Chapter Contents

• Differential Equations
• Predicting AIDS - a DEs example
• 1. Solving Differential Equations
• 2. Separation of Variables
• 3. Integrable Combinations
• 4. Linear DEs of Order 1
• 5. Application: RL Circuits
• 6. Application: RC Circuits
• 7. Second Order DEs - Homogeneous
• 8. Second Order DEs - Damping - RLC
• 9. Second Order DEs - Forced Response
• 10. Second Order DEs - Solve Using SNB
• Differential Equations Problem Solver

• Comments, Questions?

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8. Damping and the Natural Response in RLC Circuits

Consider a series RLC circuit (one that has a resistor, an inductor and a capacitor) with a constant driving electro-motive force (emf) E. The current equation for the circuit is

[or the equivalent ]

Differentiating, we have

This is a second order linear homogeneous equation.

Its corresponding auxiliary equation is with roots:

Now

α = R/2L is called the damping coefficient of the circuit

is the resonant frequency of the circuit.

m₁ and m₂ are called the natural frequencies of the circuit.

The nature of the current will depend on the relationship between R, L and C.

There are three possibilities:

Case 1: R² > 4L/C (Over-Damped)

Here both m₁ and m₂ are real, distinct and negative. The general solution is given by

The motion (current) is not oscillatory, and the vibration returns to equilibrium.

Case 2: R² = 4L/C (Critically Damped)

Here the roots are negative, real and equal,

i.e. m₁ = m₂ = -R/2L.

The general solution is given by

The vibration (current) returns to equilibrium in the minimum time and there is just enough damping to prevent oscillation.

Case 3: R² < 4L/C (Under-Damped)

Here the roots are complex where

The general solution is given by

where

α = R/2L is called the damping coefficient

In this case, the motion (current) is oscillatory and the amplitude decreases exponentially, bounded by

as we can see in the diagram above.

When R = 0, the circuit displays its natural or resonant frequency .

Example

In a series RCL circuit driven by a constant emf, the natural response of the circuit is given by

for which the initial conditions are i(0) = 2 A and at t = 0 is 4.

State the nature of response of the current and hence solve for i.

Answer