Consider a series R L C circuit with R=25 Ω, L= 6.0 mH, and C=25 μF . The circuit is connected t

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Consider a series R L C circuit with R=25 Ω, L= 6.0 mH, and...

Consider a series $R L C$ circuit with $R=25 \Omega, L=$ $6.0 \mathrm{mH}$, and $C=25 \mu \mathrm{F} .$ The circuit is connected to a $10-\mathrm{V}(\mathrm{rms}), 600-\mathrm{Hz}$. AC source. (a) Is the sum of the voltage drops across $R, L$, and $C$ equal to $10 \mathrm{~V}(\mathrm{rms}) ?$ (b) Which is greatest, the power delivered to the resistor, to the capacitor, or to the inductor? (c) Find the average power delivered to the circuit.

Key Concepts

Phasor Representation

In AC circuit analysis, voltages and currents are represented as phasors, which are complex numbers (or rotating vectors) that encode both magnitude and phase information. This method allows for the effective analysis of circuits where different elements introduce phase shifts, making simple arithmetic addition of magnitudes invalid.

Complex Impedance

Every circuit element in an AC system has an associated impedance that may have both real and imaginary parts. For resistors, the impedance is purely real, while for inductors and capacitors, the impedance is imaginary and frequency-dependent. The total impedance in a series circuit is the complex sum of individual impedances, directly influencing current and voltage phase relationships.

AC Voltage and Current Relationships

Unlike DC circuits where voltage drops add directly, in AC circuits the instantaneous voltage drops across elements must be added as vectors (phasors) since they may not be in phase. This explains why the sum of the rms voltage drops across circuit elements does not equate to the rms applied voltage despite meeting Kirchhoff’s voltage law in the phasor domain.

Average Power in AC Circuits

In AC circuits, average (or real) power is dissipated only in resistive components, since reactive components such as inductors and capacitors only store and release energy without actual energy loss over a complete cycle. The average power delivered to the circuit is calculated using the resistive part of the impedance, typically through the formula P = I²R or P = V²/R for the resistor.

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