# Oscillations and Resonance in an RLC Circuit

#### Description of the Experiment:

Electrical circuits are the most noted examples non-mechanical oscillatory behavior.  In fact, because of their great practical importance, they have become the prominent model used in describing oscillatory motion.  In this experiment we made a slight variation on the standard RLC circuit usually used to understand linear oscillatory behavior.  By replacing the capacitor in the circuit with a diode we added a non-linearity to our system.  How is this non-linearity created by a diode?  When a diode is forward biased, or conducting, it acts as a constant voltage source.  When the diode is reversed biased it acts as a capacitor.  However, the diode is not a perfect device and therefore experiences a non-zero time length for switching from forward to reverse bias and vice-versa.  The non-linearity occurs as a result of uncombined charges which had crossed the diode's p-n junction while it was in the forward bias mode.  When the diode switches to its reverse mode, these charges diffuse back over some amount of time.  Therefore, the diode conducts for a short time after it switches from forward to reverse bias.  If this amount of time is significant then then the system becomes non-linear.  This non-linearity allowed us to investigate non-linear oscillatory behavior as well as chaos in the system.  It is important to note that the amount of current through the diode affects the time it takes for the diode to return to one of its bias equilibrium states; the greater the magnitude of the current, the more significant is this effect.  The apparatus used to investigate these ideas was relatively simple.  The RLC circuit was built on a bread board with an inductor of 25 mH and a diode (1N 4005), while a Tektronix CFG 253 Function Generator provided the sinusoidal voltage.  A Tektronix TDS 320 digital Oscilloscope was used to observe and record data from the circuit.  To compare the input and output signals of our circuit we placed one probe before the inductor to measure the input voltage and another probe to read the voltage across the diode.

#### Oscillations and Resonance in the Experiment:

The standard method for studying non-linear effects in an RLC circuit such as our system, is to set the frequency of the oscillating power source near the resonant frequency of the circuit and to vary only the amplitude of the power source.  The rationale of this method purports that as the magnitude of the current increases, the non-linear effects of the diode will increase as well and the system will exhibit non-linear behavior such as period doubling and chaos.  We can observe this non-linear behavior in a phase plot that compares the input voltage with the voltage across the diode.  This occurrence of period doubling is an good example of how at resonance in a non-linear system, there is not a unique output amplitude.  Normally then, to observe chaotic behavior in the system of a diode-inductance circuit the applied voltage would be varied while the input frequency would be maintained close to that of the resonating frequency of the inductor and the reverse capacitance of the diode (wr = (1/(L*C))1/2).  However, in our experiment we arbitrarily chose to demonstrate the non-linearities of the system by varying the input frequency instead of the voltage.  In order to observe chaotic behavior in an oscillatory system one needs only to vary one parameter of the input or driving force, so while our methods were unconventional they still produced the desired results.

Indeed, using our modified approach the experiment still clearly exhibited non-linear oscillatory effects, including chaotic behavior. It is evident from the data that as the input frequency is increased from resonance, the output or diode voltage is seen to undergo period doubling, quadrupling, octupling and eventually becomes chaotic.  Again the presence of chaos is confirmation of the idea that in a non-linear oscillatory system a periodic driving force does not always yield a periodic output. If we continued to increase the input frequency the diode voltage is then seen to escape the region of chaotic behavior, descend through period octupling, quadrupling, doubling and finally return to the normal periodicity that was first exhibited at resonance.  The following phase spaces are graphs of the input voltage as a function of the diode voltage and demonstrate part of this sequence from normal periodicity to chaos and back.

The standard method for producing the non-linear behavior in our system is similar to the approach used in the pendulum experiment.  To observe non-linear behavior including chaos in the output, it is necessary to vary one non-linear parameter of the driving force while the driving frequency of oscillation is maintained near the resonant frequency of the system.  When these conditions are met it is possible to observe such non-linear behavior as period doubling and chaos.  These results demonstrate that in a non-linear system, there is no unique amplitude of oscillation at resonance and that a periodic driving force does not always produce a periodic output.  It is important to understand these differences between linear and non-linear oscillations if one wants a thorough understanding of many physical oscillatory systems.

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