RLC Circuit Response And Analysis (Using State Space Method)

IJCSNS International Journal of Computer Science and Network Security, VOL.8 No.4, April 200848RLC Circuit Response and Analysis (Using State Space Method)Mohazzab1 JAVED, Hussain1 AFTAB, Muhammad1 QASIM, Mohsin1 SATTAR1Engineering Department, PAF-KIET, Karachi, Sindh, PakistanAbstract-- This paper presents RLC circuit response andanalysis, which is modeled using state space method. Itprovides a method with the exact accuracy to effectivelycalculate the state space models of RLC distributedinterconnect (nodes) and transmission line in closedforms in time domain and transfer functions by recursivealgorithms in frequency domain, where their RLCcomponents can be evenly distributed or variouslyvalued. The main features include simplicity andaccuracy of the said closed forms of the state spacemodels {A,B,C,D} without involving matrix inverse andmatrix multiplication operations, effectiveness andaccuracy of the said recursive algorithms of the transferfunctions. The response of the RLC is examined fromdifferent input functions by using Matlab. A timevarying state-space control model was presented andused to predict the stability and voltages of the RLCseries circuit results are shown to validate the method.We find that the effects of changing the resistances andcapacitors on the systems are negligent, whereaschanging the inductor causes the output to change. Thesalient features of this algorithm are the inclusion of theparameter variation in the RLC.1. INTRODUCTIONIn practice, engineering problems are difficult to solve.Most often, numerical methods are used as analyticalsolutions to such problems may be non-existent.Numerical methods in themselves are usually iterative innature requiring several intermediate steps in order toarrive at a solution. An RLC circuit (also known as aresonant circuit or a tuned circuit) is an electrical circuitconsisting of a resistor (R), an inductor (L), and acapacitor (C), connected in series or in parallel. An RLCcircuit is called a second-order circuit as any voltage orcurrent in the circuit can be described by a second-orderdifferential equation for circuit analysis.One very usefulcharacterization of a linear RLC circuit is given by itsTransfer Function, which is (more or less) the frequencydomain equivalent of the time domain input-outputrelation. These methods do not use any knowledge ofthe interior structure of the plant, and as we have seenallows only limited control of the closed-loop behaviorwhen feedback control is used.Manuscript received April 5, 2008Manuscript revised April 20, 20082. STATE-SPACE METHOD2.1 DefinitionThe so-called state-space description provide thedynamics as a setoff coupled first-order differentialequations in asset of internal variables known as statevariables, together with a set of algebraic equations thatcombine the state variables into physical outputvariables. The concept of the state of a non-lineardynamic system [7], refers to a minimum set ofvariables, known as state variables that fully describethe system and its response to any given set of inputs.This definition asserts that the dynamic behavior of astate-determined system is completely characterized bythe response of the set of n variables xi(t), where thenumber n is defined to be the order of the system. Thesystem shown in Fig. 1 has two inputs u1(t) and u2(t),and four output variablesy1(t), . . . , y4(t). If the systemis state-determined, knowledge of its statevariables(x1(t0), x2(t0), . . . , xn(t0)) at some initial timet0, and the inputs u1(t) and u2(t) for t t0 insufficient todetermine all future behavior of the system. The statevariables are an internal description of the system whichcompletely characterize the system state at any time t,and from which any output variables yi(t) may becomputed. Large classes of engineering, biological,social and economic systems may be represented bystate-determined system models.Fig. 1: System inputs and outputsFor such systems the number of state variables, n, isequal to the number of independent energy storageelements in the system. The values of the state variablesat anytime t specify the energy of each energy storageelement within the system and therefore the total systemenergy and the time derivatives of the state variables

IJCSNS International Journal of Computer Science and Network Security, VOL.8 No.4, April 2008determine the rate of change of the system energy.Furthermore, the values of the system state variables atany time t provide sufficient information to determinethe values of all other variables in the system at that time.There is no unique set of state variables that describeany given system; many different sets of variables maybe selected to yield a complete system description.However, for a given system the order n is unique, andis independent of the particular set of state variableschosen. State variable descriptions of systems may beformulated in terms of physical and measurablevariables, or in terms of variables that are not directlymeasurable. It is possible to mathematically transformone set of state variables to another; the important pointis that any set of state variables must provide a completedescription of the system.2.2 The State EquationsA standard form for the state equations is usedthroughout system dynamics. In the standard form themathematical description of the system is expressed as aset of n coupled first-order ordinary differentialequations, known as the state equations, in which thetime derivative of each state variable is expressed interms of the state variables x1(t), . . . , xn(t) and thesystem inputs u1(t), . . . , ur(t). In the general case theform of the n state equations is:x 1 f1 (x, u, t)x 2 f2 (x, u, t): :x n fn (x, u, t)49where f (x, u, t) is a vector function with ncomponents fi (x, u, t).For an LTI system of order n, and with r inputs, Eq.(1) becomes a set of n coupled first-order lineardifferential equations with constant coefficients:x 1 a11x1 a12x2 . . . a1nxn b11u1 . . . b1rurx 2 a21x1 a22x2 . . . a2nxn b21u1 . . . b2rur.x n an1x1 an2x2 . . . annxn bn1u1 . . . bnrurEq. (3)where the coefficients aij and bij are constants thatdescribe the system. This set of n equations definesthe derivatives of the state variables to be a weightedsum of the state variables and the system inputs.Equations may be written compactly in a matrixform:x Ax BuEq. (4)where the state vector x is a column vector of length n,the input vector u is a column vector of length r, A is ann x n square matrix of the constant coefficients aij, andB is an n x r matrix of the coefficients bij that weight theinputs.2. 3 The Output EquationsEq. (1)where x i dxi/dt and each of the functionsIt is common to express the state equations in a vectorform, the set of n state variables is written as a statevector x(t) [x1(t), x2(t), . . . , xn(t)]T, and the set of rinputs is written as an input vector u(t) [u1(t),u2(t), . . . , ur(t)]T . Each state variable is a timevarying component of the column vector x(t).Thisform of the state equations explicitly represents thebasic elements contained in the definition of a statedetermined system. Given a set of initial conditions(the values of the xi at some time t0) and the inputs fort t0, the state equations explicitly specify thederivatives of all state variables. The value of eachstate variable at some time Δt later may then be foundby direct integration. The system state at any instantmay be interpreted as a point in an n-dimensional statespace, and the dynamic state response x(t) can beinterpreted as a path or trajectory traced out in thestate space. In vector notation the set of n equations inEqs. (1) may be written:x f (x, u, t)Eq. (2)A system output is defined to be any system variable ofinterest. A description of a physical system in terms of aset of state variables does not necessarily include all ofthe variables of direct engineering interest. An importantproperty of the linear state equation description is thatall system variables may be represented by a linearcombination of the state variables xi and the systeminputs ui. An arbitrary output variable in a system oforder n with r inputs may be written:y(t) c1x1 c2x2 . cnxn d1u1 . . drur(5)Eqwhere the ci and di are constants. such equations may bewritten as:y1 c11x1 c12x2 . . . c1nxn d11u1 . . . d1rury2 c21x1 c22x2 . . . c2nxn d21u1 . . . d2rur.ym cm1x1 cm2x2 . . . cmnxn dm1u1 . . . dmrurEq.(6)

IJCSNS International Journal of Computer Science and Network Security, VOL.8 No.4, April 200850Table 1: Power VariablesThe output equations are commonly written in thecompact form:y Cx DuEq.(7)where y is a column vector of the output variablesyi(t), C is an mxn matrix of the constant coefficientscij that weight the state variables, and D is an m x rmatrix of the constant coefficients dij that weight thesystem inputs. For many physical systems the matrixD is the null matrix, and the output equation reducesto a simple weighted combination of the statevariables:y Cx.Eq. (8)2. 4 State Equation Based Modeling ProcedureThe complete system model for a linear time-invariantsystem consists of (i) a set of n state equations,defined in terms of the matrices A and B, and (ii) a setof output equations that relate any output variables ofinterest to the state variables and inputs, and expressedin terms of the C and D matrices. The task ofmodeling the system is to derive the elements of thematrices, and to write the system model in the form:x Ax Buy Cx Dustate equationoutput equationEq. (9)Eq. (10)The matrices A and B are properties of the system andare determined by the system structure and elements.The output equation matrices C and D are determinedby the particular choice of output variables.3. APPLYING STATE SPACE METHOD ONRLC CIRCUIT3.1 Series RLC CircuitConsider the series RLC circuit given variableThroughvariableknowniV12V23V3giRiLiCWe reduced this circuit in the “Big Picture” handoutto yield a second order differential equation relatingthe input v and the voltage across the capacitor v .1gV1g LC(d2V3g/d2t) RC(dV3g/dt) V3g3gEq.(11)The state-space representation can be thought of as apartial reduction of the equation list to a set ofsimultaneous differential equations rather than to asingle higher order differential equation. Although thestate variables of a system are not unique anddefinition of many non-physical variables is possible,we will work with physical variables, specifically theenergy storage variables of a system. There are twoindependent energy storages in RLC circuit, thecapacitor which stores energy in an electric field andthe inductor which stores energy in a magnetic field.The state variables are the energy storage variables ofthese two elements,V3g and iL.The energy storage elements of a system are what makethe system dynamic. The flow of energy into or out of astorage element occurs at a finite rate and is describedby a differential equation relating the derivative of theenergy storage variable (a state variable) to the otherpower variable of the element. We will first formulatethe state equations to find state variables, V3g and iL.We will then formulate the output equations tocalculate all of the remaining unknowing powervariables using the input v, and the state variables V3gand iL.3.2 Important Points Fig. 2: Series RLC circuitAlways start a state equation reduction withthe elemental equation of an energy storageelement. You will have as many stateequations as there are independent energystorages in the system.