at the time instant, yields
or, solving for
(12) The solution of (12) at time at the initial time through
is related to the solution (13)
where (14) (with matrix products ordered so that smaller indices are toward the right) is the numerically computed transition matrix. has distinct If, for each , eigenvalues, i.e. it is simple, (13) can be recast as (15)
III.
EIGENFUNCTION
As anticipated in the cited bibliography and recalled in the previous section the numerical computation of the eigenfunction is the first necessary step for the computation of the
where
is a diagonal matrix and is the matrix of the related eigenvectors. Vector ,
2Similarly, 0 matrix.
denotes the
N 2 N null matrix and 1
the
M 2 M identity
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the th column of , is the numerical representation of the eigenfunction evaluated at , while vector , the th row of , is the numerical representation of the eigenfunction evaluated at . In particular, vectors are and vectors are samples of the samples of eigenfunction along one period. expressions, one for each time point Now collect for , as in (12), so that one full period is represented and add the periodicity constraint . In compact form we write (16) where components of the
.. .
.. .
matrix are defined by
.. .
.. .
.. .
..
.. .
.. .
.
..
.. .
.
and is obtained by stacking column-wise at the each column vector corresponding to the solution time instants; the vector column is similarly formed by stacking column-wise the , i.e. vectors. This vector is a solution of (16) it is in the null space of the singular matrix. Now consider the adjoint problem obtained by transposing and changing its sign (17)
B. Numerical Computation of In principle, the vector could be calculated by adopting the procedure reported at the beginning of Section II-A, but it will suffer of the drawback that we have already detailed. It may seem that another possible solution would be to directly . The numerical computation of compute the null space of is accomplished by means of the singular value matrix. Unfortunately, for high Q osdecomposition of the cillators, which have more than one Floquet multiplier close to 1, more than one singular value is very close to 0 so that the identification of the null space becomes ambiguous. Furthermore, the numerical computation of the null space of a matrix can be a CPU time consuming task especially if circuits of medium size are considered. In order to overcome the above mentioned problems, we propose a more reliable numerical procedure. To this end, reconsider the original unperturbed system (10) and determine its periodic steady-state solution (sampled in points) after having period. If added as a new unknown the value of the (10) are collected over one period and the periodicity conis added as new equation, we obtain straint an underdetermined system composed of equations unknowns. A further equation, for example, in bounding one sample of one state variable to a given admissible , is added in order to obtain a well posed value, i.e. problem (other choices are admissible see [22] for example). The nonlinear problem becomes
(19) , where or, in compact form, , and . The above nonlinear system is solved through the Newton algorithm and, at the th iteration, we have
where
(20)
.. .
.. .
.. .
..
Once the Newton algorithm has reached convergence, the term and the variable variations and will vanish and converges to the Jacobian matrix, that the matrix sequence is block partitioned as
.. .
.
(21) .. .
.. .
.. .
..
.
.. .
Note that (17) is equivalent to the integration of the adjoint system (8) with the forward Euler formula (18) with the addition in the first row of the periodicity constraint . Thus, if a vector is formed stacking eigenfunction at column-wise the column vectors of the time instants, we have that the . this vector is a solution of (17) so that
is the matrix defined in (16), is where on top of , and the vector obtained by stacking is a vector with a single non zero element representing the last equation in (19). Now consider the perturbed system, where is a vector of small signal perturbations ( was defined in (4)), we have (22) where represents the variation in amplitude of the lution and the variation of the period due to the
soper-
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turbation. Non-singularity of and some insight regarding the structure of its inverse is given by the following proposition:3 Proposition 1: Consider a matrix with and define vectors as a normalized base and as a normalized base for , i.e. for and with and . and such Take any two unit vectors and , both in that and ; write the augmented matrix as (23) where and are some known real constants. With these hypotheses, matrix is nonsingular and its inverse is (24) where and are some real scalar constants and is a and generalized inverse of , i.e. . The choice of among all possible generalized inverses of , is bound by conditions on its null space (both right and left), in particular, vector is a base for , while is a base for . Proof: We write the matrices corresponding to the prodand and prove that they are the identity ucts matrix (25) • •
Element Element
since if
is a vector in is chosen as
(27) assured by the fact that and (recall that for we have that any matrix and ). We can thus write as either or by an appro, , and . Direct calculation shows priate choice of , , and that , we have , so that . deterThe choice of vectors and that border matrix among all possible generalized inverses mines the choice of and we obtain of ; in particular, if the Moore–Penrose generalized inverse , characterized by and the additional conditions ; while, for and , the group gen, characterized by is oberalized inverse tained. More general results on generalized inverses are found in [12]. Property 1 allows us to write (22) in a more comprehensible form (28) where
is the generalized inverse of and
characterized by and, cor-
respondingly
.
. since is a base for Element so that . Before considering element let us write matrix •
(26) and note the following. • Element since is a vector in . if is chosen as • Element . since is a base for • Element so that . Let us now consider the and blocks. In order we shall left and right to prove that they are indeed equal to and prove multiply both matrices by a generic vector that the result is always itself. To this end, write as the sum of two complementary subspaces, two forms are appropriate: with , and with , . 3This result applies to general matrices that satisfy the hypothesis reported in the proposition and in particular to the matrix, that was previously defined.
A
Note that any vector can be recast in each one of the two forms defined above since
from which we see that and represent the sensitivity of the state variables and of the working period with respect to the small perturbation . In other words is the sampled obtained using BE. version of vector can be done The numerical computation of the . in a straightforward way by recalling that Since often is LU factorized, we have so can be determined as the solution of the linear algethat braic problem where is used to “select” the last row of . This represents one of the main results of this paper. The statement of (28) and its solution solve the first problem outlined in Section II-A. Furthermore, Proposition 1 shows that the vector in (28) is independent from the choice of the and vectors in (23) except for the scaling constant . Indeed this choice has a large “impact” that, however, is neglected in on the generalized inverse phase noise analysis. This yields to the important result that despite how the algorithm to compute the steady-state behavior of an autonomous circuit has been developed, the last row of is exactly , if the matrix structure (28) and hypotheses of Proposition 1 hold. We shall say that Proposition 1 gives large freedom on how to implement the shooting method, that is, on how to border
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the matrix. Indeed it shows that all the vectors implementing a suitable steady-state periodicity condition of the shooting problem sketched in (10) are equivalent since they matrix nonsingular and allow the Newton method make the to compute the solution. IV. NON-LIPSCHITZ OSCILLATORS In the time domain, the variational based method shows a drawback when the autonomous circuit is modeled through functions which do not satisfy the Lipschitz condition. Consatisfies the sider (1), it admits a unique solution only if , in Lipschitz condition is some constant. the time interval of interest, where must be at least to have the variational problem Thus, (5) well posed, therefore . In some situations, it can is discontinuous and thus it does not satisfy happen that the Lipschitz condition in an enumerable and finite number of time instants in the working period. However, in this case, the integration time interval can be time subintervals so that, in each of split in the them, (1) satisfies the Lipschitz condition. Since we suppose , the value in the that, at least, subinterval represents the initial condition for the problem in subinterval. A variational problem the subsequent can be formulated in each of the above time subintervals but, in this case, the variational problem can not be written in , since , where . By defining as and the state and transition functions computed over the subsequent intervals where is a singular time instant, we . A new state transition have should be contribution lumped at the singular time instant taken into account in order to correctly compute the vector. This aspect will be better investigated in Section IV-A. satisfies the Lipschitz condition or does not The fact that can depend on the model adopted to simulate the oscillator. For example, the ring oscillator shown in Fig. 9 can employ inverters (an odd number of them) and they can be modeled as ideal elements with infinite gain. In this case, the transition of the inverter is instantaneous and the current flowing in the capacitor loading its output is a discontinuous function of time. Despite this extreme case, there are situations where the Lipschitz constant is finite, but so large that does not satisfy the Lipschitz condition from the numerical point of view. For example, consider the ring oscillator modeled by (29), where the input/output , characteristic of each inverter is and is the conductance of the resistors in series to each capacitor .4 inverter 1 inverter 2 inverter 3. (29) 4This model was adopted in [7] so that the interested reader can compare the result shown in this paper to those there reported.
The constant can be chosen sufficiently large to cause a sharp transition of the inverter output voltage, even when the minimum relintegration time interval is reduced to the ative value allowed by the arithmetic unit (alu) of the comand puter.5 Suppose that at the time instant while . Expand in Taylor series with respect to time and truncate it at the first-order term, . Suppose that obtaining so that its value is the the integration time step is lowest admissible one on the computer, meaning that if it is furfrom the computer point of ther decreased then is sufficiently large, we have view.6 Now if
which shows that there is a sharp commutation of the inverter 1 and a current step through the corresponding capacitor. In turn this means that the variational problem can not be formulated in . In the following, we show a possible approach to deal with “non-Lipschitz” oscillators. A. Oscillators With Comparators in (6), The computation of the state transition function as shown in Section II-A, can be correctly done when is a function of time. In this case, we have (30) and . where it can be seen that (30) does On the other hand, if no longer hold. In the following, this aspect is investigated and a modification to the calculation of the state transition function (6) is described. Consider, for simplicity, an oscillator composed of a single ideal comparator.7 The dynamics of the circuit is modeled by the DAE system
(31)
represents the amplitude of the comparator where function, , : output, which is modeled by the , , : , and , . in and consider its value in Linearise (32) Suppose that in the interval and that is the threshold at which has a sharp variation; is independent from the time instant at which the comparator 5On 64-bit IEEE floating point compliant computers, as most personal com: . puters, the relative precision of the alu is 6The
+
' 2 22 2 10
operator denotes the numerical summation of two floating point numbers by the alu of the computer, that can be affected by round-off errors. 7If there are more than one comparator, we suppose that only one of them commutes at a time instant.
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switches. Define as a finite and small perturbation of by exploiting (32) we write
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;
(33) is the corresponding variation of the time interval where so that, at the time instant , the function is equal to . Considering again the third equation in (31) and write Fig. 1. Schematic of e relaxation oscillator. The comparator is assumed to be ideal.
where spect to obtain
(34) is the gradient of with reevaluated in , . From (33) and (34) we
, we obtain the state and finally, taking the limit for transition function of the variational problem between the and time instants
(35) Denote as . Define time instants
the solution of (31), assumed periodic of period and approximate in the and
(36) Define as the value assumed by in the time instant at which has a sharp variation. If we of , of and introduce the small variations of , we can write
(37)
B. Relaxation Oscillator Consider the relaxation oscillator circuit shown in Fig. 1. We is arbitrarily close to 0, so that it suppose that the value of can be neglected, and that the comparator is ideal [10], [23]. The dynamics of the relaxation oscillator is modeled by
which leads to
and finally, by subtracting equations in (37)
Assume that the and time instants are so that and thus to this leads to . The above equation can be recast as
(40) We observe that, in general, only a subset of the state variables affects the comparator input variable , these variables vector. Only these correspond to non zero entries of the elements contribute to the right-most terms of (40). In the limit , the vector case in which no state variables affect is null and the state transition function reduces to the iden, then tity, as it must obviously be. Furthermore, if and the state transition function reduces, once more, to the identity. We end this section by noting that the presented approach can be easily applied when the sharp commuting elements and their location in the circuit are known in advance and thus the structure of the DAE can be determined. In the other cases even though possible it can be difficult or too much “numerically” expensive to determine the device that undergoes a sharp transition.
(41)
,
(38) Now, substituting (35) in (38) we have (39)
During a working period of the oscillator there are two commutations of the comparator; we assume that the first is in and the other in . Since we have chosen [see (31)], we have , . Note . Being , (41) that, in our case, is considered separately in the two open subintervals (0, ) and ( , ). We have set the time reference in . In the first and second interval, the steady-state solutions are
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where we obtain
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. By imposing the condition
By adopting BE the above equation transforms to
If what has been detailed in Section II is applied to compute the we have state transition function
It is easy to see and leads to
and, being
. Now consider (40), its adoption in
and
, we have
which leads to
C. Less Extreme Cases Oscillators that fail to satisfy the Lipschitz condition could be a direct consequence of “too extreme” models of circuit elements. For example the comparator employed in the real circuit corresponding to the model of the relaxation oscillator shown in Fig. 1 is characterized by a finite input/output gain and exhibits a well defined internal dynamics so that capacitor currents in the real circuit are continuous functions of time. We could be induced to think that by adopting circuit models described by eigenfunction Lipschitz functions, the computation of the would not pose any problems. In the following we show that in general this is not true, mainly if the length of the integration time step is not suitably controlled. The positive effects of the approach proposed in this section on the accuracy in computing are shown in Section VI through the Tow–Thomas oscillator and a ring oscillator. Both circuits were used as benchmarks in the previous papers [7], [20], [21]. We show that computation was accurate in [7] only because low gain inverters were of used. Consider once more the relaxation oscillator in Fig. 1, but now with a finite value of . Suppose that the comparator is . modeled by the continuous function Equation (41) can be rewritten as
(42) capacitor does no more allow any instanThe addition of the voltage and thus of the output taneous variation of the voltage. The local truncation error introduced in solving (42) where is defined as is the exact value of the state function at the time instant is that computed by the simulator. It is obvious that and unknown is estimated in some way. being The criterion to control the integration time step and thus the accuracy of the numerical solution is in general based on the inequality (43) where defines the relative error with respect to and the absolute error. If the above inequality is not satisfied, is suitably shortened, since from (42) it can be shown that and . When condition (43) is met a further reduction of seems simply to waste CPU time, since accuracy already satisfies the required level deand constants. However if we pay attention fined by the to the variational problem (5) and to its discrete version (16), . From we see that it is mandatory to accurately sample (7) we know that time derivatives of state functions must lie in the null space of the matrix, that is, they must solve (16). If is not accurately computed and sampled, errors directly eigenfuncreflect on the Floquet multipliers and on the tion. In this case (16) will not be satisfied. Note that sampling in (14) is obtained as is quite important since matrices sampled at the time the product of the sampling the instants. To give an insight on the effects of relaxation oscillator in Fig. 1 was analyzed with a simulator that adopts the error criterion (43). Two simulations were carried out and ; the Gear method of order 2 with V, , was employed; circuit parameters are F, pF, k and k . During the second simulation the time step was upper limited so that (43) was largely satisfied. Fig. 2(a) shows two overlapping waveforms computed by state function of the rethe simulator and referring to the laxation oscillator. Fig. 2(b) shows the samples of the currents capacitor at the first commutation of the flowing through the comparator that happens at about 0.6889 ms from the beginning of the working period [Fig. 2(a)]. The segments interpolating points are not shown. We see that the two falling fronts are separated by the very small time difference of 2 ns and mainly that the rightmost front was computed by employing a larger number of time points corresponding to a shorter integration time step
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eigenfunctions computed in the two cases; it is easily seen that they are very different. In the following, we introduce an error criterion to control sampling and in the computation of the accuracy in the the derivatives of the state variables. Suppose, to adopt one of the backward difference formulas (BDF) (a subclass of the LMS methods) to solve (1) that, in the generic time instant , is formulated as (44) if the BDF is implicit and if it is explicit. where Its application transforms (1) to a nonlinear algebraic equation that we suppose solved by the Newton iterative method. The BDF method contributes entries to the Jacobian matrix of the problem
(a)
Now if we consider the variational problem (5), in order to reuse matrix computed while solving (1) we have to adopt the the same BDF method. This choice leads to (45)
(b) Fig. 2(a). Two y (t) overlapping waveforms referring to the relaxation oscillator. (b) Currents flowing through the C capacitor of the relaxation oscillator. The two falling fronts are separated by 2 ns.
Fig. 3. Two v (t) eigenfunctions referring to the computed with a small and a high number of samples.
y (t)
state function,
. Therefore by looking at Fig. 2(b), we expect the Floquet multipliers to be more accurate if more samples are used; in fact we , with few samples and , have with more samples. Fig. 3 shows the periodic
We know that should be one of the solutions of (45). As already shown, this is not always true espe. It is well known that an implicit cially when or explicit BDF method of order gives an exact solution of (1) and of (5) if the solution is a polynomial of the same order. , which is denoted by We predict the value of , with an explicit BDF method of order , equal to that of the implicit method adopted to solve (1) and (5). , will be close to the solution of the At , if variational problem, which is denoted by is well approximated by a polynomial of order . Formally, we check that the inequality
(46) holds. When the above inequality is not met, the integration time step is suitably shortened. More details on the criterion (46) are in [18], [19]. In practice when the shooting method has converged to a steady-state solution, one more transient analysis is repeated by starting from the steady-state initial conditions. This transient analysis is carried out with a variable time step that is controlled through (46) to form the variational problem with the required accuracy. We end this subsection by underling that errors in the compuinfluence in tation of the Jacobian matrices and of a straightforward way the convergence properties of the Newton method often adopted to solve the shooting problem sketched in (10). However we shall say that the Newton method can converge to a solution even though the Jacobian is incorrect and that errors in the Jacobian matrix do not necessarily reduce the
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accuracy of the computed solution as clearly evidenced by the results shown in Fig. 2(a). V. FREQUENCY-DOMAIN PHASE NOISE ANALYSIS period sensitivity is used to anIn this section, the in the frequency domain.8 alyze the oscillator phase noise To this purpose, consider the Fourier series of the T-periodic unperturbed solution
The discrete spectrum of is thus given by the beats of the fundamental of and of the angular frequency of the perturbation. The term represents the modulation index and it increases when . Consider a generic component of the spectrum de; it contributes to the spectrum noted as according to the following expression:
(47) where the perturbation
and , the solution becomes
. Now introduce
(48) (52)
where (49) and coefficients in (47) are employed unNote that all the changed in (48) since, it is assumed, as an approximation, that noise does not appreciably modify the amplitude of the unper. turbed steady-state solution is small when comUnder the hypothesis that pared to , we can approximate the argument of integral (49) with the linear term of its Taylor expansion (50) obtaining (51)
where is the modulation index and are the Bessel functions [11], which can be approximated as
when . The coefficients describe how phase noise is folded close to . For example, consider noise at and , then and two sidebands at and , with equal amplitude at will appear, showing that noise has been up-converted. In general, it can be concluded that noise sources close to are converted to sidebands close to whose amplitude depend coefficients. on the Now let us consider the particular case in which the perturbaexactly coincides with a multiple of the untion frequency . In this case, perturbed oscillator frequency , that is the phase noise contains a term that grows linearly with time
By comparing (51) to (48), we derive the phase noise term as (53)
Period sensitivity is -periodic and admits the Fourier series representation , where . To gain more insight on phase noise modulation suppose that the input perturbation is a sinusoidal small signal with angular frequency . Phase noise becomes
for 8Part
of this material was previously presented in [5].
and corresponds to a shifting oscillation frequency.
of the
VI. SIMULATION RESULTS is apIn this section, the proposed approach to compute plied to three different autonomous circuits. In particular the second circuit was described in [20] and [21] and has been later used as benchmark in [4] to simulate phase noise. Therefore, we limit our attention to the accurate computation of , that is, the key aspect in the approach presented in [4]. The third cirfunction cuit can be modeled by (29) and in this case the was already computed in [7] to which the results here presented can be compared. Indeed, the same results are obtained when of the inverters is quite low as in [7]. The only the gain “new” circuit is the first one about which there are not already
BRAMBILLA et al.: COMPUTATION OF PERIOD SENSITIVITY FUNCTIONS
Fig. 4. Schematic the Butler oscillator. R = 560 , R = 33 , R = 1.8 k , R = 2.7 k , R = 93 , C = 15 pF, C = 47pF, C = 1200 pF, C = 26 fF, C = 4.18 pF, L = 140 nH, L = 420.82496 nH, L = 6.76557 mH, V ee = 10 V.
0
TABLE I VALUES OF SPICE MODEL PARAMETERS OF T 1 TRANSISTOR
available results in literature, since it is characterized by a resonator (crystal) with a relatively high quality factor Q. Therefunction can be difficultly identified by the apfore, the proaches previously reported in literature. The first circuit considered is the Butler oscillator shown in Fig. 4 [14]. The crystal is characterized by the quality factor when it is unloaded and by the resonant frequency 120 MHz. The electrical equivalent model of the crystal is shown in Fig. 4. Table I reports the values of the Spice model [17]. The parameters of the bipolar junction transistor (BJT) transistor model is characterized by three parasitic capacitors that contribute the corresponding number of state variables [17]. The steady-state behavior of the oscillator was computed with a simulator implementing a version of the shooting method [15], [16]. The simulator adopts the modified nodal analysis, therefore Floquet multipliers refer to linear combinations of the state variables. We expect that one of the Floquet multipliers will be equal to 1 and some other ones will be very close to of the crystal. The computed Floquet 1 due to the high multipliers are reported in Table II. We have labeled as , and the first three of these that are very close to 1.0. By recalling the algorithm outlined in Section II-A, it is difand possibly refers to ficult to identify which among , (step 4 of the algorithm) and then to the eigenvector that gives the initial condition to back-integrate the adjoint variational problem. We could select , but the Floquet multipliers are affected by numerical errors introduced during their computation (we employed routines from the LAPACK library). Thus it is not ensured that is that we are looking for. This situation
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Fig. 5. Period sensitivity function dT : dVc referring to the collector internal node of the T transistor model and the dT : dIo one referring to the Io current through the L inductor.
worsens if is increased. For example, if the Driscoll oscillator in [16] is considered, we have two Floquet multipliers close to 1 that have more than 6 equal digits. The method proposed in this vector does not pose any problem. paper to compute the Fig. 5 shows the two period sensitivity functions: the first one and magnified by 1000 times) with respect (labeled as to the collector internal node of the transistor model to which one of the parasitic capacitances of the transistor is connected, with respect to the o current flowing the second through the inductor in Fig. 4.9 The second circuit considered is the Tow–Thomas oscillator shown in Fig. 6(a) [20], [21]. We chose this circuit since it has been already used as benchmark in literature [6]. The operational amplifiers are assumed to be ideal and are modeled by nullors. The inverter block connected between the and nodes, denoted as , is composed of three cascaded inverters as show in Fig. 6(b); the model of the MOSFET transistors is the BSIM3v3; values of the model parameters are not of the bandpass filter can be reported.10 The quality factor resistors. varied, for example, by changing the value of the The main aspect of this circuit is that when commutes, the o current can have a sharp variation depending on the value of resistor in series to and on the “gain” of the the inverter (three inverters were cascaded to increase gain). The Floquet multipliers and the vector must be carefully computed as we have already detailed in Sections IV-B and IV-C by considering the relaxation oscillator. As far as we know, this aspect has never been considered in any previous paper dealing with the Tow–Thomas oscillator. Fig. 7 shows the voltage waveforms at the and nodes of the circuit, respectively, and the o current (magnified by 1000 times). By looking at the and waveforms we see that their variations become sharper due to the increasing gain along the inverter chain; this implies that o also undergoes a sharp variation. Fig. 8 shows the period sensitivity functions ( 9In this case, and in some others that follow in this section the working period to which waveforms refer can be time shifted [16]. 10A version of the employed simulator pan, the netlists of the circuits shown in this paper and the transistor models are available at the URL http://brambilla.elet.polimi.it.
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TABLE II VALUES OF FLOQUET MULTIPLIERS OF BUTLER OSCILLATOR
(a)
Fig. 8. The dT : dVA and dT : dVB sensitivity functions related to the A and B nodes of the Tow–Thomas oscillator.
(b) Fig. 6(a). Schematic of the Tow–Thomas oscillator. R = 1 k , R = 2 k , 20 nF. (b) Schematic of the inverter. All MOSFET transistors have the same width w = 10m and length l = 1m. Vdd = 5 V, Vss = 5 V.
C =
0
Fig. 9. Schematic of the ring oscillator. Each inverter block encloses the circuit shown in Fig. 6(b). R = 100 k , C = 1 nF.
Fig. 7. Voltage waveforms at the input and output of the inv inverter, corresponding to node A and Z, respectively, and the Io current magnified by 1000 times (y-axis: current [A], voltages [V]).
and ) with respect to the and node voltages of the exhibits a sharp variation too. circuit. We see that This is an interesting aspect that can be qualitatively explained , say at the by noting that just before the “commutation” of
time instant, a small variation of the node voltage causes a corresponding variation in , that is, some time jitter. After a small variation of the voltage does not cause any further and, mainly, being damped by the bandcommutation of pass filter, the voltage variation less influences the subsequent . Thus, we expect that the sensitivity funccommutation of tion just before has a larger magnitude than that it has after . This aspect is clearly evidenced in the last proposed circuit that is shown in Fig. 9 and that models a three-stage ring-oscillator. Consider a time interval along which there is not any commutation of the three inverters. We can think of three identical RC circuits which are working “uncoupled” and driven by constant voltage sources corresponding to the output voltage of each inverter. Only in certain time intervals two of these three circuits are “coupled” through one of the three inverters. We dare say that the RC circuits “synchronise” when inverters commute. If the gain of the inverter is large, then the “coupling time interval” is short and current through capacitors have sharp variations. matrix in (14) is thus the result of the product The of diagonal matrices computed in the time intervals when the RC circuits are uncoupled and of matrices with off diagonal entries computed when the RC circuits are coupled through the
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are obviously wrong too. Once more the adoption of (46) leads to
and , suitably close to 1.
and
, with
VII. CONCLUSION
Fig. 10. The period sensitivity functions referring to voltages at the X, Y, and Z circuit nodes.
Fig. 11. Period sensitivity functions referring to voltages at the X, Y and Z circuit nodes when = 5.
0
inverters. These second matrices largely impact on the computation of the Floquet multiplier and of the period sensitivity functions. The adoption of (46) leads to the Floquet multiplier , and and to the sensitivity function shown in Fig. 10. The above aspect can be better evidenced by reconsidering , the ring oscillator modeled by (29) where initially 1 mS and 2 F. In this case, we have , , and and the sensitivity functions are shown in Fig. 11.11 If we set (i.e. a “very large” value) and do not adopt (46) we have that the matrix in (13) becomes
This paper presents and discusses a general approach to model phase noise in oscillators on the basis of the rigorous Floquet perturbation theory. According to this theory, it is known that the dependence of phase noise or jitter on input perturbations can be reconduced to the determination of the eigensolution of the related variational problem. In practice, the numerical computation of such eigensolution hides several difficulties. In this work we have shown that the above drawback can be overcome since the oscillator phase noise sensitivity on input perturbations can be directly computed from the time-domain steady-state periodic solution. The method we have presented is very robust and allows to handle the case of oscillators with sharp nonlinear behaviors or even with non-Lipschitz functions. We have analytically studied the extreme case in which non-Lipschitz elements undergo instantaneous commutations and we have shown how the state transition function should be corrected. Then, we have investigated the less extreme case, often encountered in practical applications, in which the oscillatory mechanism is driven by element commutations that are not just instantaneous but anyway very fast when compared to the main system dynamics. In this case, only a high accurate steady-state time-domain simulation as proposed can assure a reliable phase noise model. Our argumentation is described and supported by several numerical examples involving real life oscillator circuits. REFERENCES [1] F. X. Kaertner, “Analysis of white and f [2] [3] [4]
[5] [6] [7]
This result clearly underlines how the three RC circuits behave as if they were effectively uncoupled from the numerical point of view but this is obviously wrong. The Floquet multipliers 11This case was considered in [7]. Results shown there have been reobtained with the proposed approach.
[8] [9] [10] [11]
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[12] A. Ben-Israel and T. N. E. Greville, Generalized Inverses. New York: Wiley, 1974. [13] L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev, and L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, Part I. Singapore: World Scientific, 1998. [14] R. W. Rhea, Oscillator Design and Computer Simulation. New York: McGraw-Hill, 1995. [15] D. Frey, “On the equivalence of various methods for finding the periodic steady-state solution of nonlinear networks,” in Proc. ISCAS’99, vol. 5, pp. 322–326. [16] A. Brambilla and P. Maffezzoni, “Envelope following method to compute steady-state solutions of electrical circuits,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 50, no. 3, pp. 407–417, Mar. 2003. [17] P. Antognetti and G. Massobrio, Semiconductor Device Modeling With SPICE. New York: McGraw-Hill, 1988. [18] J. Vlach and K. Singhal, Computer Methods for Circuit Analysis and Design. New York: Van Nostrand Reinhold, 1983. [19] R. K. Bryton, F. G. Gustavson, and G. D. Hatchel, “A new efficient algorithm for solving differential-algebraic systems using implicit backward differentiation formulas,” Proc. IEEE, vol. 60, no. 1, pp. 98–108, Jan. 1972. [20] S. Pavan and Y. P. Tsividis, “An analytical solution for a class of oscillators, and its application to filter tuning,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 45, no. 5, pp. 547–556, May 1998. [21] A. Dec, L. Toth, and K. Sunyma, “Noise analysis of a class of oscillators,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl.I, vol. 45, no. 6, pp. 757–760, Jun. 1998. [22] A. I. Mees, Dynamics of Feedback Systems. New York: Wiley, 1981. [23] F. Hoppensteadt, “Asymptotic stability in singular perturbation problems,” J. Differ.Eq., vol. 15, pp. 501–521, 1974.
Angelo Brambilla received the Dr. Ing. degree in electronics engineering from the University of Pavia, Pavia, Italy, in 1986. Currently, he is full professor in the Dipartimento di Elettronica e Informazione, Politecnico di Milano, Milan, Italy, where he has been working in the areas of circuit analysis and simulation.
Paolo Maffezzoni received tha laurea degree (summa cum laude) in electronic engineering from the Politecnico di Milano, Milan, Italy, and the Ph.D. degree in electronic instrumentation from Universita di Brescia, Brescia, Italy, in 1991, and 1996, respectively. Since 1998, he has been an Assistant Professor in the Dipartimento di Elettronica ed Informazione, Politecnico di Milano. His main research interests are in the area of circuit analysis and simulation. He is the author or coauthor of more than 50 scientific papers, nearly half of which have been published in international journals.
Giancarlo Storti Gajani received the Dr. Ing. degree in electronic engineering and the Ph. D. degree in electronic systems from Politecnico di Milano, Milan, Italy, in 1986, and 1991, respectively. Since 1992, he has been an Assistant Professor, and since 2002, an Associate Professor of Circuit Theory at Politecnico di Milano. His reseach interests have been initially focused on the development of architectures for signal processing (in particular for audio and music applications) and neural networks. More recently, his research interests include nonlinear circuits and nonlinear dynamics.
