^{1,a)}and Matthew J. Lanctot

^{1,b)}

### Abstract

We investigate a simple variation of the series RLC circuit in which antiparallel diodes replace the resistor. The result is a damped harmonic oscillator with a nonlinear damping term that is a maximum at zero current and decreases inversely with the current for currents far from zero. Unlike the standard RLC circuit, the behavior of this circuit is amplitude dependent. The transient response makes a transition from underdamped to overdamped behavior, and the resonance response of the steady-state driven oscillator becomes sharper as the source amplitude increases. A set of nonlinear differential equations is derived for the circuit and integrated numerically for comparison with measurements. The equipment is inexpensive and common to upper level physics labs.

This research was supported by an award from the Research Corporation. M.L. was supported in part by the Undergraduate Research Assistantship Program at UNCG. We thank David Birnbaum for valuable suggestions.

I. INTRODUCTION

II. CIRCUIT ANALYSIS

III. METHODS

IV. RESULTS AND DISCUSSION

### Key Topics

- Capacitors
- 20.0
- Oscillators
- 12.0
- Electrical resistivity
- 10.0
- Electric measurements
- 6.0
- Resistors
- 5.0

## Figures

The nonlinear circuit obtained from antiparallel diodes (1N4148), an inductor , and capacitor . The resistor is included to account for the inherent resistance of the inductor. The voltage source is either a low frequency square wave for the transient response experiments or sinusoidal for the steady-state driven oscillator experiments.

The nonlinear circuit obtained from antiparallel diodes (1N4148), an inductor , and capacitor . The resistor is included to account for the inherent resistance of the inductor. The voltage source is either a low frequency square wave for the transient response experiments or sinusoidal for the steady-state driven oscillator experiments.

The calculated dynamic resistance (solid line) for antiparallel diodes using Eq. (7). Also shown is the inverse current approximation (dashed line). The dimensionless current is , where is the current in amperes and is the diode’s reverse saturation current.

The calculated dynamic resistance (solid line) for antiparallel diodes using Eq. (7). Also shown is the inverse current approximation (dashed line). The dimensionless current is , where is the current in amperes and is the diode’s reverse saturation current.

Standard push-pull driver used with the signal generator in the antiparallel diodes-LC circuit. This use is necessary because the output impedance of most signal generators is too high to drive the circuit. General purpose op amps and transistors suffice. We used LF411, 2N3904, and 2N3906. A simpler driver omits the diodes and the and resistors and connects the op amp output directly to the bases of the transistors. This simplification adds a small cross-over glitch to the signal from the signal generator.

Standard push-pull driver used with the signal generator in the antiparallel diodes-LC circuit. This use is necessary because the output impedance of most signal generators is too high to drive the circuit. General purpose op amps and transistors suffice. We used LF411, 2N3904, and 2N3906. A simpler driver omits the diodes and the and resistors and connects the op amp output directly to the bases of the transistors. This simplification adds a small cross-over glitch to the signal from the signal generator.

Transient response of the capacitor voltage when the square wave source makes a transition from at . The numerical result (smooth line) and measured response (noisy line) are nearly indistinguishable. Note that a transition from underdamped to overdamped behavior occurs near . As described in the text the overdamped decay to zero is comparatively slow and is not complete in the plot.

Transient response of the capacitor voltage when the square wave source makes a transition from at . The numerical result (smooth line) and measured response (noisy line) are nearly indistinguishable. Note that a transition from underdamped to overdamped behavior occurs near . As described in the text the overdamped decay to zero is comparatively slow and is not complete in the plot.

The resonant frequency as a function of the source amplitude. The resonance was identified by adjusting the frequency to maximize the capacitor voltage amplitude for each source amplitude. The numerical results are lines and measured data are circles.

The resonant frequency as a function of the source amplitude. The resonance was identified by adjusting the frequency to maximize the capacitor voltage amplitude for each source amplitude. The numerical results are lines and measured data are circles.

The maximum voltage gain as a function of the source amplitude. is the ratio of the voltage amplitudes of capacitor and source at the resonant frequency. The solid line is the result using the single parameter set for the diode characteristics and the dashed line is the prediction when an additional set of diode parameters was included for high currents as described in Sec. III The dot-dashed line includes for the capacitor’s equivalent series resistance.

The maximum voltage gain as a function of the source amplitude. is the ratio of the voltage amplitudes of capacitor and source at the resonant frequency. The solid line is the result using the single parameter set for the diode characteristics and the dashed line is the prediction when an additional set of diode parameters was included for high currents as described in Sec. III The dot-dashed line includes for the capacitor’s equivalent series resistance.

Calculated resonant frequency (dashed line), Eq. (13), and maximum gain (solid line), Eq. (14), for the standard RLC circuit versus and . The resonant frequency drops to zero and at .

Calculated resonant frequency (dashed line), Eq. (13), and maximum gain (solid line), Eq. (14), for the standard RLC circuit versus and . The resonant frequency drops to zero and at .

The frequency response of the voltage gain for two source amplitudes: (lower) and 1.2 (upper) V. The numerical results used the two-part parametrization for the diodes corresponding to the dashed line result in Fig. 5.

The frequency response of the voltage gain for two source amplitudes: (lower) and 1.2 (upper) V. The numerical results used the two-part parametrization for the diodes corresponding to the dashed line result in Fig. 5.

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