Equation of current in vector method is

Question

Examoogle · Accepted Answer

$I^2 = X^2 + Y^2$

Explanation: In an A.C. parallel circuit, the total current is determined by the vector sum of the individual branch currents. Since branches typically consist of resistive (in-phase) and reactive (quadrature) components, the total current is found using the Pythagorean theorem based on the orthogonal components.

Working Principle: In a parallel circuit, total current $I$ is defined as $I = I_a + jI_r$, where $I_a$ is the active current component and $I_r$ is the reactive current component. By taking the magnitude of this complex current, we obtain $|I| = \sqrt{I_a^2 + I_r^2}$, which simplifies to $I^2 = I_a^2 + I_r^2$. The variables X and Y in the question represent the quadrature components of the total current.

Key Formulas: $I = \sqrt{I_a^2 + I_r^2}$ — Magnitude of complex current; $I^2 = X^2 + Y^2$ — Relationship between quadrature components

Key Points: Total current in A.C. circuits is a phasor quantity.; The active component is in-phase with voltage.; The reactive component is $90^\circ$ out-of-phase with voltage.; The vector method treats current components as sides of a right-angled triangle.

Advantages: Simplifies calculation of complex impedance networks.; Provides a clear visual representation of phase relationships.

Disadvantages: Requires understanding of complex numbers.; Cannot be used directly for non-sinusoidal waveforms without Fourier analysis.

Applications: Power factor correction analysis.; Design of parallel RLC resonance circuits.

Additional Info: Option A ($I=V/R$) is valid only for purely resistive circuits.. Option C is derived from the geometric interpretation of phasor addition where the active and reactive currents are orthogonal.

Answer: C is correct — The total current in a parallel AC circuit is the hypotenuse of a right-angled triangle formed by the active and reactive current components.

Exam Tip: Always verify if the current components are in quadrature before applying the squared sum formula; otherwise, you must include the phase angle term ($2XY \cos	heta$).

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