# AN 14 Errors in Parallel Plate Measurements (The Effect of Surface Texture)

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1 AN 14 Errors in Parallel Plate Measurements (The Effect of Surface Texture) Summary This application note will examine the effect of a textured sample surface similar to that of a fabric. The resulting measurement of permittivity is less than the true permittivity of the material and results from the contribution of air pockets created by the textured surface. This application note will model the effect of these air pockets to develop an understanding of the measurement error. Good parallel plate measurements of dielectric constant ε (also known as the real component of complex permittivity or, simply, relative permittivity) uses an excitation and a sense electrode with a guard electrode to eliminate fringing electric fields around the sense electrode. This guarded electrode configuration allows accurate calculation of test cell capacitance. Consequently, ASTM D states that the guarded electrode (three terminal method) is to be used as the referee method unless otherwise agreed upon. Careful attention to measurement technique will yield good results, but it is often easy to overlook sources of error due to non-ideal samples. An ideal sample will have the following characteristics: An area large enough to completely cover the sense electrode Smooth, flat surfaces for the sides contacting the electrodes Parallel surfaces for the sides contacting the electrodes PARALLEL PLATE MEASUREMENTS Dielectric instrumentation measures electrical properties of the Material Under Test (MUT) between a pair of electrodes, which can be modeled as a conductance in parallel with a capacitance, as shown in Figure 1. Version 0.9 AN 14-1

2 I RES Driven electrode + V EXC - G MUT A C MUT D (MUT) Grounded electrode Figure 1 Electrical model of dielectric The raw measurements at a given frequency f are: G MUT = conductance (ohms -1 ) C MUT = capacitance (farads) With the known quantities of: ω = 2πf ε 0 = 8.86 x F/cm A/D = ratio of area to distance for electrodes then it is possible to calculate the resistance: (eq. 1) R MUT = 1/G MUT (resistance) and the following material properties: (eq. 2) ρ = R MUT *A/D (resistivity or ion viscosity) (eq. 3) σ = G MUT / (ε o * A/D) (relative conductivity) (eq. 4) ε' = C MUT / ( ε o * A/D) (relative permittivity) (eq. 5) ε" = σ / ω (loss factor) Dissipation, or tanδ, at measurement frequency f is the ratio of a material s relative loss to its relative permittivity, and is given by the relationship: (eq. 6) tanδ = ε / ε = 1 / (ω C MUT R MUT ) Version 0.9 AN 14-2

3 In the case of a solid material which can be fabricated as a laminate or a panel, a parallel plate electrode configuration is often used. The guarded parallel plate electrodes of the Lambient Technologies LT-4203 Test Fixture are diagrammed below in Figure 2. LT-451 or LCR meter Excitation Excitation electrode Response (0.025 cm to 0.50 cm Thick TYP) Sense electrode (6 cm diameter, cm 2 area) Figure 2 Diagram of LT-4203 guarded parallel plate configuration For the configuration and electrode dimensions of Figure 2, the air capacitance of the test cell, and the capacitance when filled with a material of relative permittivity ε = 4.0 (typical for polyimide-glass composites) are listed below in Table 1: Electrode Separation D (cm) Table 1 Parameters of Example Parallel Plate Configuration A/D Ratio (cm) Air Capacitance (ε = 1.0) Material Capacitance (ε = 4.0) pf 400 pf pf 40.0 pf Version 0.9 AN 14-3

4 CONTACTING ELECTRODE MEASUREMENTS The contacting electrode method requires only one measurement with the electrodes in direct contact with the MUT as shown in Figure 3. The surface of the MUT must be flat to prevent an air gap between the sample and the electrodes, which can cause a measurement error. The MUT should also be incompressible so the separation between the electrodes is the same as the true thickness of the sample. LT-451 or LCR meter Excitation Excitation electrode t a = thickness of (MUT) Response Response electrode (Area = A) Figure 3 Configuration for contacting electrode measurements Dielectric properties at frequency of measurement f are calculated below: (eq. 7) ε' = C P / ( ε o * A / t a ) (relative permittivity) (eq. 8) tanδ = ε / ε = 1 / (ω C P R P ) (dissipation) (eq. 9) ε" = ε' * tanδ (loss factor) Where: ω = 2π * f ε 0 = 8.86 x F/cm C P = Capacitance of measurement R P = Resistance of measurement If the sample has an irregular surface which creates air gaps, then the results of this measurement will have an error because the electrodes do not make close and uniform contact with the sample. Version 0.9 AN 14-4

5 Sample with Textured Surface The following discussion will examine the errors in measuring relative permittivity caused by a sample with a textured surface, as shown in Figure 4. Figure 4 Sample with fabric-like textured surface The surface of this sample has a regular texture created by the weave of the fiberglass substrate. The combination of the fiberglass and the polymer which impregnates the weave has an expected relative permittivity of approximately 3.0. When using the setup and assumptions of Figure 3, measurement of relative permittivity in a parallel plate test fixture such as the Lambient Technologies LT-4203 results in a value of approximately 1.9--significantly lower than expected. The (MUT), however, is not a homogeneous slab, and so equation 7, which was used to calculate relative permittivity, cannot apply. A more realistic cross section of the MUT is shown in Figure 5, showing air pockets resulting from the weave of the sample. Air pockets (TYP) Excitation electrode t b = thickness of material between air pockets t a = thickness of (MUT) Response electrode (Area = A) Figure 5 Cross section of textured sample between parallel plate electrodes Version 0.9 AN 14-5

6 The electrical model of this sample is not that of Figure 1, but the more complex one shown below in Figure 6. A 2 C2 C1 A 1 t b = thickness between air gaps A 1 = A 0 ( 1 A 2 / A 0 ) t a = thickness of MUT C MEAS = C1 + C2 Where: A 0 = Area of sense electrode A 1 = Area of material contacting sense electrode A 2 = Area of air pockets Figure 6 Electrical model of sample with fabric-like textured surface The model of Figure 6 is two capacitors in parallel. One capacitor (C1) is filled with the dielectric material and the other capacitor (C2) is only partly filled with the dielectric material. The measured capacitance of this model is C MEAS = C1 + C2 and the resulting apparent relative permittivity is ε MEAS. They are given by the expressions: (eq. 10) C MEAS = ε 0 ε [( D / B ) + ( 1 D )] [ A 0 / t a ] (eq. 11) ε MEAS = C MEAS / ( ε 0 *[ A 0 / t a ]) = ε [( D / B ) + ( 1 D )] Where: ε = True relative permittivity of MUT (unitless) t a = Full thickness of sample (cm) = Electrode separation (cm) for contacting electrode method t b = Thickness of sample between air gaps (cm) A 0 = Area of sense electrode (cm 2 ) A 2 = Area of air pockets covered by sense electrode (cm 2 ) B = ε [( t b / t a ) * (ε 1)] D = A 2 / A 0 = ratio of area of air pockets to area of sample Version 0.9 AN 14-6

7 Discussion Using the model of Figure 6, equation 11 and the following parameters: ε = 3.0 (true relative permittivity of MUT) D = 0.75 (ratio air pocket area to sample area) t b / t a = 0.6 (ratio sample thickness between air pockets to sample thickness) a value of 2.0 was calculated for the expected measured relative permittivity. This value is of course incorrect and illustrates how a textured surface can reduce the measured relative permittivity from its true value. Conclusion A properly prepared sample is necessary when using a parallel plate test fixture. The sample surface must be smooth and flat to allow close contact of the parallel plate electrodes over their entire area. Any surface irregularity such as a texture which creates air gaps will result in inaccurate measurements. Version 0.9 AN 14-7

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