Cap9 Laplace Transformation

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Laplace TransformationA special category of methods is represented by the operational methods. For these, the algorithm to solve the problem in transient regime is like the algorithm to solve the analysis for sinusoidal regime. The differential equations of network theory involve time as the independent variable and the differential equation is said to be in the time domain. The Laplace transformation and its use in a prescribed procedure take a linear differential equation with constant coefficients in the time domain and transforms it as it follows: (1) Transforms the differential equation into complex frequency (or s domain), where the independent variable is the complex frequency s= W j[ . (2) Permits the use of algebraic methods to obtain the Laplace transform of the solution. (3) Allows the solution in the time domain to be recovered by an inverse transformation. The Definition of the Laplace Transform. Original and Image Functions The Laplace transformation, or Laplace transform, of a function f(t) is defined as:

L[ f (t )] ! F ( s) ! f (t )e st dt0

g

(1)

where, in general, s is a complex number ( W j[ ). If the integral in the equation (1) is written as :

F (s ) ! f (t )e st dt0

T

(2)

then it is seen that the Laplace transform will exist as long as the limit given by the equation (2) exists. When this limit exists then equation (1) is said to converge, and the Laplace transforms exists and has meaning. If the limit of the integral does not exist the Laplace transform does not exist. In the equations (1) and (2), it should be noted that functions in the time domain are indicated by a lowercase letter [f(t),g(t),] and their corresponding Laplace transforms are designated by the corresponding uppercase [F(s),G(s),]. The inverse Laplace transformation:

f (t ) ! L1[ F ( s)] !recovers it from F(s).

1 W 0 j[ st W 0 j[ F (s)e ds 2Tj

(3)

Some basic operations 1.The Laplace Transform of the Sum of Two Functions (Addition) Let L[f(t)]=F(s) and L[g(t)]=G(s) then:

L[ f (t ) g (t )] ! [ f (t ) g (t )]e st dt ! f (t )e st dt g (t )e st dt ! F ( s ) G ( s)0 0 0

g

g

g

(It is shown that because integration is a linear operator, the Laplace transform is also a linear operator). Thus, the Laplace transform of the sum of two functions is the sum of the individual transforms of the functions: L[f(t)+g(t)]=F(s)+G(s) 2. The Laplace Transform of a Funtion Multiplied with a Constant Let L[f(t)]=F(s) theng g

L[Cf (t )] ! Cf (t )e st dt ! C f (t )e st dt0 0

or: L[Cf(t)]=CL[f(t)] The Laplace transform of a constant times a function(?...) is equal to the constant times the Laplace transform of the function. It is demonstrated that the Laplace transform is a linear operator:

L[ Ck f k (t )] ! Ck L[ f k (t )]k !1 k !1

n

n

3. The Laplace Transform of the Derivate of a Fuction Let L[f(t)]=F(s). Then, assuming that f(t) possesses a derivative g d d L[ f (t )] ! [ f (t )]e st dt 0 dt dt An integration by parts with: d u ! e st ; dv ! [ f (t )]dt; du ! s e st dt ; v ! f (t ) in: dt

g

0

g u dv ! u v / 0 vdu 0

g

gives:

L[or:

g d g f (t )] ! f (t ) e st / 0 s f (t )e st dt 0 dt

d f (t )] ! sF ( s) f (0) (*) dt Again let L[f(t)]=F(s). By equation (*): L[ f d)] ! sF ( s ) f (0) (t Thus: d L[ f d )] ! sL[ f d) f d )] ! s[ sF ( s ) f (0)] f d ) and hence: (t (t (0 (0 d L[ f d )] !! s 2 F ( s) sf (0) f d ) (t (0 These results may be extended by mathematical induction to the Laplace transform of the n-th derivative: L[ f n (t )] ! s n F ( s) s n1 f (0) s n 2 f d ) ... f ( n1) (0) (0 L[The Laplce Transform of the Integral Function With L[f(t)]=F(s) the Laplace transform of the integral with the upper limit t replaced by T where g e a e 0 will be: g T T L[ f (t )dt ] ! f ( z )dz e st dt a 0 a An application of integration by parts with T 1 u ! f ( z )dz; dv ! e st dt; du ! f (t )dt ; v ! e st a s gives: T T 1 1 g L[ f (t )dt ] ! e st f ( z )dz / g f (t )e st dt 0 a a s s 0 or with the limits substituted, T 1 1 0 L[ f (t )dt ] ! F ( s ) f (t )dt a s s a The integral

0

a

f (t )dt is often called the accumulation from t=a to

t=0- and is represented as the value of a quantity at t=0-

0

a

f (t )dt ! f (t )dt / t !0

The accumulation represents the value of the integral at t=0-. For example, the charge that has accumulated on the plates of a capacitor during the time interval between some time t=a and t=0- is the value at t=0-:

q(0) ! i (t )dt ! idt / t !0a

0

The flux set up around an inductor during the tie interval 0