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.

Answer 2

The transfer function of a linear system is a specific, formally defined quantity, so each option can be checked against that formal definition rather than just recalled.

1. Option "ratio of the output \( V_o(t) \) and input \( V_i(t) \)": This describes a ratio of time-domain signals directly, but such a ratio is not generally constant or even well-defined for arbitrary inputs and outputs (it depends on \( t \) and on the specific waveform), so it cannot serve as a fixed system characteristic; the transfer function must be independent of the particular input applied.
2. Option "ratio of the derivatives of the output and the input": Differentiating both signals does not eliminate the dependence on the specific input waveform either, and this is not how the transfer function is formally defined in system theory; there is no standard result equating a transfer function to a ratio of derivatives.
3. Option "ratio of the Laplace transform of the output and that of the input with all initial conditions zero": By definition, for a linear time-invariant system with zero initial conditions, the transfer function is \( H(s) = Y(s)/X(s) \), where \( Y(s) \) and \( X(s) \) are the Laplace transforms of the output and input respectively. This ratio is independent of the specific input signal (as long as conditions are zero) and is a fixed property of the system itself, exactly matching the standard definition.
4. Option "none of these": Since one of the listed statements (the Laplace-domain ratio with zero initial conditions) is precisely the standard textbook definition, this catch-all option does not apply.

Only the option built on Laplace-domain quantities, with the zero-initial-condition qualifier, is independent of the particular input and thus qualifies as the system's transfer function.

So the correct answer is ratio of the Laplace transform of the output and that of the input with all initial conditions zero.