Chapter 8 NOVEL STATE SPACE APPROACH TO ELECTRIC CIRCUITS WITH VOLTAGE SOURCE

Transcription

1 Chapter 8 NOVEL STATE SPACE APPROACH TO ELECTRIC CIRCUITS WITH VOLTAGE SOURCE 25

2 NOVEL STATE SPACE APPROACHES TO ELECTRIC CIRCUITS WITH VOLTAGE SOURCE The modern control theory which has developed to meet the stringent requirements of complex systems with multiple inputs and multiple outputs is based on state space representation approach. State space analysis of a system is a simple task provided the system variables are assigned with correct state variables. It is used in classical dynamics, mechanical systems and all engineering systems. Nevertheless for an electric circuit, assigning state variables and state space modeling is a difficult task. In this thesis novel methods of assigning state variables particularly for electric circuits have been suggested. This makes the state space representation of electric circuit a routine one that does not require any specific substitution. 8. State space representation of Electric circuits State space method of analysis of a multivariable system is an appropriate tool for finding complete solution of system variables even with initial conditions. Hence state space representation of a system has become popular. There are numerous ways of representing a system in state space. The application of state space techniques to electric circuits is relatively tedious work compared to mechanical and other systems. For electrical circuits, Ogata has tried a method [6] based on choosing energy variables, such as current through the inductor and voltage across the capacitor as state variables and that is adopted for electric circuits. To use such variables the circuit should be simple enough with just one capacitor in the shunt branch and an inductor in the series branch. In pedagogical point of view and as new teaching methods, interesting techniques have been evolved and reported here, which will be applicable even when circuit is complicated. 26

3 These methods do not need any critical imagination in choosing the state variables and they follow sequential steps. For simple electric circuits with shunt capacitor and series inductor il and vc> may be ideal state variables. When the capacitor in a circuit has resistance or inductance in series the selection of state variables becomes difficult. Nise [4] tries to solve the circuit equations for the voltage across the inductor and current through the capacitor to find the state space representation in a round about manner. Francois [9] suggests admittance method, which is also difficult to evaluate the state variables. Here effective methods of state space approach to electric circuits with voltage source are developed and presented. The proposed method is based on converting integro-differential equations into ordinary linear differential equation. This idea is brought out by comparing with mechanical systems. In mechanical and other such systems one can notice that system equations are not in integro-differential form but represented by ordinary linear differential equation. When we adopt ordinary linear differential equation and variables other than il and vc as state variables, state space modeling becomes easy even for complicated circuits. Nise[4] has tried to solve the circuit equations for the voltage across the inductor and current through the capacitor to find state space representation. But this method becomes quite involved when the number of loops is more. Francois suggests admittance method which is also difficult to evaluate the state variables. When the circuit has more loops with energy storage elements, the numbers of simultaneous equations are equally increasing which further complicates the analysis. Francois in analysing an electric circuit has straight away differentiated the integro differential equation to eliminate the integral term. This leads to derivative term present in the input variable that will also complicate the analysis. Our method explained below has 27

4 high teaching value and becomes an additional simple technique along with the existing methods. 8.2 Circuits with voltage source A typical electric circuit with voltage source is shown in Fig.8..The state variables for this circuit are assigned by skillfully selecting the electrical parameters as state variables. In the conventional method, current through the inductor and voltage across the capacitor are chosen as state variables, i.e. physical variables are considered as state variables. The circuit equations are written such that each equation is a first order differential equation. Here state variables are selected by listing voltage current relationship between energy storage elements. Applying Kirchhoff s Current law to the node in the circuit U + i2 + i3 = dv i.e., i.+i,+c - =...(8.) ' 2 dt ' Applying Kirchhoff s Voltage law for loop Li - vc + i R i +e(t) = at...(8.2) Applying Kirchhoff s voltage law for loop 2...(8.3) Here the variables ii(t), i2(t) and vc(t) are chosen as state variables xi(t), x2(t) and X3(t) respectively because in each equation one variable contains first order differential term. The equations in terms of state variables are Lx x, = -i^x, + x3 - u{t)...(8.4) 28

5 L2 x2 = -R2x2 + x3...(8.5) Cx3 = x, -Xj...(8.6) Fig.8. Electric circuit with branch current The corresponding state space representation is given by the expression ' ' X Lx X, ~L\ R, x2 = 2 x7 + l l2 t2 x3 X, ~c L J...(8.7a) where u(t) = e(t). If the output y(t) is concerned with potential drop across the resistor R2. Then the output equation in state space form is y(/)=[ R2...(8.7b) When the circuit becomes somewhat complicated as shown in Fig.8.2, the analysis becomes lengthy and cumbersome as the selection of proper set of state variables becomes a difficult task and the conventional method fails. 29

6 8.3 Proposed Simultaneous Equation Method (Method - ) Here a simultaneous equation method is suggested to retain il and vc as state variables. It is possible to retain the physical state variables using simultaneous equations. The method is developed for circuit shown in Fig.8. as follows. Consider i, \2 and ic as outgoing branch currents, while ii is flowing through Lj, h is flowing through L2 and ic is flowing through the capacitor. VL + vl2 + ic = -it Ri+h + vc ej(t)...(8.8) vli + vl2 + ic = Oil ~~h R2 + vc...(8.9) vli + vl2 + ic = -ii - i2 + vc...(8.) It is clear from the above equations the variables ij, i2 and vc are the state variables xi(t), x2(t) and X3(t) respectively written in terms of vli, vl2 and ic which will contribute first differential of the state variables. The equations (8.8), (8.9) and (8.) are solved to evaluate vli, vl2 and ic in terms of i2 and vc. The resulting solution is vl] =-Riii+vc-ei(t)...(8.) vl2 = - R2j2 + vc...(8.2) ic = -ii -h...(8.3) where vli = Ljdii/dt, vl2 = L^dh/dt and ic = Cdvc/dt. Hence it, 2 and vc are chosen as state variables xi(t), x2(t) and x3(t) respectively and the equations (8.), (8.2) and (8.3) are written in state variable form. vli =-RiXi+x3-ej(t)...(8.4) vl2 = - R2X2 + x3...(8.5) ic = -xi - x2...(8.6) The state space representation is obtained as given in equation (8.7). This new method utilising simultaneous equations is routine but time consuming. 3

7 For the slightly modified circuit shown in Fig.8.2 with outgoing branch currents ii through ili, b through il2 and ic through C. The loop equations are written in terms of ij, i2 and vc as VLI +ii R]+ e;(t) - icr2- vc =...(8.7) Vl2 + b R3 - icr2 - vc =...(8.8) ic = -ii -b...(8.9) The three unknowns vli, vl2 and ic are written in terms of i )2 and vc which are chosen as state variables. Writing in proper form vli + vl2 - icr2 = -ii Ri + i2 + vc - e;(t)...(8.2) vli + vl2 - icr2 = Oil - i2 R3 + vc...(8.2) vli + vl2 + ic = -ii - b + vc...(8.22) where, vli = Lidii/dt, vl2 = L2di2/dt and ic = Cdvc/dt. Hence ij, i2 and vc are chosen as state variables xi(t), X2(t) and X3(t) respectively and the equations (8.), (8.2) and (8.3) are written in state variable form. VLi = -(R+R2) XI- R2X2 + x3 - e,(t) vl2 = - R2X - (R2+R3) x2 + x3 ic = -xi - x2...(8.23)...(8.24)...(8.25) The resulting state space equation is as follows VL (R+R2) R 2 X f VL2 = -R2 -(R2+R3) x2 + *c L - - _x3 _...(8.26) In a bid to retain the physical variables as state variables, the solution of state space representation based on simultaneous equation becomes slightly complicated even though routine. The addition of new element like resistance here or a loop of elements will increase the complexity but the method is routine and based on sequential steps. 3

8 L, L2 r3 Fig.8.2 Electric circuit with two elements in the central branch 8.4 Proposed charge equation Method (Method - 2) The electric charge equation method of state space representation, which does not require any prior assumption is as follows. The loop equations are written for the electric circuit shown in Fig The loop equation based on the reference current ii is written as 'A +L\^r + ~ R'i -h)dt = e{t) at c J...(8.27) The loop equation based on the reference current i2 is written as j(/2 -/,)dt + h^r + hr3 = dt...(8.28) r2 The integration term in equations (8.27) and (8.28) are eliminated by changing current as rate of charge. Hence equations (8.27) and (8.28) become L, ~ + R^ + ~(qi -q2) = e(t) dt2 dt CWl H2...(8.29) "2 ;i' + Ri At dq2 + 7; (#2 ~ 9i) - dv dt C...(8.3) 32

9 The state variables are directly chosen as q,=xt, qi=x2, q2 = x3, q2 = x 4 x, = x2...(8.3) The equation (8.29) becomes x, = x2. X-, H X, + I CZ,, (x, -x3) e(/)...(8.32) x3 x4...(8.33) The equation (8.3) becomes. R3,. n x4 + -^x4+ (x3-x,) =...(8.34) The equations are written in vector matrix form r ] * X, CL, CL, x2 *3 *4. cl2 cl2 R j z.2_ X, " ' + L, x3 LX4j...(8.35a) The output is chosen as current q2 i.e. X4 through R2. x, j; = [ 2 X 3 x4...(8.35b) If the output is chosen as potential drop across R2 due to current X4, then x, y = [ R 2 3 *4...(8.36) Suppose the state variables are chosen in the other way as 33

10 qi=xi, q2 = x2, qi=x3, q 2=x4 Then equation (8.29) and equation (8.3) are written in state vector matrix form as R, CL, CZ, -- R, L*L2 (LL 7...(8.37a) y = [o l'...(8.37b) The integro differential Kirchhoff equations for the circuit shown in Fig.8.2 are written quit easily. Then the current is chosen as rate of charge q and integro differential equations become ordinary differential equations. The loop equation based on the reference current ii is written as /, R{ +L, f(i, - i2 )dt + (/, - i2 )R2 = e(t) at c :...(8.38) The loop equation based on the reference current i2 is written as (/, - /, )R2 + f(j, - /, )dt + L2 + /2i?3 = n * /It dt...(8.39) The integration terms in equations (8.38) and (8.39) are eliminated by choosing current as rate of charge as d2q,, D dq,,dq, dq: ) = «(<) ' ~U'...(8.4) d2q2,ddq2 dq2 dqx L " 7? H R2 ( ) + 7^2-9,i) = 2 dt2 dt 2 dt dt C...(8.4) The state variables are directly chosen as 34

11 Ql ^, q^ X2, Q2 X3, q2 *^4 The equation (8.4) becomes x, = x....(8.42) Rl, V. x2 + x2 + (x, -x3) + -^(x2 -x4) = e( Li X-<j i-<j...(8.43) The equation (8.4) becomes...(8.44) x4 + x4 *3 + ^2 (x4, -*2) \ + ^2 The resulting state vector matrix is -Xi) =...(8.45) r * r (A+tf2) r2 * x2 Ck A Ck k *2 *3 R2 *3 (R\+R2) x4_ CL, A> CL, h J X4 LH...(8.46a) If the output is the potential drop e across R3 e<> i2 R-3 q 2 -^3 i.e. y = R3 X4 The output equation is X\ y=[ R, x2 x3 x4...(8.46b) Suppose when the state variables are chosen in the other way as qi=x], q2 = x2, <?,=x3, q2 = x4 The circuit equations become 35

12 X, =x3...(8.47) x2 = X4...(8.48) Rl +R X, X, + x3 CLX Cl, 2 k...(8.49) CL. -x, + CL, R2+R, -x, + L2 L....(8.5) The corresponding state space equations is X, *2 x3 X, J, (R)+R2) *2 *2 + CLy CZ, k k r2 (R2+R3) Xa x4_ Ch CL2 l2 L2 x3 A J...(8.5a) The output e= 2R3 = q2 R3= R3X4 The output equation is y = [ R, x, x2 x3 X4...(8.5b) 8.5 Analysis of Typical Circuit The application of charge variable method for perfect nodal electric circuit is explained here. The circuit shown in Fig.8.4 is simple, but choosing the state variables by the existing method is tedious one. The state equations are determined by nodal analysis. e- x x, - x. C, Xi = h R, R,...(8.52) c, x 2 = R2...(8.53) The state vector matrix equation is 36

13 r- Xi _x2 L... + **, -^2^ R2c2 R2c xl x2 + *...(8.54a) The output e = X2 Hence y = [o if*" _*2 J...(8.54b) 8.6 Proposed method The charge variable method is convenient even for nodal type circuits. According to the proposed method the circuit equations are written with reference to loop currents. Ri R2 Fig.8.4 Simple RC circuit e(t)=rlii +-J-J(/, -i2)dt...(8.55) ci 7T f(»2 ~h)dt + R2h +~r[i2dt...(8.56) C, J C2 J The above equations are simplified by replacing current with rate of charge as XiVt + yrivi -42) = e(...(8.57) Jr(g 2-9i) + ^2?2 + 7 ^2 =...(8.58) c, c2 Choosing q, = x,, q2- x2 it is obtained 37

14 i x' + ^7T"(x< ~x2) = --u(t) \ j (_^ j iv]...(8.59) v _l_ v (x -x ) d r* DC' K2C2 K2(^ j.(8.6) The state equation is _ * _*2. m i r2cx RXC{ (,, \ + r2cx r2c 2 J L*2j + RX...(8.6a) The output is e = \i2dt C, J i-e. e= -?2 y C, The output equation y : O _ j~ L c2j H CM H _ i (8.6b) 8.7 Analysis by state variable diagram In this method the input function is differentiated, to convert integrodifferential equation into ordinary differential equation. The state variable diagram technique explained in the previous chapters is applied to determine state space representation. The state variable diagram of any one of the four categories shall be used to analyse electric circuit. R L C -yw «_] U(t) Fig.8.5 Electric circuit with single input 38

15 The equation for the electric circuits shown in the Fig.8.5 is Ri + L + f idt = i dt C J Further differentiating it is obtained ndi Td2i.,... R f- L h / dt = u{t) dt dt C i.q., d2i Rdi.\.,x y+ +» (=-r«( dt L dt LC L...(8.62) Taking Laplace Transform on both sides 2 R {s+ts+ic){s)=tsu{s)...(8.63) Following the standard form for second order system (s2 + axs + a2.v)/(i)= (bs2 +b]s + b2)ll(s) The equation for the circuit becomes I(s) _ Os2 +6,5 + U(s) s2+axs + a2...(8.64) Here b = and b2 = The state variable diagram is given in Fig.8.6 Then x] = x2 x2 = a2x] axx2 bo^--* 39

16 The state vector matrix equation is * + _*2_ -a2 ~a,_ " The output equation format for second order system is y = [b2-ba2 bx-ba{] xi *2 i.e., _y = [-a2 6,-a,] y = [ bx] x, X 8.8 Conclusion An easy and direct method of selection of state variables for electrical circuits is proposed. The method is simple. The state variables are selected in a sequential order. There is no need of prediction and critical selection of state variables. The analysis also becomes routine. The method is simple and important in pedagogical point of teaching. Even though this method is novel one it has a limitation that if the element representing highest derivative term in the circuit equation becomes common to two loop currents the method fails. This can be averted by solving for the variables of the equation and the state space representation for each variable shall done using phase variable technique. Here it is proved that state variable diagram of any one of the four categories shall be used to analyse electric circuit. 4