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GATE 2016 EC – Question 41

Networks, Signals and Systems · Sinusoidal Steady State Analysis · 2 marks · Multiple choice

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

$$\sum_{k=1}^{3} a_k \cos(k\omega_0 t), \text{ where } a_k \neq 0, \omega_0 \neq 0.$$

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. $\sum_{k=1}^{3} b_k \cos(k\omega_0 t + \phi_k)$, where $b_k \neq a_k, \forall k$
  2. $\sum_{k=1}^{4} b_k \cos(k\omega_0 t + \phi_k)$, where $b_k \neq 0, \forall k$
  3. $\sum_{k=1}^{3} a_k \cos(k\omega_0 t + \phi_k)$
  4. $\sum_{k=1}^{2} a_k \cos(k\omega_0 t + \phi_k)$

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Correct answer: (A) $\sum_{k=1}^{3} b_k \cos(k\omega_0 t + \phi_k)$, where $b_k \neq a_k, \forall k$

Explanation

A linear time-invariant network can only change the amplitude and phase of each frequency component. It cannot create a new frequency, which rules out the fourth term in B. Because the source has non-zero impedance, each component is reduced by a voltage divider, so each amplitude changes to a different value $b_k \neq a_k$, which rules out C and D. The output is three cosines at $\omega_0$, $2\omega_0$ and $3\omega_0$ with changed amplitudes and phases.