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Nonlinear Damping of the LC Circuit using Anti-parallel Diodes
Edward H. Hellen a) and Matthew J. Lanctot b)
Department of Physics and Astronomy, University of North Carolina at Greensboro,
Greensboro, NC 27402
Abstract
We investigate a simple variation of the series RLC circuit in which anti-parallel diodes
replace the resistor. This results in a damped harmonic oscillator with a nonlinear
damping term that is maximal at zero current and decreases with an inverse current
relation for currents far from zero. A set of nonlinear differential equations for the
oscillator circuit is derived and integrated numerically for comparison with circuit
measurements. The agreement is very good for both the transient and steady-state
responses. Unlike the standard RLC circuit, the behavior of this circuit is amplitude
dependent. In particular for the transient response the oscillator makes a transition from
under-damped to over-damped behavior, and for the driven oscillator the resonance
response becomes sharper and stronger as drive source amplitude increases. The
equipment is inexpensive and common to upper level physics labs.
Keywords : nonlinear oscillator; LC circuit; nonlinear damping; diode
I. Introduction
The series RLC circuit is a standard example of a damped harmonic oscillator-one of the
most important dynamical systems. In this paper we investigate a simple variation of the
2
RLC circuit in which the resistor is replaced by two anti-parallel diodes. This oscillator
circuit is fundamental in the sense that it is constructed from a small number of the most
basic passive electrical components: inductor, capacitor, and diodes. The result is that
oscillator characteristics that have no amplitude dependence for the standard RLC circuit
have strong amplitude dependence for the oscillator presented here. Transient response,
voltage gain, resonant frequency, and sharpness of resonance exhibit threshold behavior
due to the nonlinear current-voltage characteristics of the diodes. We show that
measured behavior agrees remarkably well with numerical prediction using standard
circuit component models.
The use of anti-parallel diodes creates a current-dependent damping term. Section II
shows that this oscillator is described by the homogeneous nonlinear differential
equation:
2
d
x
dx
(
()
)
+
b
+
b
x
+
x
=
0
(1)
0
1
d
τ
2
d
τ
where
b
⎧
max
x
>>
1
⎨
() ⎩
b
x
=
x
.
(2)
1
b
x
→
0
max
Here x is the dimensionless current in the circuit, τ is dimensionless time, b 0 is the
familiar constant damping term and b 1 ( x ) is the current-dependent damping term. Thus
damping is largest at the equilibrium position x = 0, and decreases inversely and
symmetrically about it. The nonlinearity of Eq. (1) is due solely to b 1 ( x ). Our interest is
840449650.004.png 840449650.005.png
3
in the case when damping is dominated by b 1 ( x ). This term is shown to be responsible for
the amplitude dependent behavior of the oscillator.
Equation (1) is nonlinear, but when expressed as a system of first order equations is
easily solved numerically using standard methods such as an adaptive Runge-Kutta. We
compare these numerical predictions with experimental measurements for both the
transient response and the steady-state response to a sinusoidal source.
The circuit investigated here looks similar to the extensively studied chaotic oscillator
constructed from a sinusoidal signal generator driving the series combination of a
resistor, inductor, and diode. 1-5 In those investigations interesting dynamics including
chaos can occur when the reverse-recovery time of the diode is comparable to the
circuit’s natural oscillation period. For the circuit in this paper the reverse-recovery time
is assumed to be zero since it is much shorter than the oscillation period. In addition we
include a capacitor so that the diode’s capacitance is negligible. It is the voltage
dependence of the diode’s dynamic resistance that causes interesting behavior.
This oscillator circuit has analogy with the “eddy-current damped” pendulum, often used
as a demonstration of Faraday’s and Lenz’s Laws. In this demonstration a metal disk
pendulum passes through the poles of a magnet placed at the bottom of the swing. Thus
the damping occurs near the equilibrium position x = 0 when the speed
dx /
dt
is highest.
For the oscillator presented here this corresponds to maximum damping at zero current
4
and maximum voltage across the capacitor and inductor. However the analogy does not
extend to the mathematical form of the position dependence of the damping.
II. Circuit analysis
The nonlinearly damped LC circuit using anti-parallel diodes is shown in Figure 1.
Summing voltages around the circuit and then taking the time derivative gives the
equation
2
dV
dV
d
I
dI
1
L
+
R
+
d
+
I
=
s
(3)
2
dt
dt
dt
C
dt
where V d is the voltage across the anti-parallel diodes. The resistor R accounts for the
intrinsic resistance of the inductor in addition to any explicit resistor. For the transient
response the right hand side is zero. [In actual measurements a square wave source V s
makes its transition at time zero from V 0 to the square wave’s other value V f . This
provides the initial conditions of zero current and V 0 for the capacitor voltage, and thus
(
) L
dI
/
dt
=
V
−
V
/
.] For the steady state response V s is a sinusoidal source whose
f
0
amplitude and frequency are varied.
The voltage V d across the anti-parallel diodes is a function of current, so its time
derivative is written as
dV
dV
dI
dI
() dt
d
=
d
=
R
I
(4)
d
dt
dI
dt
()
where
R
I
=
dV
/
dI
is the dynamic resistance of the anti-parallel diodes. This is
d
d
derived as follows.
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5
The standard current-voltage relation for a diode is
⎛
eV
⎞
⎛
⎞
()
I
V
=
I
⎜
⎝
exp
d
−
1
⎟
⎠
(5)
⎜
⎝
⎟
⎠
d
0
mkT
where e is the elementary charge, k is Boltzmann’s constant, T is the absolute
temperature, and V d is the voltage across the diode. 6 At room temperature
kT
/
e
=
V
≈
25
mV. Quantity I
0 is the reverse saturation current and m is a correction
th
factor. Typical values for a silicon diode are m = 2 and a few nanoamperes for I 0 .
Applying Eq. (5) to anti-parallel diodes as in Fig. 1 results in the current-voltage relation
⎛
⎞
V
()
I
V
=
2
I
sinh
⎜
⎝
d
⎟
⎠
.
(6)
d
0
mV
th
We define dimensionless quantities for voltage and current:
V
I
v
=
d
i
=
.
(7)
mV
2 I
th
0
The current-voltage relation and the dynamic resistance for the anti-parallel diodes are
then
( )
( ) ( )
( )
i
v
=
sinh
v
v
i
=
sinh
−
1
i
(8)
dV
mV
mV
R
dv
()
R
i
=
d
=
th
=
th
=
max
.
(9)
(
)
(( )
()
d
−
1
dI
2
I
di
2
I
cosh
v
i
cosh
sinh
i
0
0
( )
Putting in values for m , V th , and I 0 = 3 nA gives
R
=
mV
/
2
I
=
8
M
Ω
. Figure 2
max
th
0
shows the dynamic resistance, Eq. (9), as a function of dimensionless current. The
behavior of the dynamic resistance in Eq. (9) for large and small currents is
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