Solve the given problems. Sketch an appropriate figure,...

Solve the given problems. Sketch an appropriate figure, unless the figure is given. In a series $R L$ circuit, the impedance$Z$ can be represented as the hypotenuse of a right triangle that has legs $R$ (resistance) and $X_{L}$ (inductive reactance). The angle $\phi$ between $R$ and $Z$ is called the phase angle. See Fig. $4.48 .$ If $R=12.0 \Omega$ and $X_{L}=15.0 \Omega,$ find the impedance and the phase angle.

Solve the given problems. Sketch an appropriate figure, unless the figure is given.
In a series $R L$ circuit, the impedance$Z$ can be represented as the hypotenuse of a right triangle that has legs $R$ (resistance) and $X_{L}$ (inductive reactance). The angle $\phi$ between $R$ and $Z$ is called the phase angle. See Fig. $4.48 .$ If $R=12.0 \Omega$ and $X_{L}=15.0 \Omega,$ find
the impedance and the phase angle.

Key Concepts

RL Circuit

An RL circuit is an electrical circuit that contains resistors (R) and inductors (L) connected in series or parallel. In a series RL circuit, the current flows through both components consecutively, and the total impedance of the circuit is determined by both resistance and the inductive reactance, which depends on the frequency of the current and the inductance.

Impedance

Impedance is the total opposition that a circuit presents to the flow of alternating current (AC). In a series RL circuit, the impedance is the vector sum of the resistance and the inductive reactance. It is calculated as the hypotenuse of a right triangle where one leg represents the resistance (R) and the other represents the inductive reactance (XL), using the formula Z = ?(R² + XL²).

Resistance

Resistance is a measure of the opposition to the flow of electrical current through a conductor and is represented by the component R in the circuit. In RL circuits, resistance affects the magnitude of the current as well as the overall impedance but does not change with frequency.

Inductive Reactance

Inductive reactance (XL) is the opposition to the change in current provided by an inductor in an AC circuit. It increases with both the frequency of the AC signal and the inductance. Inductive reactance is represented along the imaginary axis in the impedance triangle of an RL circuit.

Phase Angle

The phase angle in an RL circuit is the angle between the total impedance and the resistance. It indicates the phase difference between the voltage and the current in the circuit. The phase angle is determined using the arctangent of the ratio of the inductive reactance to the resistance, ? = arctan(XL/R), and provides insight into the circuit’s reactive characteristics.

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