Homotopy Method for Finding the Steady States of ... - IEEE Xplore

IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 33, NO. 6, JUNE 2014

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Homotopy Method for Finding the Steady States of Oscillators Hans Georg Brachtendorf, Robert Melville, Member, IEEE, Peter Feldmann, Fellow, IEEE, Siegmar Lampe, and Rainer Laur

Abstract—Shooting, finite difference, or harmonic balance techniques in conjunction with a damped Newton method are widely employed for the numerical calculation of limit cycles of (free-running, autonomous) oscillators. In some cases, however, nonconvergence occurs when the initial estimate of the solution is not close enough to the exact one. Generally, the higher the quality factor of the oscillator the tighter are the constraints for the initial estimate. A 2-D homotopy method is presented in this paper that overcomes this problem. The resulting linear set of equations is underdetermined, leading to a nullspace of rank two. This underdetermined system is solved in a least squares sense for which a rigorous mathematical basis can be derived. An efficient algorithm for solving the least squares problem is derived where sparse matrix techniques can be used. As continuation methods are only employed for obtaining a sufficient initial guess of the limit cycle, a coarse grid discretization is sufficient to make the method runtime efficient. Index Terms—Continuation, homotopy, oscillator simulation, path following method, quartz crystal oscillators, steady state.

I. Introduction

T

HERE is a great deal of interest in simulating the initial transient response [1]–[3] or the limit cycles of oscillators, e.g., [4]–[8] for a short overview. Transient simulation is a reliable method; however, the runtime is prohibitive when the quality factor Q of the oscillator is high. The limit cycle or periodic steady state solution (PSS) calculation is required for phase noise analysis [9], [10], which is crucial for the design of communication systems. The somewhat classical methods for calculating the steady state employ the harmonic balance (HB) technique (based on a trigonometric polynomial), time-domain finite differencing Manuscript received August 30, 2013; revised November 26, 2013; accepted January 15, 2014. Date of current version May 15, 2014. This work was supported in part by the ICESTARS Project under Grant 214911, in part by the ENIAC Research Project ARTEMOS under Grant 829397, and in part by the FWF under Grant P22549. This paper was recommended by Associate Editor A. Demir. H. G. Brachtendorf is with the Department of HW/SW Design, University of Applied Science of Upper Austria, Steyr 4400, Austria (e-mail: [email protected]). R. C. Melville is with the Center for Solar-Terrestrial Research New Jersey Institute of Technology, Newark, NJ 07103 USA. P. Feldmann is with IBM John Watson Research Center, Yorktown Heights, NY, USA. S. Lampe was with the University of Bremen, Bremen 28359, Germany, and is currently with Sikora AG, Bremen 28307, Germany. R. Laur is with the University of Bremen, Bremen 28359, Germany. Digital Object Identifier 10.1109/TCAD.2014.2302637

or shooting methods. The resulting set of nonlinear equations is usually solved by Newton-like methods. Unfortunately, Newton-like methods often fail to converge toward the limit cycle of high Q oscillators employing quartz crystal or cavity resonators when the initial condition is not close enough to the desired solution and converges to the unstable DC-operating point solution instead. This result is independent of the specific method employed, e.g., finite differencing, shooting or HB techniques. One method to avoid convergence to the trivial DC solution is the use of deflation techniques [11]; however, there is no guarantee that when applying Newton’s method to the deflated system convergence to the limit cycle will be achieved. Deflation techniques are methods which try to eliminate known solutions from the system of algebraic equations. Computational experience suggest that deflated systems often diverge or still converge to the trivial solution; because of the deflation the condition number [12] of the system’s Jacobian matrix grows beyond any bound and convergence stalls. A specific problem encountered with autonomous circuits is that there is an infinite number of time-shifted solutions, making the system of equations under determined because the time period T is an additional unknown. One solution for dealing with this problem is to add an equation which specifies the solution. The technique used in [13] fixes the amplitude of one unknown at t = 0. Alternatively, one can specify the solution employing the Poincaré map method, see [3], [14] and references therein. This paper addresses this problem of nonconvergence by employing homotopy or continuation techniques for finding the limit cycle directly. Homotopy techniques for calculating all operating points are well-established methods [15]–[18], especially for calculating all DC operating points of a circuit [4], [15], [16], [18]–[23] whereas to the authors’ knowledge only a few attempts for using continuation techniques for determining limit cycles have been published yet [7], [13]. For increasing the range of convergence of Newton-type methods the concept of a probe voltage has been introduced by different authors [5]–[8], [14], [24]. The idea of a probe is as follows: The probe voltage charges up the circuit by supplying periodic energy into the system. When the steady state is found, the probe current is zero, i.e., no additional energy is required anymore for ramping up the oscillator and the probe can be removed without any change of the steady state waveforms. The probe voltage source supplies the circuit with additional energy such

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that convergence to the trivial DC solution may be avoided. The probe technique may be understood as pumping additional energy into the circuit, charging up the circuit and pulling away from the operating point. The steady state oscillation is obtained when the probe current is zero. These methods vary both the probe voltage waveform and the frequency of the probe until the limit cycle will be found. Though this method seems to be more reliable than directly calculating the steady state, i.e., the probe current is zero, convergence is still not always obtained for high-Q oscillators as long as the initial estimate is not close enough to the steady state solution. In [7], a homotopy method is used in conjunction with the probe technique. Ma et al. [13] employed a homotopy technique by adding an augmenting equation (first proposed by Kundert et al. [4]) in conjunction with transient analysis. The choice of the augmenting equation is left to the user. Adding an algebraic equation transforms the system of (implicit) ordinary differential equations (ODEs) to a system of DAEs. Transient methods may run into severe problems when the index of a system of DAEs is two or higher. Unfortunately, the paper does not give any hint which index the augmented system has in practice. The paper presented here proposes a different homotopy method. First, a finite differencing method, sampling balance, or HB technique is used. Second and more important, no algebraic augmenting equation has been added. Therefore, a one parameter homotopy method cannot be applied. Instead, a two-parameter homotopy method is employed which differs from known multiparameter continuation techniques [17], [25], [26]. The method presented here is well suited for oscillators with a high quality factor such as quartz crystal and cavity oscillators. For high Q oscillators it is relatively easy to obtain a good estimate of the frequency. The technique from [27] based on an eigenvalue analysis is summarized in the Appendix. However, an estimate of the signal’s amplitude is very difficult. One explanation is as follows: the condition number of Jacobian matrix at the DC solution is extremely high. This means that the sensitivity with respect to perturbations of circuit parameters is very high too. Consequently, an a priori estimate is very difficult or even impossible. This paper is organized as follows. Section II gives a short review of periodic limit cycle calculations as a boundary value problem (BVP) of a system of ordinary differential algebraic equations (DAEs). Section III is a short summary of predictor– corrector homotopy techniques. The methods differ in the ways the corrector is realized: pseudo-arclength, hyperspherical and minimal least squares correctors are illustrated. For a more detailed analysis of homotopy techniques, the authors refer to standard textbooks such as the one of Allgower and Georg [28] and Seydel [29]. However, as we will see, standard techniques are not well suited for oscillators. The reason is that they are autonomous DAEs where any time-shifted solution also solves for the DAEs, i.e., if xss (t) is a steady state solution, xss (t + t0 ), t0 ∈ R is a solution too. This leads to an under-determined system where additional constraints are required. Therefore, Section IV presents a novel pathfollowing technique that is suited for the oscillator problem

Fig. 1.

Poincaré map with poincaré hyperplane .

under study. Section V illustrates the homotopy method that is particularly useful for oscillator circuits. Section VI presents the simulation results. We note here that our method can be used for forced oscillators as well and then is even simpler to implement than the autonomous case.

II. Limit Cycle Calculation of Oscillators We consider the autonomous circuit DAE of dimension N d f (x(t)) = i(x(t)) + q(x(t)) = 0 (1) dt where x(t) is the vector of unknown voltages and currents, i : RN → RN the vector sums of currents representing Kirchhoff’s current law and some vector sums of voltages representing Kirchhoff’s voltage law. Furthermore, q : RN → RN is the vector of charges and magnetic fluxes. We assume that: 1) the DC operating point is unstable, i.e., i(xDC ) = 0 is an unstable solution of (1); 2) the circuit DAE exhibits a nontrivial (i.e., not DC) periodic limit cycle xss (t) = xss (t + T ) ∀ t where the period T is a priori unknown. If xss is a periodic steady state solution of (1), i.e., xss (t) = xss (t + T ) it can be shown easily that xss (t + t), t, t ∈ R is a solution of (1) as well by employing the chain rule. Fig. 1 shows the phase portrait of an oscillator with Poincaré hyperplane. In a well conditioned Poincaré section the normalized flow vector at x0 and the normalized vector to the hyperplane have an inner product equal to one. The direction vector of the flow at x0 and the normal vector of the hyperplane coincide. An approximation of the hyperplane plays an important role in determining a specific solution. We assume that (1) has a unique solution for any set of consistent initial conditions x(t) = (y, t), where (y, t) is the so called state transition function and y an arbitrary initial condition. Hence, the boundary value problem can be rewritten as (y, T ) − y = 0. This fixed-point equation might be solved by (multiple) shooting techniques. The resulting set of equations is still under determined. Therefore, the solution will be defined by ˆ (Fig. 2) such that the flow vector of choosing a hyperplane

BRACHTENDORF et al.: HOMOTOPY METHOD FOR FINDING THE STEADY STATES OF OSCILLATORS

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Here, F, X, I, Q are vectors of the size N · K and ∇ represents the discretized time derivative which has a block diagonal form ∇ := [∇ nm ] , 1 ≤ n, m ≤ N ∇nn if n = m ∇nm := 0 if n = m.

Fig. 2.

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Under the assumptions of the periodicity of the solution and an equidistant grid spacing, ∇nn is a circulant matrix 1 circ (a0 , a1 , . . . , aK−1 ). (4) ∇nn = t The coefficients ai are independent of the time step. One choice for the finite difference scheme (among others) are Gear’s higher order backward differentiation formulas (BDF) [31]. For backward Euler one gets a0 = 1, a1 = −1 and for BDF-2 the coefficients a0 = 23 , a1 = −2, a2 = 21 and zero otherwise. However, in [32], it has been shown that BDF methods based on algebraic polynomials are not suited for oscillators with large quality factor such as quartz crystal oscillators because they do not preserve but dissipate energy. Modified BDF methods have therefore been proposed based on trigonometric polynomials which are nearly dissipation free [33]. BDF type methods lead to highly sparse matrices. Alternatively high order Rung–Kutta method are suited too [34]. In [35], Simpson’s rule has been applied for discretizing the PSS. Alternatively, the coefficients ai might be chosen from a trigonometric expansion of the waveforms. Let F be the coefficient matrix of the discrete Fourier transform and

Illustration of the shooting method for autonomous systems.

= j diag {. . . , −2ω, −ω, 0, ω, 2ω, . . . } ∈ CK×K Fig. 3.

Illustration of the shooting method by Aprille et al. [30].

the time derivative in the frequency domain where ω = Then, ∇nn is obtained from

the initial state x(j) of the jth iterate and the normal vector coincide. The initial value problem of the DAE (1) is solved ˆ is crossed and for the iterate y(j) at the jth iteration until ˆ and the vector of the flow x˙ (t) have the normal vector of the same direction (i.e., their inner product is > 0). From the sensitivity information one updates the initial vector of y until convergence is achieved, i.e., an initial state is found which fulfills the boundary value problem too. An alternative method has been presented by Aprille and Trick [30], fixing the value of a selected waveform at a particular gridpoint (Fig. 3). Unfortunately, this technique can fail if the fixing value is outside the amplitude range of the selected waveform in steady state. For further details see [4], [30]. Alternatively, the boundary value problem might be solved by finite difference or the HB techniques. For solving the DAE (1) numerically, it might be discretized using some sort of finite difference schemes choosing K nodes T on an equidistant mesh with the grid spacing t = K , where T is the a priori unknown period of the limit cycle

F (X) = I(X) + ∇ Q(X) = 0.

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2π . T

∇nn = F −1 F. This method is sometimes referred to as sample balance method. The matrix ∇nn is full. Alternatively, a spline basis may be chosen [36]. The eigenvectors of the circulant matrix ∇nn are the column vectors of F −1 . Only the eigenvalues differ for different ∇nn . As T or alternatively ω is an additional unknown, it is common practice to define a solution by fixing the phase of one selected waveform, without loss of generality x1 (t) 1 T 2π x1 (t) cos t dt = 0. (5) T 0 T The augmentation (5) reduces the continuum of time-shifted solutions to two distinct solutions which are 180° phase shifted. As already discussed above, (5) may lead to an index increase of the DAE system which should be avoided due to numerical instabilities. Alternatively, in [35] (approximations of), bi-orthogonal eigenvectors of the Jacobian corresponding to the generalized characteristic Floquet multiplier λ = 1 are determined and used as an augmenting equation, see also [37] for their numerically efficient calculation. For deriving the homotopy method in Section V, it is advantageous to rewrite a circulant matrix as a Toeplitz matrix.

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Fig. 4.


Fig. 5.

Simple continuation.

Pseudo-arclength continuation technique.

III. Homotopy Methods A Toeplitz matrix is the sum of a symmetric and a skew symmetric matrix ⎡

c 1 r2 ⎢ .. ⎢ c2 . ⎢ ⎢ T (c, r) := ⎢ ... . . . ⎢ ⎢. ⎣ ..

... .. . .. . .. . cp . . . . . .

⎤ . . . rp .. ⎥ .⎥ ⎥ . . .. ⎥ . . .⎥ ⎥ ⎥ .. . r2 ⎦ c2 c1

f (x(λ), λ) = 0 f : Rn × R → Rn . (6)

The column vector c and the row vector r uniquely define the Toeplitz matrix. Choosing c = [a0 , . . . , ap−1 ]T and r = [ap−1 . . . , a1 ] gives a circulant matrix circ (a0 , a1 , . . . , ap−1 ) = T (c, r).

We consider the solution of parameter dependent systems of nonlinear equations

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Unfortunately, HB or finite difference techniques when applied to oscillators often fail to converge to the limit cycle when the initial estimate is not in the vicinity of the limit cycle.1 Nonconvergence is significantly a problem for high Q oscillators. Whereas it is relatively easy to estimate the frequency from the solution and its Jacobian at the operating point, the PSS waveform is extremely difficult to estimate. Therefore, in Section V a homotopy technique is presented with improved convergence radius. As standard continuation methods are not well suited for oscillator circuits, due to the additional unknown period, we start with a summary of homotopy techniques in Section III followed by a generalization in Section IV that is suited for the system under investigation. 1 An estimate for the frequency specifically for high Q oscillators has been presented in [27].

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Engineers are often interested in tracking the path x(λ). Typical examples include oscillators where the parameter λ might be the amplification of the feedback. It is well-known from the Andronov–Hopf theorem that when the absolute value of the open loop amplification exceeds unity a limit cycle bifurcates from the operating point. Another application comes from homotopy or continuation techniques which where successfully applied e.g., for calculating all operating points [15]–[17]. Assume that (8) is easy to solve for λ = 0 and the solution of interest is obtained at λ = 1. Continuation techniques are efficient methods for tracking the path within the range 0 ≤ λ ≤ 1 by calculating the zero curve for a finite number of λν , ν = 1, 2, . . . . In this paper, we assume that: 1) the solution at λ = 0 is unique; 2) x(λ) does not bifurcate; 3) x(λ) stays bounded.2 The implicit function theorem states if the zero curve x = x(λ) is locally parametrizable by λ. Theorem 1: Let f : Rn × R → Rn , f = f (x, λ) a continu ously differentiable function and f (x0 , λ0 ) = 0. If ∂f ∂x (x0 ,λ0 ) is of full rank n, then there exists locally a continuously differentiable function x = x(λ) such that x(λ0 ) = x0 and ∂f dx ∂f f (x(λ), λ) = 0 and + = 0. ∂x dλ ∂λ Proof: The proof can be found in [38] and [39]. 2 This condition is fulfilled in physical systems because the power consumption is bounded.


Fig. 7.

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Moore–Penrose pseudoinverse homotopy.

that the Newton corrector is orthogonal to the direction vector, namely

Fig. 6.

Pseudo-arclength homotopy method.

T

T

Let y := [x λ] and ds be the step size of the continuation method. Consider the simple continuation technique illustrated in Fig. 4. The predictor is given by

xν yˆ ν+1 = λν + dsν where the direction vector is tν = [0, . . . , 0, 1]T . The under determined system (8) is augmented by demanding that the search direction must be orthogonal to the predictor, i.e., λν+1 = const. It is obvious that this simple homotopy method does not allow continuation of a branch past folds because the implicit function theorem is violated. As long as a solution path exists that does not bifurcate and stays bounded, the requirements of the theorem can be fulfilled by a reparametrization. Pseudo-arclength continuation is one method that allows ˙ T be the normalpath-following beyond folds. Let t := [˙xT λ] 3 ized direction or tangent vector of the solution branch, i.e., t2 = 1. The predictor is given by

x + dsν x˙ . yˆ ν+1 = ν λν + dsν λ˙ The corrector solves (8) by Newton’s method where the under determined system will be augmented by demanding 3 Please note that in this Section t is a vector in Rn+1 and has no connotation of time. Furthermore, x˙ , λ˙ represent derivatives to s.

f (x(sν+1 ), λ(sν+1 )) = 0 (9)

x˙ (s ) x˙ (s ) x(sν+1 ) x(sν ) − + sν ˙ ν ⊥ ˙ ν . λ(sν+1 ) λ(sν ) λ(sν ) λ(sν ) A graphical interpretation of the pseudo-arclength homotopy method is given in Fig. 5. The tangent vector spans the 1-D kernel or nullspace of the Jacobian of (8) at yν+1 tν+1 ∈ ker fx (yν+1 ) fλ (yν+1 ) . A stable calculation of the nullspace4 of fx (yν+1 ) fλ (yν+1 ) might be performed either by a QR-factorization or singular value decomposition (SVD) that unfortunately requires O(n3 ) operations. Furthermore, QR-factorization and SVD do not exploit the sparsity structure of the Jacobian. However, for large systems the same result can be achieved by solving fx (yν+1 ) fλ (yν+1 ) x˙ (sν+1 ) 0 ˙ ν+1 ) = 1 ˙ ν) x˙ T (sν ) λ(s λ(s for tν+1 . The first equation guarantees that the solution is within the nullspace of [fx fλ ]; the second equation fixes the length of the solution directional vector. Note that tν+1 must finally be normalized to unity, i.e., tt . However, the 2 solution is not unique; if tν+1 is a solution then −tν+1 is a solution too. Therefore, for breaking the tie the direction with the smaller comprising angle with the previous search direction T is chosen, i.e., tν+1 tν > 0, because in the limit for ds → 0 both vectors coincide. Fig. 6 shows the pseudo-code of the arclength homotopy. An alternative augmenting idea has been presented in [40]. First, the predictor is along the tangent t as for pseudoarclength and hypersphere homotopy. The Newton corrector y(j) at any iteration j is calculated by demanding that the 4 The nullspace or kernel of a matrix A is defined as: ker(A) = {x | A x = 0}. If A is a square matrix of full rank then the kernel is a singleton: ker(A) = {0}.

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corrector is orthogonal to the nullspace (where fy is the Jacobian matrix of f ) of ∂f (j) (j) (j) (j) (yν+1 ) := fy (yν+1 ) := fx (yν+1 ) fλ (yν+1 ) ∂y namely

ˆ of A := A ˆ z is regular, 3) If the leading n × n matrix A the solution is efficiently calculated by

−1

−1 ˆ b ˆ z xT t A A , t := , α := − T0 x0 := 0 −1 t t x = x0 + α t.

(j) yν+1

⊥ ker

(j) fy (yν+1 ).

Please note that according to our assumptions the nullspace is 1-D. The requirement is fulfilled if and only if (j) yν+1 2

is minimized over all y that are solutions of the under determined equation that can be proven easily: Lemma 1: Let x = A+ b be the least Euclidean norm solution of the under determined system A x = b, A : Rn+1 → Rn where the superscript + denotes the Moore– Penrose pseudoinverse and let rank A = n. Then, x ⊥ ker A. Proof: Let x0 be any solution of the linear system and t, t2 = 1 the unit basis vector spanning the nullspace of A, i.e., A t = 0. By assumption the kernel is 1-D. Hence, the set of solutions is given by x(α) = x0 +αt. The solution with minimal xT t Euclidean norm is obtained for α = − t T0 t , α ∈ R. Multiplying the minimal norm solution with t T proves the orthogonality. Formally, the least Euclidean norm solution can be written as (j)

(j)

(j)

yν+1 = −fy (yν+1 )+ f (yν+1 ) (j)

where fy (yν+1 )+ denotes the Moore–Penrose inverse of the Ja (j) cobian matrix. This process is to be repeated until f (yν+1 ) 2 becomes sufficiently small. The process starts from the predictor yˆ ν+1 . A graphical interpretation is shown in Fig. 7. Remark 1: Note that the least Euclidean norm solution is not invariant to a scaling of the coefficient matrix. On calculating the Moore–Penrose pseudoinverse x = A+ b, A : Rn+1 → Rn . 1) Singular value decomposition: Let A := U 0 V T the singular value decomposition (SVD) of A. Then, x =

−1 U T b gives the solution with least Euclidean V 0 norm. 2) QR-decomposition: Let A := Q R S the QRfactorization of A. As A has full rank, R is an upper regular matrix of dimension n × n and S a column

−1 T R Q b vector. Let x0 = a solution of A x = b,

−1 0 R S the vector t := spans the nullspace of A. Let −1 xT t

α := − t T0 t , then x = x0 +αt is the minimal norm solution of the under determined system. Alternatively, if AT is T available, T from aT QR-decomposition of A one obtains: A := R 0 Q where R is an upper matrix of full rank. The minimal Euclidean norm solution is therefore

−T R b. x=Q 0

ˆ is sparse in most applications this technique As A leads to much smaller CPU time than an SVD or QRfactorization. 4) Alternatively, a fast algorithm can be achieved by first calculating

the nullspace. Let ˆt be a column vector such A that ˆT has full rank, i.e., the last search direction of t the homotopy technique. Calculate the nullspace of A by solving for t

A 0 ˆt T t = 1 and finally scale t to unity, i.e., tt . Obtain x by solving 2

A b x= tT 0 where also sparse matrix techniques might be employed. Note that both matrices only differ in the last row. Therefore, the second equation for x can be solved efficiently by employing the Sherman–Morrison–Woodbury

A formula [41] when an LU-decomposition of ˆT is t available. Hence, this method leads to a similar CPU time as the technique presented before. Note that the latter two techniques lead only to a small overhead of CPU time compared with the pseudo-arclength and hypersphere homotopy technique when the LU-factorization is stored because at each Newton iteration only one additional solve with an already decomposed system must be performed. As, in general, fewer Newton iterations must be performed, this technique is superior to the ones shown before. A generalization for a map A : Rn+m → Rn is given in the Appendix.

IV. Generalized Homotopy Technique In this Section, we generalize the single-parameter homotopy method shown in Section III to a two-parameter continuation procedure that is better suited for limit cycle calculations of autonomous systems as we will see in the next Section. We consider the solution of a system of nonlinear equations that depends on two parameters f (x(λ1 , λ2 ), λ1 , λ2 ) = 0 f : Rn × R × R → Rn .

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Let y := [xT λ1 λ2 ]T and ds be the step size of the continuation method and let t := [˙xT λ˙ 1 λ˙ 2 ]T , t2 = 1 a normalized direction vector that is tangent to the zero plane. We assume that fy has always full rank n. Note that t is not


Fig. 8.

2-D homotopy with least squares.

Fig. 9.

2-D homotopy combining pseudo-arclength homotopy and least squares.

unique because the nullspace is 2-D. The calculation of t is shown below. The predictor is given by ⎤ xν + dsν x˙ ν = ⎣λ1ν + dsν λ˙ 1ν ⎦ . λ2ν + dsν λ˙ 2ν ⎡

yˆ ν+1

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The under determined system might be augmented by requiring that the specific solution with least Euclidean norm will be chosen (j) y = min 2 that is y(j) = −fy (y(j) )+ f (y(j) )

When solving (10) by Newton’s method one obtains an under determined system with a 2-D nullspace fy (y(j) ) y(j) = −f (y(j) ),

y(j+1) = y(j) + y(j) .

where the superscript + denotes again the Moore–Penrose pseudoinverse and (j) (j) (j) fy (y(j) ) := fx (yν+1 ) fλ1 (yν+1 ) fλ2 (yν+1 ) .

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The proof that y(j) ⊥ ker fy (y(j) ) is similar to lemma (1). A graphical interpretation is depicted in Fig. 8. An alternative might be obtained by combining the concepts of pseudoarclength continuation and minimal least squares homotopy (Fig. 9). First, we restrict the search of the zero curve to the hyperplane that is orthogonal to the search direction tν : tν ⊥ yν+1 − yˆ ν+1 . For sufficiently small step-sizes ds there is always a hyperplane that cuts the 2-D zero curve because t is tangent to it. Second, the resulting set of equations from Newton’s method have a 1-D kernel. The under determined system is therefore solved in a least squares sense. Fig. 9 illustrates this method. The zero curve, that lies in the hyperplane = {yν+1 | tν ⊥ yν+1 − yˆ ν+1 , yν+1 , yˆ ν+1 ∈ Rn+2 } spanned by the orthogonality condition, is 1-D. The Newton updates lie y(j) both T in the hyperplane and are orthogonal to the kernel fyT t . Now, we return to the problem of obtaining the tangent vector. A new tangent vector t := tν is chosen such that it comprises the smallest angle with the previous one. Let ¯t := tν−1 the previous vector with unit length. We therefore have to choose t such that the inner product t T ¯t is maximal. The following lemma shows how to calculate t efficiently when the Jacobian is sparse. Lemma 2: Let A be an n × (n + 2) matrix of full rank T and let ¯t a vector of dimension (n + 2). Assume that AT ¯t is of full rank (n + 1). Then

+ 0 A t := ¯T 1 t yields the vector with the smallest comprising angle with ¯t that lies in the nullspace of A, where the superscript + denotes the Moore–Penrose pseudoinverse. Furthermore, let t1 , t2 be two orthogonal vectors of unit length that span the kernel5 of A. Then, t is obtained from t = α t1 + β t2 with α=

¯t T t1 ¯t T t2 , β= . (¯t T t2 )2 + (¯t T t1 )2 (¯t T t2 )2 + (¯t T t1 )2

Proof: First, by construction t lies in the nullspace of A. ¯t T t = t1 . The Second, because ¯t T t ≡ 1 by construction t 2 2 condition for a smallest comprising angle, i.e., maximizing the inner product of two vectors of unit length, is here equivalent with minimizing t2 . Solving for t by the Moore–Penrose inverse leads by definition to the solution with least Euclidean norm that proves the first part of the lemma. The second part can be proven as follows. From the augmenting equation, we get ¯t T t ≡ 1, namely

T α ¯t t1 ¯t T t2 = 1. β Minimizing t2 is equivalent with the under minimizing T determined equation. Therefore α β must be orthogonal 5 An orthogonal basis of the kernel might be obtained by a QR-factorization or SVD.

Fig. 10. 2-D homotopy combining pseudo-arclength and least squares continuation methods.

to the The nullspace T of the equation above is nullspace. ker ¯t T t1 ¯t T t2 = −¯t T t2 ¯t T t1 . Hence, solving

T ¯t t1 ¯t T t2 α 1 = 0 −¯t T t2 ¯t T t1 β yields the desired solution with least Euclidean norm ⎤ ⎡ ¯t T t1

⎢ (¯t T t2 )2 +(¯t T t1 )2 ⎥ α ⎥ =⎢ ⎦ ⎣ β ¯t T t2 (¯t T t2 )2 +(¯t T t1 )2

that proves the second part of the lemma. Fig. 10 illustrates the pseudo-code of this homotopy technique.

V. Homotopy for Oscillators We assume that the oscillator’s DAE (1) has a unique solution for any initial condition x(0) = x0 and that there is only one stable limit cycle of interest. Furthermore, suppose that x0 = xDC , where xDC is the DC or operating point solution, and that the operating point solution is unstable, i.e., the Poincaré maps P(x(j) ) = x(j) , T (j) , x(0) = x0 (11) converge asymptotically toward a single point on the limit cycle, that is lim x(j) = xss (ˆt ), ˆt ∈ R

j→∞


where xss (t) = xss (t + T ), t ∈ R is the limit cycle of the oscillator. In practice, the fixed point equation above shows poor convergence for oscillators with high quality factors. Computational experience shows that applying (damped) Newton techniques to the fixed point problem (11) exhibits either poor convergence, convergence toward the trivial DC operating point, or even divergence. We therefore suggest to solve the boundary condition x − (λ1 x + (1 − λ1 ) x0 , T ) = 0

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Fig. 11.

Simple quadrature circuit.

Fig. 12.

20 MHz Clapp oscillator schematic.

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where λ1 is a real homotopy parameter 0 ≤ λ1 ≤ 1. Note that for λ1 ≡ 0 one obtains the solution of an initial value problem with initial condition x0 whereas for λ1 ≡ 1 the periodic steady state of interest is obtained. (12) is still under determined in the autonomous case because T, f or alternatively ω are additional unknowns of the oscillator problem.6 We therefore employ the 2-D homotopy outlined in Section IV where λ2 = T/T0 or vice versa λ2 = f/f0 . T0 = 1/f0 is the initial estimate of the period;7 hence x − (λ1 x + (1 − λ1 ) x0 , λ2 ) = 0. Simulation results suggest that the choice of the initial period is not overly crucial for the method. For obtaining the solution of (12), (multiple) shooting methods can be employed [30], [42]. Using preconditioned Krylov subspace techniques [43] the solution can be obtained efficiently. Alternatively, finite difference methods in conjunction with Krylov subspace techniques can be used as well. Let t0 = K2πω0 , ω0 = 2π the T0 equidistant8 initial grid spacing of a mesh with K knots, X0 the numerical solution of the initial value problem of the DAE (8) for the initial condition x(0) = x0 on the specified mesh. Let ∇nn from (4) be written as the sum of two Toeplitz matrices ∇ˆ nn and ∇¯ nn [compare (6) and (7)] 1 (13) T (c, 0), c = [a0 , . . . , aK−1 ] ∇ˆ nn (ω) := t 1 ∇¯ nn (ω) := T (0, r), r = [aK−1 , . . . , a1 ] t 1 circ (a0 , a1 , . . . , aK−1 ) ∇nn (ω) := ∇ˆ nn (ω) + ∇¯ nn (ω) = t Note that ∇ˆ nn is a lower triangular where t = K2πω , ω = 2π T ¯ and ∇nn a strictly upper triangular matrix. As noted above, ∇nn is a highly sparse banded matrix when (modified) BDF methods are employed [32]. The homotopy technique sweeps from an initial value problem (λ1 = 0) to a periodic boundary value problem (λ1 = 1) ˆ ¯ ˆ =0 Q(X) + ∇(ω) Q(X) F (X, λ1 , ω) = I(X) + ∇(ω) ˆ = λ1 X + (1 − λ1 ) X0 , 0 ≤ λ1 ≤ 1 X (14) where X0 is an initial estimate of the waveforms. Note that in (14) F is a map F : Rm × R × R → Rm , where m = N · K. Associate now the normalized angular frequency ω/ω0 with the second homotopy parameter λ2 in Section IV. Either of 6 This is true only for the autonomous case. If T is known a priori, as it is the case for some driven oscillators, the homotopy technique of Section III might be used for tracking (12). 7 For an estimate of the period or frequency see [27] and the Appendix. 8 Equidistant spacing is only assumed for ease of presentation.

the 2-D homotopy methods are therefore started with the initial conditions (λ1 , λ2 ) = (0, 1). In practice, the homotopy method is used for evaluating sufficiently good initial estimates for a (damped) Newton method. Therefore, only a coarse estimate of the limit cycle is necessary before starting the final Newton run; hence, K can be chosen relatively small, speeding up the simulation. For the final Newton iteration an interpolation to a fine grid is performed, i.e., KNewton > Khom . VI. Results The circuit examples were taken from the testbench of the European Research Project ICESTARS [44]. Though the emphasis of the paper lies in high Q oscillators, we consider first the quadrature oscillator Fig. 11 that is of very low Q.9 Due to the symmetry of the circuit the waveforms at the output are phase-shifted by ∠ 90 with regard to the input waveforms. First the operating point is calculated and an eigenvalue analysis is performed, from which one obtains an initial estimate of the frequency f0 = 1.3096·108 Hz [27]. The method is summarized in the Appendix. Alternatively, in [14] an FFT based method is proposed based on an initial transient simulation. For oscillators with low Q this method would lead to better initial estimates. The differential algebraic equations are discretized by K = 64 grid points. Due to the low Q we expect that the initial guess and the final value differ drastically. Indeed, we finally got a ratio f/f0 = 0.056086, i.e., roughly a factor of 18 between the estimate and the numerical result of the frequency of the limit cycle. A better initial 9 Oscillators with a low or medium Q can efficiently be simulated by alternative techniques discussed in this paper as well.

876


TABLE I

TABLE II

Results for Several Oscillators

Simulation Results for Oscillators in MOS and Bipolar Technology

estimate might be obtained employing Kurokawas’s condition [24]; however, the method found the limit cycle within a reasonable number of iterations. The homotopy technique of Section V required 29 steps. The initial X0 has been obtained by performing a transient analysis from a slight perturbation of the operating point (x0 − xDC )2 = 3.0·10−3 V on the time interval [0, T0 ]. A possible improvement might be to perform an initial transient calculation for a few periods such that the rapidly decaying states vanish completely and therefore cannot disturb the continuation method.10 Alternatively, we simulated as a high Q example the Clapp oscillator depicted in Fig. 12. The advantage of two-parameter homotopy method comes in for high Q oscillators where e.g. Kurokawa-based techniques [8] fails to converge unless the initial conditions are highly accurate. The initial estimate of the frequency is f0 ≈ 20 MHz. Due to the circuit’s high Q we obtained with the method in [27] the ratio f/f0 = 9.9998·10−1 as expected for a high Q oscillator. The higher the Q of the oscillator the better is the initial estimate of the free running frequency by the eigenvalue method outlined in the Appendix [27]. In contrast the estimate of the amplitude of the oscillation is very difficult to achieve which is reflected by the extremely high condition number of the Jacobian matrix at the operating point. Results were obtained after 16 steps. A Colpitts quartz crystal oscillator shows similar results compared with the Clapp circuit: The initial frequency obtained by an eigenvalue analysis is f0 ≈ 20 MHz and is in good agreement with the final solution f/f0 = 9.9996 · 10−1 . Again 16 steps were needed. Table I shows initial estimate f0 , final ratio f/f0 and number of iterations for various oscillators. The initial frequency estimates are accurate for high Q oscillators only. However, the algorithm can find the steady state solution even in reasonable steps for inaccurate initial guesses. The Table II shows simulation results for several industry relevant circuits of a medium quality factor. The number of iterations is acceptable for all circuits under test. It has been already mentioned that the pseudoinverse is not invariant with regard to scaling. Though first tests suggest that the number of iterations is only slightly influenced by scaling, this topic needs some additional investigation, both theoretically and practically. For very low Q systems, such as the quadrature oscillator the estimate of the frequency by 10 It should be noted that this low Q circuit is better simulated by standard transient analysis because of its short settling time, hence continuation is not necessary in this case and proves mainly the usefulness of the algorithm.

an eigenvalue analysis is extremely poor so that one has to look for alternatives for initializing the frequency such as Kurokawa’s condition [24] or the technique from [35]. VII. Conclusion This paper shows that 2-D homotopy techniques in conjunction with a predictor-corrector scheme is a useful means for obtaining the limit cycles of oscillators. A continuation method is proposed that sweeps continuously from an initial value to a periodic boundary value problem of the underlying circuit’s DAEs. The under determined systems are solved by a Moore–Penrose pseudoinverse. The main assumptions are as follows: 1) instability of the operating point; 2) uniqueness of the initial value problem for given initial conditions; 3) the solution path is both bounded and does not bifurcate. The path following method has been initialized by first calculating the operating point, second an eigenvalue analysis for obtaining an initial estimate of the frequency and finally an initial transient simulation over one estimated period for a perturbed operating point as the initial solution. The eigenvalue analysis exhibits good results when the oscillator is of sufficiently high Q but is rather poor for low Q oscillators. Kurokawa’s condition might be preferred for these circuits. The initial waveforms can be improved by calculating the transient response for a few estimated periods because small time constants decay rapidly and can therefore not contribute to convergence problems of the homotopy method. The Moore– Penrose pseudoinverse is not invariant with regard to scaling. Though tests suggest that the number of iterations is only slightly sensitive to scaling this topic has to be investigated in more detail. Appendix A Initialization of Homotopy Method The idea behind the initialization is provided by a theorem of Grobman and Hartman [45]. First, the unstable operating point is calculated. The theorem emphasizes the fact that orbits of a nonlinear ODE system are similar to the orbits of the linearized system, known as small signal analysis. Therefore, the initialization is performed in two steps, calculation of the unstable operating point first and the solution manifold of the linearized system second. Let x¯ 0 be the operating point of


(1), i. e. i(¯x0 ) = 0. Furthermore let G := ix (x) x=¯x0 and C := qx (x) x=¯x0 be the Jacobian matrices evaluated at x¯ 0 . One obtains the homogeneous linear differential equation G x + C x˙ = 0.

The second equation guarantees that x has no component in the kernel of A, i.e., exhibits least Euclidian norm solution.

(15) Acknowledgment

A solution of the generalized eigenvalue problem GY +CY =0

877

(16)

can be obtained by the QZ algorithm. is a matrix in Jordan form of the generalized eigenvalues λi and Y a matrix of all generalized eigenvectors, ordered column wise. In practice,

reduces to a diagonal matrix. The general solution of the homogeneous linear system is therefore given by X(t) = Y exp t. Hence, the column vectors of X span the solution space of the linearized DAE. X(t) is referred to as the fundamental or Wronski matrix. The state transition matrix (t) is therefore (t) = X(t) X−1 (0) = Y exp t Y −1 . For a given initial condition x0 = x¯ 0 , where x¯ 0 is the operating point, the solution of the linear system is calculated by (t) x0 for any t. In what follows x0 = x¯ 0 , i. e. the unstable operating point shall not be the initial value. Moreover, let x0 = x0 − x¯ 0 2 . Due to the instability of the operating point there exists at least one pair of conjugate complex eigenvalues λ, λ∗ with corresponding conjugate complex eigenvectors y, y∗ , y2 = 1 with positive real part. Therefore it makes sense to initialize partial DAE x0 the by (i) ω (0) = Im{λ} and (ii) x (t ) = x ¯ y exp (j t1 ) + + 0 0 1 0 2 x0 ∗ √ + y exp (−j t ), x ∈ R , j = −1. 1 0 2 Appendix B On Calculation of Moore–Penrose Pseudoinverse Let x = A+ b, A : Rn+m → Rn where m n. Below two equally efficient techniques are presented which solve the under determined system efficiently employing sparse matrix techniques without an SVD or QR decomposition. ˆ of A := A ˆ z is regular, 1) If the leading n × n matrix A the solution is given by

−1

−1 ˆ z ˆ b A A x0 := , α := −(t T t)−1 (t T x0 ) , t := 0 −Im x = x0 + t α where Im is the m × m identity matrix and t spans the m dimensional kernel of A. 2) The solution is calculated in two steps: computing a basis of the kernel of dimension m first and solving an augmented system Let ˆt be an Rn+m×m column

second. A matrix such that ˆT has full rank. Calculate the kernel t of A by solving for t : Rn+m×m

A 0 t = ˆt T Im and finally scale the column vectors of t to unity. Obtain x by solving

A b x= . tT 0

The authors would like to thank the reviewers for their valuable comments on the improvement of the quality of the paper and further references.

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Hans Georg Brachtendorf graduated in electrical engineering from the RWTH Aachen, Aachen, Germany, in 1989, and received the Dr.-Ing. degree in electrical engineering from the Institute for Electromagnetic Theory and Microelectronics, University of Bremen, Bremen, Germany, in 1994. From 1994 to 2001, he was an Assistant Professor (C1) also with the University of Bremen and obtained the Venia Legendi (Habilitation) from the same university in 2001. From 1997 to 1998, he was affiliated with the Wireless Laboratory of Bell Laboratories/Lucent Technologies in Murray Hill, NJ, USA, where he performed research in circuit simulation and design. In 2001, he joined the Fraunhofer Institute for Integrated Circuits, Erlangen, Germany. His focus there was on system design and simulation for satellite broadcasting systems (WorldSpace, XM radio) and transceiver designs. Since 2005, he has been a Full Professor with the Fachhochschule Oberösterreich, Steyr, Austria, for system design and simulation, communications, and signal processing. He is the author and co-author of one book and numerous technical papers dealing mainly with circuit simulation and device modeling. He holds four patents in various fields of circuit analysis and design, including a patent for a novel image reject filter, a subsampling receiver architecture and on multirate simulation techniques. His current research interests include circuit design, modeling and simulation, as well as signal processing and digital communication. Robert Melville (M’98) received the undergraduate training at the University of Delaware, Newark, DE, USA, then went on to graduate work at Cornell University, Ithaca, NY, USA, culminating in the Ph.D. degree in computer science, in 1981. He was a Junior Faculty Member with Johns Hopkins University, Baltimore, MD, USA, before joining the Bell Laboratories, Murray Hill, NJ, USA, in 1985. He was with the Bell laboratories for 15 years in the areas of computer-aided design, numerical simulation of electronic circuits, and design and fabrication of RF integrated circuits. Most recently, he has taught electrical engineering at Columbia University, New York, NY, USA, and served with the United States Antarctic Program at the Amundsen-Scott base at the South Pole doing engineering work in support of geophysics experiments. Dr. Melville has served as a Professional Referee for various IEEEsponsored journals and conferences, and the Society for Industrial and Applied Mathematics. He co-organized a conference on numerical circuit simulation at Sandia National Laboratories and participated in the AT&T Teachers and Technology Enrichment Program for high-school mathematics and science teachers. Peter Feldmann (F’00) was born in Timisoara, Romania. He received the B.Sc. degree (summa cum laude), in computer engineering, and the M.Sc. degree in electrical engineering from the Technion, Israel, in 1983 and 1987, and the Ph.D. degree from Carnegie Mellon University, Pittsburgh, PA, USA, in 1991. He was with Zoran Microelectronics, Haifa, Israel, involved in designing digital signal processors. Subsequently, he served as a Distinguished Member of Technical Staff with Bell Laboratories, Murray Hill, NJ, USA, in the Design Principles Research Department and then, as a Research Staff Member with the IBM John Watson Research Center, Yorktown Heights, NY, USA. He is currently with D. E. Shaw Research, New York, NY, USA, engaged in hardware accelerated molecular dynamics simulation. In the past, he also served as a Vice President of VLSI and Integrated Electro-optics with Celight, a fiber-optic communications start-up, and taught as an Adjunct Professor at the Department of Electrical Engineering, Columbia University, New York, NY. He has authored over 100 papers and patents. His current research interests include analysis, modeling, design, and optimization methods for integrated electronic circuits and communication systems, analysis and modeling for timing, power, and noise in digital VLSI circuits, and also on the large scale dynamic analysis, and modeling of the electricity transmission grid. Rainer Laur received the Dipl.-Ing. and Dr.-Ing degrees in electrical engineering from the University of Aachen, Aachen, Germany, in 1970 and 1977, respectively. He was a Project Manager for Research and Development Projects with Philips, Hamburg, Germany, in 1977. In 1978, he became a Professor for electronics with the University for Applied Sciences, Dortmund, Germany. From 1989 to 2009, he was a Full Professor for microelectronics and electrical engineering with the University of Bremen, Bremen, Germany, and was the Head of the Institute for Electromagnetic Theory and Microelectronics. In 2009, he retired but still has a contract with the University of Bremen. His current research interests include simulation, modeling, design of microelectronic circuits, and microsystems.