Effective permittivity model of insulating presspaper is built on the basis of the microstructure of the material. Due to the essentially layered structure in *z*-direction of presspaper, air voids inside the mixture can be treated as right prismatic inclusions. Analytical formula for the prediction of the effective permittivity of insulating presspaper is derived. Interestingly, the derived formula equals to the mixing equation applied for dielectrics in series. Numerical simulation was used to validate the analytical results by considering the air voids as cubical inclusions. Results show a good agreement between the analytically and numerically calculated effective permittivity values. Furthermore, dielectric permittivity results of commercial kraft paper and laboratory-made presspaper at 50 Hz were measured and compared with modeled data. It turns out that the deduced results give a good accuracy for the effective permittivity determination.

Cellulose materials including presspaper and pressboard have been used together with mineral oil in power transformers for more than 70 years.^{1} The mismatch of relative permittivity between insulating presspaper and mineral oil, which will lead to an electric field distortion inside the composite system, is one of the main drawbacks for the combination.^{2} Therefore, a better understanding of the effective permittivity of presspaper, which are composed of cellulose fibers and air voids, will be of theoretical importance and practical significance to enhance the electrical performance of oil-paper composite insulation.

Studies on the effective permittivity of a two-component mixture started as early as the theory of electromagnetism.^{3} First, laminated structures were investigated. Equations (1) and (2) are applied for dielectrics in parallel and in series, respectively.

Where ε_{eff} represents the effective permittivity of the mixture; ε_{f} and ε_{a} are permittivity values of inclusion phase and environment phase; *θ*_{f} is the volume ratio for the inclusion phase. Wiener gives a generalized formula for the laminar mixtures,^{4}

When *μ* = ∞ and *μ* = 0, Eq. (1) and Eq. (2) are obtained. For the case of spherical inclusions randomly embedded in the isotropic dielectric matrix, the Maxwell-Garnett mixing rule is widely used,^{5}

Another form for Eq. (4) is

Equation (5) is also called the Rayleigh mixing formula. In addition to the Maxwell-Garnett formula, the Bruggeman formula is also a commonly used mixing rule.^{5}

Where *ν* is a dimensionless parameter. When *ν* = 0 and *v* = 2, Eq. (4) and Eq. (6) are recovered. For powder and granular materials, power-law models are generally used. The effective permittivity of the mixture is given as^{7}

Where *β* is also a dimensionless parameter. When *β* = 1/3, the Looyenga equation is obtained. When *β**i*s close to zero, the Lichtenecker formula is got by expanding the terms with exponents into an infinite series and neglecting the terms of higher order.^{4}

By combining the simple series model and parallel model, engineers in EHV-Weidmann put forward a famous formula to predict the effective permittivity of oil-paper composite insulation.^{8}

Where cellulose fiber is regarded as environment phase, and mineral oil is treated as inclusion phase. Although there are plenty of formulas for dielectric mixtures, whether or not the above models are suitable to predict the effective permittivity of insulating presspaper is unknown. Even though there exists a model that works well, the reason behind that is also unknown.

In this letter, we present that the air voids in presspaper can be treated as right prismatic inclusions. Based on this assumption, the analytical mixing formula, which turns out to be the simple series equation, is derived. We also demonstrate a convenient and fast numerical simulation method to validate the analytical results by considering the air voids as cubical inclusions. Furthermore, effective permittivity results of commercial kraft paper and laboratory-made presspaper are measured and compared with modeled permittivity values.

The effective permittivity of a mixture is determined by the permittivity of one of the constituents and the topology of the mixture.^{9} Therefore, to model the macroscopic permittivity of presspaper, we have to first get insight into its microstructure.^{6} Porous structure of presspaper can be divided into horizontal structure (in the plane of the sheet) and vertical structure. If we have a look at the cross-section of presspaper, namely the vertical structure, we will find that nearly no fiber traverses the thickness of the sheet from bottom to top although the fiber length (2 mm to 6 mm for most softwood fiber^{1}) can be much larger than the paper thickness (e.g. 0.13 mm). Further studies conclude that presspaper has an essentially layered structure due to the nature of the sheet-forming mechanism.^{10} Based on this fact, the air voids in presspaper can be assumed as right prismatic inclusions. Figure 1(a) shows the scanning electron microscope image of insulating presspaper made from unbleached softwood pulp. We can see that width of softwood fiber is about 40 μm. As cell wall thickness of softwood fiber is about 6 μm,^{11} the layer thickness in *z* direction is about 12 μm. Thus we can use Fig. 1(b) to simulate the layered structure shown in Fig. 1(a). Note that cubes are used in Fig. 1(b) just as an example. We can definitely use other right prisms.

As cellulose fibers and air voids can be considered as linear, homogeneous, and isotropic dielectrics, we can then deduce the mixing rule applied for presspaper by assuming that air voids are randomly distributed throughout presspaper. First, let us define effective permittivity ε_{eff} of presspaper as the relation between the volume average electric field and flux density

Then we can rewrite the average field and flux density by weighing the fields with corresponding volume fractions

Where ε_{air} and ε_{f} are the relative permittivity values of air voids and cellulose fibers; *θ*_{air } is the ratio of void volume to total volume of presspaper. As **E**_{air} and **E**_{f} can be assumed as constants, effective permittivity ε_{eff} is obtained.

Where *A* is the ratio of **E**_{air} to **E**_{f}. Note that presspaper has a layered structure, and air voids can be treated as right prismatic inclusions.

It is easy to derive that

Equation (17) is the analytical formula for the calculation of effective permittivity of presspaper. Interestingly, it is another form of Eq. (2), which means we can just use the simple series model to predict the macroscopic permittivity value of presspaper. However, the difference between the real structure of presspaper and the laminated structure corresponding to Eq. (2) is dramatic.^{9} To validate the analytical results, we demonstrate a numerical approach to calculate the effective permittivity of presspaper.

As shown in Fig. 1, we can use one layer of cubes to model a single layer structure in presspaper. Therefore, multiply layers of cubes can be used to simulate the real structure of presspaper. Considering each cube as a capacitor, then we can obtain the effective permittivity by calculating the total capacitance of the capacitor network.^{12} Suppose we use *m* × *m* × *n* cubes to model a presspaper sample. Capacitance of one of the cubes represent air voids is defined as

Where *a* is the edge length of the cube; ε_{0} is the permittivity of the vacuum; ε_{air} is the relative permittivity of air void. Capacitance of one of the cubes represent cellulose fibers is expressed as

Where ε_{f} is the relative permittivity of cellulose fiber. then

*C*_{total} is the equivalent capacitance of the network composed of *m* × *m* × *n* capacitors. As air voids are assumed to be randomly distributed in presspaper, porosity of the presspaper sample represents the probability of a particular cube being occupied by air void. For a given small cube, a random number between 0 and 1 is generated. If the value is lower than the porosity, the cube is supposed to be air void, and the color is white. Otherwise, the cube is assumed to be cellulose fiber, and the color is red. Figure 2 demonstrates a numerical model composed of 15 × 15 × 10 cubes to represent the topology of a presspaper sample with a porosity of 35%.

Before applying the model above to calculate effective permittivity of presspaper, we need to determine the edge length of the cube. Since most voids have a diameter ranging from 0.05 μm to 2.5 μm, air voids in presspaper are assumed to be entirely of one size.^{10} To balance the simulation time and accuracy, we consider 10 μm as the edge length of the cube. Two types of presspaper are simulated, each of them has a length of 20 mm and a width of 20 mm. Thicknesses of them are 130 μm and 400 μm. Relative permittivity and density of cellulose fiber are 6.5 (at 50 Hz) and 1.53 g/cm^{3}, respectively.^{4,13} Those corresponding to air void are assumed to be 1 and 0 g/cm^{3}. Figure 3 shows the comparison of the numerically calculated permittivity values and the analytical solutions for presspaper with a bulk density from 0.7 g/cm^{3} to 1.2 g/cm^{3}. It can be seen that the numerical simulation results are in good agreement with the analytical values. This proves that the above derived analytical formula is effective. With the increase of sample thickness, the difference between numerical results and analytical values becomes smaller. If we reduce the edge length of the small cube used in the simulation, the same trend will also happen. As analytical formula shown in Eq. (17) is the lower bound for the predication of effective permittivity,^{14} numerical results is slightly higher than analytical solutions.

To further confirm the accuracy of the modeled effective permittivity values, relative permittivity results at 50 Hz of commercial kraft paper and laboratory-made presspaper were measured. The tests were performed with a Novocontrol broadband dielectric spectrometer.^{15} Magnitude of the applied sinusoidal voltage was 1 V. Prior to the measurements, test samples were thermally treated at 105° C for more than 24 hours. Figure 4 presents the modeled and measured effective permittivity values of presspaper with a bulk density from 0.7 g/cm^{3} to 1.2 g/cm^{3}. We can see that the mixing formula deduced in this study and the proposed numerical method give a good prediction of the effective permittivity of presspaper. As a comparison, results of other commonly used mixing formulas are also presented in Fig. 4. Obviously, they overestimate the effective permittivity values. The difference between the calculated and measured effective permittivity values is mainly attributed to the variation of local thickness and bulk density of presspaper^{16} and the residual moisture in the composite.^{17,18}

In summary, analytical mixing formula and numerical approach are proposed on the basis of the layered microstructure of presspaper in this work. The model can provide a better understanding and a good prediction of the effective permittivity of insulating presspaper, which will contribute to the manufacture of low-permittivity presspaper and the design of ultra-high voltage power transformer.

This work was supported by the Science and Technology Project of China Southern Power Grid (No. KY2014-2-0016).