An alternating emf source with a variable frequency fd is...

An alternating emf source with a variable frequency $f_{d}$ is connected in series with an $80.0 \Omega$ resistor and a $25.0 \mathrm{mH}$ inductor. The emf amplitude is $6.00 \mathrm{~V}_{-}$(a) Draw a phasor diagram for phasor $V_{R}$ (the potential across the resistor) and phasor $V_{L}$ (the potential across the inductor). (b) At what driving frequency $f_{d}$ do the two phasors have the same length? At that driving frequency, what are (c) the phase angle in degrees, (d) the angular speed at which the phasors rotate, and (e) the current amplitude?

Key Concepts

Angular Frequency and Driving Frequency

Angular frequency (?) is related to the driving frequency (f) by the equation ? = 2?f. It represents the rate at which the phasors rotate in the complex plane, and is crucial for converting between time-domain and phasor (frequency-domain) representations in AC circuit analyses.

Frequency Dependency in RL Circuits

The magnitude of the inductor’s reactance is directly proportional to the frequency of the applied AC source. This dependency means that as the driving frequency changes, so does the relative magnitude of the inductor’s voltage, which can be compared to the resistor’s voltage to find specific operating conditions, such as when they have equal amplitude.

Phase Relationships in RL Circuits

In resistor-inductor (RL) circuits, the resistor’s voltage and the inductor’s voltage are out of phase: the resistor’s voltage is in phase with the current while the inductor’s voltage leads the current by 90 degrees. Knowing these phase differences is fundamental in accurately representing and analyzing the circuit’s behavior.

Resistive Voltage in AC Circuits

In AC circuits, the voltage across a resistor follows Ohm’s law and is in phase with the current. This means that its phasor points along the reference axis (typically the horizontal axis), making it a key reference point for comparing phase differences of other circuit elements.

Phasor Representation in AC Circuits

Phasors are a mathematical tool used to represent sinusoidally varying quantities, such as voltages and currents, in AC circuits. They simplify analysis by converting differential equations into algebraic equations, representing each sinusoid as a complex number (or rotating vector) with a magnitude and a phase angle.

Inductive Reactance

Inductive reactance is the opposition offered by an inductor to a change in current, and it increases with frequency according to the relation X_L = ?L, where ? is the angular frequency and L is the inductance. In a phasor diagram, the voltage drop across an inductor leads the current by 90 degrees.

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