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Ch 32: AC Circuits
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 32: AC CircuitsProblema 49a
Capitolo 32, Problema 49a

Use a phasor diagram to analyze the RL circuit of FIGURE P32.49. In particular, Find expressions for I, VR, and VL.

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Step 1: Understand the components of the RL circuit. An RL circuit consists of a resistor (R) and an inductor (L) connected in series with an alternating current (AC) source. The current (I) is the same through both components, but the voltage across the resistor (V_R) and the inductor (V_L) will differ in phase due to the inductive reactance.
Step 2: Represent the voltages and current as phasors. In an RL circuit, the voltage across the resistor (V_R) is in phase with the current (I), while the voltage across the inductor (V_L) leads the current by 90 degrees. Use a phasor diagram to represent these relationships, with the current phasor as the reference (horizontal axis).
Step 3: Write the expressions for the voltages. The voltage across the resistor is given by \( V_R = I R \), where \( R \) is the resistance. The voltage across the inductor is given by \( V_L = I X_L \), where \( X_L = \omega L \) is the inductive reactance, and \( \omega \) is the angular frequency of the AC source.
Step 4: Use the phasor diagram to find the total voltage. The total voltage \( V \) is the vector sum of \( V_R \) and \( V_L \). Since \( V_R \) and \( V_L \) are perpendicular, the magnitude of the total voltage is \( V = \sqrt{V_R^2 + V_L^2} \). Substitute the expressions for \( V_R \) and \( V_L \) to get \( V = \sqrt{(I R)^2 + (I X_L)^2} \).
Step 5: Solve for the current \( I \). The current can be expressed as \( I = \frac{V}{\sqrt{R^2 + X_L^2}} \). Once \( I \) is known, substitute it back into the expressions for \( V_R \) and \( V_L \) to find their respective values. This completes the analysis of the RL circuit using a phasor diagram.

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Phasor Diagrams

Phasor diagrams are graphical representations of sinusoidal functions, where each phasor represents the amplitude and phase of a sinusoidal voltage or current. In the context of AC circuits, phasors simplify the analysis of circuit elements by converting time-dependent sinusoidal signals into complex numbers, allowing for easier calculations of voltages and currents in the circuit.
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Phasors

RL Circuit

An RL circuit consists of a resistor (R) and an inductor (L) connected in series or parallel. The behavior of an RL circuit is characterized by its impedance, which combines resistance and inductive reactance. The inductor introduces a phase shift between the voltage and current, affecting the overall circuit response to alternating current (AC) signals, which is crucial for determining the expressions for current and voltages across the components.
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LR Circuits

Impedance

Impedance is a measure of how much a circuit resists the flow of alternating current, combining both resistance (R) and reactance (X). In an RL circuit, the impedance is given by Z = R + jωL, where j is the imaginary unit and ω is the angular frequency of the AC source. Understanding impedance is essential for calculating the current (I) and the voltages across the resistor (VR) and inductor (VL) using Ohm's law and Kirchhoff's laws in the phasor domain.
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Impedance in AC Circuits
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