GATE EC 2016 Set 1 — Question 41
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Network Theory → Circuit Fundamentals & Theorems → Sinusoidal Steady-State
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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 formThe source has nonzero impedance. Which one of the following is a possible form of the output measured across a resistor in the network?
Correct answer
(A) Σₖ₌₁³ bₖ cos(kω₀ t + φₖ), where bₖ ≠ aₖ, ∀ k
Solution
The given network consists of linear resistors, inductors, and capacitors, making it a Linear Time-Invariant (LTI) system.
1.Frequency Preservation: For any LTI system, if the input is a sum of sinusoids at frequencies , the steady-state output will contain sinusoids at the exact same frequencies. No new frequencies can be generated (eliminating option B), and existing frequencies cannot be removed unless the transfer function is zero at that specific frequency (making option D less general).
2.Transfer Function Effects: The input is . The output across a resistor will have the form:
where is the transfer function of the network.3.Amplitude and Phase Change: Because the network contains reactive elements (L and C) and the source has nonzero impedance, the magnitude of the transfer function will generally vary with frequency and will not be equal to unity. Thus, the output amplitudes will generally be different from the input amplitudes (eliminating option C).
Therefore, the most general possible form is given by option (A), where the amplitudes are modified () and phases are introduced.Continue learning with Success Tracker
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