Driven RLC Circuits, Phasors, Impedance, and Resonance in AC Circuits

Driven RLC Circuits

Introduction to Driven RLC Circuits

A driven RLC circuit consists of a resistor (R), inductor (L), and capacitor (C) connected in series with an alternating voltage source. The analysis of such circuits involves solving a second-order differential equation, similar to those encountered in differential equations courses. The presence of a driving voltage leads to both transient (damped oscillations) and steady-state solutions.

• Kirchhoff's Law for RLC Circuit: The sum of voltage drops across each element equals the applied voltage.

• Differential Equation: The general equation for a driven RLC circuit is: $L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{1}{C}q = \mathcal{E}(t)$

• Solution Structure: The total solution is the sum of the transient (homogeneous) and steady-state (particular) solutions.

• Steady-State Guess: For a sinusoidal driving voltage, guess $q(t) = A \sin(\omega t)$ for the steady-state solution.

Additional info: The steady-state solution persists due to the continuous energy input from the voltage source.

Phasors and Phase Relationships in AC Circuits

Phases in the AC Driven RLC Circuit

In AC circuits, voltages and currents are sinusoidal and may have different phases. Phasor diagrams provide a graphical method to analyze these phase relationships.

• Current and Charge: If $q(t) = A \sin(\omega t)$, then $i(t) = \frac{dq}{dt} = A\omega \cos(\omega t)$.

• Voltage Drops:

  • Capacitor: $V_C = \frac{q}{C} = \frac{A}{C} \sin(\omega t) = \frac{A}{C} \cos(\omega t - \frac{\pi}{2})$

  • Resistor: $V_R = Ri = RA\omega \cos(\omega t) = RA\omega \sin(\omega t + \frac{\pi}{2})$

  • Inductor: $V_L = L \frac{di}{dt} = -LA\omega^2 \sin(\omega t) = LA\omega^2 \cos(\omega t + \frac{\pi}{2})$

• Phase Relationships:

  • The voltage across the capacitor lags the current by $\frac{\pi}{2}$.

  • The voltage across the inductor leads the current by $\frac{\pi}{2}$.

  • The resistor's voltage is in phase with the current.

• Phasor Representation: Each voltage can be represented as a vector (phasor) rotating in the complex plane with angular frequency $\omega$; their projections on the x-axis give instantaneous values.

• Mnemonic: "CIVIL" helps remember: for a Capacitor, I (current) Voltage Is Lagging; for an L inductor, Voltage Is Leading.

Reactance and Impedance in AC Circuits

Reactance of Circuit Elements

Reactance quantifies the opposition to AC current by capacitors and inductors, analogous to resistance for resistors. It is defined as the ratio of the amplitude of voltage to the amplitude of current for each component.

• Resistor: $X_R = R$

• Capacitor: $X_C = \frac{1}{\omega C}$

• Inductor: $X_L = \omega L$

Reactances are frequency-dependent, except for resistors.

Impedance of Series RLC Circuits

Impedance ($Z$) generalizes resistance to AC circuits, combining resistance and reactance (with phase differences) into a single quantity.

• Impedance Formula: $Z = \sqrt{R^2 + (X_L - X_C)^2}$

• Ohm's Law for AC Circuits: $V_{rms} = I_{rms} Z$

• Phase Angle: The phase difference $\phi$ between the total voltage and current is given by: $\tan \phi = \frac{X_L - X_C}{R}$

Table: Impedance of Common AC Circuits

Element	Impedance $Z$
R	$R$
C	$X_C$
L	$X_L$
RC	$\sqrt{R^2 + X_C^2}$
RL	$\sqrt{R^2 + X_L^2}$
LC	$|X_L - X_C|$
RLC	$\sqrt{R^2 + (X_L - X_C)^2}$

Example: Calculating RMS Voltages in RLC Circuits

Given the RMS voltages across the resistor ($V_R$ = 30 V), capacitor ($V_C$ = 90 V), and inductor ($V_L$ = 50 V), the total RMS voltage is not simply their sum due to phase differences. Instead, use impedance:

• $V_{rms} = \sqrt{V_R^2 + (V_L - V_C)^2}$

• Plugging in values: $V_{rms} = \sqrt{(30\ \mathrm{V})^2 + (50\ \mathrm{V} - 90\ \mathrm{V})^2} = 50\ \mathrm{V}$

Additional info: This demonstrates the importance of considering phase when adding voltages in AC circuits.

Resonance in RLC Circuits

Resonant Frequency and Its Effects

Resonance occurs in an RLC circuit when the inductive and capacitive reactances are equal in magnitude but opposite in sign, causing them to cancel each other. At resonance, the impedance is minimized, and the current reaches its maximum value for a given applied voltage.

• Condition for Resonance: $X_L = X_C$

• Resonant Frequency: $\omega_0 = \frac{1}{\sqrt{LC}}$

• Impedance at Resonance: $Z = R$

• Current at Resonance: $I_{rms} = \frac{V_{rms}}{R}$ (maximum value)

• Applications: Resonant circuits are used in radio and communication receivers to select specific frequencies from a range of signals.

Additional info: The resonance phenomenon in RLC circuits is analogous to the driven damped harmonic oscillator in mechanics.