1.9 Summary

• Circuits that contain energy storing devices can be described by integrodifferential equations and upon differentiation can be simplified to differential equations with constant coefficients.

• A second order circuit contains two energy storing devices. Thus, an RLC circuit is a second order circuit.

• The total response is the summation of the natural and forced responses.

• If the differential equation describing a series RLC circuit that is excited by a constant (DC) voltage source is written in terms of the current, the forced response is zero and thus the total response is just the natural response.

• If the differential equation describing a parallel RLC circuit that is excited by a constant (DC) current source is written in terms of the voltage, the forced response is zero and thus the total response is just the natural response.

• If a circuit is excited by a sinusoidal (AC) source, the forced response is never zero.

• The natural response of a second order circuit may be overdamped, critically damped, or underdamped depending on the values of the circuit constants.

• For a series RLC circuit, the roots s ₁ and s ₂ are found from

  
  

  or

  
  

  where

  
  

  If ? _S ² > ? ₀ ², the roots s ₁ and s ₂ are real, negative, and unequal. This results in the overdamped natural response and has the form

  
  

  If ? _S ² = ? ₀ ², the roots s ₁ and s ₂ are...