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## Harmonic LRC Lab: Instructor's Guide

### Main Ideas

• LRC circuit
• Driving force
• Resonant frequency and phase shifting
• Lab experiences

Estimated Time: 70 mins

1. To measure the current response of the series LRC circuit to a sinusoidal voltage input at a series of frequencies about resonance frequency.
2. To determine the complex admittance of the LRC circuit as a function of frequency, in amplitude/phase form.
3. To estimate the quality factor $Q$ and the resonance frequency $\omega_1$ from admittance data.
4. To compare the experimental result to the model discussed in class.

### Prerequisite Knowledge

The free, damped oscillator: students should have had their first in-depth encounter in previous lectures in this course (some may also have “seen the damped oscillator” in introductory physics).

The relationship between current (determines the resistor voltage), its derivative (charge, determines the capacitor voltage) and its integral (determines the inductor voltage).

A nodding acquaintance with an oscilloscope. If at least half the class has used a scope before (as is the case for our students), those students are paired with inexperienced peers, and the lab works well.

### Props/Equipment

• Experimental stations (one per three students) with oscilloscope, function generator, circuit components and simple breadboard, coax cables and connectors. Equipment description
• A handout describing the task for each student hand out

### Activity: Introduction

A general description of the system, equipment at hand, and the task has been assigned as reading before the lab. We discuss as a group what the task is and I try to elicit student responses that the sinusoidal voltage supply is the function generator and that we will measure the response of the circuit as determined by the current and will therefore measure the voltage across the resistor. (Charge is an equally good proxy for response, and some students might measure that if they finish early). Students are instructed to build the damped harmonic oscillator from the circuit components, and to explore what happens when they pick different frequencies for the driving voltage. The only pointer that are given is to ensure that one side of the resistor is connected to the oscilloscope ground (otherwise ground loops can become a problem and interfere with the message of the lab). Students are encouraged to “play” before recording data, because it is important to get an idea of the entire scope of the experiment before focusing on detailed measurements.

### Activity: Student Conversations

Students work in groups of 2-4, and often consult other groups. They should be encouraged to do so.

Many students try to take detailed data before they know the general behavior of the system and the expected extremes. They need to be reminded to “play” for a few minutes, and to recognize that such “play” can save time later. For example, many do not think about how many measurements they will need, do not change the frequency enough between each measurement and fail to get to the interesting region.

Once the students have established that the voltage across the resistor is (i) at the same frequency as the driving voltage, (ii) in phase with, and equal to, the driving voltage at a particular frequency, and (iii) out of phase with, and smaller than, the driving voltage at other frequencies, they can proceed to make quantitative measurements. Some students with little experience in electronics wonder why the current (which is proportional to the voltage across the resistor) is not the same at al frequencies despite the fact the driving voltage is kept constant and the value of the resistor does not change. Some students are able to respond with a discussion of reactance and its frequency dependence, though the discussion is rarely sophisticated. This topic is taken up in class later.

The discussion of phase is always lively. Confronted with two sinusoids slightly displaced in time on an oscilloscope screen, students invariably point to the trace with the peak or zero-crossing to the right (larger time) as the signal that is “ahead” because it looks like it “will win the race”. In fact that trace is behind or lagging. Some Socratic questioning is in order about which function reached a particular value earlier in time to help the students resolve the issue. Further questions are necessary about whether leading or lagging corresponds to a positive or negative sign for the phase angle (it depends how the angle has been defined in the statement of the problem). In addition, there is the issue of the 90-degree phase difference between charge and current, and current and dI/dt.

It is important to ask students (calling the class together is most efficient) once they have some experience with the circuit, to check that the time base control is set so that the time base is properly calibrated. Most scopes allow non-calibrated time bases, which renders true frequency readings impossible. It's safest to set the voltage scale to fully calibrated, too, in case the scale is changed. Ask the students why it's important. Many students read the frequency from the function generator, and not the scope. They can get away with this, but the scope is far more accurate.

Students, entirely reasonably, want to know how they can find the “true” values of R, L, and C. Should they read them from the values printed on the components? Measure them on other devices like an ohmmeter or capacitance bridge? Useful discussion can come from this apparently straightforward question. Resistors and capacitors are nearly ideal elements in the low frequency range we use (up to 1-2 kHz) and one can use either method. But the large inductors we use have an inductance (about 0.1 mH) that is frequency dependent, and the values we measure at low frequency don't reflect those printed on them (which are high-frequency values), and they seem to vary from one another, too. This leads the discussion in the direction of what is really “known”, and perhaps our measurement is really an inductance measurement assuming the model is correct rather than a validation of a model assuming the given inductance is correct. We treat the lab as a means to find the exact value of the inductor.

### Activity: Wrap-up

Most of the conversations in the lab focus around issues of measurement and not so much on physics. The wrap up sets the stage for the following day's discussion. The following session is a discussion of the predictions of the model. Students MUST have their data plotted by the next class period, and they are to list all the questions they have (some of these can be elicited at this point to be recorded and addressed in the next session). Students bring two copies of their plotted data to class: one with a name and date on it, for (minimal) grading, and another with a number on it for anonymity so that feedback from peers can be obtained.

### Extensions

Students who wish to investigate further can repeat the measurements measuring the voltage amplitude and phase across the capacitor (proxy for charge) and the inductor (proxy for dI/dt), as a function of the driving frequency.

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