Which of the following are valid methods to determine a circuit's frequency response H(f)?

• A. Replace capacitor C with 1/(jωC) and inductor L with jωL, then compute Vout/Vin.
• B. From the circuit's differential equation, find the impulse response h(t) and then Fourier transform it to obtain H(f).
• C. Take the differential equation and transform it directly to the frequency domain (e.g., using s = jω).
• D. All of the above.

Answer: (D)

The idea being tested is how to obtain a circuit’s frequency response for linear time-invariant systems, which tells you how Vout relates to Vin at each frequency through a complex transfer function H(f). In steady state, a sinusoidal input of frequency f is scaled and phase-shifted by H(f), and ω = 2πf.

Using impedances is the straightforward frequency-domain route: replace each capacitor with its impedance 1/(jωC) and each inductor with jωL, then solve the algebraic network to get Vout/Vin as a function of ω (i.e., H(jω)). This directly yields the frequency response.

If you know the time-domain picture, you can find the impulse response h(t) of the circuit and then take its Fourier transform to obtain H(f). The impulse response encapsulates the system’s complete response to a delta input, and its transform is precisely the frequency response.

Transforming the differential equation itself into the frequency domain is another valid path. By substituting derivatives with jω (or using s = jω for steady-state), you turn the differential equation into an algebraic relation between Vin and Vout at each ω, from which H(jω) follows.

Since all three methods describe the same linear relationship between input and output across frequencies, they are all valid ways to determine the circuit’s frequency response.