If two sinusoids of the same frequency but of different amplitudes and phase angles are subtracted t

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If two sinusoids of the same frequency but of different amplitudes and phase angles are subtracted the resultant is

Examoogle · Accepted Answer

sinusoid of same frequency

Explanation: When two sinusoids of the same frequency are added or subtracted, the resultant is always another sinusoid of the same frequency. This occurs because sinusoids of the same frequency are linearly additive in the phasor domain.

Working Principle: The operation can be represented using phasors, where each sinusoid $A_i \cos(\omega t + \phi_i)$ is represented by $A_i \angle \phi_i$. Subtraction is simply $A_1 \angle \phi_1 - A_2 \angle \phi_2 = (A_1 \cos \phi_1 - A_2 \cos \phi_2) + j(A_1 \sin \phi_1 - A_2 \sin \phi_2) = A_r \angle \phi_r$. The result $A_r \cos(\omega t + \phi_r)$ maintains the original angular frequency $\omega$.

Key Formulas: $v(t) = V_m \sin(\omega t + \phi)$ — General expression for a sinusoidal wave; $A_1 \angle \phi_1 \pm A_2 \angle \phi_2 = A_r \angle \phi_r$ — Phasor addition/subtraction property

Key Points: Linearity property: The set of sinusoids with a fixed frequency $\omega$ forms a vector space.; The resultant amplitude and phase depend on the magnitudes and individual phase angles.; Frequency conversion (doubling or halving) occurs only with non-linear operations like multiplication (e.g., modulation) or rectification.

Advantages: Simplifies analysis of linear AC circuits using complex numbers; Allows usage of Superposition Theorem in AC networks

Disadvantages: Only applicable when frequencies are strictly identical; Does not apply to non-sinusoidal periodic waveforms (like square waves) directly

Applications: AC Power system load calculation; Phasor analysis in electrical engineering; Interference pattern study in wave optics and acoustics

Additional Info: If frequencies were different, the resultant would be a complex signal, specifically a beat frequency signal or an amplitude-modulated-like waveform, but not a pure sinusoid.. Option B and C are incorrect because they imply frequency multiplication/division, which is a non-linear process.

Answer: A is correct — The resultant of any linear combination of sinusoids with the same frequency is always a sinusoid of that same frequency.

Exam Tip: Always remember that in linear systems (like RLC circuits), frequency never changes; only magnitude and phase of the response shift based on impedance.

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