UNIT 1 LAPLACE TRANSFORM 1. Introduction In this chapter, we will introduce Laplace transform. This is an extremely important technique. For a given set of initial conditions, it will give the total response of the circuit comprising of both natural and forced responses in one operation. The idea of Laplace transform is analogous to any familiar transform. For example, Logarithms are used to change a multiplication or division problem into a simpler addition or subtraction problem and Anti logs are used to carry out the inverse process. This example points out the essential feature of a transform: They are designed to create a new domain to make mathematical manipulations easier. After evaluating the unknown in the new domain, we use inverse transform to get the evaluated unknown in the original domain. The Laplace transform enables the circuit analyst to convert the set of integro-differential equations describing a circuit to the complex frequency domain, where they become a set of linear algebraic equations. Then using algebraic manipulations, one may solve for the variables of interest. Finally, one uses the inverse transform to get the variable of interest in s time domain. Also, in this chapter, we express the impedance in domain or complex frequency domain 2. Definition of Laplace transforms A transform is a change in the mathematical description of a physical variable to facilitate computation. Keeping this definition in mind, Laplace transform of a function) is defined as Here the complex frequency is . Since the argument of the exponent e in equation (1) must be dimensionless, it follows that s has the dimensions of frequency and units of inverse seconds (sec_1). The notation implies that once the integral has been evaluated, a time domain f (t), function is transformed to a frequency domain function. f (s), If the lower limit of integration in equation (1) is then it is called the bilateral Laplace - ∞, transform. However for circuit applications, the lower limit is taken as zero and accordingly the transform is unilateral in nature. The lower limit of integration is sometimes chosen to be 0_ to permit to (t) or its derivatives. Thus we should note immediately that the f (t) include δ
UNIT 1 LAPLACE TRANSFORM 1. Introduction In this chapter, we will introduce Laplace transform. This is an extremely important technique. For a given set of initial conditions, it will give the total response of the circuit comprising of both natural and forced responses in one operation. The idea of Laplace transform is analogous to any familiar transform. For example, Logarithms are used to change a multiplication or division problem into a simpler addition or subtraction problem and Anti logs are used to carry out the inverse process. This example points out the essential feature of a transform: They are designed to create a new domain to make mathematical manipulations easier. After evaluating the unknown in the new domain, we use inverse transform to get the evaluated unknown in the original domain. The Laplace transform enables the circuit analyst to convert the set of integro-differential equations describing a circuit to the complex frequency domain, where they become a set of linear algebraic equations. Then using algebraic manipulations, one may solve for the variables of interest. Finally, one uses the inverse transform to get the variable of interest in time domain. Also, in this chapter, we express the impedance in s domain or complex frequency domain 2. Definition of Laplace transforms A transform is a change in the mathematical description of a physical variable to facilitate computation. Keeping this definition in mind, Laplace transform of a function) is defined as Here the complex frequency is . Since the argument of the exponent e in equation (1) must be dimensionless, it follows that s has the dimensions of frequency and units of inverse seconds (sec_1). The notation implies that once the integral has been evaluated, f (t), a time domain function is transformed to f (s), a frequency domain function. If the lower limit of integration in equation (1) is - ∞, then it is called the bilateral Laplace transform. However for circuit applications, the lower limit is taken as zero and accordingly the transform is unilateral in nature. The lower limit of integration is sometimes chosen to be 0_ to permit f (t) to include δ (t) or its derivatives. Thus we should note immediately that the
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