Which operation transforms a circuit's time-domain differential equation into a frequency-domain representation by substituting s with jω?

Which operation transforms a circuit's time-domain differential equation into a frequency-domain representation by substituting s with jω?

• A. Take the Fourier transform of the impulse response
• B. Use the Z-transform
• C. Use the discrete-time Fourier transform
• D. Apply the Laplace transform to the differential equation and substitute s = jω

Answer: (D)

The essential idea is to use the Laplace transform to convert the time-domain differential equations that describe a circuit into algebraic equations in the complex frequency variable s. Derivatives become multiplications by s, so the entire relationship between voltages and currents becomes an equation in s. By then evaluating that s-domain expression on the imaginary axis, substituting s = jω, you obtain the frequency-domain representation H(jω) that characterizes how the system responds across frequencies. This is the standard route from time-domain dynamics to a frequency-domain transfer function.

The other options don’t fit this exact process. Taking the Fourier transform of the impulse response gives the frequency response directly but doesn’t describe converting the governing differential equation itself. The Z-transform and the discrete-time Fourier transform apply to discrete-time signals or systems, not to continuous-time differential equations.