A series R L C circuit has a 0.20-mF capacitor, a 13-mH inductor, and a 10.0-Ωresistor, and is c

step 1 So we have R1 or R equals 10 om and L equals 0 .013 Henry and we also have a capacitor in the ci

A series R L C circuit has a 0.20-mF capacitor, a 13-mH...

A series $R L C$ circuit has a $0.20-\mathrm{mF}$ capacitor, a $13-\mathrm{mH}$ inductor, and a $10.0-\Omega$ resistor, and is connected to an ac source with amplitude $9.0 \mathrm{V}$ and frequency $60 \mathrm{Hz}$ (a) Calculate the voltage amplitudes $V_{L}, V_{C}, V_{R},$ and the phase angle. (b) Draw the phasor diagram for the voltages of this circuit.

A series $R L C$ circuit has a $0.20-\mathrm{mF}$ capacitor, a $13-\mathrm{mH}$ inductor, and a $10.0-\Omega$ resistor, and is connected to an ac source with amplitude $9.0 \mathrm{V}$ and frequency $60 \mathrm{Hz}$
(a) Calculate the voltage amplitudes $V_{L}, V_{C}, V_{R},$ and the phase angle. (b) Draw the phasor diagram for the voltages of this circuit.

Key Concepts

Phasor Diagrams

Phasor diagrams are graphical representations that show the relationships between the magnitudes and phases of sinusoidal quantities like voltage and current in an AC circuit. By depicting these quantities as vectors in a plane, phasor diagrams make it easier to visualize the phase differences and the cumulative effect of multiple circuit elements, thereby simplifying the analysis of circuit dynamics.

Voltage Distribution in Series Circuits

In series circuits, the overall voltage is divided among the resistor, inductor, and capacitor in accordance with their individual impedances. The voltage across each element can be calculated by multiplying the current by its respective voltage drop. This distribution shows that while the resistor’s voltage is in phase with the current, the inductor’s and capacitor’s voltages are shifted, leading to complex interactions that are best understood through phasor analysis.

Phase Angle

The phase angle in an AC circuit quantifies the time difference between the voltage and the current waveforms. It is determined by the relative contributions of the resistive and reactive (inductive and capacitive) components. The phase angle indicates whether the current leads or lags behind the voltage, which is essential for predicting circuit behavior under different operating conditions.

RLC Circuit Analysis

RLC circuit analysis involves studying circuits that contain resistors (R), inductors (L), and capacitors (C). These circuits exhibit behaviors such as resonance, damping, and phase shifts that depend on the frequency of the applied AC source. Understanding RLC circuits is fundamental to analyzing how energy is stored and dissipated within the system.

Impedance and Reactance

Impedance in an AC circuit is the total opposition to current flow and is a combination of resistance and reactance. Reactance comes in two forms—inertial (or inductive) reactance and capacitive reactance—which are frequency-dependent and introduce phase differences between current and voltage. Calculating the overall impedance requires combining these components using vector addition.

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