Equation of current in vector method is

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

{"type":"technical","methodBadge":"Concept & Principle","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.","workingPrinciple":"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.","diagramSvg":"Active (X)Reactive (Y)Resultant (I)","keyFormulas":["$I = \\sqrt{I_a^2 + I_r^2}$ — Magnitude of complex current","$I^2 = X^2 + Y^2$ — Relationship between quadrature components"],"keyPoints":["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."],"comparisonTable":[],"additionalInfo":["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.","correctedOptions":["$I = \\frac{V}{R}$","$I = \\frac{X}{Y}$","$I^2 = X^2 + Y^2$","$I = X - Y$"],"coreConcepts":["Phasor Algebra","Quadrature Components","Pythagorean Theorem in Circuits"],"crossTopicTip":"Always verify if the current components are in quadrature before applying the squared sum formula; otherwise, you must include the phase angle term ($2XY \\cos\\theta$)."}