Question

The transfer function of a linear system is the

• A. ratio of the output $V_o(t)$ and input $V_i(t)$
• B. ratio of the derivatives of the output and the input
• C. ratio of the Laplace transform of the output and that of the input with all initial conditions zero
• D. none of these

Hint:

Transfer functions are always defined in the Laplace domain under zero initial conditions.

Answer 1

The Correct Option is C

Step 1: Definition of transfer function.
The transfer function of a linear time-invariant (LTI) system is defined in the Laplace domain. It relates the output to the input under zero initial conditions.
Step 2: Mathematical representation.
If $Y(s)$ is the Laplace transform of the output and $X(s)$ is the Laplace transform of the input, then the transfer function $H(s)$ is given by:
\[ H(s) = \frac{Y(s)}{X(s)} \quad \text{(with all initial conditions zero)} \]
Step 3: Elimination of incorrect options.
Option (A) considers time-domain signals, not Laplace transforms.
Option (B) is not a standard definition.
Option (D) is incorrect because option (C) is correct.
Step 4: Final conclusion.
Hence, the transfer function is the ratio of the Laplace transforms of output and input with zero initial conditions.