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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 ∑ₖ₌₁³ 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. ∑ₖ₌₁³ bₖ cos(kω₀ t + φₖ), where bₖ ≠ aₖ, ∀ k
2. ∑ₖ₌₁⁴ bₖ cos(kω₀ t + φₖ), where bₖ ≠ 0, ∀ k
3. ∑ₖ₌₁³ aₖ cos(kω₀ t + φₖ)
4. ∑ₖ₌₁² aₖ cos(kω₀ t + φₖ)

Correct answer: ∑ₖ₌₁³ aₖ cos(kω₀ t + φₖ)

Solution

The correct option reflects that the output across the resistor can maintain the same frequency components as the input, but with possible phase shifts, which is typical in linear systems. Since the network is linear and the input consists of multiple frequency components, the output can also include these same components with adjusted phases.

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