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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 Version 1.0 AN 4-1 http://www.lambient.com Lambient TechnologiesTM 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) Version 1.0 CTOT-AIR + CTOT-HEXANE = CMUT-AIR + CMUT-HEXANE + 2 * CBASE AN 4-2 http://www.lambient.com 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. Version 1.0 AN 4-3 http://www.lambient.com