Question

Phasor analysis is valid for

• A. Transient signals
• B. Steady state sinusoidal signals
• C. both transient and steady state
• D. exponential signals

Hint:

Phasor domain operations map time differentiation directly to complex algebraic multiplication: \[ \frac{d}{dt} \;\Longleftrightarrow \; j\omega \] This converts tedious integro-differential circuit equations into simple linear equations, but this elegant mapping is only possible under steady-state single-frequency conditions!

Answer 1

The Correct Option is B

Concept: Phasor analysis is a mathematical technique used to transform time-domain differential equations governing electrical circuits into simple algebraic equations in the frequency domain. A phasor is a complex number that represents the amplitude and initial phase angle of a sinusoidal function of time. Consider a time-dependent sinusoidal signal given by: \[ x(t) = X_m \cos(\omega t + \phi) \] Using Euler's identity, this real-valued signal can be expressed as the real part of a rotating complex exponential: \[ x(t) = \text{Re}\left\{ X_m e^{j(\omega t + \phi)} \right\} = \text{Re}\left\{ X_m e^{j\phi} \cdot e^{j\omega t} \right\} \] The phasor representation $\vec{X}$ drops the structural time dependence $e^{j\omega t}$, retaining only the constant magnitude and phase: \[ \vec{X} = X_m e^{j\phi} = X_m \angle\phi \] Detailed Core Requirements for Valid Phasor Analysis:
• Constant Frequency ($\omega$): In phasor transformation, the factor $e^{j\omega t}$ is completely suppressed because every voltage and current response in a linear time-invariant (LTI) system oscillates at that exact same frequency. If the frequency changes or isn't constant, the common factor cannot be cancelled out.
• Steady-State Conditions: Phasor analysis assumes the circuit has been connected to the source for an infinitely long time ($t \to \infty$), so all initial switching disturbances or transients have naturally decayed to zero. Evaluating Option Categories:
• Transient Signals: Transient behavior occurs right after a structural switch or disturbance in the circuit. These signals change rapidly over time and contain a broad spectrum of frequencies rather than a single steady frequency. Thus, standard phasor analysis cannot be applied; instead, differential equations or Laplace transforms must be used.
• Steady State Sinusoidal Signals: These signals possess a completely constant amplitude, a fixed phase, and a singular unchanging frequency $\omega$. This fulfills all conditions required for phasor analysis.
• Exponential Signals: Purely exponential signals ($e^{-\alpha t}$) do not possess a steady-state oscillatory behavior at a fixed real frequency $\omega$, rendering standard phasor modeling inapplicable. Hence, phasor analysis is strictly valid for steady-state sinusoidal signals.