A series R L C circuit is driven in such a way that the...

A series $R L C$ circuit is driven in such a way that the maximum voltage across the inductor is 1.50 times the maximum voltage across the capacitor and 2.00 times the maximum voltage across the resistor. (a) What is $\phi$ for the circuit? (b) Is the circuit inductive, capacitive, or in resonance? The resistance is $49.9 \Omega,$ and the current amplitude is 200 $\mathrm{mA} .$ (c) What is the amplitude of the driving emf?

Key Concepts

Series RLC Circuit Analysis

A series RLC circuit consists of a resistor (R), an inductor (L), and a capacitor (C) connected in a single loop. The analysis of such a circuit involves understanding how these elements interact when driven by an alternating current (AC) source. In this configuration, the total voltage is distributed among the three components, and the current in the circuit is the same through each element. This analysis forms the basis for exploring aspects such as impedance, phase relationships, and resonance.

AC Impedance and Reactance

In AC circuits, the opposition to current flow is characterized by impedance, which is a combination of resistance and reactance. The resistance (R) opposes current regardless of frequency, whereas the reactances of the inductor and the capacitor (?L and 1/?C, respectively) vary with the frequency of the applied AC signal. Understanding these reactive components is crucial because the net reactance (the difference between inductive and capacitive reactance) contributes, along with resistance, to the total impedance of the circuit through the relation Z = ?(R² + (X_L - X_C)²).

Phase Angle in AC Circuits

The phase angle (?) in AC circuits indicates the phase difference between the voltage and the current and is determined by the relative magnitudes of resistance and net reactance. In a series RLC circuit, this phase angle is given by tan(?) = (X_L - X_C)/R. A positive phase angle indicates an inductive behavior (voltage leads current), while a negative angle indicates a capacitive behavior (voltage lags behind current). Accurately determining ? is essential for understanding the timing and distribution of energy within the circuit.

Resonance in RLC Circuits

Resonance in an RLC circuit occurs when the inductive reactance equals the capacitive reactance (?L = 1/?C), which results in the cancellation of the reactive effects and minimizes the circuit’s impedance to just the resistance (R). At resonance, the current is maximized, and the voltages across the inductor and capacitor, despite potentially being large individually, are out of phase and can cancel each other in the overall voltage balance. Recognizing whether a circuit is in resonance, inductive, or capacitive is key to predicting its behavior under different driving frequencies.

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