nonlinear_capacitor_model_appnote.pdf

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APPLICATION NOTE
A Nonlinear Capacitor Model for Use in the PSpice
Environment
December 2009
THEORY
The charge and current formulas for a linear capacitor are:
Q = C * V
(1a)
I = C * dV(t)/dt
(1b)
For a nonlinear (voltage-dependent) time-independent capacitor these formulae become:
Q = C(V) * dV
(2a)
I = C(V) * dV/dt
(2b)
This applies to cases where the capacitance has been measured at different bias voltages.
Some would argue that for a nonlinear capacitor,
Q = C(V) * V
(3a)
where V is a function of time. Therefore,
I = dQ/dt = C(V(t)) * dV(t)/dt + dC(V(t))/dt * V(t)
(3b)
This is not correct. The flaw in this argument is equation (3a). Although (1a) holds true for linear
capacitors, the generalized definition of charge is (2a). Capacitance is the partial derivative of
Q
with
respect to
V
; which means
I = dQ/dt = Q/V * dV/dt = C(V) * dV/dt
(4)
Given this relationship between the current through a nonlinear capacitance and the voltage applied to it,
analog behavioral modeling can be used to model any nonlinear capacitor whose capacitance,
C(V)
, is a
function of the voltage applied to it.
THE MODEL
The nonlinear capacitor is modeled by using ABM (Analog Behavioral Modeling). The capacitor is
replaced by a controlled current source, Gout, whose current is defined by (2b). The time derivative,
®
®
dV(t)/dt
, is modeled by using the DDT( ) function in the Cadence PSpice environment. A voltage
dependent capacitance can be specified by using a look-up table, or by using a polynomial.
LOOK-UP TABLE EXAMPLE
If a Capacitance vs. Voltage curve is available, a look-up table can be used in the ABM expression. This
table contains (Voltage, Capacitance) pairs picked from points on the curve. The voltage input is
nonlinearly mapped from the voltage values in the table to the capacitance values. Linear interpolation is
used between table values. This voltage dependent capacitance is the multiplied by the time derivative of
the voltage to obtain the output current.
Figure 1: Nonlinear capacitor with look-up table expression
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APPLICATION NOTE
POLYNOMIAL EXAMPLE
Figure 2: Nonlinear capacitor with polynomial expression
The value of the nonlinear capacitor model in this example has a second order polynomial dependence
on its voltage. This is equivalent to the standard PSpice capacitor model, whose linear and quadratic
coefficients,
VC1
and
VC2
, can be defined in a .MODEL statement. This model is parameterized so that
users can specify the polynomial coefficients in a parameter block (the PARAM symbol), or on the symbol
in the schematic editor.
Example Simulation
A transient analysis can be run to verify the model.
Figure 3: Test circuit for voltage-controlled capacitor
Figure 4: PSpice plot of transient analysis results
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APPLICATION NOTE
The transient analysis slowly varies the voltage across the capacitor from 1V to 50V. The effective
capacitance can be viewed in PSpice as the expression:
- I(V1)/D(V(V1:+))
, which is derived from
equation (2b).
D( )
is the derivative with respect to time. The minus
(-)
sign is required because PSpice
measures voltage source currents as flowing from the positive node to the negative. PSpice will show that
the capacitance varies from
1.02u(F)
to
1.78u(F)
.
SETTING INITIAL CONDITIONS
The IC1 and IC2 symbols in
special.olb
(or the
.ic
statement in an Include file) can be used to specify
initial conditions for the capacitor.
© Copyright 2009 Cadence Design Systems, Inc. All rights reserved. Cadence, the Cadence logo, and Pspice are registered trademarks of Cadence
Design Systems, Inc. All others are properties of their respective holders.
21309 12/09 MK/DM/PDF
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APPLICATION NOTE

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