Sensitivity Analysis of Resonant Circuits

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1 1 Sensitivity Analysis of Resonant Ciruits Olivier Buu Abstrat We use first-orer perturbation theory to provie a loal linear relation between the iruit parameters an the poles of an RLC network. The sensitivity matrix, whih efines this relationship, is obtaine from the systems eigenvetors an the erivative of its eigenvalues. In general, the sensitivity matrix is relate to the equilibrium flutuations of the system. In partiular, it may be use as the basis for a statistial moel to effiiently preit the sensitivity of the iruit response to small omponent variations. The metho is illustrate with a alulation of onitional probabilities by Monte Carlo Simulation. Inex Terms Perturbation Theory, Linear Response, Resonant system, Statistial Moel, Monte Carlo Simulation. S I. INTRODUCTION ENSITIVITY analysis is an integral part of omputeraie iruit esign. Effiient statistial analysis algorithms are available to simulate the iruit response at fixe frequenies [1], from whih the sensitivity to omponents variation may be obtaine by regression. In the ontext of resonant iruits, however, the esigner is primarily intereste in the poles an the iruit response on-resonane. Traking the resonanes at eah trial requires an extra omputational step that unermines the effiieny of existing methos. In this paper, we follow the reverse proeure: we first etermine the loal linear relationship between the iruit parameters an the poles an response of the system, then arry out primitive Monte Carlo simulations. For yiel preitions, whih require a large number of trials, the fixe ost assoiate with the etermination of the sensitivity matrix is lower than the reurring osts in existing approahes. We efine the sensitivity matrix as the Jaobian of the transformation between the iruit parameters an the poles an system response [2]. We present a general metho to alulate the sensitivity matrix base on the solution of the eigenvalue problem assoiate with the iruit. This metho is illustrate with a simple example. II. BACKGROUND Consier the transfer funtion of an arbitrary network of resistanes, inutors, an apaitors. The observable response of the iruit is entirely haraterize by the poles an the values of the transfer funtion on-resonane 1. Formally, there exists a ompliate relationship between the iruit parameters an the olumn vetor forme by the real an imaginary parts of the poles an eah inepenent omponent of the transfer funtion at eah resonane frequeny. To stuy the sensitivity to small omponent variations, we follow [2] an linearize this relation: (1) where enotes the matrix transpose. In pratie, only a subset of the observable parameters may be uner speifiation, an the size of the sensitivity matrix is reue aoringly. Although the alulation of the Jaobian may be omputationally ostly for large systems, the simple linear relation (1) allows for effiient statistial analysis of the iruit. In some ases, the Jaobian gives iret aess to the multivariate istribution funtion. In partiular, if the relation between an is bijetive, the probability ensity funtion assoiate with is known loally from the relation: (2) where is the probability ensity funtion assoiate with the ranom vetor. More generally, when the seon moment of the istribution of the iruit parameters exists, the ovariane matrix for the ranom vetor is given by: (3) where is the o-variane matrix of the vetor. III. SENSITIVITY MATRIX CALCULATION The sensitivity matrix is assemble from the erivatives of the poles an the transfer funtion with respet to the iruit parameters. In this setion, we review the perturbation metho use to alulate these erivative terms base on the solution of the iruit eigenvalue problem. The eigenvalue problem is formulate from the iruit state equation. A. State Equation The iruit equation for a Linear Time-Invariant network is assume to take the stanar form: (4) 1 This list may inlue a pole at infinity.

2 2 where the vetor inlues the noe voltages at the apaitors terminals an the urrents flowing through the inutors. is the input matrix an is the exitation vetor. is a matrix ompose of apaitane an inutane values. The matrix inlues the resistane values an the noe-branh iniene matrix esribing the network uner stuy. Both an may be written as sums of sparse matries orresponing to the iniviual omponent ontributions. These sparse matries, sometimes alle omponent stamps in the literature [3], are useful for the sensitivity analysis presente below. Provie oes not ontain any linearly-epenent variables, then (4) is a state equation an the matrix is fullrank. We will assume this onition to be fulfille in the rest of this paper. The transfer funtion is etermine by the output equation: where is the output vetor, is the output matrix, an is the transmission matrix. By taking the Laplae transform of (3) an (4) an applying the efinition of the transfer funtion to the zero-state output vetor, we obtain: where is the Laplae variable. B. Generalize eigenvalue problem The iruit is entirely haraterize by the eigenvalues an eigenvetors of the state equation. The square matries of right eigenvetors an left eigenvetors are solutions of the generalize eigenvalue problem: (5) (6) an (7) where is a iagonal matrix of eigenvalues. Sine an are real matries, the eigenvalues are either real or omplex onjugate pairs. For passive networks, all the eigenvalues are loate in the left half of the omplex plane. From (7), it an be shown that the eigenvetors are biorthogonal. Sine is assume to be non-singular, we an always fin a normalization suh that: an (8) where is the ientity matrix. Using (5) an (7) we obtain the following expression for the transfer funtion: Together with the eigenvalues, this last expression forms the basis of the sensitivity analysis esribe in the next setions. (9) C. Derivative of Eigenvalues For simple eigenvalues, the erivatives with respet to a iruit parameter is [4]: (10) As note above, the matrix erivatives an are sparse an losely relate to the stamp for the iruit omponent parameterize by. The ase of multiple eigenvalues is aresse in [5]: for an eigenvalue of multipliity with assoiate eigenvetors an, there are erivatives whih are the eigenvalues of the matrix. D. Derivative of the Transfer Funtion The erivative of the on-resonane transfer funtion inlues two terms: (11) where is the imaginary part of the n th pole. To obtain these two terms we introue some intermeiate alulation steps: (12) (13) (14) In the previous expressions, we have assume that the resonane of interest is ampe, so the matrix is non-singular. Note that (12) oes not require a full matrix inversion. The right-han-sie terms of (11) follows from erivatives of (6): (15) (16) IV. COMPUTATIONAL COST The most ostly step of the sensitivity matrix alulation is the solution of the eigenvalue problem, whih sales as operations. If the probability ensity funtion an be obtaine, from equ. (2) or otherwise, the ost of a Monte Carlo trial is the ost of sampling the istribution. In the worst ase, an aitional matrix multipliation (equ. (1)) is require,. By ontrast, the ost of setting up a quarati approximation of the response funtion at a given frequeny is. Sine the number of egrees of freeom is equal the number of inepenent ative elements, this ost is equivalent to. Eah simulation involves a matrix multipliation, with a

3 ost. However, the response funtion has to be evaluate at ifferent frequenies to haraterize the iruit, so the fixe an reurring osts are, respetively an in this metho.

For a ~10-2 auray on simulation results, we assume a 10 4 trials run. Consiering a iruit with 50 eigenmoes, the sensitivity matrix metho woul require ~10 7 operations.

3 3 ost. However, the response funtion has to be evaluate at ifferent frequenies to haraterize the iruit, so the fixe an reurring osts are, respetively an in this metho. Moreover, there is a ost assoiate with traking the resonane frequenies at eah trial. The stanar eviation of yiel preitions sales as the inverse of the square root of the number of trials. For a ~10-2 auray on simulation results, we assume a 10 4 trials run. Consiering a iruit with 50 eigenmoes, the sensitivity matrix metho woul require ~10 7 operations. The same simulation woul ost ~10 9 operations to ahieve the same auray with the quarati approximation. V. EXAMPLE Fig. 1 shows a iruit example use in Magneti Resonane instruments [6]. The inutive transuer L oil is embee in a two-port mathing network, where port 1 is tune to 200 MHz, an port 2 to 50 MHz. We use Matlab to generate the eigenvalues an eigenvetors, whih are reporte in table 1. In aition to the nominal resonanes at 200 MHz an 50 MHz, there is a spurious resonane at 179 MHz an a pole at DC. TABLE I EIGENVALUES AND EIGENVECTORS Channel 1 a Channel 2 a Spurious a DC Frequeny b MHz 50.0 MHz MHz j10.8 -j8.2 j j2.1 j7.9 j2.4 0 j8.9 -j8.4 j2.7 0 a Right eigenvetor from the eigenvalue of positive natural frequeny. b Eah frequeny orrespons to a omplex-onjugate pair of poles. Voltage noes in V. Inutor urrents in ma. Beause we eliminate the reunant variables from the state vetor, the riving term epens on the time erivative on the input vetor [7] an takes the form with: an (22) Similarly, the transmission term in the output equation takes the form beause the DC moe is not observable [7]. The oeffiients of the output equation are: In this RF appliation, the S-parameters are the preferre signal representation. However, it is onvenient to first obtain the 2x2 port-amittane matrix an omponent stamps by Moifie Noal Analysis. In this ase, the matries an are: an (17) The sub-matries are obtaine by inspetion: (18) (19) (20) (21) an (23) (24) These moifiations to the stanar iruit equation a terms to (15) but o not rastially alter the sensitivity analysis. The sattering matrix follows from the wellknown relation: (25) where 50 Ohm is the harateristi port impeane. Differentiating (25) yiels: In this example the speifiations are: (26) (27) (28) (29) (30)

4 Corresponingly, we haraterize the iruit response with the vetor: (31) where enotes the real part an the imaginary part. The orresponing sensitivity matrix is ompile in table 2.

4 4 Corresponingly, we haraterize the iruit response with the vetor: (31) where enotes the real part an the imaginary part. The orresponing sensitivity matrix is ompile in table 2. Component TABLE II SENSITIVITY MATRIX a a a a a a a a a a a a a a a a b 0.14 e 0.0 b 0.00 e 0.0 b 0.01 e 0.0 b 0.00 e -0.4 b 0.02 e 0.0 b 0.00 e f f a In units of MHz/pF. b In units of MHz/nH. In units of MHz/Ohm. In units of 1/pF. e In units of 1/nH. f In units of 1/Ohm. In this example, the etaile yiel analysis points to the return loss on hannel 1 as the largest risk of failure. The sensitivity matrix suggests that reuing the variane of,, an woul improve the yiel. Replaing by a trimmer apaitor an aing a tuning proess is another solution to the yiel issue. These ifferent assumptions an their eonomi impliations may be teste by re-alulating the yiel with the Monte Carlo metho. The experimental valiation of the moel may be one by Design of Experiment (DOE) base on the sensitivity matrix alulate above. Assuming a 5% variane an no orrelation between the omponent values, we an alulate the seon moment of the hosen engineering parameters from the iagonal elements of the o-variane matrix obtaine from (3). The values are liste in table 3. TABLE III PREDICTED YIELD a b Variane 4.54 MHz MHz 0.25 Partial Yiels 73% 35% 79% 79% Combine Yiels 26% 62% Total Yiel 19% a b To alulate the yiel, we further assume the iruit parameters to be normally istribute. The istribution of the ranom vetor is then multivariate normal. Instea of integrating the 6-imensional probability ensity funtion over the speifiation omain, we foun it more aurate an faster to use a Matlab routine to sample the istribution. The various onitional probabilities, estimate by averaging 10 9 trials, are reporte in table 3. The oeffiient of variation on these figures is ~10-3. A onvenient representation of the results is the area-proportional Venn iagram, whih was reate with the VennEuler algorithm [10] an is shown on Fig. 2. VI. CONCLUSION In this paper we have shown that the sensitivity matrix metho is an effiient way to arry out the statistial analysis of resonant iruits. Despite the omputational ost of etermining the sensitivity matrix, its value may be realize in the iret alulation of the o-variane matrix, DOE stuies, or substantiating ausal analyses use in six-sigma quality ontrol frameworks. Our approah may be improve with more effiient eigenvetors alulation algorithms [8] or faster Monte Carlo methos [9]. A low-ost approximation of the sensitivity matrix may also be obtaine by the quarati approximation, if the eigen-frequenies are known in avane. Finally, this approah is appliable to any linear time-invariant system, an may be expane to other network haraterizations, as esribe in the appenix. VII. ACKNOWLEDGMENTS This work is an offshoot of Dr. M. A. Smith s Six-Sigma green belt projet. We also thank Bob Taber for rawing our attention to the importane of eigenmoes in resonant iruits.

5 5 APPENDIX ALTERNATE CIRCUIT CHARACTERIZATIONS Engineering speifiations may inlue iruit haraterizations other than the quantities we onsiere in the analysis presente above. In this appenix, we provie the elements of perturbation theory that may be use with other well-known haraterizations. For simpliity, we restrit this setion to the ase of iruits with single eigenvalues. The ase of multiple eigenvalues has been worke out [10] but is outsie the sope of this paper. A. Time Domain Analysis The iruit natural response may be alulate in terms of eigenvetors from (4) an (7): (32) where is the vetor of initial onitions. Sine is iagonal, the alulation of the exponential term presents no numerial iffiulty. The ifferentiation of (30) with respet to a iruit parameter involves the erivative of eigenvetors. Ref. [11] gives their expression as linear ombinations of the un-perturbe eigenvetors in the ase of istint eigenvalues: (33) An alternate expression involving only one un-perturbe eigenvetor is given by [12]. B. Resiues The matrix funtions: may be expane as a sum of rational (34) The poles an resiue matries provie a omplete haraterization of the observable response. By expaning (8) we an express the resiue matries in terms of eigenvetors: (35) C. Zeros Single Input Single Output systems are often analyze in terms of pole-zero loi. The first-orer erivative of a zero may be obtaine by ifferentiating the impliit relation ( )=0 with respet to the iruit parameter : (36) The numerator an enominator are obtaine similarly to (15) an (16). In this ase, the vetors an may be interprete as the respetive solutions of the iret an ajoint systems at. REFERENCES [1] R. Biernaki, J.W. Banler, J. Song, Q. Zhang, Effiient quarati approximation for statistial esign, IEEE Trans. Ciruits an Systems, vol. 36, no. 11, pp, , Nov [2] A.S. Cook, T. Downs, Estimating manufaturing yiel by means of Jaobian of transformation, IEE Pro. G, vol. 129, issue 4, pp , Aug [3] F. N. Najm, Ciruit Simulation, Hoboken, NJ, Wiley, 2010, p 35. [4] S. B. Haley, The Generalize Eigenproblem: Pole-Zero Computation, IEEE Pro., vol. 76, no. 2, pp , Feb [5] A. Papoulis, Perturbations of the Natural Frequenies an Eigenvetors of a Network, IEEE Trans. Ciruit Theory, vol. 13, Issue 2, pp , Jun [6] M. Lupu, A. Briguet, J. Mispelter, NMR Probeheas: For Biophysial an Biomeial Experiments: Theoretial Priniples & Pratial Guielines, 1 st e., Imperial College Press, 2006, pp [7] S. Natarajan, A systemati metho for obtaining state equations using MNA, IEE Pro. G, vol. 138, no. 3, pp , Jun [8] A. Expósito, A. B. Soler, J. A. R. Maías, Appliation of Generalize Phasors to Eigenvetor an Natural Response Computation of LTI Ciruits, IEEE Trans. Ciruits an Systems, vol. 53, no. 7, pp, , Jul [9] M. Keramat an R. Kielbasa, Moifie latin hyperube sampling Monte Carlo (MLHSC) estimation for average quality inex, Int. Jour. Analog Integrate Ciruits Signal Proess., vol. 19, no. 1, pp , Apr [10] L. Wilkinson, "Exat an Approximate Area-Proportional Cirular Venn an Euler Diagrams," Visualization an Computer Graphis, IEEE Transations on, vol.18, no.2, pp.321,331, Feb [11] M. I. Friswell, The Derivatives of Repeate Eigenvalues an Their Assoiate Eigenvetors, ASME J. Vibration an Aoustis, vol. 118, Issue 7, pp , Jul [12] R. H. Plaut, Derivatives of eigenvalues an eigenvetors in non-selfajoint systems, AIAA J., vol. 11, Issue 2, pp , Feb [13] R. B. Nelson, Simplifie Calulation of Eigenvetor Derivatives, AIAA J., vol. 14, Issue 9, pp , Sep Similarly to the time-omain analysis, the perturbation of the resiue matries involves the erivative of eigenvetors.

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