Hennig, Eckhard

Krauße, Dominik

Schäfer, Eric

Sommer, Ralf

Trunk, Carsten

Winkler, Henrik

Digitale Bibliothek Thüringen

       Frequency compensation for a class of DAE's arising in electrical circuits  Preprint No. M 11/11 Eckhard Hennig, Dominik Krauße, Eric Schäfer, Ralf Sommer, Carsten Trunk, Henrik Winkler 2011 Impressum: Hrsg.: Leiter des Instituts für Mathematik Weimarer Straße 25 98693 Ilmenau Tel.: +49 3677 69-3621 Fax: +49 3677 69-3270 http://www.tu-ilmenau.de/math/ Technische Universität Ilmenau Institut für Mathematik FREQUENCY COMPENSATION FOR A CLASS OFDAE'S ARISING IN ELECTRICAL CIRCUITSECKHARD HENNIG, DOMINIK KRAUSSE, ERIC SCHÄAFER,RALF SOMMER, CARSTEN TRUNK, AND HENRIK WINKLERAbstract. Structured perturbations of regular pencils of the formsE ¡A, E;A 2 Rn£n; s 2 C; are considered which model the ad-dition of a capacitance c in an electrical circuit in order to improvethe frequency response.1. IntroductionIn this note we consider structured perturbations of regular pencilsof the form(1) sE ¡ A; E;A 2 Rn£n; s 2 C;which are related to design problems of electrical circuits. Linear elec-trical networks can be described by DAE's of the form E _x = Ax, withE;A 2 Rn£n, where the entries of E are in particular determined bythe capacitances of the network. The poles of the transfer functionof the corresponding system are given by the eigenvalues of the pen-cil sE ¡ A. To get a desired frequency response, the poles should beproperly located in the left open halfplane C¡ with a su±cient distancefrom the imaginary axis. A location on the rays f¡t§ it : t > 0g wouldimply appropriate properties of the frequency response, see [4],[2]. Itis the aim to improve the frequency behavior of the circuit by insertingadditional capacitances between certain nodes. This gives, within themodel, a structured perturbation of the matrix E by a matrix of theformc(ei ¡ ej)(ei ¡ ej)T ;where ei, ej are the standard unit vectors. The problem is to ¯ndthe appropriate place, i.e., the node determined by ei, ej, and thenumerical value of the capacitance c > 0, such that the eigenvalues ofthe perturbed pencils(E + c(ei ¡ ej)(ei ¡ ej)T )¡ Aare in a position to improve the desired properties of the frequencyresponse. It is shown in this note how the parameter c can be deter-mined in terms of a rational function, see relation (6) below, which is2000 Mathematics Subject Classi¯cation. Primary 15A22, 34A09; Secondary47A56.12 HENNIG, KRAUSSE, SCHÄAFER, SOMMER, TRUNK, AND WINKLERin particular feasible when the matrix E is singular with a low rank.As a corollary we determine nodes which are not suitable.2. The generalized Weierstrass formConsider a pencil of the form (1). It is assumed that the pencil isregular, that is, the characteristic polynomial P (s) = det(sE¡A) doesnot vanish identically on C. Moreover, it is assumed that the pencilis of index 1, meaning that the degree of the characteristic polynomialP (s) is equal to the rank of E. To be more speci¯c, any regular pencilcan be put into the so-called Weierstrass form, see [3], that is, to anyregular pencil sE¡A with E;A 2 Rn£n there exist invertible matrices~S and ~T in Cn£n such that the matrices ~SE ~T and ~SA ~T are of blockform(2) ~SE ~T =µIm 00 N¶; ~SA ~T =µJ 00 Ir¶with identity matrices Im 2 Rm£m and Ir 2 Rr£r such that r = n¡m,a nilpotent matrix N of index v, where v is the smallest nonnegativeinteger such that N v = 0, and a matrix J 2 Cm£m which is in Jordanform. The number v is uniquely determined by the pencil sE ¡A andit is called the index of the pencil. If E is invertible, then n = m andv = 0. Instead of the Weierstrass form, where, in particular, in thecalculation of the Jordan matrix J , numerical problems may occur, inthis note the generalized Weierstrass form is used, which is numericallystable, see [1].It is shown in [1] that there exist invertible matrices S; T 2 Rn£nwhich transform the pencil into the so-called generalized Weierstrassform such that E^ = SET and A^ = SAT are of the block form(3) E^ =µIm 00 0¶; A^ =µD 00 Ir¶with some matrix D 2 Rm£m.3. Main resultAssume now that the pencil sE ¡A undergoes a structured pertur-bation of the form scbb¤ with c 2 R and b 2 Rn is a standard unitvector or the di®erence of two standard unit vectors, that is b = ei orb = ei ¡ ej, i; j 2 f1; : : : ; ng . Let p = Sb and q = T ¤b, then(4) S(sE ¡ A+ scbb¤)T = sE^ ¡ A^+ scpq¤;where E^ and A^ are in the generalized Weierstrass form (3).Note that the relation (4) implies thatdet(sE^ ¡ A^+ scpq¤) = detS det(sE ¡ A+ scbb¤) detT;FREQUENCY COMPENSATION IN ELECTRICAL CIRCUITS 3and since detS and detT are nonzero constants, it follows that thepencils sE^ ¡ A^+ scpq¤ and sE ¡A+ scbb¤ have the same eigenvalues.Theorem 3.1. Assume that the pencil sE^ ¡ A^ is in the generalizedWeierstrass form (3). Let p; q 2 Rn be of the form p = (pT0 ; pT1 )T andq = (qT0 ; qT1 )T with p0; q0 2 Rm; p1; q1 2 Rr, then(¡1)r det(sE^ ¡ A^+ scpq¤) =det(sIm ¡D) + sc(det(sIm ¡D + p0q¤0)¡(1 + q¤1p1) det(sIm ¡D)):(5)The next result shows some particular cases of perturbations of theoriginal pencil. Note that the ¯rst item shows which perturbationshave no in°uence on the eigenvalues of the original pencil.Corollary 3.2. For a pencil as in Theorem 3.1, withAdj(sIm ¡D) := (sIm ¡D)¡1 det(sIm ¡D);the following hold.(1) If p0 = 0 or q0 = 0, then(¡1)r det(sE^ ¡ A^+ scpq¤) = det(sIm ¡D)(1¡ scq¤1p1):(2) If q1 ? p1, then(¡1)r det(sE^ ¡ A^+ scpq¤) = det(sIm ¡D) + scq¤0Adj(sIm ¡D)p0:In order to ¯nd appropriate perturbations, denoteQ(s) = det(sIm ¡D + p0q¤0)¡ (1 + q¤1p1) det(sIm ¡D)andP (s) = det(sIm ¡D);then the left hand side of (5) vanishes for some c > 0 if and only ifP (s) + scQ(s) = 0: One has to ¯nd values s 2 f¡t§ it : t > 0g forwhich(6) c = ¡ P (s)sQ(s)> 0:For such values, in particular, Im sQ(s)P (¹s) has to vanish. Appropriatevalues can be found by checking the sign changes of Im sQ(s)P (¹s) onthe ray f¡t+ it : t > 0g.Example 3.3. Assume that a pencil of the form (3) in R3£3 is givenwith m = 2, r = 1 andD =µa b¡b a¶4 HENNIG, KRAUSSE, SCHÄAFER, SOMMER, TRUNK, AND WINKLERwith a < 0, b > 0. ThenP (s) = (s¡ a)2 + b2 = (s¡ s0)(s¡ ¹s0)with s0 = a + ib. Let the perturbation of E^ be of the form c(e1 ¡e2)(e1 ¡ e2)T , then q = p = e1 ¡ e2, and via Corollary 3.2 it can beeasily seen thatQ(s) = 2(s¡ a) = (s¡ s0 + s¡ ¹s0):It follows from (6) that1c=ss0 ¡ s +s¹s0 ¡ s:A straightforward calculation implies that Im (P (s)=sQ(s)) vanisheson the ray s 2 f¡t+ it : t > 0g if and only if b ¸ ¡a. Actually, aquadratic equation for the appropriate values of t givest1;2 = ¡ 12a¡a2 + b2 § (b4 ¡ a4)1=2¢ ;and an investigation of Re (¡P (s)=sQ(s)) shows that only fors = ¡t2 + it2the value of c in relation (6) is positive.References[1] T. Berger, A. Ilchmann and S. Trenn, "The quasi-Weierstrass form for regularmatrix pencils", Lin. Alg. Appl., (2011) in press, preprint available online, In-stitute for Mathematics, Ilmenau University of Technology, Preprint Number09-21.[2] D. Krau¼e, J. Nowak, E. SchÄafer, R. Sommer, E. Hennig, "Frequency Com-pensation by Automated Topology Modi¯cation Using Mixed Analytical andNumerical Methods for Design of Fast TIA for HDTV Applications" in Proc.10th International Workshop on Symbolic and Numerical Methods, Modelingand Applications to Circuit Design (SM2ACD'08), Erfurt, Germany, Oct. 2008[3] P. Kunkel and V. Mehrmann, Di®erential-Algebraic Equations, Analysis andNumerical Solution, EMS Publishing House, ZÄurich, Switzerland, 2006[4] R. Sommer, D. Krau¼e, E. SchÄafer, E. Hennig, "Application of Symbolic CircuitAnalysis for Failure Detection and Optimization of Industrial Integrated Cir-cuits" in Design of Analog Circuits through Symbolic Analysis, to be publishedBentham Science Publisher.FREQUENCY COMPENSATION IN ELECTRICAL CIRCUITS 5Eckhard Hennig, Institut fÄur Mikroelektronik- und Mechatronik(IMMS), Erfurt, GermanyE-mail address: eckhard.hennig@imms.deDominik Krau¼e, Fachgebiet Elektronische Schaltungen und Sys-teme, Technische UniversitÄat Ilmenau, PF 100565, D-98684 Ilmenau,GermanyE-mail address: dominik.krausse@tu-ilmenau.deEric SchÄafer, Fachgebiet Elektronische Schaltungen und Systeme,Technische UniversitÄat Ilmenau, PF 100565, D-98684 Ilmenau, Ger-manyE-mail address: eric.schaefer@tu-ilmenau.deRalf Sommer, Fachgebiet Elektronische Schaltungen und Systeme,Technische UniversitÄat Ilmenau, PF 100565, D-98684 Ilmenau, Ger-manyE-mail address: ralf.sommer@tu-ilmenau.deCarsten Trunk, Institut fÄur Mathematik, Technische UniversitÄatIlmenau, PF 100565, D-98684 Ilmenau, GermanyE-mail address: carsten.trunk@tu-ilmenau.deHenrik Winkler, Institut fÄur Mathematik, Technische UniversitÄatIlmenau, PF 100565, D-98684 Ilmenau, GermanyE-mail address: henrik.winkler@tu-ilmenau.de

Frequency compensation for a class of DAE's arising in electrical circuits

https://www.db-thueringen.de/servlets/MCRFileNodeServlet/dbt_derivate_00023411/IfM_Preprint_M11_11.pdf

Frequency compensation for a class of DAE's arising in electrical circuits

Abstract

Similar works

Full text

Available Versions

Digitale Bibliothek Thüringen