0195136047.pdf

142 TIME-DEPENDENT CIRCUIT ANALYSIS

The form of the natural response due to an impulse can be found by the methods presented

in this section. As for the initial capacitor voltages and inductor currents, notice that a current

impulse applied to a capacitance provides an initial voltage of

vC

(

0 +

)

=

q

C

(3.2.45)

whereqis the area under the current impulse. Similarly, a voltage impulse applied to an inductance

provides an initial current of

iL

(

0 +

)

=

λ

L

(3.2.46)

whereλis the area under the voltage impulse. The impulse function allows the characterization

of networks in terms of the system’s natural response, which is dependent only on the network

elements and their interconnections. Once the impulse response is known, the system response

to nearly all excitation functions can be determined by techniques that are available, but not

disclosed here in view of the scope of this text.

3.3 Laplace Transform

Many commonly encountered excitations can be represented by exponential functions. The

differential equations describing the networks are transformed into algebraic equations with

the use of exponentials. So far, methods have been developed to determine the forced re-

sponse and the transient response in Sections 3.1 and 3.2. Theoperational calculuswas de-

veloped by Oliver Heaviside (1850–1925) based on a collection of intuitive rules; the trans-

formation is, however, named after Pierre Simon Laplace (1749–1827) because a complete

mathematical development of Heaviside’s methods has been found in the 1780 writings of

Laplace. The Laplace transformation provides a systematic algebraic approach for determining

the total network response, including the effect of initial conditions. The differential equa-

tions in the time domain are transformed into algebraic equations in the frequency domain.

Frequency-domain quantities are manipulated to obtain the frequency-domain equivalent of the

desired result. Then, by taking the inverse transform, the desired result in the time domain

is obtained.

The single-sided Laplace transform of a functionf(t) is defined by

L[f(t)]=F(s)=

∫∞

0

f(t)e−stdt (3.3.1)

wheref(t)=0 fort<0, andsis a complex-frequency variable given bys=σ+jω. The

frequency-domain functionF(s) is the Laplace transform of the time-domain functionf(t). When

the integral of Equation (3.3.1) is less than infinity and converges,f(t) is Laplace transformable.

Note that forσ> 0 ,e−stdecreases rapidly, making the integral converge. The uniqueness of the

Laplace transform leads to the concept of thetransform pairs,

L[f(t)]=F(s)⇔L−^1 [F(s)]=f(t) (3.3.2)

which states that theinverse Laplace transformofF(s)isf(t). It should be noted that the Laplace

transform is a linear operation such that

L[Af 1 (t)+Bf 2 (t)]=AF 1 (s)+BF 2 (s) (3.3.3)

in whichAandBare independent ofsandt, andF 1 (s) andF 2 (s) are the Laplace transforms off 1 (t)

andf 2 (t), respectively. Using the definition of Equation (3.3.1), Table 3.3.1 of Laplace transform

pairs is developed for the most commonly encountered functions. Note that it is assumed that