JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II CONTROL SYSTEMS Previous Question Paper (November/December - 2020 - R16)

To find the transfer function of a typical RC network (often represented in such problems as a series resistor R followed by a shunt capacitor C), we apply the voltage division rule in the s-domain. Let V_i(s) be the input voltage and V_o(s) be the output voltage across the capacitor.

The impedance of the resistor is Z₁ = R, and the impedance of the capacitor is Z₂ = 1 / (sC). The transfer function G(s) is defined as the ratio of the output voltage to the input voltage:

Substituting the impedance values into the formula:

Multiplying the numerator and the denominator by sC simplifies the expression:

This result represents a first-order low-pass filter with a time constant τ = RC. If the network configuration in the figure differs (e.g., an RLC circuit or an operational amplifier circuit), the transfer function is derived by applying Kirchhoff's Voltage Law (KVL) or nodal analysis to obtain the differential equation, taking the Laplace transform assuming zero initial conditions, and solving for the ratio of output to input.

To obtain the transfer function X(s)/F(s) for a mechanical system, we apply Newton's second law of motion to the mass element. For a standard translational mechanical system consisting of a mass M, a spring with constant K, and a viscous damper with coefficient B, the force balance equation is given by:

Applying the Laplace transform to the differential equation, assuming zero initial conditions, we obtain:

Factoring out X(s) from the left side of the equation:

The transfer function is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input. Thus, the transfer function is:

In cases where the system involves multiple masses or interconnected components, the procedure involves writing individual nodal equations for each mass, representing the forces acting on each body (inertial force, damping force, and spring force), and solving the resulting system of algebraic equations in the s-domain to isolate the ratio of the specific output displacement to the input force.