A network consisting of a finite number of linear resistor (R), inductor (L), and capacitor (C) ...

RLC Network Question and Solution

Question

A network consisting of a finite number of linear resistor (R), inductor (L), and capacitor (C) elements, connected all in series or all in parallel, is excited with a source of the form:

Σ (k = 1 to 3) aₖ cos(kω₀ t), 
where aₖ ≠ 0, ω₀ ≠ 0.
                

The source has nonzero impedance. Which one of the following is a possible form of the output measured across a resistor in the network?

1. Σ (k = 1 to 3) bₖ cos(kω₀ t + φₖ), where bₖ ≠ aₖ for all k
2. Σ (k = 1 to 3) bₖ cos(kω₀ t + φₖ), where bₖ ≠ 0 for all k
3. Σ (k = 1 to 3) aₖ cos(kω₀ t + φₖ)
4. Σ (k = 1 to 2) aₖ cos(kω₀ t + φₖ)

Solution

The network contains only linear R, L, and C elements. Therefore, it is a Linear Time-Invariant (LTI) system.

An LTI system:

• Does not generate new frequencies.
• Processes each frequency component independently.

Since the input consists of harmonics at frequencies kω₀ (k = 1, 2, 3), the output across a resistor must also contain only these same frequencies. However:

• Amplitudes may change.
• Phase shifts may occur.
• Some harmonics may be completely attenuated.

Therefore, a possible output form is:

Σ (k = 1 to 2) aₖ cos(kω₀ t + φₖ)
                

This corresponds to option (d), where one harmonic is completely suppressed.

Correct Answer: (d)