A Periodically Forced RLC Circuit In this lab we continue the study of simple RLC circu...

A Periodically Forced RLC Circuit

In this lab we continue the study of simple RLC circuits that we began in a lab in Chapter 3 (see page 381). The circuit is shown in Figure 4.41. The parameters are the resistance R, the capacitance C, and the inductance L. The dependent variables we use are v_C, the voltage across the capacitor, and i , the current. (We follow the engineering convention of representing current as i . This variable does not represent the square root of −1.) In this lab we consider circuits in which the voltage source v_T = v_T (t) is a time-dependent forcing term.

From the lab in Chapter 3, we know that the voltage v_C and current i satisfy the system of differential equations

which is more commonly written as the second-order equation

In this lab, we consider the possible behavior of solutions to this equation (or the corresponding system) when v_T (t) = a sin ωt .

Consider the following questions:

Your report: Address each of the items above, justifying all statements and showing all details. Give graphs of solutions as appropriate.

In a typical circuit R is on the order of 1000, C is on the order of 10⁻⁶, and L is on the order of 1. Does this information help in limiting the possible behaviors of solutions?