• Know how to find the second-order differential equation describing a parallel RLC circuit by applying Kirchhoff's Current Law.
• Learn how to find the characteristic equation of a second-order differential equation by assuming an exponential solution.
• Be able to form the natural response for a variable using the two roots of the characteristic equation.
• Learn what the characteristic roots are, and their dimensionality. Also learn the meaning of the neper frequency and the resonant radian frequency.
• Learn the conditions which determine the damping of the second-order circuit, namely, whether the response is overdamped, critically damped, or underdamped. and know the resulting general forms of the response variable.
• Learn how the values of the two coefficients in the expression for the response variable may be found from the values of the initial conditions on the inductor and the capacitor.
• Learn the meaning of the damped radian frequency and how it relates to the neper frequency and the resonant radian frequency. Learn the definition of the damping factor. Know why the term ringing is applied to an underdamped response.

Skills: In studying the material in these sections, you should make certain that you develop the following skills:

• Be able to find the differential equation for a second-order parallel RLC circuit which is excited only by initial conditions.
• Be able to find the characteristic equation from the differential equation for a second-order parallel RLC circuit.
• Be able to determine whether the response will be overdamped, critically damped, or underdamped, and know what the general form of the expression for the response will be for these three cases.
• Be able to find the response by relating the initial conditions on the capacitor and inductor to the values of the undetermined constants in the general form of the expression for the response.

Review Questions: Test your understanding of the material in these sections by answering the following review questions:

1. In the parallel RLC circuit, if the element values are such that the response is critically damped, what happens to the damping if the value of the resistor is increased?

2. In the parallel RLC circuit, if the element values are such that the response is critically damped, what happens to the damping if the value of the inductor is increased?

3. In the parallel RLC circuit, if the element values are such that the response is critically damped, what happens to the damping if the value of the capacitor is increased?