Classifications

• • G—PHYSICS
  • G01—MEASURING; TESTING
  • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
  • G01R15/00—Details of measuring arrangements of the types provided for in groups G01R17/00 - G01R29/00, G01R33/00 - G01R33/26 or G01R35/00
  • G01R15/14—Adaptations providing voltage or current isolation, e.g. for high-voltage or high-current networks
  • G01R15/24—Adaptations providing voltage or current isolation, e.g. for high-voltage or high-current networks using light-modulating devices
  • G01R15/247—Details of the circuitry or construction of devices covered by G01R15/241 - G01R15/246
• • G—PHYSICS
  • G01—MEASURING; TESTING
  • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
  • G01R1/00—Details of instruments or arrangements of the types included in groups G01R5/00 - G01R13/00 and G01R31/00
  • G01R1/02—General constructional details
  • G01R1/18—Screening arrangements against electric or magnetic fields, e.g. against earth's field
• • G—PHYSICS
  • G01—MEASURING; TESTING
  • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
  • G01R15/00—Details of measuring arrangements of the types provided for in groups G01R17/00 - G01R29/00, G01R33/00 - G01R33/26 or G01R35/00
  • G01R15/14—Adaptations providing voltage or current isolation, e.g. for high-voltage or high-current networks
  • G01R15/24—Adaptations providing voltage or current isolation, e.g. for high-voltage or high-current networks using light-modulating devices
  • G01R15/241—Adaptations providing voltage or current isolation, e.g. for high-voltage or high-current networks using light-modulating devices using electro-optical modulators, e.g. electro-absorption
  • G01R15/242—Adaptations providing voltage or current isolation, e.g. for high-voltage or high-current networks using light-modulating devices using electro-optical modulators, e.g. electro-absorption based on the Pockels effect, i.e. linear electro-optic effect
• • G—PHYSICS
  • G01—MEASURING; TESTING
  • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
  • G01R15/00—Details of measuring arrangements of the types provided for in groups G01R17/00 - G01R29/00, G01R33/00 - G01R33/26 or G01R35/00
  • G01R15/14—Adaptations providing voltage or current isolation, e.g. for high-voltage or high-current networks
• • G—PHYSICS
  • G01—MEASURING; TESTING
  • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
  • G01R29/00—Arrangements for measuring or indicating electric quantities not covered by groups G01R19/00 - G01R27/00
  • G01R29/12—Measuring electrostatic fields or voltage-potential

Definitions

• the present invention relates to a voltage sensor for measuring voltage and is particularly suitable for measuring high voltages such as on high-voltage power transmission lines.
• the list of known high-voltage voltage transducers includes inductive transformers, capacitive dividers or capacitive voltage transformers, and transducers using bulk-optic electric field sensors.
• the first two transducers suffer from bandwidth restrictions, expensive failure, extensive maintenance, heavy weight, and output variations. They also suffer from the need for substantial insulation that is both expensive and potentially hazardous to the environment (e.g oil and/or SF5 gas).
• Pockels cells are known devices that have been used for measuring voltage particularly high voltage, for example see US patent 5477134 issued to H. Hamada and US patent 5731579 issued to G.K. Woods.
• a preferred form of Pockels cell for use in the present invention is an integrated optics Pockels cell such as that described in US patent 5,029,273 issued July 2, 1991 to Jaeger the disclosure of which is incorporated herein by reference. Brief description of the present invention
• the present invention reduces significantly the deficiencies inherent to the existing high- voltage transducer technology.
• the simple structure of the invention removes the need for expensive and/or environmentally unfriendly insulation; may be made lighter, allowing for less expensive transportation, installation, and maintenance; and is compatible with existing standoff structures, allowing for easy construction.
• compact electric field sensor technology such as the integrated optics
• Pockels cell the invention offers wide bandwidth and easy interfacing with emerging digital technology.
• the invention relates to a method of measuring the value of voltage difference between two points to provide a measured value V of said voltage difference comprising measuring electric field at at least one location in space to provide a measured value of electric field E for each of said at least one location and using a mathematical combination of the value of electric field E for each of said at least one location, said combination being arranged and said at least one location being chosen so that for any given value of said voltage difference between said two points any practical disturbance in electric field influencing the measured value E does not significantly change the measured value N of said voltage difference.
• the present invention also relates to a method of measuring value of voltage difference between two points to provide a value V of said voltage difference comprising choosing at least two locations spaced distances x, from one of said two points, measuring and providing values of electric fields Ei at said at least two locations, applying weighting factors ⁇ ; to said measured electric fields E; at said corresponding positions spaced their respective distance Xi from said one point to obtain a value V based on
• V _ a t E
• n the number of electric field sensors and is at least 2
• Xi and cti are selected so that dE; representing any changes in E; measured at said at least two locations spaced their respective distance Xj from said one point, due to external disturbances, are compensated for so that they do not materially affect said value N, so that the value of
• Xi and ⁇ are obtained by a method selected from the group consisting of trial - and-error, mathematical or computer modeling.
• x, and ctj are determined using a quadrature method or an integration formula.
• the quadrature method is a Gaussian quadrature.
• the present invention relates to an apparatus for measuring voltage comprising an electrically isolating section between a pair of spaced conductors defining opposite ends of said isolating section between which voltage difference is to be measured, at least one electric field sensor which measures electric field at at least one location within said isolating section, said isolating section having a relative dielectric permittivity and geometry which provides sufficient screening of said at least one location from other electric field sources external to the isolating section so that said external electric field sources of practical strength do not materially change said at least one electric field measured at said at least one location.
• the present invention relates to an apparatus for measuring voltage comprising an electrically isolating section between a pair of spaced conductors defining opposite ends of said isolating section between which voltage difference is to be measured, at least one electric field sensor which measures electric field at at least one location within said isolating section, said isolating section having a relative dielectric permittivity and geometry which provides sufficient screening of said at least one location from other electric field sources of practical size external to the isolating section so that the error in voltage difference measured under presence of said external sources of electric field is less than 6%.
• the error in said voltage difference measured is less than 1%, more preferably less than 0.3%.
• the present invention relates to a method and apparatus for measuring voltage comprising an essentially electrically isolating section between a pair of spaced conductors between which voltage difference is to be measured. At least one electric field sensor is positioned within the isolating section.
• the isolating section has a permittivity and size sufficient to provide screening of the field sensor from other electric field sources external to the isolating section so that the other electric field sources do not materially affect the voltage measurement.
• the electric field sensor is an integrated optics Pockels cell.
• the relative dielectric permittivity of the section is greater than 2, more preferably greater than 20.
• the isolating section is a hollow isolating section in which the electric field sensor is mounted.
• the electric field sensors positioned in the isolating section in spaced relationship along a longitudinal axis of the isolating section between the two conductors.
• the number of electric field sensors in the isolating section is two, a first sensor positioned in said isolating section spaced from the one conductor by a distance of between 50 and 100% of distance L between the pair of conductors and a second electric field sensors positioned in the isolating section spaced from the one conductor by a distance of between 0 and 50% of distance L between the pair of conductors.
• the number of said electric field sensors in the isolating section is three, one positioned in said isolating section spaced from the one conductor a distance of between 70 and 100% of distance L between the conductors, another electric field sensor positioned in the isolating section spaced from the one conductor by a second distance of between 30 and 70% of distance L between the pair of conductors and yet another electric field sensor positioned in the isolating section spaced from the one conductor a third distance of between 0 and 30% of distance L between the pair of conductors.
• the number of sensors in the isolating section will not exceed 9.
• said electric field sensors collectively occupy less than 10%, more preferably less than 2% of the distance L between the conductors.
• FIG 2 illustrates a basic unperturbed Electric Field Sensor Voltage Transducer (EFS NT) structure of the present invention in front and side views as it may be installed in a high- voltage substation.
• Figure 3 shows three basic EFS NT structures each using three field sensors in a three- phase high- voltage transmission system.
• EFS NT Electric Field Sensor Voltage Transducer
• Figure 5 is similar to Figure 3 but further schematically illustrating an arrangement wherein a vertical semi-infinite ground plane is placed spaced from Phase A of a three-phase transmission system (by a distance of one meter in this example).
• Figure 6 is a plot illustrating the affect of relative permittivity ( ⁇ ) of the dielectric shielding on electric field distribution inside the isolating section.
• Figure 7 is a section along the line 7 - 7 of Figure 1. showing the sensor positioned on the axial center line of the dielectric column.
• Figure 8 is a section through the sensor (e.g. an integrated optics Pockels cell) along the line 8 - 8 of Figure 7
• Figure 9 is a side elevation of the sensor looking from the right in Figure 7.
• Figure 10 is a view similar to Figure 1 showing a modified version of the invention
• Figure 11 shows another embodiment of the invention showing the use of a solid rod (as opposed to the hollow column) to form the isolating section. Description of the preferred embodiments
• the invention as schematically shown in Figure 1 and 2 applied to a standoff 11 is formed by a column or isolating section 10 supported on a cylindrical metallic (current conducting) stand 12.
• Electric field sensors (EFS) 14 (three shown but only one is necessary and more may be used if desired) are mounted in the column 10 and are protected by the column 10 from the outside environment.
• the isolating section 10 is interposed between (in the illustrated arrangement of Figure 2, 3 and 5) a power line 18 and the metallic stand 12 connected to ground schematically indicated at 15 between which the voltage is to be measured.
• the isolating section 10 containing electric field sensors may be positioned between and connected to any two conductors between which the voltage is to be measured.
• the column 10 has been shown as a hollow cylinder with an outside diameter d 0 , an inside diameter d divide and a length L. Obviously if the column 10 is not hollow, the inside diameter dj is zero (0).
• the thickness d of the dielectric material between the sensor 14 and the ambient surroundings in effect defines the thickness of the dielectric material shielding the sensor 14 which coupled with the permittivity of the dielectric material defines the shielding and as discussed below with respect to Figure 6 influences the structure of the electric field.
• the isolating section 10 is formed by a laminate of a number of dielectric layers 145, 150, 155, 160, 165 and 170 (in the illustrated case, hollow cylindrical layers) that may have the same or different permittivities as indicated by the relative permittivity symbols ⁇ i, ⁇ 3 , 84, ⁇ 5 , ⁇ 6 , and ⁇ .
• the layers 145, 155 and 165 may be air.
• the number of layers and the permittivity of each layer or the combined permittivity of all the layers may be changed to suit the application, as desired.
• ⁇ i, and ⁇ 4 will be greater than ⁇ o, ⁇ 3 , and ⁇ 5 .
• the relative permittivity ⁇ 0 is the relative permittivity of the surrounding atmosphere and ⁇ 2 that of the material in which the sensor is encapsulated.
• outer layer 170 may be of non-uniform thickness demonstrating that the thickness of the isolating section 10 need not be uniform.
• One of the simplest ways to build the isolating section 10 is to stack a plurality of discrete axial portions (measured parallel to length L) some of which termed modules will incorporate the EFS(s) and simply fix the portions and modules together to construct the section 10 with the modules containing the EFS(s) in the desired locations along the length of the section 10 i.e. the section 10 could be made of portions i.e. slices (different thickness and/or hollow) with the module(s) being slices in selected locations along the axial length of the section 10.
• the length L for any installation depends on many factors the most important is safety and then accuracy. The length L should be held to a minimum for accuracy, however for safety it cannot be too short.
• the values e.g.
• the distance from high voltage (HN) line to ground) will be chosen so that the electric field anywhere inside, near, and outside the structure is not greater than the break-down strength (field) of the material(s) present at that location under any reasonable operating condition as provided by governing bodies such as the Institute of Electrical and Electronics Engineers (IEEE), the American National Standards Institute (ANSI), the International Electro-technical Commission (LEC), and/or other local and international standards, 1 e the length L will be set to meet the required local and international standards
• the dielectric material from which the section is constructed causes a structured elect ⁇ c field distribution in and around the column 10
• Measurement of the elect ⁇ c field by each of the at least one strategically placed elect ⁇ c field sensor 14 is delivered via lines schematically indicated at 63 (e g a pair of lines 63 for each sensor 14 (see figure 1)) to a suitable computing means 61 which may be m the form of a suitable chip or the like which as will be explained herein determines the voltage difference N between point b at one end (top 16) and point a at the opposite end of the column or section 10 (top of plate 30)
• the sensors 14 are strategically placed in the isolating section 10 to tend to minimize error in the measurement (determination) of the voltage difference
• the preferred placement of the sensor(s) can be determined by any suitable method for example by t ⁇ al-and-error but the quadrature method as will be described herein below is the preferred method of determining the placement of the sensors
• the end result of proper placement and selected compensation for distortions in the electric field distribution is that for the present invention typical distortions in the elect ⁇ c field distribution do not significantly affect the reported voltage measurement value N
• Typical causes for distortions or perturbations include the presence of other conductors at other potentials, e g , in a high-voltage three-phase system, the presence of other two phases can be considered a cause of a perturbation
• These distortions also include the effects of pollution on and around said isolating section or column 10 and its sheds, if any, and other nearby structures, conducting or non-conducting, mobile or stationary
• these other structures are usually located at a distance determined using various relevant LEEE, ANSI, EEC, and/or other standards or guidelines on each voltage class.
• NTs are typically required to have errors less than 0.3%; so, in such a case where a NT is to be used as a part of a revenue metering system, the statement “do not significantly affect the reported voltage measurement value” means that "the reported voltage measurement value is within ⁇ 0.3% of the actual voltage.” Obviously, for other applications, or other standards, the terms “significantly” or “sufficiently” or “materially” correspond to other numerical values.
• Another example is a NT that has to meet class 3P relaying standard according to EEC standard 60044-2 (1997-02); basically, the NT is allowed to have ⁇ 3% error in measuring the voltage magnitude and ⁇ 2° phase angle error in measuring the phase of the power frequency voltage, typically a 60Hz or 50Hz signal, (of course there are many other requirements in the standard and are out of the scope of this brief example); so, in this case, the statement “do not significantly affect the reported voltage measurement value” means "the reported voltage measured is within ⁇ 3% of the actual voltage and the phase angle is within ⁇ 2° of the actual phase angle;” in other words, it means that "the NE meets all class 3P accuracy requirements according to EEC standard 60044-2 (1997-02).” In general, "significant” change or error refers to a change or error that is not acceptable to the user of the equipment as far as the relevant application s) or case(s) or requirement(s) is concerned.
• EFS electric field sensors
• the total number of EFSs positioned in the isolating section 10 may, for example, be set as high as 200, but generally will not exceed 9 and normally will be less than 6.
• EFS electric field sensor
• Other locations are also possible depending on the accuracy required, but for higher accuracy the above-defined location is preferred particularly for installation in 3-phase HV substations.
• a first of the two sensors positioned in said isolating section is normally spaced from one end of the isolating section 10 i.e. from plate 30 (point a) between 50 and 100% of distance L between the bottom plate 30 (point a) and top 16 (point b) and a second of the two electric field sensors 14 positioned in the isolating section 10 is normally spaced from plate 30
• the number of said electric field sensors 14 in the isolating section 10 is three, one positioned in said isolating section 10 spaced from plate 30 (point a) by a distance xi of between 70 and 100% of distance L between top 16 and plate 30, another sensor 14 positioned in said isolating section 10 spaced from plate 30 a distance x 2 of between 30 and 70% of distance L and the third of the three electric field sensors 14 positioned in said isolating section 10 spaced from plate 30 by a distance x 3 of between 0 and 30% of distance L.
• EFS electric field sensors
• IO EFS Integrated Optics Electric Field Sensors
• IOPC Integrated Optics Pockels Cell
• Mach-Zehnder type field sensors such as those with domain inversion in one branch (see for example N.A.F. Jaeger and L. Huang "Push-Pull Integrated- optics Mach-Zehnder Interferometer with Domain Inversion in One Branch" Optics Letters, vol. 20, no. 3, pp. 288-290, February 1995 or a sensor as described in US patent 5,267,336 issued November 30, 1993 to Sriram et al. may be used, however these sensors are not as effective as the IOPC EFS referred to above.
• EFS Error-optic electric field sensor with piezoelectric body sensor
• the sensor holder 24 has a major diameter d g substantially equal to the inside diameter dj of the column 10 assuming the isolating column 10 is a hollow column so that the sensor holder 24 may easily be secured in position mechanically or by a suitable adhesive.
• IOPC 32 is indicated as l pc and is significantly shorter than the diameter d g and the IOPC is centered on the longitudinal axis 34 of the column 10 (see Figure 9).
• the thickness of the IOPC measured along the axis 34 i.e. in the x direction is indicated as thickness t.
• the thickness t is short relative to the length L of the isolating section 10.
• the IOPC 32 is positioned so that it measures the electric field vector parallel to the axis 34 and preferably is centered on the axial centerline 34 and extends substantially perpendicular to the axis 34.
• a suitable point field sensor is any sensor that measures electric field in a region between two points that are very close together as compared to the distance between the two points between which the potential/voltage is to be measured.
• t 1 millimeter (mm) versus a distance between the points a and b i.e. length L of 4,000 mm is satisfactory.
• the ratio of L/t is typically larger than 20, but this is not an absolute minimum.
• the collective length of the voltage sensor elements will not exceed 10% of the length L and preferably will not exceed 2% of length L between the conductors 16 and 30.
• the sensor preferably should have the additional property that it does not significantly disturb the electric field distribution by its presence, or if it does, it should do so in a well known manner).
• DS dielectric shielding
• QM quadrature method
• the electric field near the center of the isolating column increases as the permittivity of the high-permittivity column 10 increases, and, therefore, the IOPCs or other field sensors near the center of the column require less sensitivity, i.e., there exists enough signal (electric field) amplitude to be measured by the sensor.
• the DS may be minimal depending on the accuracy required, i.e., the relative permittivity ⁇ i of the high-permittivity material can be as low as 1.
• the high-permittivity material can be air or free-space (not observable), and the device can still be effective (especially if many IOPC sensors are used inside the column 10).
• the high permittivity material may need to have very high relative permittivity value ⁇ i such as several 1000s (e.g., barium titanate) for high accuracy and very strong shielding.
• very high relative permittivity value ⁇ i such as several 1000s (e.g., barium titanate) for high accuracy and very strong shielding.
• relative permittivity values from 2 to 100, preferably greater than 20.
• the QM is a mathematical technique that allows for significant improvement in accuracy for a given number of EFSs. There are, possibly, many other mathematical techniques that are useful. Some are in essence the same as the QM, but have different names. Some are different in their methodology but result in the same thing: improvement in accuracy.
• the purpose of using a QM is to determine optimal or nearly optimal positions for the EFS(s) in isolating section 10, and to determine an algorithm for combining the measured values of the electric field(s) at the locations of the EFS(s) so that the voltage measurement obtained using said combination algorithm is sufficiently and/or highly accurate, regardless of the presence of certain external influences disturbing the electric field distribution in the isolating section 10.
• the QM can improve the accuracy of the measurement, and the result is to a degree DS dependent as DS affects the w, (weighting factor) and the x, distance between the lower plate 30 and the specific sensor being positioned (See equation 5) due to the effect of DS on the electric field distribution
• QM is preferably used to determine the optimal locations for the EFSs to be mounted inside the column 10, and in this way QM affects the physical structure of column (including EFSs) 10.
• the system may be used to measure any reasonable voltage e.g. voltages from 0.001 volts to 1,000,000 volts and higher.
• the economic (and safety) benefits, i.e. practical range of usage, will be at higher voltages in the several thousand-volt ranges.
• IOPCs could be used as the sensors by being placed somewhere in a standoff structure 11 attached to the transmission line.
• a standoff 11 is as above described made up of, from top to bottom, a conducting cap 16, an insulating column or isolating section 10, and a conducting bottom 30 and typically a conducting electrical ground column or stand 12.
• the output of each IOPC is used to deduce the voltage on the transmission line. Since the IOPC output depends on the electric field, as described previously, the standoff structures must be modified in such a way so as to reduce the mutual coupling effects of other HV transmission lines and any other structures in the vicinity.
• the length L of the column, the relative permittivity ⁇ i of the shielding material, and the thickness d of the dielectric shield are all related and their choice depends on the accuracy and safety requirements for the voltage sensor.
• the proper design procedure involves modeling the voltage sensor column, considering the restrictions provided under various relevant standards (such as those mentioned above), considering the accuracy requirements, and considering the availability of high-permittivity materials, their permittivity, weight, and cost to provide an economical, accurate (for the intended application), and safe voltage sensor.
• Quadrature Method (QM)
• EFSVT EFS Voltage Transformer
• an EFSVT design consists of one or more EFSs, depending on measurement accuracy requirements, and relies on a suitable method, e.g. a Gaussian guadrature method, to determine the positions x, of (distance from plate 30) and weights w, or ⁇ , on the outputs of the
• EFSs which are preferably positioned along or close to the axial centerline of the column 10
• the positioning x, and weighting w, are optimal in the context of polynomial approximations and both theory and design procedures are described.
• path of integration r aD is any path in space from a to b.
• E x is the component of the electric field E parallel to the x-axis and is a function of x.
• a is taken to be a point on conducting plate 30, electrode A, and b is taken to be a point on the conducting plate 16, electrode B (in Figure 2 the points a and b where a is connected to ground column 12 and b is connected to high- voltage transmission line 18).
• the unperturbed system is arbitrarily defined as any practical system preferably including geometry and material properties of the voltage sensor.
• the electrodes produce a unique electric field, the unperturbed electric field E P(x, y, z).
• the x-component of this electric field along the x-axis is written as
• a perturbed system is defined as any system that is a variation of the unperturbed system. Examples of a variation include the introduction of other voltage sources in space and inhomogeneities in the medium. The resulting electric field for such a system is the perturbed electric field E?(x,y,z). The x-component of this electric field along the x-axis is written as
• the positions of these sensors could be determined and the outputs of these sensors could be weighted as dictated by the quadrature method to obtain a very good approximation of the voltage between the cap 16 and the bottom plate 30. This is accomplished in the following two-step procedure.
• Step 1 Obtaining E" x p (x) using a model of the unperturbed system
• the modeling consists of solving Poisson's equation for the electric potential ⁇ (x,y,z) from which the electric field E(x,y,z) can be computed by use of the definition
• Step 2 Determining the abscissas xj and weights i using quadrature method with E u x p (x)
• the basic unperturbed EFSVT structure is shown in Figure 2.
• Step 2 E/" (x) is then used to compute the optimal positions xi and weights wi ( 0 r ⁇ ;)-
• Test simulations can now be set up to measure the performance of the EFSVT designs in a perturbed environment.
• the NT is used to measure the line-to-ground voltage on each of three buses constituting a three-phase system i.e. the perturbed system.
• the voltages applied to the lines are sinusoidal with a frequency of 60 Hertz, but Phase A leads Phase B by 120°, and Phase B leads Phase C by 120° resulting in a balanced system as shown in Figure 3.
• Phase B has been arbitrarily chosen as the reference for the phase information given in this example (see Tables 1 to 6).
• the voltage on a particular line can be represented by its amplitude and phase as follows:
• V t + ⁇ ba ) b,. V J( ⁇ a ba (6)
• ⁇ is the angular frequency in radians per second
• Vbal is the voltage amplitude or magnitude
• ⁇ ba is the phase
• each component of the electric field E is sinusoidal if voltage sources are sinusoidal, and E can be represented as
• HN line and the output of the respective NT will be expressed in terms of an amplitude error and a phase error. Though in general the output of the NT gives an instantaneous reading of the voltage continuously in time. Substituting (6) and (7) into (2) gives
• Ii and Io are integrals that represent the in-phase (real) and out-of-phase (imaginary) parts of the voltage Nba- The magnitude and phase of Nba are then given by
• each EFS is weighted and then superimposed (double-EFS and triple-EFS designs) resulting in a signal that is linearly proportional to the voltage, and subsequently the magnitude and phase of the voltage can be computed.
• each of the integrals Ii and I 0 is being approximated by the Gaussian quadrature developed earlier.
• equations for the quadrature approximations to the integrals for the case of a double-EFS VT design are expressions for the quadrature approximations to the integrals for the case of a double-EFS VT design:
• the calibration constants above are constants of proportionality and are determined based on the actual voltage that is to be measured on the line; in these examples here, we are looking for normalized or percentage errors and we are not concerned with the actual voltage class. Also, the percentage errors given are relative to (percentage of) the amplitude of the
• an EFS e.g. an IOPC
• the EFS may have to be mounted on a flange or other small section of either insulating or conducting material, which in turn can then be mounted inside the dielectric column during construction.
• Such a structure may significantly alter the electric field that would otherwise exist in the column, thereby making the problem of finding the Gaussian quadrature points a more complex one. In this case, (5) becomes
• shielding due to high dielectric constants reduces the gradient in the vertical electric field component E x " p (x) .
• increased shielding could be used to reduce the sensitivity of the accuracy of the quadrature to an error in positioning.
• calibration would also be employed to minimize measurement error.
• increased shielding reduces the effects of any external influences on the field inside the dielectric column. In terms of the discussion in this report, this means that increased shielding would reduce the nonlinearity in the variation p(x) due to external influences. In other words, a lower degree polynomial could better approximate p(x), and the quadrature would be more accurate.

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