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  PAN STABILITY TESTING OF DC CIRCUITS USING VARIATIONAL METHODS XVIII - SPETO pod patronatem. Summary
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  1 PCE SEMINIUM Z PODSTW ELEKTOTECHNIKI I TEOII OBWODÓW 8 - TH SEMIN ON FUNDMENTLS OF ELECTOTECHNICS ND CICUIT THEOY ZDENĚK BIOLEK SPŠE OŽNO P.., CZECH EPUBLIC DLIBO BIOLEK MILITY CDEMY, BNO, CZECH EPUBLIC XIII - SPETO pod patonatem PN STBILITY TESTING OF DC CICUITS USING ITIONL METHODS Summay The pape compises consideations concening some thought souces of cicuit theoy. It is shown that some etun to the categoies of analytical mechanics and nonequilibium pocess themodynamics can give a hand with the solution of such poblems as the stability of DC cicuits with nonecipocal elements. t fist sight, analysis of cicuits with contolled souces seems to be the closed pat of cicuit theoy. Howeve, putting the accustoming pocedues into pactice can lead to the paadoxical situations as shown fo example in "school" poblem with the independence of tansfe function on the input polaity of opeational amplifie. In ou pape, the oots of some paadoxical situations in cicuit theoy ae investigated in moe details. We investigate two estictions of Newton's mechanics and analytical dynamics that have also influenced cicuit theoy.
  2 . INTODUCTION nalysis of cicuits with contolled souces belongs to the well-known pats of cicuit theoy. Howeve, putting the accustoming pocedues into pactice can lead to the paadoxical situations as shown in following "school" example. Detemine the voltage gain F = 2 / accoding to Fig.. Conside infinite input amplifie esistance and zeo output esistance. Gain is abitay nonzeo eal numbe. Utilization of abitay method based on Kichhoff's laws and Ohm's law leads to the esult K 2 = F., F =, 0, (.) K2 2 whee K = and K2 =. 2 2 It is emakable that lim F = lim F = 2. (.2) Equation (.2) imposes the idea that the function of amplifie in Fig. does not depend on the polaity of input banch of Opmp. Howeve, this conclusion does not coespond to the pactical expeiences. Paadox oigin inside the thought system can point at cetain estictions of this system. Fo example, the well-known antique paadox about the chilles and totoise has aisen in consequence of the fact that the late knowledge of the aea of infinitesimal calculus could not be included to the consideations. Next paadox - when the pile of stones stops to be the pile taking away stone by stone - stops be the paadox consideing some pinciples of fuzzy sets. Ou "Opmp" paadox is caused due to peculia position of stability in the cicuit theoy. It is conditioned histoically because this doctine has been unde the influence of the Newton's mechanics and analytical dynamics. Equation (.) namely does not compise infomation if the equilibium point is stable o unstable. This equation only ensues that Kichhoff's laws and coupling condition 2 = will be complied. Thee is no sense in definition of voltage gain F in case of unstable equilibium point. In this way, the paadox (.2) is explained. It emains to investigate the stability conditions of ou DC solution. s usual in cicuit theoy, the simple model in Fig.2 can be compiled using equations those have led to the esult (.). The sign of open loop gain detemines the stability, e.g. 2 K. 2 < 0 o <. (.3) Let us exploe the oots of the paadox (.2) in moe details. We investigate two estictions of Newton's mechanics and analytical dynamics those have also influenced cicuit theoy. We concentate ou attention on the contolled souces fom this athe unusual point of view. 2. NEWTON'S SPECTS IN THE CICUIT THEOY It has been poved duing 50th yeas that Kichhoff's voltage law (KL) in the cicuit theoy K 2 coesponds to the 2nd Newton's postulate of the classical physics and consequently to the basic equation Fig.2 of vaiational calculus, i.e. Eule equation []. Choice of electical chage q as genealized Fig. K
  3 coodinates has been the assumption of this analogy. Then cuents i = q& have coesponded to genealized speeds, electical voltages to genealized foces and magnetic fluxes to momentum. In this way, the ingenious tools of analytical dynamics could be used mainly fo the analysis methods based on the KL, e.g. fo the little-known method of loop chages and fo the method of loop cuents. It is less known that this pinciple is also utilized to the cicuit analysis using nodal analysis method (NM). Utilizing the dual popeties of functions and cofunctions those descibe electical, magnetic and dissipative fields of used elements, the cicuit Lagange function can be constucted as a function of nodal voltages and thei deivatives L( v, v& ) = T, whee T () v & and () v ae time deivatives of electical field coenegy of chaged capacitos and the magnetic field enegy of inductos, v is the vecto of nodal voltages. In the aea of NM, the Hamilton vaiational pinciple can be fomulated as follows: The tajectoy connecting points v( t0 ) and v( t) that system chooses fo its motion, minimizes the time integal of system Lagange function t (2.) t 0 t δ L( v, v & )dt thus L dt = 0, (2.2) t0 whee δ means the symbol of vaiation of the whole tajectoy between points v( t ) and v( t ) aound the actual tajectoy. Eule equation is then equivalent to equation (2.2) d L L 0 & =. (2.3) dt v v The eal tajectoy v () t is the solution of Eule equation fo given initial conditions. The eal tajectoy is the extemale of vaiational task (2.2). The cicuit compising also esistos and excited by cuent souces can be descibed using extended Eule equation d L L d & I = dt v& v dt v (2.4) whee I is the vecto of cuent souces connected to cicuit nodes, is the ayleigh dissipative cofunction that maps the enegy dissipation on esistos. If the cicuit is chaacteized by conductance matix G, then T T v = I dv~ = G v~ dv~ (2.5) () ( ) Γ is the linie integal along the contou Γ fom the oigin of coodinates to the point v. In case of cicuits compising linea ecipocal esistos, the ayleigh function is as follows: () v v T = G v. 2 If the cicuit contains the elements those set coupling of type f ( v, v &) = 0 (i.e. contolled souces, contolled switches etc.), this condition will be included to the Lagange system function using the method of indefinite coefficients λ: L = Lλ. f. Γ 0
  4 The physical intepetation of coefficient λ is the system esponse to the coupling f. This is the genealized foce that the system has to expend to keep condition f. Moe infomation s can be found in the classical wok [2]. If the cicuit consists only of the esistive elements those ae coupled by condition f, the simplified fom of extended Eule equation may be used: ( λf ) = I. (2.6) v In case of linea ecipocal esistos we can wite ( λ f ) G. v = I. (2.7) v Neglecting coupling element, the equation (2.7) epesents the basic equation of NM. We have seen that algoithms commonly used in the cicuit theoy ae stongly influenced by the methods developed duing past centuies in the aea of classical physics. We have paticulaly followed the influence of fist two Newton's postulates. Howeve, the cicuit theoy has own peculiaities concening 3d Newton's postulate of eaction. 3. POSTULTE OF ECTION s mentioned in the opening example, the infomation about the cicuit stability does not follow automatically the analysis method. This infomation must be obtained independently of the method additionally using special test. It looks to be something infeio not included to the main theoy. This sepaation of the stability test and the est analysis is typical not only fo the cicuit theoy but also fo many othe banches like contol theoy. This is connected with the fact that these banches have assumed the conception of fist two Newton's laws of classical mechanics but they have modified the thid Newton's postulate of eaction accoding to own needs. Contol theoy has modified this postulate to the fom "eaction is equal to zeo". This fomulation has enabled to decompose oiginally two-way couplings in the system to the unilateal ones and to intoduce the system as oientated "block" diagam (the sample of such diagam consisting exclusively of the elements with the unilateal fowad coupling is in Fig.2). Moe infomation can be found in [3]. Cicuit theoy let the eaction chaacte to the natue of individual elements that can be eithe ecipocal (action = eaction) o nonecipocal (action eaction). Due to modification of the postulate of eaction, vaious theoies aise. Some thei conclusions seem paadoxically fom the classical physics point of view. Fo instance, the pepeetum mobile can exist in the cicuit and contol theoy (oscillato etc.). It is inteesting fo ou suggestion that invalidity of the law of enegy consevation is one of the implications of the 3d Newton's postulate modification. This fact is pactically pojected to the possibility of system instability. This possibility then follows natually the axiomatic of given theoy. Hee thee is necessay to seek explanation of why the stability test seems to be alien element in the aea of such theoy. Let us then ty to analyze ou cicuit in Fig. fom the positions mentioned in chapte 2. Fig.3 shows the same cicuit pepaed fo the NM desciption. The input voltage souce has been eplaced by the cuent souce I = G. pplying equation (2.7) yields G G G 2 2 G G I =.λ 0 λ (3.) Equilibium in the esistive netwok with the coupling condition f = = 2 0 (3.2)
  5 is eached by means of compensation cuent souces affecting 2 2 both nodes and!. On the othe hand, it is known that ideal voltage amplifie ensues condition (3.2) only using its output, I i.e. affecting only node!. This contadiction has aose because of the egulation (2.7) comes out the validity of the Newton's - postulate of the equality of action and eaction. Howeve, the voltage amplifie as nonecipocal element violates this Fig.3 egulaity by its own law that "eaction is equal to zeo". Taking into account afoementioned facts, equation (2.7) can be genealized as follows ( λ f ) G. v = I K (3.3) v whee K is the matix of weight coefficients that adapts elation 2 2 between the action and eaction accoding to eality. Let us then inset infomation that ou voltage amplifie has infinite input esistance to equation (3.3). We obtain I λ G v = I λ. (3.4) Fig.4 Meaning of the undetemined coefficient λ is then the cuent supplied to the netwok by the output of amplifie. Equation (3.4) along with the condition (3.2) give diections fo computation of both nodal voltages and unknown cuent λ. Now let us investigate how to test stability of cicuits without enegy stoage elements using vaiational pinciples. 4. POBLEM OF ENEGY DISSIPTION If the system has no stoage elements, then the most well-known citeion cannot be used to stability test because the time facto is not entiely pesented. Fo instance, this poblem is solved in [4] in espect of the algoithms fo computation of DC opeating points in the simulation pogam SPICE. The tem "potential stability" is intoduced in this wok. The pesented algoithms opeate in this way that paasitic inductances o capacitances ae added to the cicuit with the aim to each cetain dynamics. s shown in afoementioned chapte, the alone possibility of state instability is paadox as a matte of fact which is caused by the modification of the 3d Newton's postulate of eaction. s egads cicuit without inetia, this poblem is still moe complicated because alone Newton's dynamics is oiginally the doctine about consevative systems. esisto as the element dissipating enegy into heat is the alien element in the theoy based on the Newton's conception. This fact is well evident in the extended Eule's equation (2.4): the ayleigh's function is violently placed hee to expess eal esistos. esulting tajectoy of such system is not extemale of vaiational pinciple (2.2). Hamilton's pinciple is in its essence valid only in case of consevative systems. This is the sign of the peiod in which the poblems of celestial mechanics and othe phenomena wee solved. In these phenomena the system dynamics played dominant ole in compaison with the compaatively negligible dissipation of kinetic enegy. Howeve, ideal esisto as an element geneating heat has became the cental point of elatively young doctine - nonequilibium themodynamics. This doctine also epots to the tems as capacito and inducto, gives diections fo the compilation of system motion equations and it also has its own vaiational pinciples those mostly coespond to the pinciples of analytical dynamics. The nonequilibium themodynamics eveals that the integal Hamilton's pinciple cannot be efomed fo the systems with dissipative elements as esistos those themodynamic essence baffles Newton's conception.
  6 It is possible to seach answes to the stability poblems of pue dissipative systems in the themodynamical vaiational pinciples. mong them, the diffeential pinciples dominate, i.e. pinciples efeing to the instantaneous system behavio in the concete tajectoy point. One of the fom of least enegy dissipation pinciple given by Glansdoff and Pigogine [5] is suitable fo the cicuit desciption using NM. This pinciple can be loosely fomulated as follows: Duing motion, system minimizes the expession v v I T Φ =. (4.) () () v in view of the possible instantaneous vaiations of nodal voltages. Function is the ayleigh's system cofunction defined accoding to (2.5). This pinciple can be fomulated in moe pactical way: Stable DC opeating point occus in the local minimum of function (4.). emak: s egads nonecipocal systems, the choice of the contou Γ of integation is impotant fo the ayleigh cofunction definition (2.5). Fig.5 Let us attempt to analyze stability of ou voltage amplifie cicuit using the function Φ. fte disconnection of outside excitation to node " accoding to Fig.3, we can only constuct ayleigh cofunction using (4.). We disconnect output of amplifie. Ou task is to seach function of cicuit with open feedback loop (see Fig.5): ( ) I( v ~ ) dv ~ =. It can be easily poved that I = whee = 2. Thus ( ) 2 2 =. If > 0 then ayleigh cofunction has local minimum in the opeating point = 0. Then < 2. Howeve, this is the elation (.3) deived in the fist chapte. 5. CONCLUSION Equation (2.3) shows that KCL and Eule's equation of vaiational calculus ae equivalent if the suitably chosen cicuit voltages ae selected as system coodinates. This fact enables to use vaiational methods fo the cicuit analysis using NM. Equation (3.3) is univesal. Fo instance, DC cicuits with nonideal Opmps those models ae given using conditions f ( v, v & i ) = 0 can be analyzed using this equation. Themodynamical stability citeion (4.) can be also used fo the stability testing of lage cicuits with nonecipocal elements as Opmps. The suitable selection of integation contou fo the constuction of ayleigh cofunction is impotant key to success. EFEENCES - 2 I [] MYSLÍK,J.: nalysis of electical cicuits using vaiational methods. Electical Engineeing Jounal, No.4, 972. Czechoslovakia. [2] LNCZOS,C.: The vaiational pinciples of mechanics. Mathematical expositions, No.4, Toonto Pess, 962. [3] GLPEIN,I.I.: utomatics as one-sided mechanics. Moskva 964. [4] GEEN,M.M.-WILLSON,.N.: How to Identify Unstable DC Opeating Points. IEEE Tans. on CS I, vol. 39, No.0, 992, pp [5] GYMTI,I.: Non-equilibium themodynamics. Spinge-elag, New Yok
  7 SPŠE OŽNOĚ P.. Ing. Zdeněk Biolek Školní 60 alašská OŽNO P.., CZECH EPUBLIC OŽNO P.., CZECH EPUBLIC MILITY CDEMY BNO Depatment of Electical Engineeing K30 ssoc.pof. Ing. Dalibo Biolek, CSc. Kounicova 65, PS 3 Hoácké nám. 9/ BNO, CZECH EPUBLIC BNO, CZECH EPUBLIC