A. What is the relation between unit step, unit ramp and unit impulse functions?
B. Explain use of Partial Fraction Method in determining the Inverse Laplace Transform.
C. State and explain the initial and final values theorems .
D. Obtain the domain (Laplace transform) equivalent circuit diagram of an inductor and capacitor with initial condition?
The Correct Answer and Explanation is :
A. Relation between Unit Step, Unit Ramp, and Unit Impulse Functions
The unit step function (denoted as ( u(t) )), unit ramp function (denoted as ( r(t) )), and unit impulse function (denoted as ( \delta(t) )) are fundamental concepts in signal processing and system analysis, each representing different types of signals.
- Unit Impulse Function ( \delta(t) ): This function is zero everywhere except at ( t = 0 ), where it is infinitely high such that its integral over all time equals 1. It represents an instantaneous impulse at time zero.
- Unit Step Function ( u(t) ): This function is zero for ( t < 0 ) and one for ( t \geq 0 ). It can be considered as the integral of the unit impulse function:
[
u(t) = \int_{-\infty}^t \delta(\tau) \, d\tau
] - Unit Ramp Function ( r(t) ): This function is zero for ( t < 0 ) and increases linearly with time for ( t \geq 0 ) (i.e., ( r(t) = t ) for ( t \geq 0 )). The ramp function is the integral of the unit step function:
[
r(t) = \int_{0}^t u(\tau) \, d\tau = \int_{0}^t \tau \, d\tau
]
Thus, the relationship among these functions can be understood in terms of differentiation and integration: the step function is the integral of the impulse function, and the ramp function is the integral of the step function.
B. Use of Partial Fraction Method in Determining the Inverse Laplace Transform
The partial fraction method is a technique used to simplify the process of finding the inverse Laplace Transform, particularly when the Laplace Transform is given as a rational function ( \frac{P(s)}{Q(s)} ) where ( P(s) ) and ( Q(s) ) are polynomials and the degree of ( P(s) ) is less than that of ( Q(s) ).
By expressing this rational function as a sum of simpler fractions (known as partial fractions), each corresponding to simpler, more manageable inverse transforms, the overall inverse can be found by summing these individual inverses. This approach is especially useful when ( Q(s) ) can be factored into linear or simple quadratic factors.
C. Initial and Final Value Theorems
- Initial Value Theorem: This theorem helps predict the initial behavior of a time-domain function based on its Laplace Transform. The theorem states that:
[
f(0^+) = \lim_{s \to \infty} sF(s)
]
provided the limit exists and ( f(t) ) does not have a pole at ( s = 0 ). - Final Value Theorem: This theorem predicts the steady-state behavior of a time-domain function and is given by:
[
f(\infty) = \lim_{s \to 0} sF(s)
]
This holds provided ( sF(s) ) has no poles in the right-half plane other than possibly at ( s = 0 ).
D. Domain Equivalent Circuit Diagram of an Inductor and Capacitor with Initial Conditions
For an inductor ( L ) with initial current ( i_L(0^-) ), the Laplace transform equivalent includes the inductance ( L ) in series with a voltage source ( Li_L(0^-)s ).
For a capacitor ( C ) with initial voltage ( v_C(0^-) ), the Laplace transform equivalent includes the capacitor ( C ) in parallel with a current source ( Cv_C(0^-)s ).
Let’s now generate the image depicting this equivalent circuit diagram for both an inductor and capacitor with initial conditions.
Here is the electronic circuit diagram in the Laplace domain for an inductor and capacitor, each with initial conditions. The diagram includes an inductor in series with a voltage source, representing the inductance and initial current impact, as well as a capacitor in parallel with a current source, representing the capacitance and initial voltage impact. This visualization should help in understanding how these components are represented in the Laplace domain.