Lecture: Response of a circuit to a sinusoidal driving force (30 minutes)

lecture notes

This discussion adds a sinusoidal driving force to the restoring force and the resistive force present in a series RLC circuit. Rather than discuss the rigorous mathematics (homogeneous equation, particular solutions, etc.), a choice is made to propose a sinusoidal solution with the frequency of the driving force. Make physical arguments about why this is reasonable after a time scale long compared with the damping time. The object is to get the students to work with the proposed solution in complex form and solve for the proposed amplitude and phase (which are functions of the driving frequency) on terms of the parameters of the system.

There is rich discussion to be had. In particular, students must have a solid grasp of the “known” or “given” physical parameters of the system, which are considered fixed (L,R,C, V0) and which are variable parameters (driving frequency). What placeholder variables have been temporarily introduced (amplitude and phase of charge or current) that can be replaced by expressions involving physical parameters? Why are the amplitude and phase of charge dependent on the driving frequency?

Working with the complex solutions is a new application of the recently-learned complex numbers. Students work in groups to accomplish the manipulations necessary to identify the amplitude and phase of charge with the physical parameters.

Wrap-up discussion involves connecting to the lab of the previous day where the amplitude and phase of the response current have been measured. It's important to discuss that the “response” of a series RLC circuit could be viewed as induced charge on the capacitor or as induced current in the resistor and the 90-degree difference in phase between current response and response is important. Limits at high, low and resonance frequencies should be discussed.

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