In this lab, non-linear behavior was investigated using a diode circuit.  The three main passive components of electronic circuits are resistors R, capacitors C, and inductors L.  In an LRC circuit, the voltage across them depends on charge q.  If a single sine wave voltage at frequency υ is applied, the output voltage across any of the three devices is also a single sine wave at frequency υ.  If the capacitor C is replaced by a diode D, then many frequencies can be generated simultaneously.  The voltage across the diode has non-linear terms in the charge q (eg q², etc) and its derivatives.  I investigated the non-linear electronic response of an LRD circuit.

The setup of the circuit involves an Agilent DSO-X2014A Oscilloscope and Agilent 3320-A Waveform Generator.  The waveform generator can generate a wave up to a frequency of 10 MHz.  By keeping the frequency at 1 kHz, the amplitude of the sine curve can be varied.

First I explored an LRC circuit, since that was simpler and much easier to understand.  There are three components: a resistor, a capacitor, and an inductor.  The input and output signals are viewed simultaneously in y-t mode.  When switching to the x-y mode on the oscilloscope, the image on the screen shows a perfect circle, because the input and output are both sine waves.  The Fast Fourier Transform (FFT) mode shows the Fourier transform of the output curve.  In an LRC circuit, there is only one peak in FFT mode since it is only a sine curve.  By varying the frequency, the amplitude of the output curve and the phase difference between the input and output curves can be measured and the following graphs resulted:

Phase Difference (phi) vs Frequency (kHz).  This graph shows how as the frequency changes, the phase difference also varies.

Amplitude vs frequency graph.  At around 60 kHz, the amplitude hit its peak and then began to decrease.  This means that at around this frequency, the circuit hit resonance.

Next I explored the LRD circuit, which replaced the capacitor with a diode.  DC offset is now a variable because the diode is asymmetric.  A diode is a two-terminal electronic component with an asymmetric transfer characteristic.  A diode has a low resistance to current flow in one direction and high resistance in the other.  The most common function of a diode is to allow an electric current to pass in one direction while blocking current in the opposite direction.  The input amplitude also becomes a variable because the degree of non-linear response depends on it.  The frequency response may or may not be different depending on whether frequency is increased or decreased and it may or may not depend on the starting frequency.  Because the diode only lets current pass in one direction, that is why there is a flat plane where the positive part of the sine curve should be.

Now, the interesting part is that the FFT mode shows lots of peaks instead of one peak.  The general trend seems to be

and f₀ is the input frequency.  At low amplitudes, these peaks are the only ones in the FFT.  However, at 1 V_pp, and f = 3 kHz, other peaks suddenly popped up at 1 kHz, 2 kHz, 4 kHz, 5 kHz, etc.  There is really no discernible pattern with this, and that is why this circuit is chaotic.  However, what stays the same are the peaks that occur at nf₀, which are islands of stability.  What is cool is that at a frequency of 5 kHz and 10 kHz, no extra peaks turn up even when the amplitude is increased up to the maximum.  Unfortunately, I did not get a chance to finish the lab, and this was the furthest I got, but what I explored so far in the lab was fascinating.