Chaos in a Diode

observing period-doubling and chaotic behavior in a driven diode circuit

Chris Osborn Daniel Richman

The Diode• Passive (not self-powered) circuit component• Restricts current flow to one direction• Modern diodes are based on semiconductor p-n junctions

– Situation when diode is not connected to a circuit (a voltage):Electron and hole recombination in vicinity of junction, depletion of

mobile charges; remaining negative charges in p-type and positive charges in n-type cause potential difference across junction (field points from n-type to p-type)

– Connection methods:

• Forward bias—positive emf terminal to p-type, negative to n-typeHoles and electrons are pushed toward non-conducting center and

oppose built-in potential barrier; current tunnels through the barrier

• Reverse bias: negative emf terminal to p-type, positive to n-typeReinforces built-in potential barrier; current is blocked

The Circuit

Characteristics (why nonlinearity)

• Diode has threshold voltage required for conduction– Forward voltage drop must reach this value for conduction

to occur

• Diode has residual conduction during reverse-bias part of cycle(forward bias means diode “on”, reverse bias means diode

“off”)– Exponential decay of current—recovery time– Frequency of cycle is faster than recovery time

Affects starting condition for next forward-bias part of cycle

Experimental Procedures

• Function generator output: sine– Frequency is circuit’s resonance frequency (73.7 kHz)– Amplitude adjustable from millivolts to tens of volts

• Inductor set to ten mH• Oscilloscope reads generator voltage and diode

voltage

1. Fix the driving frequency2. Start at a low voltage amplitude3. Increase amplitude in steps of tenths4. Observe multiplying curves on diode voltage

display

Observing the CircuitWhat are we looking at?

Diode voltage versus time(# of periods is # of distinct peaks)

Diode voltage versus driving voltage(# of periods is # of loops)

Chaos

Experimental Data[Diode voltage versus driving voltage bifurcation graph]

brief chaos between red and blue

chaos between purple and gold

Simulation

Original computation in Matlab for the longtime convergence of xn

• produces a sequence of terms whose long-time convergence is the bifurcation diagram

• xo = 0.7, r = {2.5 : 0.015 : 4.0}

xn+1 = r xn (1 – xn)