Jawaharlal Nehru Technological University Hyderabad B.Tech III Year CONTROL SYSTEMS ENGINEERING Previous Question Paper (August - 2022 - R13)

The transfer function of a linear time-invariant (LTI) system is defined as the ratio of the Laplace transform of the output variable to the Laplace transform of the input variable, assuming all initial conditions are zero.

For an RLC series circuit with a resistor R, inductor L, and capacitor C in series connected to an input voltage source V(t), the governing differential equation based on Kirchhoff's Voltage Law (KVL) is:

Since the output is the voltage across the capacitor, v₀(t) = (1/C)q(t), we can express the equation in terms of v₀(t):

Taking the Laplace transform of both sides, assuming zero initial conditions:

The transfer function H(s) is the ratio of output V₀(s) to input V(s):

Alternatively, using the voltage divider rule in the s-domain, the impedance of the capacitor is Zc = 1/(Cs) and the total impedance of the series circuit is Z(s) = R + Ls + 1/(Cs). The transfer function is:

Mason's Gain Formula is given by the expression:

where P_k is the gain of the k-th forward path, Δ is the determinant of the graph, and Δ_k is the cofactor of the k-th forward path.

To solve for the transfer function, follow these systematic steps:

1. Identify all forward paths from the input node to the output node and calculate their gains (P_k).
2. Identify all individual loop gains (L_i) in the signal flow graph.
3. Identify all possible combinations of two, three, or more non-touching loops to calculate the determinant Δ, defined as: Δ = 1 - (sum of individual loop gains) + (sum of gain products of all combinations of two non-touching loops) - (sum of gain products of all combinations of three non-touching loops) + ...
4. Calculate Δ_k for each forward path by removing the loops that touch the k-th forward path from the determinant expression.
5. Substitute these values into the Mason's Gain Formula to obtain the overall transfer function T = C(s)/R(s).

The final transfer function is obtained by simplifying the algebraic expression resulting from the ratio of the sum of path gains weighted by their respective cofactors to the system determinant.