Application Note 4—Base Capacitance and A/D Ratio
Lambient TechnologiesTM
AN 4--Base Capacitance and A/D Ratio for Interdigitated
Electrodes
Introduction
Researchers studying dielectric properties often use parallel plate electrodes
because of their simplicity. However, the plate separation can change with pressure or
expansion and contraction of the material between them. The ratio of plate area A and
the distance D between the plates—the A/D ratio—therefore may not be precisely
known. As the scaling factor between conductance and conductivity (or capacitance and
permittivity), uncertainty in A/D leads to corresponding inaccuracies in determining these
dielectric properties.
A common alternative geometry is the interdigitated electrode as shown in Figure
AN 4-1. The electrodes are supported by a rigid substrate so the planar structure does
not change with pressure, or expansion and contraction of the Material Under Test.
While the A/D ratio is classically applied to parallel plates, it may be generalized to other
structures such as interdigitated electrodes. In this case, A is not necessarily the area of
the electrodes and D is not necessarily the distance between them, but also accounts for
fringing electric fields and other factors. Two-dimensional numerical simulations of
electrodes in lossy media have validated this generalization across a wide range of
conductivity. Since the geometry of the interdigitated electrode is fixed, the
corresponding A/D ratio is constant, and this structure can accurately measure both
conductivity and permittivity.
Interdigitated electrodes—
Geometry is fixed regardless of conditions
D
Parallel plates—
Separation can
change with
conditions
A
Figure AN 4-1
Comparison of parallel plate and interdigitated electrodes
While conceptually similar to parallel plates, interdigitated electrodes have a
substrate which introduces an additional component into the measurement system. The
cross section of Figure AN 4-2 shows that capacitance CTOT has a contribution CMUT from
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AN 4-1
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Application Note 4—Base Capacitance and A/D Ratio
the Material Under Test above electrodes. However, there is also a contribution CBASE
from the substrate supporting the electrodes. This second component is called the base
capacitance.
Material
Under
Test
GMUT
Driven
Electrode
CTOT
Response
Electrode
CMUT
Substrate
CBASE
Figure AN 4-2
Cross section of interdigitated electrode structure
The total capacitance measured by the interdigitated electrodes is:
(eq. D-1)
CTOT = CMUT + CBASE
The capacitance of the Material Under Test is calculated from equation E-2:
(eq. D-2)
CMUT = εo * ε’MUT * A/D
Calculating Base Capacitance with Known A/D
It is possible to determine the base capacitance of a sensor by measuring its
response in two different, non-conducting materials of known permittivity. To determine
base capacitance, measure the sensor capacitance in air.
(eq. D-3)
CTOT-AIR = CMUT-AIR + CBASE
Then measure the sensor capacitance in a second, non-conducting material such as
hexane.
(eq. D-4)
CTOT-HEXANE = CMUT-HEXANE + CBASE
Sum together eq. D-3 and eq. D-4.
(eq. D-5)
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CTOT-AIR + CTOT-HEXANE = CMUT-AIR + CMUT-HEXANE + 2 * CBASE
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Application Note 4—Base Capacitance and A/D Ratio
Lambient TechnologiesTM
CTOT is measured in each case, and CMUT is calculated in each case from equation E-2
using the known permittivity of each Material Under Test and the A/D ratio of the sensor.
Then CBASE can be calculated using equation D-6:
(eq. D-6)
CBASE = 1/2 [(CTOT-AIR + CTOT-HEXANE) - (CMUT-AIR + CMUT-HEXANE)]
Calculating Base Capacitance and A/D When Both are Unknown
Both the A/D ratio and the base capacitance are required to fully describe a
sensor. If they are both unknown then the two measurements described previously can be
used to create a system of two equations in two unknowns which can be solved with basic
algebra. Equations D-3 and D-4 are repeated below:
(eq. D-3)
(eq. D-4)
CTOT-AIR = CMUT-AIR + CBASE
CTOT-HEXANE = CMUT-HEXANE + CBASE
The capacitance for the Material Under Test for equations D-3 and D-4 can be rewritten
using equation D-2:
(eq. D-7)
(eq. D-8)
CTOT-AIR = (εε’MUT-AIR * εo ) * A/D + CBASE
CTOT-HEXANE = (εε’MUT-HEXANE * εo ) * A/D + CBASE
Equations D-7 and D-8 make up a system of two equations in two unknowns, where the
unknowns are the A/D ratio and CBASE. First, these equations can be solved for the A/D
ratio:
(eq. D-9)
A/D = (CTOT-AIR - CTOT-HEXANE) / (εεo * (εε’MUT-AIR - e'MUT-HEXANE ) )
Knowing the A/D ratio, the base capacitance can then be calculated from equations D-6
and D-2.
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