When computing totals for series RLC circuits (voltages or impedances), which method correctly accounts for the phase relationships among resistive and reactive components?

Introduction:
Series RLC analysis hinges on the fact that resistive and reactive elements introduce phase shifts between voltage and current. Therefore, totals cannot be computed with simple arithmetic sums of magnitudes; one must respect phase using vector (phasor) methods.

Given Data / Assumptions:

• Sinusoidal steady state.
• Series connection of R, L, and C components.
• Voltages across elements can be larger than the source due to phase relations.

Concept / Approach:
Represent voltages and impedances as complex numbers: R on the real axis, X_L = +jωL and X_C = −j/(ωC) on the imaginary axis. Resultants are obtained by vector (phasor) addition, i.e., adding real and imaginary components separately to form a complex total, then taking magnitude/angle as needed.

Step-by-Step Solution:
Write impedances: Z_R = R, Z_L = jX_L, Z_C = −jX_CTotal impedance: Z_total = R + j(X_L − X_C)Voltage totals: V_S = V_R + V_L + V_C as phasors, not scalar magnitudesMagnitude: |Z_total| = sqrt(R^2 + (X_L − X_C)^2)Angle: φ = arctan((X_L − X_C) / R)

Verification / Alternative check:
Phasor diagrams graphically confirm that orthogonal components must be combined vectorially. Numerical examples (e.g., V_R = 3 V, V_L = 30 V, V_C = 18 V) produce |V_S| = sqrt(3^2 + (30 − 18)^2) = sqrt(153) ≈ 12.37 V, which differs fundamentally from any simple sum/difference of magnitudes.

Why Other Options Are Wrong:

• Subtracting / multiplying / averaging magnitudes: Disregard phase, leading to large errors.
• Graphing the angles: A visualization aid, not the computation method; one must still perform vector addition.

Common Pitfalls:

• Assuming the source voltage equals the arithmetic sum of branch voltages.
• Ignoring that V_L and V_C oppose along the imaginary axis.
• Confusing rms with peak values; consistency is required for phasor math.

Final Answer:
Adding vectors (phasor addition)