^{1}, M. Josebachuili

^{1}, P. Zurita

^{1}and S. Gil

^{2,a)}

### Abstract

We discuss an experiment on coupled RLC circuits with variable coupling strength. The inductive coupling can be easily varied and measured for different configurations. The experiment allows us to explore the variation of the resonance frequencies as a function of the coupling strength between the oscillators. The system illustrates the effect of eigenfrequency repulsion when an interaction couples different modes of oscillation. The experiment is conceptually simple, and its results can be compared quantitatively with theoretical predictions.

The authors would like to express their acknowledgement of the other students who have explored the physics of coupled circuits with different experimental techniques. They also express their gratitude to Prof. Carlos Acha for his assistance in the setting and programming of the lock-in amplifier and to the valuable comments and suggestions made by Dr. E. Batista and Prof. C. Grosse. We are also grateful to Dr. A. Schwint for a careful reading of the manuscript.

I. INTRODUCTION

II. THEORETICAL CONSIDERATIONS

III. THE EXPERIMENT

IV. RESULTS AND DISCUSSION

V. CONCLUSIONS

### Key Topics

- Amplifiers
- 12.0
- Coils
- 12.0
- Coupled oscillators
- 6.0
- Electric measurements
- 5.0
- Inductance
- 4.0

## Figures

Inductively coupled LC circuits.

Inductively coupled LC circuits.

Inductively coupled RLC circuits. The resistance is the internal resistance of the AC source, and and are the internal resistances of inductors having inductances and , respectively. The voltage drop in the resistances of the primary and secondary circuits is measured directly by the lock-in amplifier. is the voltage drop in the resistance and is used to monitor the currents in each network.

Inductively coupled RLC circuits. The resistance is the internal resistance of the AC source, and and are the internal resistances of inductors having inductances and , respectively. The voltage drop in the resistances of the primary and secondary circuits is measured directly by the lock-in amplifier. is the voltage drop in the resistance and is used to monitor the currents in each network.

Schematic of the experimental setup. The two coils move along a wooden rod. Here is the separation between the coils. An AC source is connected to the primary circuit.

Schematic of the experimental setup. The two coils move along a wooden rod. Here is the separation between the coils. An AC source is connected to the primary circuit.

Experimental results for the mutual inductance as a function of the separation . The diamond symbols are the results for obtained by direct measurements of . The continuous line is an empirical fit to the data using an exponential function.

Experimental results for the mutual inductance as a function of the separation . The diamond symbols are the results for obtained by direct measurements of . The continuous line is an empirical fit to the data using an exponential function.

The observed current amplitude in the primary circuit as a function of frequency for different separations between the coils. The frequency separation of the maxima of these curves increases for smaller distances, that is, stronger coupling.

The observed current amplitude in the primary circuit as a function of frequency for different separations between the coils. The frequency separation of the maxima of these curves increases for smaller distances, that is, stronger coupling.

The observed current amplitude in the secondary circuit as a function of frequency for different separations between the coils. As in Fig. 5, the frequency separation of the maxima increases for smaller distances.

The observed current amplitude in the secondary circuit as a function of frequency for different separations between the coils. As in Fig. 5, the frequency separation of the maxima increases for smaller distances.

The observed current amplitude as a function of the frequency in the primary (crosses) and secondary (circles) circuits for . The continuous lines are fits to the data using Eq. (A10) with the mutual induction coefficient as the only adjustable parameter. The values of obtained for each value of are shown in Fig. 4.

The observed current amplitude as a function of the frequency in the primary (crosses) and secondary (circles) circuits for . The continuous lines are fits to the data using Eq. (A10) with the mutual induction coefficient as the only adjustable parameter. The values of obtained for each value of are shown in Fig. 4.

Observed maximum response frequencies as a function of . The heavy lines are the result of Eq. (10). The dotted horizontal lines indicate the uncoupled eigenfrequencies. The square symbols and triangles represent the eigenfrequencies (frequencies of the maxima in the secondary circuit of Fig. 6).

Observed maximum response frequencies as a function of . The heavy lines are the result of Eq. (10). The dotted horizontal lines indicate the uncoupled eigenfrequencies. The square symbols and triangles represent the eigenfrequencies (frequencies of the maxima in the secondary circuit of Fig. 6).

## Tables

Values of the parameters used. The second column shows the values of the parameters obtained by direct measurements. The third column shows the results obtained from a fit of the data, as illustrated in Fig. 6. In this case the model provides only the total resistance of the circuit.

Values of the parameters used. The second column shows the values of the parameters obtained by direct measurements. The third column shows the results obtained from a fit of the data, as illustrated in Fig. 6. In this case the model provides only the total resistance of the circuit.

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