Determination of Averaged Field Characteristics Using a Graphical-Analytical Method of Flow Path Variation
Abstract Generated abstract
This paper describes the use of a graphical analytical method of flow path variation for estimating averaged characteristics of electric fields, including conductance, resistance, capacitance, and field nonuniformity. The method divides field flux into tubes whose boundaries are varied to maximize total conductance, with individual tubes approximated by uniform, cylindrical, or spherical model fields. From the tube conductance and cross-sectional area, the paper derives expressions for averaged field strength along a tube, the maximum averaged field strength at the narrowest section of the shortest tube, and a corresponding nonuniformity coefficient. It concludes that although these averaged characteristics differ from point values, they can provide useful approximations, especially for smooth electrode geometries.
Full Text
PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE NAMED AFTER S. M. KIROV,
ORDER OF THE OCTOBER REVOLUTION AND ORDER OF THE RED BANNER OF LABOR
Vol. 227 1972
DETERMINATION OF AVERAGED FIELD CHARACTERISTICS USING THE GRAPH-ANALYTICAL METHOD OF FLOW PATH VARIATION
P. P. Galinsky
(Presented by the scientific seminar of the Department of Theoretical Foundations of Electrical Engineering
and the laboratory of rotating ESG of the YaF Research Institute)
The graph-analytical method of flow path variation is intended for determining integral field characteristics—conductances, resistances, and capacitances.
For the exact determination of the differential characteristics of a field (electric-field intensities, current densities, electric displacement), in the general case it is necessary to know analytical expressions for the potential as a function of the coordinates or to have an exact field pattern obtained on a model in an electrolytic bath. Nevertheless, the graph-analytical method of flow path variation indicated above makes it possible to find approximate differential field characteristics and thereby to obtain some idea of the nature of the field distribution in the volume under consideration.
In this method, the flux of the field vector is divided into tubes, the beginnings of which are distributed arbitrarily over the area of one of the electrodes and thereafter remain unchanged. Subsequently, by varying the position of the ends of the tubes, the flux is redistributed in such a way that the total conductance is greatest. The division of the flux into tubes is performed so that the field inside individual tubes is close in configuration to one of the model fields: uniform, cylindrical, or spherical.
In practice, the following quantities are used to characterize the field distribution over the volume:
the average field intensity of the gap
\[ E_{\mathrm{av}}=\frac{U}{l}, \]
where \(U\) is the potential difference between the electrodes,
\(l\) is the distance between the electrodes,
the field nonuniformity coefficient
\[ f=\frac{E_{\max}}{E_{\mathrm{av}}}, \]
which is the ratio of the maximum field intensity to the average.
In practice, the reciprocal quantity is also used, called the insulation utilization coefficient,
\[ \eta=\frac{E_{\mathrm{av}}}{E_{\max}}. \]
Knowing the indicated characteristics, one can determine the maximum field strength
\[ E_{\max}=\frac{U}{l}\cdot f=\frac{U}{l}\cdot \frac{1}{\eta} \]
and the breakdown field strength
\[ E_{\text{br}}=\frac{U_{\text{br}}}{l}\cdot f=\frac{U_{\text{br}}}{l}\cdot \frac{1}{\eta}, \]
where \(U_{\text{br}}\) is the voltage corresponding to breakdown of the dielectric.
For approximate calculations, the values of the average and maximum electric-field strengths can be determined approximately by the graph-analytical method of flow-path variation. The method makes it possible to determine the field strength averaged over the cross section of a tube; moreover, the greatest field strength will evidently be in the shortest tube, in its narrowest part.
As is known, the differential resistance over an element of length of a tube of current-density flow in a conducting medium is equal to
\[ dR=\frac{dx}{\gamma S(x)}, \]
where \(dx\) is an element of the length of the mean field line,
\(\gamma\) is the specific conductivity of the medium,
\(S(x)\) is the cross-sectional area of the tube.
The voltage over an element of length \(dx\) of the tube is
\[ dU=I_{\text{tr}}\,dR, \]
where \(I_{\text{tr}}\) is the current inside the tube,
\[ I_{\text{tr}}=U G_{\text{tr}}, \]
\(U\) is the voltage between the ends of the tube,
\(G_{\text{tr}}\) is the conductance of the tube, determined in the method of flow-path variation after the boundaries of the tube have been established.
The field strength averaged over the cross section of the tube is
\[ E(x)=\frac{dU}{dx}=\frac{I_{\text{tr}}\,dR}{dx}=\frac{U}{\gamma}\cdot \frac{G_{\text{tr}}}{S(x)}. \]
The maximum averaged field strength is determined from this formula for the shortest tube at the location of its minimum cross section:
\[ E'_{\max}=\frac{U}{\gamma}\cdot \frac{G_{\text{tr}}}{S_{\min}}. \]
The minimum field strength, conversely, corresponds to the maximum cross section of the tube.
The average field strength in this method can be determined by the relation
\[ E_{\text{av}}=\frac{U}{l_{\text{av}}}, \]
where \(l_{\text{av}}\) is the length of the mean field line of the tube.
After this it is not difficult to calculate the field nonuniformity coefficient corresponding to the maximum field strength averaged over the tube cross section:
\[ f'=\frac{E'_{\max}}{E_{\text{av}}}=\frac{G_{\text{tr}}}{\gamma}\cdot \frac{l_{\text{av}}}{S_{\min}}. \]
Although this characteristic, averaged over the tube cross section, differs from the point characteristic, for smooth electrode shapes both characteristics are close to one another.