RLC Circuits and Second-Order Differential Equations

Introduction to RLC Circuits

RLC circuits are fundamental electrical circuits consisting of a resistor (R), inductor (L), and capacitor (C) connected in series or parallel. The analysis of these circuits involves solving second-order ordinary differential equations (ODEs) with constant coefficients, a key application of calculus in engineering and physics.

• Parallel RLC Circuit: All elements share the same voltage.

• Series RLC Circuit: All elements share the same current.

Parallel RLC Circuits: Natural Response

Formulating the Differential Equation

The natural response of a parallel RLC circuit is found by applying Kirchhoff's Current Law (KCL) and the constitutive relations for each element:

• KCL:

• Capacitor:

• Inductor:

• Resistor:

After substituting and differentiating to eliminate the integral, the governing equation becomes:

• Or, after dividing by :

This is a second-order, linear, homogeneous ODE with constant coefficients.

Solving Second-Order Circuits

• Assume a solution of the form , where and are constants.

• Substitute into the ODE to obtain the characteristic equation:

• This is a quadratic equation in .

General Solution and Characteristic Roots

• The general solution is , where and are the roots of the characteristic equation.

• Roots are found using the quadratic formula:

• Define Neper frequency and resonant frequency .

• Thus,

Classification of Solutions

The nature of the roots determines the circuit's response:

• Overdamped: (roots real and distinct)

• Underdamped: (roots complex conjugate)

• Critically damped: (roots real and repeated)

Initial Conditions

• Two initial conditions are required to solve for and :

• : Initial voltage across the capacitor

• : Initial rate of change of voltage, related to initial current in the capacitor

Types of Voltage Responses in Parallel RLC Circuits

Overdamped Response ()

• Roots , are real and negative.

• General solution:

• Initial conditions determine and .

• Response decays to zero without oscillation.

Underdamped Response ()

• Roots are complex: , where

• General solution:

• Oscillatory response with exponentially decaying amplitude.

• Constants and are determined from initial conditions.

Critically Damped Response ()

• Roots are real and repeated:

• General solution:

• Fastest non-oscillatory return to equilibrium.

Summary Table: Parallel RLC Natural Response

Step Response of a Parallel RLC Circuit

Formulation

The step response is the solution when a DC current source is applied at . The governing equation is:

• After substitution and differentiation:

This is a nonhomogeneous second-order ODE. The complete response is the sum of the natural response and the forced response (due to the constant source).

General Solution Structure

• Complete response:

• : Natural response (homogeneous solution)

• : Forced response (particular solution, usually a constant for DC input)

Step Response Cases

• Overdamped:

• Underdamped:

• Critically damped:

Where is the final (steady-state) current, typically equal to the current source value.

Initial and Final Conditions

• For initial current, consider the inductor:

• For initial voltage, consider the capacitor:

• Final condition: As , the capacitor acts as an open circuit, and the inductor as a short circuit.

Worked Examples and Drill Exercises

Example: Overdamped Parallel RLC Circuit

• Given , , , ,

• Calculate and :

• Since , the response is overdamped.

• Find roots: ,

• Use initial conditions to solve for and .

• Final solution:

Example: Underdamped Parallel RLC Circuit

• Given , , , ,

• Calculate and :

• Since , the response is underdamped.

• Find

• General solution:

• Voltage:

Key Definitions and Formulas

• Neper Frequency:

• Resonant Frequency:

• Damped Frequency:

• General Second-Order ODE:

• Characteristic Equation:

• Quadratic Formula:

Summary Table: Types of Damping in RLC Circuits

Study Suggestions

• Make summary notes with equations for natural and step responses for both parallel and series RLC circuits.

• Practice by working through all provided examples and drill exercises.

Additional info: The mathematical techniques used here (solving second-order ODEs, classifying roots, and applying initial conditions) are directly relevant to Calculus II/III and Differential Equations courses, especially in the context of physical systems modeling.