|
Nonlinear Damping of the LC Circuit using Anti-parallel Diodes
|
| A simple variation of the RLC circuit with nonlinear
damping is obtained by replacing the resistor with anti-parallel diodes
as shown below. The resulting circuit still consists of the most basic passive
electronic components. Its behavior is described by the homogeneous nonlinear
differential equation shown below. Note that damping is maximal at equilibrium
(zero current), and decays inversely with current away from equilibrium.
For complete description, see eprint: Hellen and Lanctot, AJP 2007 |
|
|
where
 |
Here i is the dimensionless current
in the circuit and b0 is the usual constant
damping term. Source Vs is taken to be zero for the homogeneous equation.
Time is dimensionless, having been normalized by sqrt(LC).
The dynamic resistance (basically b1(i))
of the anti-parallel diodes is shown in the graph below. The dashed lines
show the inverse current behavior. |
Dynamic Resistance of Anti-parallel Diodes
|
| The current dependent resistance shown in the graph is responsible
for interesting amplitude dependent behavior of the circuit. For example
the transient response makes a transition from an under-damped to an over-damped
oscillator. The graph below shows predicted and measured transient response
for the capacitor voltage for initial conditions 8.2 volts and zero current.
Note that at t = 0.075 ms the behavior changes from under-damped
to over-damped. |
Transient Response
|
| For the driven steady-state response there is a strong dependence
on the amplitude of source Vs. As the source amplitude increases,
the resonance response becomes stronger and sharper as shown in the figure
below. This is because the effective resistance of the anti-parallel diodes
is less for large amplitude currents, as shown above in the dynamic resistance
graph. |
Steady-State Response
|
| The voltage gain A is the ratio of the amplitudes of the capacitor voltage and source voltage. Thus this resonant oscillator circuit behaves similar to a RLC circuit where R is a decreasing function of the source amplitdue. |
