JNTU B.Tech III CONTROL SYSTEMS ENGINEERING Previous Question Paper (February - 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.

Mathematically, for an input X(s) and output Y(s), the transfer function H(s) is:

The relationship with the impulse response h(t) is established by the fact that the transfer function is the Laplace transform of the system's impulse response. If the input is a unit impulse function δ(t), its Laplace transform is 1. Therefore, the output Y(s) becomes equal to H(s), meaning the impulse response is the inverse Laplace transform of the transfer function:

Mason's Gain Formula provides the transfer function T(s) of a system represented by a Signal Flow Graph (SFG) as:

To obtain the transfer function, follow these steps:

1. Identify all forward paths from the input node to the output node. Let P_k be the gain of the k-th forward path.
2. Identify all individual feedback loops in the graph. Let L_i be the gain of the i-th loop.
3. Calculate the graph determinant Δ using the formula: Δ = 1 - (sum of individual loop gains) + (sum of gain products of all possible combinations of two non-touching loops) - (sum of gain products of all possible combinations of three non-touching loops) + ...
4. For each forward path P_k, calculate the path factor Δ_k, which is the value of Δ evaluated for that part of the graph not touching the k-th forward path.
5. Substitute these values into the formula to compute the final transfer function.