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

This is a definitional question, so instead of steps, here is why each option does or does not match the standard definition of a transfer function.
A transfer function must be a fixed property of the system alone, not something that changes with whatever input waveform happens to be applied; a plain ratio of time-domain output to input, or of their derivatives, generally varies with the input signal itself and so cannot serve this role.
Working in the Laplace domain and setting all initial conditions to zero removes this input-dependence: \( H(s) = \dfrac{Y(s)}{X(s)} \) becomes a fixed characteristic of the system's differential equation coefficients alone.
This is exactly the standard, textbook definition of the transfer function, and it is precisely the option describing a Laplace-domain ratio with zero initial conditions. \[ \boxed{H(s) = \dfrac{Y(s)}{X(s)},\ \text{initial conditions} = 0} \]

Answer 2

A complementary way to confirm this is to think about what property a transfer function must have (uniqueness and input-independence) and test each option against that requirement.

1. Option "ratio of the output \( V_o(t) \) and input \( V_i(t) \)": If two completely different input waveforms are applied to the same system, this time-domain ratio would, in general, come out different for each one; a quantity that changes depending on which input is used cannot represent a fixed characteristic of the system, so this fails the uniqueness requirement.
2. Option "ratio of the derivatives of the output and the input": The same objection applies: differentiating both signals still leaves a quantity that depends on the shapes of the particular input and output waveforms, not on the system alone.
3. Option "ratio of the Laplace transform of the output and that of the input with all initial conditions zero": Under zero initial conditions, the Laplace-domain relationship \( Y(s) = H(s)X(s) \) holds for every possible input \( X(s) \), meaning \( H(s) = Y(s)/X(s) \) is the same fixed function of \( s \) no matter which input is used; this uniquely characterizes the system rather than any single input-output pair, satisfying the requirement for a true transfer function.
4. Option "none of these": Since the Laplace-domain definition satisfies the required uniqueness and input-independence, this option is unnecessary.

Testing which candidate stays the same regardless of the input signal used isolates exactly one option as the valid definition.

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