Series RLC totals: When combining impedances or voltage phasors in a series RLC circuit, which method must always be used to obtain correct totals?

Introduction:
Series RLC circuits contain resistive and reactive elements whose voltages and impedances have phase differences. To compute accurate totals, we must account for both magnitudes and angles, which requires vector (phasor) addition rather than simple arithmetic addition or subtraction.

Given Data / Assumptions:

• Series connection: the same current flows through R, L, and C
• Voltages across R, L, and C differ in phase: 0°, +90°, and -90° with respect to current
• Goal: total voltage or total impedance calculation

Concept / Approach:

Represent voltages and impedances as complex numbers (phasors). The total series impedance is Z = R + j(XL - XC). The total supply voltage is V = VR + VL + VC, but these are vector sums because of their phase offsets. Therefore, the correct method is vector (phasor) addition in the complex plane.

Step-by-Step Solution:

Verification / Alternative check:

Graphical phasor diagrams or complex-number calculations both yield identical results. Arithmetic addition of magnitudes would overestimate totals, especially when VL and VC partially cancel.

Why Other Options Are Wrong:

• graphing the angles: A visualization tool, not a calculation method.
• multiplying the values: No basis in phasor summation for series totals.
• subtracting the values: Partial cancellation occurs vectorially, not by blind subtraction of magnitudes.
• adding magnitudes arithmetically: Ignores phase; produces incorrect totals.

Common Pitfalls:

• Summing magnitudes of VL and VC without recognizing their opposite phases.
• Forgetting to convert degrees to radians when using complex arithmetic programmatically.

Final Answer:

adding values vectorially