LAPLACE TRANSFORM AS AN USEFUL TOOL IN TRANSIENT STATE ANALYSIS

1. LAPLACE TRANSFORM AS AN USEFUL TOOL IN TRANSIENT STATE ANALYSIS Oana Mihaela Drosu Dr. Eng. , Lecturer POLITEHNICA University of Bucharest Department of Electrical Engineering LPP Erasmus+

2. PART IINTRODUCTION TO LAPLACE TRANSFORM THEORY

3. Definition The Laplace transform is a linear operator that switched a function f(t) to F(s), where s = s+wj. (Go from time argument with real input to a complex angular frequency input). Note that the real part s of the complex variable smust be large enough for the integral to converge.

4. There are two governing factors that determine whether Laplace transforms can be used: • f(t) must be at least piecewise continuous for t ≥ 0 • |f(t)| ≤ Meγt where M and γ are constants Restrictions

5. Conditions • to be limited and integrable on any interval (t1, t2), where 0< t1< t2 ; • to be absolute integrableon the interval [0, t0 ], wheret0 >0; • at least one value s=s0, to exist for the integral to have sense; • if it is absolute convergent for s=s0,then it will be generally absolute convergent :

6. In these conditions we can find a minimum value of Re{s}, denoted , for the Laplace transform of f(t)to exist (this is simple convergence abscissa); • The definition domain for F(s)is the complex semiplane at the right of :

32. BIBLIOGRAPHY: • Norman Balabanian: Electric Circuits; McGraw-Hill,Inc., USA, 1994 • K.E. Holbert: Laplace transform solutions of ODE’s, 2006 • Walter Green: The Laplace transform presentation; University of Tennessee, Electrical and Computer Engineering Dep. Knoxville, Tennessee • E.Cazacu, O.Drosu, G.Epureanu, Theory and applications of electric circuits: vol.1 Transient state analysis Matrix Rom, Bucharest, 2005.

33. PART II SOLVING TRANSIENT STATE CIRCUITS USING LAPACE TRANSFORM METHOD

34. INVERSE LAPLACE TRANSFORM Mellin-Fourier(orBromwich-Wagner) formula is a general case of Fourier integral transformation. It establishes that, for each Laplace transform F(s)There is a coresppondance original (time variable) function f(t) given by: Most of the time in the common applications, the function F(s) is expressed as the ratio of two polynomial functions, the denominator having the higher degree:

35. Performing the inverse transform is straightforward when using partial fractions expansion . Whereskare the multiple solutions of order mkfor the polynomial function N(s), and the coefficients Cklare given by:

36. Applying inverse Laplace transform we obtain the general Heaviside formula for the original function:

37. If the denominator has only simple roots, then the general Heaviside formula can be simplified for the following cases: 1) If the denominator does not have also the solution s=0, then the Laplace image F(s) can be written as : And the corresponding original function will be given by Heaviside I formula:

38. 2) If the denominator has also the solution s=0, then the Laplace image F(s can be written as : In this case : N(s)=sP(s), Then, the original function f(t) is given by Heaviside II formula:

39. Kirchhoff Equations in time-domain: Kirchhoff Equations after Laplace transform is applied:

40. The ideal resistor

41. The inductor

42. The ideal capacitor

43. The coupled coils

44. Ideal current and voltage sources

45. The algorithm for solving transient circuits using Laplace transform • We determine the initial conditions of the circuit (current through the inductors, voltages on capacitors) before switching action. • We draw the equivalent operational circuit, containing the Laplace transform of given sources, the sources corresponding to initial conditions, operational impedances corresponding to the R, L,C elements. • We calculate the images of the given time variable functions (usually voltage of current sources) using direct transformation formula or transform pairs tables. The expressions will depend on the complex variable s.

46. We apply the operational expresions of Kirchhoff Equations (or different methods ussually applied to circuits: loop method, node potentials methods, etc). These equations will be solved with respect to Laplace images of the unknown functions. • After finding the Laplace transforms of the unknown quantities, the original functions (time dependent) are determined using inversion methods (Mellin-Fourier, Heaviside formulas or tables with transform pairs);

47. Example 1 Let’s consider the circuit given in Fig.1-a, with the following values of the circuit elements : R1 = 6 , R2 = 3 , inductance L = 0.8 mH, and constant value of the input voltage source E = 36V. At moment t =0 the switchKcloses. Using Laplace transform method determine the variations of the currents through the three elements of circuit and variation of the inductor voltage. Fig.1-a

48. Fig. 1-b The initial condition of the circuit is determined by the current through the inductor and it can be easily calculated from the circuit presented in the Fig. 1-b: iL(0–) = 0

49. After switching (closing K), the equivalent operational circuit is given in the Fig. 1-c Fig. 1-c Equivalent circuit after switching Taking into consideration the equivalent circuit, we can determine the total equivalent operational impedance and, then, using Kirchhoff II Theorem, we can calculate the current through the resistor R1.

50. Using current divider formula, we can determine the current through resistor R2, respectively through the inductor L: