On generating compact, passive models of frequency-described systems

On Generating Compact, Passive Models of Frequency-Described Systems ∗

Carlos P. Coelho [email protected]

†

Joel R. Phillips [email protected]

∗

INESC ID/ Cadence European Laboratories Dept. of Electrical and Computer Engineering Instituto Superior T´ecnico Lisboa, 1000 Portugal

L. Miguel Silveira [email protected] †

∗

Cadence Berkeley Laboratories Cadence Design Systems San Jose, CA, 95134 U.S.A.

Abstract Increasing system integration and tighter design constraints and specifications make it necessary to account for the non-ideal behavior of all the elements in a system. For high-speed digital, and microwave systems, it is increasingly important to model previously neglected frequency domain effects. In this context, the problem of creating time domain models from tabulated frequency data, representing a device’s input-output relationship is of paramount importance. Such models must possess acceptable accuracy and also be very compact as they may be extensively used in later simulations. An important class of devices that are often described by tabulated frequency domain data is that of passive devices. In this paper we discuss several two-step algorithms based on a combination of rational interpolation with standard model order reduction procedures that is aimed at generating accurate multiport models from tabulated s-parameter data. We show that important system properties such as passivity can be maintained while generating compact models that are amenable to direct inclusion in standard time-domain simulators.

1

Introduction

As system integration evolves and tighter design constraints must be met, it becomes necessary to account for the non-ideal behavior of all the elements in a system. For high-speed digital, and microwave systems, it is increasingly important to model previously neglected frequency domain effects. Especially in high-frequency applications, certain devices are usually described and studied in the frequency domain. Devices such as coil inductors, SAW filters, non-ideal transmission lines and high-frequency transistors are commonly described by manufacturers and designers by their frequency dependent scattering parameter or admittance matrices. Other effects, such as those introduced by package parasitics or substrate induced couplings might also be frequency-dependent and available only as generic data. This frequency domain data can be obtained either through measurement or through physical simulation. In either case, the available data is typically sampled, incomplete, noisy, and covers only a finite range of the spectrum. Figure 1 schematically depicts such a problem by showing a portion of an RF front-end project involving sev-

Figure 1. Schematic depiction of an RF frontend design involving several modules. eral different blocks, possibly of different design origin and using different representations. Typically, some of these blocks are better, or exclusively, described in the frequency domain. Full verification of the correct functionality of systems such as the one depicted in Figure 1 usually requires simulation at various levels including, invariably, time-domain simulation of the system. Unfortunately it is not trivial to generate accurate circuit-level models for all the devices or modules used. Such models are however necessary for the simulation of larger designs and to account for the non-ideal characteristics of the devices. Although harmonic balance simulators can easily handle devices described by their frequency response, their use is restricted to systems that exhibit behavior and they cannot adequately deal with highly non-linear designs such as oscillators and mixers. On the other hand, time-domain simulators using state-space model integration, that can deal with highorder non-linearity, and perform transient analysis as well as periodic-steady-state analysis, require time-domain models. It is necessary that these models have frequency responses that match the available data but, they must also possess stability and passivity properties similar to those of the physical system that they represent. Several algorithms for stable rational approximation exist and can be used for frequency domain identification of stable systems (see for example [5] and the references

Proceedings of the 15 th Symposium on Integrated Circuits and Systems Design (SBCCI’02) 0-7695-1807-9/02 $17.00 © 2002 IEEE

therein). In [9, 4, 3] norm-bounded and positive-real models are generated using rational interpolation and approximation algorithms and convex programming. More recently, a general-purpose rational approximation algorithm based on multivariable Nevanlinna-Pick interpolation was also presented [5]. Such a procedure, while general, due to its constructive nature may lead to models of very high order, which may contain redundancies. This may cause problems in later time-domain simulation as the size of the models directly impacts the verification cost. In this paper, we discuss several two-step algorithms that are aimed at generating passive yet compact multiport models from tabulated s-parameter data. The algorithms are based on a combination of the Nevanlinna-Pick rational interpolation procedure presented in [5] with, after appropriate state-space transformation, standard model order reduction. The models produced by these algorithms thus should have the appropriate physical properties and be conveniently compact and amenable to direct inclusion in standard time-domain simulators. This paper is structured as follows. In section 2, background information is provided. In section 3 the matrix Nevanlinna-Pick interpolation problem from [5] is briefly reviewed, together with an incremental rational approximation algorithm using boundary Nevanlinna-Pick interpolation which was also proposed in [5]. This procedure is aimed at a first reduction in model size while maintaining accuracy and passivity. Then, in section 4, model order reduction procedures are reviewed and their applicability for further compacting the generated models is discussed. In section 5, the algorithm is applied to several real-world data-sets and the results are discussed. Finally, in section 6 conclusions and suggestions for further work are drawn.

2

Background

Passivity is an important property of certain physical systems. Networks composed by resistors, capacitors and inductors are passive, they do not generate energy. Systems that always consume energy are called strictly passive. Interconnected (strictly) passive systems are (strictly) passive. Stable systems do not possess this closure property. Stable systems loaded by stable, and even passive systems, may not constitute an overall stable system. Thus, it is imperative that passive systems be represented by passive models. The properties of the transfer function of a model representing a passive system depend on the physical interpretation of the model’s inputs and outputs. There exist several standard system representations such as the impedance parameter matrix Z, the admittance parameter matrix Y or the scattering parameter matrix S. For higher frequency systems, impedances and admittances cannot be accurately measured because the required short-circuit and open-circuit tests are difficult to achieve over a broad range of frequencies. The scattering matrix, s-parameter, representation is used for higher-frequency by characterizing the n-ports through relations between incident, a, and reflected traveling waves b, satisfying b = S a. Fortunately, it is

fairly trivial to obtain one representation given any other, assuming all information is known. See [5] for a discussion of such transformations. The admittance, impedance and scattering parameter matrices of passive systems satisfy a set of physically important conditions. In [2], Brune proved that the admittance and impedance parameter matrices of passive electrical networks are positive real matrix rational functions. A matrix function H(s) is positive real if, H(s) = H(s) H(s) is analytic in Re[s] > 0

(1) (2)

H(s) + H(s)H ≥ 0 in Re[s] > 0.

(3)

A matrix rational function is positive real if and only if (1) and (2) hold and H(jw) + H(jw)H ≥ 0 for w ∈ R.

(4)

Any pole on the imaginary axis must be simple and its residue matrix must be Hermitian and nonnegative definite. If H(s) has no poles on the closed right-half plane, it is positive real if and only if (4) holds. Since a passive system is necessarily stable and has a power gain that is less than or equal to one, the scattering matrix of a passive system is bounded-real. A matrix function H(s) is bounded-real if it satisfies (1), (2) and ||H(s)|| ≤ 1 for Re[s] > 0.

(5)

A matrix rational function is bounded-real if it satisfies, (1) and (2) and ||H(jw)|| ≤ 1, ∀w ∈ R.

3

Matrix Nevanlinna-Pick Interpolation

An algorithm for determining an approximant to a frequency-described system, using Nevanlinna-Pick interpolation, was introduced in [5]. Here we summarize the main steps and issues of that algorithm. Let ∆ represent the open right-half plane. Given a set of N data matrices, s ∈ ∆N and H ∈ CN (no ×ni ) , the no × ni matrix rational function F interpolates the data set (s, H) if F(sk ) = Hk , for k = 1, · · · , N .

(6)

The Nevanlinna-Pick problem is to describe all interpolating functions F, that are analytic in ∆ and that satisfy sup F(s) < 1.

s∈∆

(7)

Since the scattering parameter matrix of a passive system also satisfies (7), using the Nevanlinna-Pick interpolation algorithm for passive multivarible system identification seems to be a natural choice. The matrix Nevanlinna-Pick problem may be formulated as follows (see [5] for further details). Let     0 s1 Ini C− = Ini · · · Ini   .. Aπ =   .   C+ = H1 · · · HN , 0 sN Ini Theorem 3.1 (Matrix Nevanlinna-Pick) There exists a rational function F(s) that interpolates a given

Proceedings of the 15 th Symposium on Integrated Circuits and Systems Design (SBCCI’02) 0-7695-1807-9/02 $17.00 © 2002 IEEE

data set (s, H), that is analytic in ∆ and satisfies sups∈∆ F(s) < 1, if and only if the Pick matrix,

I − HH p Hq . (8) Λp,q = sp + sq

surely guarantees that much redundancy will be present in the model especially if many data samples are taken in a short frequency range. It would be preferable that the chosen model order, bear no direct connection to the number of samples and instead be related only to the physical prop1≤p,q≤N erties of the system and to the desired accuracy. To that is positive definite. In this case, there is a 2 × 2 block matrix end, an iterative algorithm, was presented in [5] aimed at function  producing smaller models. The basic idea is to choose an   C+ Θ(s) = Ino +ni + (sIN ni −Aπ )−1 Λ−1 −CH CH + − initial set of points randomly, manually, or through some C− heuristic procedure. Then by carefully checking the error (9) such that the solutions of the Nevanlinna-Pick interpolation between the unused data points and the current approximaproblem are given by tion, a new set of data points is added such that that error is −1 F(s) = [Θ1,1 (s)G(s) + Θ1,2 (s)] [Θ2,1 (s)G(s) + Θ2,2 (s)] , minimized at each step. The iterative procedure is stopped when the error reaches a certain threshold or the order ex(10) where G(s) is an arbitrary no × ni rational function that is ceeds a predetermined limit. The overall idea is to pick only analytic on ∆ and that satisfies sups∈∆ G(s) < 1. a subset of the original data set, sufficient to produce an interpolant with appropriate accuracy. Using this incremental 3.1 Boundary Interpolation fitting strategy, the size of models can be reduced without noticeable deterioration in accuracy and while guaranteeing In Theorem 3.1, the data points are considered to lie passivity in the resulting model. For a detailed description in the interior of ∆. Since we intend to use Nevanlinnasee [5]. Pick for frequency domain identification, the available data points will usually be placed on the imaginary axis. Fortunately this problem is easily solved with an appropriate 4 Model Order Reduction shifting. Let Fσ be the Nevanlinna-Pick interpolant of the The incremental fitting algorithm mentioned in Secdata set (s + σ, H), with σ real satisfying σ > 0. The intertion 3.2 is based on an interpolatory algorithm to generate polating function F(s) = Fσ (s + σ) interpolates the shifted a rational function model from a subset of the given fredata set and is analytic for Re[s] > −σ which contains ∆. quency data. Although significant reductions are reported The Nevanlinna-Pick interpolant exists if the Pick matrix

over the standard Nevanlinna-Pick algorithm, it is expected H I − Hp Hq that some redundancy may still be present and that model (11) Λp,q = sp + sq + 2σ order reduction can be successfully applied. However, it is 1≤p,q≤N required that the bounded realness of the model’s frequency is positive definite. If Hk  < 1 for all k, it is always response, and therefore the passivity of the model itself, be possible to choose a small enough σ so that (11) is positive preserved. In this section we review three model order redefinite. Although it is always possible to generate an interduction techniques that often accomplish just that. In order polant by choosing a small enough σ the behavior between to apply these procedures the approximant must first be repdata points is severely affected by these parameters. As σ resented as a state-space model. The transformation to this tends to zero, the poles of the rational function are allowed representation is also necessary since that is usually the preto approach the imaginary axis or the unit circle resulting in ferred representation for time domain simulation models. a highly oscillatory frequency response. Fortunately, such a transformation is fairly simple to derive (see [5] for details and a discussion). In the following we 3.2 An Incremental Fitting Strategy will assume that the interpolation algorithm was used to obFrom Theorem 3.1, obtaining a rational interpolant to a tain a guaranteed bounded-real frequency response model, given set of frequency data point reduces to constructing the S(s), from which a corresponding state space representaPick matrix in (11), inverting it, computing Θ(s) as given tion, [A, B, C, D] has been computed. Results from the by (9), and building the interpolant (10) with an appropriate application of the various techniques will be presented in choice of G(s). Unfortunately, while simple and guaranSection 5. teed to produce models with the desired properties, namely passivity, this procedure presents some important difficul4.1 Truncated Balanced Realization ties. The main problem, for practical purposes, is related to the size of the final model. In fact, from Eqn. (10), one In [11, 8, 6], truncation of the balanced realization (TBR) can easily see that the generated interpolant will have at of a model is proposed as a method for eliminating its least 2N ni poles, where, we recall, N is the size of the weakly observable and controllable states. Let Wo and data set, and ni the number of inputs of the system. UnWc represent the observability and controllability Gramian, fortunately, even for scalar data sets of moderate size, the given by the solution of the Lyapunov equations models generated by direct application of the boundary in(12) AH Wo + Wo A + CH C = 0 terpolation algorithm become prohibitively large for pracH H = 0. (13) AWc + Wc A + BB tical use. Furthermore, the form of the algorithm almost

Proceedings of the 15 th Symposium on Integrated Circuits and Systems Design (SBCCI’02) 0-7695-1807-9/02 $17.00 © 2002 IEEE

The Hankel singular values of the system transfer func−1 tion, H(s) = D + C (sI − A) B are defined as 1 σk (H(s)) = λk (Wc Wo ) 2 , where, by convention, σk (H(s)) ≥ σk+1 (H(s)). Although the gramians depend on the coordinate system, the Hankel singular values are invariant to state-space transformations. The balanced realization [11] therefore amounts to determining a non-singular change of coordinates T, such that the trans o = T−H Wo T−1 , and controllaformed observability, W H

c = W

o = bility, Wc = TWc T , gramians are such that W diag(σ1 , . . . , σn ). The Hankel singular values are particularly important not only because of their invariance under coordinate changes but also because they provide accurate information regarding both the gain and complexity of stable systems [8]. Balancing a system, basically orders the states from an input-output point of view in terms of their controllability and observability. Once balanced, order reduction can be accomplished simply by truncating the resulting model and keeping the modes corresponding to the more controllable and observable states. Furthermore, the following results provides a nice error bound: Theorem 4.1 Let H(s) be a stable rational no ×ni transfer function with Hankel singular values σ1 > σ2 > . . . >  k (s) be obtained by truncating the balanced σn , and let H realization of H(s) to the first k states. Then     k (jw) H(jw) − H  ≤ 2(σk+1 + · · · + σn ). (14) ∞

The model obtained by truncating the balanced realization of a bounded real system is stable. In addition, under the strong condition that the sum of the Hankel singular values that correspond to the eliminated states is less than ε which is given by, H(s)∞ = 1 − 2ε,

with 0 ≤ ε ≤ 1. (15)

where H(s)∞ is the infinity norm of the original model, then clearly the reduced system can be guaranteed to be bounded real. This bound is quite conservative and should not be used as the only criteria for the choice of the number of states that are to be truncated. In practice, for the models generated by the Nevanlinna-Pick algorithm, it was found that a significant reduction in model order, well beyond what the bound predicts, can usually be achieved. Even though passivity is not guaranteed in general, often it turns out that the truncated models are indeed passive. Thus in many cases it pays to use this procedure, generate a model, check its passivity, keep it if it is found to be passive and sufficiently accurate or discard it and choose a slightly larger/smaller model if necessary.

4.2

Transformed PRIMA

Another procedure that one can attempt to use is the PRIMA algorithm [12]. In recent years, the PRIMA algorithm has received much attention and prominence in the field of interconnect model generation and order reduction. In this context, the frequency response of the model represents a system’s impedance or admittance. It is therefore

necessary to transform the state space representation associated with the scattering parameter matrix to another state space representation,  x˙ E y

 + Bu  = Ax   = Cx + Du

(16)

that represents the positive-real admittance, Y(s), or impedance, Z(s). The PRIMA algorithm then operates on the transformed model by applying congruence transformations x = Qz,  z˙ QH EQ y

= =

  QH AQz + QH Bu   CQz + Du

= =

 ru  rz + B A   Cr z + Dr u. (17)

In the domain of interconnect modelling and order reduction, the matrices obtained from modified nodal analysis are  A  are positive definite matrices.  =C  T and E, such that B It was proved in [10], that any state space model with this internal structure and a positive real frequency response, that the congruence transformed system also has a positive real frequency response, thus preserving the passivity of the original model. This result does not actually depend on the choice of Q, although the choice of Q can impact the accuracy of the model. Therefore, for this interconnect models in this formulation, PRIMA preserves passivity. Furthermore, PRIMA is a very efficient algorithm which makes it very attractive for order reduction of very large systems. Unfortunately it turns out that the internal structure of the state space representation associated with the NevanlinnaPick rational interpolant is not in the form required by PRIMA and therefore neither passivity nor stability are guaranteed and are not usually obtained. However, we have observed in practice, that if PRIMA is applied to a previously balanced realization of the system, very often passive and accurate models are generated. This, of course, has the inconvenient of having to calculate the balanced realization of the model. For further details regarding PRIMA and other Krylov subspace and moment matching based model order reduction algorithms see [12, 7, 14]

4.3

Passive Truncated Balanced Realization

None of the previously discussed techniques can guarantee passivity of the final model. Recently, however, an algorithm similar to TBR was presented, BR-TBR, where a different type of balancing is performed and where it can be guaranteed that any truncated reduced model is boundedreal and thus passive [13]. Like TBR, this BR-TBR algorithm also determines a non-singular change of coordinates T that diagonalizes two quantities similar to the gramians introduced in Eqns. (12) and (13). This transformation is however sought while enforcing the bounded-realness of the model which leads to passivity preservation. Computation of this model is more expensive than TBR which in turn can be quite more expensive than application of PRIMA. Also, in general, TBR possesses some near-optimality properties that make TBR-based models more accurate. Interestingly enough, for the BR-TBR algorithm, an error bound quite similar to Eqn. (14) is also available. The interested reader

Proceedings of the 15 th Symposium on Integrated Circuits and Systems Design (SBCCI’02) 0-7695-1807-9/02 $17.00 © 2002 IEEE

is encouraged to look up the original reference for further details and discussion of this algorithm.

Experimental Results

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Figure 2. Order 60 approximation of the s1,1 parameter of a coil inductor.

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Figure 3. Reduced order models of the s1,1 scattering parameter of the coil inductor. 0

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In this section several experimental results are presented. The examples were chosen to illustrate the properties of the models generated by the algorithms described in Section 4. The first example is the s1,1 parameter of a coil inductor. In Figure 2-a), the original data is approximated by a 30 point interpolant, the value of σ was manually chosen: σ = 0.0125 was found to be adequate. The out band behavior is quite smooth as can be seen from Figure 2-b). Since 30 interpolation points were used, the model has order 60 which is clearly excessive. By determining the poles of the system, we can check that the approximation is stable (it was verified that, as expected, the real part of the poles is always less than −σ = −0.0125). The infinity norm of the system’s frequency response, calculated to an accuracy of 10−3 , using the method in [1], was determined to be H = 0.935 < 1 which proves the model is passive. The model order reduction algorithms, proposed in Section 4 were used to generate reduced order models of all orders from 1 to 59. The direct application of the transformed PRIMA algorithm resulted in only 16 passive models. However, balancing the realization before applying the transformed PRIMA algorithm yielded 51 passive models. From the 59 models generated through truncated balanced realization 57 were passive. In this case, the bound given by (15) guarantees that any model of order higher than 6 would be passive. In practice, accuracy constraints requires the use of orders 15 or higher. starting from about 15. Of course, using the algorithm mentioned in Section 4.3, all models were found to be passive, as expected! In Figure 3-a) a 20th order approximation generated with the prebalanced transformed PRIMA scheme is compared with an approximation of the same order generated by the truncated balanced realization algorithm. For a further analysis of relative model accuracy, Figure 3-b) compares two 15th order approximations using a linear scale for the frequency range. These results and the high number of passive models validate the schemes proposed in Section 4. In the next example, the 2 by 2 scattering parameter matrix of the same coil inductor is approximated. The generated model uses 50 interpolation points which means that

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Figure 4. Reduced order models of the s1,1 and s1,2 scattering parameters of the coil inductor. it has 200 states. The poles of the system matrix are all in the left half plane and have a real part smaller than −σ, where σ = 2.5 × 10−3 . The infinity norm, calculated to a tolerance of 10−3 , is H(s)∞ = 0.991 which proves that the model is passive. The behavior of the model outside the data set is also seen to be smooth as can be seen from Figure 4. The model order reduction algorithms presented in Section 4 were used to generate multivariable reduced order models of sizes from 1 to 199. The transformed PRIMA based methods were used to generate all models of even order (for reasons having to do with moments matched; see [12]). The direct application of the transformed PRIMA algorithm resulted in 45 passive models out of 99 models. As before, balancing the realization before applying the transformed PRIMA algorithm dramatically increased the number of successful reductions, in this case 98 of a 99 models were passive and the only failure had an infinity norm very close to 1. From the 199 models generated through truncated balanced realization 191 were passive. In this case, the bound given by (15) only guaranteed passivity for models of order higher than 193, hardly a reassuring result. Again, using the algorithm mentioned in Section 4.3, all models were found to be passive. In Figure 4 the s1,1 and s1,2 elements of the frequency response of the order 30 reduced order models generated with balanced transformed PRIMA and truncated balanced realization are compared with the model generated by the incremental fitting algorithm. In this case, for both algorithms, a significant reduction of model order was achieved without very significant quality deterioration and while maintaining the passivity of the models.

Proceedings of the 15 th Symposium on Integrated Circuits and Systems Design (SBCCI’02) 0-7695-1807-9/02 $17.00 © 2002 IEEE

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Figure 5. Multivariable reduced order models of coil inductor. In the next example, the s1,1 parameter of SAW filter was approximated. Using 61 interpolation points and a shift of 2e-4, an order 122 model was generated with the procedure described in Section 3.2. The result is shown in Figure 5-a). Again we used the algorithms from Section 4 to generate all reduced order models from 1 to 122. Using the PRIMA algorithms only 22 passive models were obtained. However using balancing followed by PRIMA, 121 of the 122 models were found to be passive. Using just TBR, 118 of the models were passive. In Figure 5-b) approximations of order 50 using PRIMA and TBR are shown to indicate the accuracy obtained with such order reductions. For our final example we computed an approximation to the 2 × 2 saw filter parameters using 80 interpolation points (order 320) and a shift of 3e-4. We generated all reduced models of even order using the algorithms in Section 4. Of the 159 models generated by PRIMA, only 2 were found to be passive. This clearly indicates that using PRIMA is not a good strategy in general. However, of the 159 models generated by balancing followed by PRIMA, all were found to be passive. Using TBR by itself lead to only two non passive models of very low order. Using the algorithm mentioned in Section 4.3, all models were also found to be passive.

6

Conclusions and future work

In this paper we discussed and analyzed several twostep algorithms based on a combination of rational interpolation with standard model order reduction procedures aimed at generating compact and accurate multiport models from tabulated s-parameter data. The interpolation procedure forms a robust method for generating a passive state space model from a frequency domain tabulated characterization of a device’s multiport description. However, the models produced are typically large and contain redundancies. To further compact the resulting models, a secondstep model order reduction scheme was introduced and several techniques were investigated. The examples shown demonstrate that the strategy of using balancing followed by PRIMA or direct truncation (TBR), does in general lead to considerable compactness of the models while often maintaining the passivity of the models. Such property cannot however be guaranteed. Application of the model order reduction algorithms recently published in [13] always generate passive models albeit its accuracy may sometimes lag

that of TBR, which is near-optimal. Given this passivity preservation property this algorithm should in principle be the algorithm of choice. By means of the introduction of two-step algorithms we illustrated the ability to compact models without much accuracy deterioration leading to models that have the desirable passivity properties but are also compact and amenable to direct inclusion in standard time-domain simulators.

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