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Building heat load contributions from medium and low voltage switchgear--part I: solid rectangular bus bar heat losses.

INTRODUCTION

Accurate prediction of building heat gains relies on accurate data and/or equipment models. For three decades, the paper by Rubin (1979) has served as a primary tool for estimating building heat gain caused by electrical distribution equipment. Owing to a built-in conservatism, heat gain estimations based on Rubin's work exceeded the heat gains occurring in practice. White, Pahwa, and Cruz (2004, 2004a) provided new data and procedures for better estimation of the heat gain. The publications by White et al. stemmed from the work performed in ASHRAE RP 1104 and represented the first step in determining a better means of heat gain prediction.

Part of the effort in RP 1104 involved spreadsheet models of low and medium voltage switchgear. The data used in the switchgear models were taken from unverified manufacturer heat loss figures found in catalogues and displayed on websites. One of the goals of ASHRAE RP 1395 is to provide verification of that published manufacturer data used in the switchgear models. This two-part paper describes an analytical and experimental approach taken to develop the required verification.

The first of these two papers describes an analytic approach taken to develop a loss model of the electrical bus. Not only is the bus important in estimating heat losses occurring in switchgear, it also plays a significant role in heat loss production occurring in other equipment such as motor control centers and panelboards. The bus is the conduit for power transfer within distribution equipment as well as between equipment pieces. As a result, accurate bus heat losses require an accurate model for prediction. It is well to spend time discussing the bus model because it plays a major roll in electrical distribution equipment heat losses. Parts of the calculation will be verified by comparing basic results with those obtained experimentally. By comparing analytically produced heat loss values to both measured data and data obtained through other analytical means, the bus loss model will be verified.

RP 1104 provided information on medium and low voltage circuit breaker heat losses. Some of this information stemmed from measurements and some from manufacturer published data. In the second paper, circuit breaker heat losses determined by both measurements and manufacturer data are used with the model to be presented in order to predict the switchgear heat loss. In some switchgear applications, fused switches are employed. The bus and breaker (fused switch) are the two leading heat-producing components of switchgear. There is a collection of special equipment employed in switchgear for tasks such as metering and climate adjustment having loss values that are much smaller than those associated with buses and breakers. The heat losses associated with the special equipment are included in the switchgear model. Another topic of the second paper is the general construction techniques of switchgear. The second paper concludes with examples of the spreadsheet use.

Types of Buses

Busways and bus bars are used to transmit large electrical currents. Although the bus conductors can consist of flexible cables, buses usually consist of copper or aluminum bars or tubes. The busway usually houses a three phase supply which is in contrast to cables and cable trays where several three phase circuits could be laid side by side. There are three types of bus configurations, all found in industrial plants. The first configuration is the non-segregated phase bus where all conductors are enclosed in a common structure with no barriers between the phases. If the size of the enclosing structure is allowed to grow and the bus is not symmetrically placed inside the structure, then this configuration approximates the bus inside electrical equipment where the bus might be passing close to a sheet metal wall. The next configuration is the segregated phase bus where all conductors are enclosed by a common structure in addition to barriers that are placed between the phase conductors. The final category is the isolated-phase bus where each conductor is surrounded by an electrically grounded metal housing that is separate from the other phases. Figure 1 illustrates the different configurations.

[FIGURE 1 OMITTED]

Instances of isolated phase buses include factories and power plants, however inside electrical equipment, bus bars are usually bare or have a thin coating of electrical insulation. The losses of isolated phase buses are well documented in the standard IEEE C37.23-2003. As stated earlier, the non-segregated bus best approximates the bus configuration inside electrical equipment for the purpose of determining heat losses.

Ohmic Heating Loss Mechanisms

The discussion of heat losses in electrical power conductors centers on losses in either copper or aluminum materials. Because electrical conductors consist of nonmagnetic material, magnetic hysteresis is excluded from consideration as a loss mechanism. The discussion that follows divides the heat generation into three parts. The first part is skin effect which increases the electrical resistance of the conductor. The second part is proximity effect which can both decrease and increase the heat losses. The final part is stray loss which involves currents being induced in surrounding structures. Each of these loss mechanisms will be described in the following text.

Ohmic heat dissipation in a conductor carrying a DC current is well understood, however when alternating current flows, the changing magnetic field created by the alternating current induces voltages in the conducting material that cause other currents to also flow in the conductor. The net result is that the current crowds to the edges of the conductor while little current flows in the center of the conductor. Recall that the resistance of a conductor is given by

R = [rho][L/A] (1)

where

R = the electrical resistance in ohm,

[rho] = the electrical resistivity of the material in ohm-m,

L = the length of the conductor in m, and

A = the cross-sectional area in [m.sup.2].

Because the current flows through an effectively smaller area, equation (1) shows that the resistance will be greater. This larger resistance provides for proportionally larger heat losses. This phenomenon is called skin effect and depending upon the size of the conductor its contribution to the heat losses is to increase the electrical resistance of the conductor, thus providing greater heat loss than that caused by a DC current of the same magnitude as the RMS (root mean square) AC current.

A current carrying conductor sets up a magnetic field. When this current is an alternating current, the current created magnetic field is able to induce voltages in surrounding metallic objects that can cause currents to flow. When a surrounding metallic object is another conductor, then the induced voltage will cause a current to flow and there is ohmic heating associated with that induced current. If the nearby conductor is carrying its own current, then the induced current can alter the current distribution over the cross-section. This rearrangement of current can both increase and decrease the total ohmic losses. Likewise, the alternating current in the nearby conductor also sets up a magnetic field that can induce currents in the original conductor and redistribute the current there. The influence the conductors have on the current distributions in adjacent conductors is called "proximity effect." The result of proximity effect can change the ohmic heating loss. Whether the change is a greater, a smaller, or an unvarying amount depends on the conductor shape, current, and relative placement of the conductors.

In some published literature regarding proximity effect, such as IEC 60278-2002 (also, see the references cited in Chapter 1of this IEC standard), the proximity effect is separated from skin effect. It can be argued that owing to linearity the total magnetic field of the collection of conductors is the superposition of the individual magnetic fields, which is true. However, the losses do not superimpose because the losses depend upon the square of the current density integrated over the cross-sectional area of the conductor. In order to develop an estimate of the ohmic losses, we need to consider all conductors at the same time. The bus model presented later in this paper does take into account all current carrying conductors at the same time, thus, skin and proximity effect are considered together.

As stated earlier, voltages (and thus currents) can be induced in surrounding metal structures in the vicinity of alternating current carrying conductors. The surrounding structures could be beams for switchgear cabinets and conductor supports and metallic cabinet walls. The currents induced in these structures cause heating and the ohmic heating associated with these structures is called "stray loss." Another term used to describe this type of heating is "enclosure loss." The prediction of stray loss values is complicated owing to the geometry of surrounding structures. The most common attribute of the geometry is that the bus bars parallel a conducting plane over most of switchgear enclosed conductor length.

The model of bus bar heating losses to be presented accounts for each of the three ohmic loss mechanisms just described.

Single Phase Model

The succeeding development shows the model used to predict the eddy current augmented power production in a single phase, rectangular bus bar. The model to be presented is a numerical one that is sufficiently simple to allow spreadsheet implementation. Because a single conductor is being considered, skin effect provides the only means of increasing the resistive heating relative to that occurring for DC current. The calculated results will be compared to measured values as a test of the validity of the model.

During the early part of the 20th century, the increase in ohmic losses caused by skin effect was investigated in hollow cylindrical, hollow square, and solid rectangular conductors. The analysis of hollow circular and square conductors covered the range from very thin tubes up to and including solid conductors. The IEEE Standard C37.23-2003 covers the cylindrical and square conductors very well. Notable contributions to this investigation of single phase losses are Dwight (1947) and the references cited in Dwight's paper. Figure 1 of Dwight's paper consisted of a compilation of all known measurement results of the ratio of AC to DC resistance for nonmagnetic solid rectangular conductors as a function of frequency, resistivity, and width to height ratio. Dwight was able to reduce the dependence of the resistance ratio to two parameters being the width to height ratio and the dimensionless quantity P given by

P = [square root of [[8[pi]fab]/[rho]]] (2)

where

P = dimensionless parameter,

f = the alternating current frequency in Hz,

a = the conductor height in meters,

b = the conductor width in meters, and

[rho] = the conductor electrical resistivity in ohm-m.

The use of P with absolute (cgs) units dates back to the earlier work of Dwight (1918). The width to height ratio (b/a) ranged from unity for square conductors to several hundred for wide flat conducting straps. Arnold (1938) has adequately treated the problem of square conductors producing an analytical formula that forms the basis for the results presented in IEEE Standard C37.23-2003. In general, for rectangular conductors there are three regions of Dwight's Figure 1 of interest being the low, mid, and high value regions for P. Dwight (1918) treated the low P region where the width to height ratio played only a minor role in determining the AC to DC resistance ratio. Thus, the analysis results, which consisted of an analytical formula, were widely applicable. Cockcroft (1929) treated the high P region of the curve where the skin effect limits the current to a thin layer at the conductor surface. Cockcroft was able to produce an analytical result. The mid P range of Dwight's Figure 1 presents problems in that the simplifying approximations invoked by Dwight (1918) and by Cockcroft are not applicable. Furthermore, the analysis must be approached from the point of view of solving the partial differential equations that govern the magnetic field inside the conductor. The partial differential equation solution is complicated by the lack of known boundary conditions at the conductor surface. Silvester (1967) points out this complication in a paper where he presented a numerical approach for determining the AC to DC resistance ratio for rectangular conductors. Mocanu (1975) presented an approximate analysis for arbitrary cross section based on an iteration method similar to that used by Dwight (1918). For mid-range frequencies, numerical methods have essentially provided the way of predicting the AC to DC resistance ratio.

Lacking an analytical formula for the midrange P values, a numerical method must be used to predict the AC to DC resistance ratio. For small P values, Dwight (1918) can be used whereas high P values are usually not applicable for power frequencies. The overall goal of this presentation is the production of a heat loss model for low and medium voltage switchgear. It is desired for this model to be part of a spreadsheet for ease of use. Finite element and finite difference methods have a long history of solving magnetic field equations, however the spreadsheet constraint precludes the sophistication of these numerical techniques. To be presented is a numerical technique that is tailored to both skin and proximity effect in rectangular conductors.

Silvester (1968) derives an integral equation for the current density inside a conductor carrying AC current. The current density J in amp/[m.sup.2] at a point (x, y) on a conductor cross-section is given by

J(x, y) = [[j[omega][[mu].sub.o]]/[2[pi][rho]]] [integral][integral]J([xi], [eta]) ln[square root of [[(x - [xi]).sup.2] + [(y - [eta]).sup.2]]]d[xi]d[eta] + [1/[rho]][E.sub.o] (3)

where

J(x,y) = current density J in amp/[m.sup.2] at a point (x, y),

(x,y) = coordinates in meters of a point on the conductor cross-section,

j = [square root of -1]

[omega] = the AC frequency in rad/sec,

[[mu].sub.o] = the permeability of free space in Henry/meter,

[E.sub.o] = the applied electric field strength in the conductor in volt/m, and

([xi],[eta]) = the variables of integration, each having the units of meters.

The integration is carried out over the cross-section of the conductor. Note that there is a sign difference between equation (3) and Silvester's text. The current density in equation (3) flows in a direction normal to the conductor cross-section. Silvester's approach is to divide the conductor cross-section into squares or cells as illustrated in Figure 2. The integral equation in equation (3) can be simplified by considering the current density to be essentially constant in a given cell. The average current density in cell m is [J.sub.m] in amp/[m.sup.2] given by

[FIGURE 2 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where

[J.sub.m] = average current density of cell m in amp/[m.sup.2],

([x.sub.m], [y.sub.m]) = coordinates inside cell m, having the units of meters.

[DELTA]w = cell width in m, and

[DELTA]h = cell height in m.

Because each cell is considered to have a constant current density then equation (4) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where n is the dimensionless integer for counting the cells on a conductor cross-section. The integral in equation (5) is familiar and it is the product of the cell cross sectional area squared and ln([D.sub.mn]), the natural log of the geometric mean distance in meters between cell m and cell n. Substituting [D.sub.mn] into equation (5) produces

[J.sub.m] = [[j[omega][[mu].sub.o]]/[2[pi][rho]]][DELTA]w[DELTA]h[summation over (n)][J.sub.n] ln([D.sub.mn]) + [1/[rho]][E.sub.o] (6)

where [D.sub.mn] is the geometric mean distance in m between cells m and n. There is a total of N cells into which the conductor cross-section is divided and a total of N unknown values of current density, one for each cell. An equation similar to equation (5) can be written for each cell, thus, there are N equations for the N unknowns.

Define [alpha] as the quantity

[alpha] = j[[[omega][[mu].sub.o]]/[rho]] (7)

where [alpha] is a simplifying constant in [m.sup.-2]. Then the N equations in the form of equation (6) can be written in matrix form as

[I - [alpha]K]J = G (8)

where

I = an N by N identity matrix,

G = an N by one vector where each element is [E.sub.o]/[rho] in amp/[m.sup.2],

J = the N by one solution vector of current densities in amp/[m.sup.2]

K = a symmetric matrix whose m,n element is given by

[K.sub.m, n] = [DELTA]w[DELTA]h ln([D.sub.mn]), (9)

where [K.sub.m,n] is the element m,n of matrix K in [m.sup.2]. Note that the value of [E.sub.o]/[rho] is unknown at this point. The purpose of this analysis is to find the AC resistance of the conductor. The reason for not knowing [E.sub.o]/[rho] is because the AC resistance of the conductor is not known and it is uncertain what strength of electric field, [E.sub.o], to apply to the conductor in order to get a particular level of electric current. In order to find [E.sub.o]/[rho], a constraint to be satisfied by the current density, will be applied. This constraint is that the current density must produce the total current when integrated over the crosssectional area. Berleze and Robert (2003) have pointed this out. By summing the current flowing in each cell we arrive at

[DELTA]w[DELTA]h[summation over (n)][J.sub.n] = [I.sub.T] (10)

where [I.sub.T] is the total electrical current in RMS amp. Should equation (10) not be true, then all of the current densities are scaled by the same factor so that equation (10) is true. To see why this statement is true consider the following argument. Solving equation (8) for the current density vector shows that

J = [[I - [alpha]K].sup.-1]G. (11)

Because the elements of the vector G are all the same, when equation (11) is substituted into equation (10) the result is

[DELTA]w[DELTA]hH[[E.sub.o]/[rho]] = [I.sub.T] (12)

where H is the sum of all of the elements of [[I-[alpha]K].sup.-1] which can be expressed as

H = [N.summation over (m = 1)][[[N.summation over (n = 1)][I - [alpha]K]].sub.[m, n].sup.[-1]] (13)

where the m, n subscript denotes the m, n element of the matrix [[I-[alpha][KAPPA]].sup.-1].

Note that H is a dimensionless quantity. Using equation (12), the value of [E.sub.o]/[rho] can be determined as

[[E.sub.o]/[rho]] = [[I.sub.T]/[[DELTA]w[DELTA]hH]] (14)

Given [E.sub.o]/[rho], the current density vector J can be found from equation (11).

This information is presented here because Silvester's (1968) presentation did not include the constraint on the current density and included several sign mistakes. The presentation of Berleze and Robert (2003) does not make clear the need or the method of invoking equation (10), especially for three phase conductors.

The use of squares for the subdivision of the conductor cross-section is preferred because it does not create a bias to either the vertical or horizontal direction of the conductor model. This point will be made clearer in a subsequent paragraph.

The rate at which the current density changes with position is governed by the magnitude of the quantity [alpha] defined in (7). In fact, the skin depth of the material is given by

d = [square root of [[rho]/[[omega][[mu].sub.o]]]] (15)

where d is the skin depth of the material in m.

For a very wide, flat conductor, the skin depth is the distance at which the magnitude of the current density will be reduced from its conductor surface value by a factor of 63%. Copper at room temperature has a resistivity of approximately 1.7 x [10.sup.-8] ohm-m and the permeability of free space is 4[pi] x [10.sup.-7] Henry/m. At 60 Hz, equation (15) produces a value of 0.006 m or slightly more than a quarter of an inch. A rule of thumb is to choose a cell size that is no greater than one fifth of a skin depth. If one were to choose a rectangular cell as opposed to a square cell, then the coordinate direction parallel to the rectangle side having the smallest dimension does a better job modeling the physics than the other coordinate direction. This is what is meant by "creating a bias" in the conductor model. In this work, only square cells were used in the calculations.

Once the cell current densities are developed, the conductor ohmic heat losses can be evaluated. The loss per unit length of conductor is given by

Q = [[rho]/[[DELTA]w[DELTA]h]][N.summation over (n = 1)][([DELTA]w[DELTA]h[[absolute value of [J.sub.n]]).sup.2] = [rho][DELTA]w[DELTA]h[N.summation over (n = 1)][[[absolute value of [J.sub.n]].sup.2] = [I.sub.T.sup.2][R.sub.AC] (16)

where

Q = the conductor heat loss in watts and

[R.sub.AC] = the AC resistance per unit length of conductor in ohm/m.

This calculation computes the square of the cell current to get [([DELTA]w[DELTA]h[absolute value of [J.sub.n]]).sup.2], multiplies it by the resistance per unit length of conductor cell [rho]/([DELTA]w[DELTA]h), and finally the heat loss of each cell is summed. The AC resistance is found by dividing the power dissipation, Q, in equation (16) by the square of the total current. It is convenient to report the AC to DC resistance ratio and this is determined as

[[R.sub.AC]/[R.sub.DC]] = [[[rho][DELTA]w[DELTA]h[N.summation over (n = 1)][[[absolute value of [J.sub.n]].sup.2]]/[I.sub.T.sup.2]] [[w * h]/[rho]] = [[w * h * [DELTA]w[DELTA]h[N.summation over (n = 1)][[[absolute value of [J.sub.n]].sup.2]]/[I.sub.T.sup.2]] (17)

where [R.sub.AC]/[R.sub.DC] is the AC to DC resistance ratio which is dimensionless. In the implementation of the calculation, the total current [I.sub.T] was always chosen as unity so that the heat loss defined in equation (16) is always numerically equal to the AC resistance.

As a test of the procedure just presented, the AC to DC resistance ratio was calculated for a copper bus bar having a height of a = h = 0.25 in. (0.0063 m). In this calculation, the ratio of conductor widths to height ranged from one (square conductor) to 16. The specific values of the b/a ratio are shown in Figure 3. The value of the parameter P ranged from one to five in increments of 0.5. By setting the parameter P to a particular value, equation (2) was used to determine the frequency of the alternating current. The results of the calculation are shown in Figure 3. A very similar curve was presented by Silvester (1967) where the resistance ratio was found by a different method and compared to available measured data.

[FIGURE 3 OMITTED]

Three Phase Model

In computing the power loss in a three phase bus, it must be appreciated that for the purpose of losses a three phase bus is not three single phase problems. The reason for this statement is proximity effect, i.e. the magnetic fields produced by each bus interact with one another. That this is true will be illustrated in examples to be shown at the end of this section.

Figure 4 illustrates a three phase bus. In this analysis, it is assumed that the phase currents are balanced. The procedures of the previous section are applied to this calculation. The differences between the single and three phase cases are: Phase angles must be assigned to each conductor. As an example, should the phase A conductor be given a phase of zero degrees then the vector G defined in equation (8) is as stated in the text, i.e. each element of G for phase A would be [E.sub.o]/[rho]. If the phase sequence was A--B--C, then for phase B each element of G would be [E.sub.o]/[rho] * [e.sup.-j2[pi]/3] because phase B lags A by 120[degrees]. Similarly, each element of G for phase C would be [E.sub.o]/[rho] * [e.sup.j2[pi]//3] because phase C leads phase A by 120[degrees]. Let

[FIGURE 4 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

where

[^.i] = a vector of length N where each entry of the vector is one and

[E.sub.oA] = the electric field strength in V/m of phase A, etc. Similar to the text following equation (9) where it was stated that [E.sub.o]/[rho] was not known, the quantities [E.sub.oA], [E.sub.oB], and [E.sub.oC] in equation (18) are not known at this time.

The constraint that the cell currents in any one phase add to the total current is implemented differently for the three phase case. This work follows the procedure presented by Berleze and Robert (2003) and a summary of the constraint implementation is presented here. Suppose that there are a total of three conductors for which there is now a total current vector, [[bar.I].sub.T], defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

where

[[bar.I].sub.T] = the total current vector in amp and

[I.sub.T.sup.A] = the total current in amp of phase A, etc.

The elements of [[bar.I].sub.T] are the phase currents. Because each conductor is the same size, the subdivision of each conductor will be the same and there will be a total of 3N cells. The current density vector defined by equation (11) is arranged so that the first N entries correspond to phase A, the second N entries correspond to phase B, etc. The three phase equivalent of equation (11) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

where each of the sub-matrices [H.sub.AA], [H.sub.AB], etc. are N by N in size, dimensionless, and are formed by the elements of the inverted matrix [[I--[alpha]K].sup.-1]. The current densities in each phase are defined by equation (20). The three phase equivalent of equation (12) and equation (13) for phase A is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

where [H.sub.AAnm] is the dimensionless n, m element of the sub-matrix [H.sub.AA], etc.

Including all phases gives us the three by three matrix equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

Solving for the unknown electric field strengths [E.sub.oA], [E.sub.oB], and [E.sub.oC] in equation (22) allows for the vector G in equation (18) to be evaluated, and finally equation (20) provides the current densities in each cell. By evaluating equation (16) for each conductor, the power loss in each conductor is can be determined. This development is slightly different than that presented by Berleze and Robert (2003). It is believed that this presentation is less abstract.

A search of the literature reveals very little information on proximity effect, especially in rectangular bars. In order to verify the results produced, two different analysis programs were developed. The first calculation stems from the information presented here and is referred to as the "integral equation solution method." The second approach is based on the paper by Barr (1991) and is referred to as the "circuit analysis method." Each of these techniques is based on different approximations and is completely different in the way the problem is approached. In both cases, the rectangular conductor is broken into a collection of square sub-conductors. Of the two techniques, the integral equation solution method has fewer approximations.

In the following examples, the influence of proximity effect on the losses of a three phase system of conductors can be seen. Three conductors are considered each of a height of 3.0 inches (0.0762 m) and a width of 0.25 inches (0.00635 m). The frequency of the alternating current is 60 Hz. and the conductors are copper. The AC to DC loss ratio for a three phase bus was determined. The geometry of the conductors is illustrated in Figure 5.

[FIGURE 5 OMITTED]

Figure 6 shows the AC to DC heat loss ratio as a function of conductor separation. The loss ratio is the ratio of the three phase AC losses to the loss of a single conductor carrying a DC current equal in magnitude to the RMS AC phase current. In the calculation, two sizes of grids were used. The 8 by 96 grid results (involving 2304 unknown current densities) are shown as a dashed line for the separation range of 0 to 0.01 m. It is initially seen that the loss ratio decreases slightly and then continues to increase until a steady value is reached. In this example, we see that the proximity effect reduces the losses from what would occur for three separate single-phase bars. By moving the bus bars closer, we reduce the losses. Practical considerations, such as arching or flashover, limit the closeness of the conductors without using electrical insulation. That this configuration shows a reduction in skin effect losses is significant because this type of bus arrangement is usually found in panelboard construction.

[FIGURE 6 OMITTED]

For a single bar, the AC to DC resistance ratio was seen to be 1.1025 (integral equation method). For the same calculation, the circuit analysis method provided a ratio of 1.1019. For a separation of 4 m, the AC to DC loss ratio was 3.3075 when using the integral equation method, three times the single phase results. Under the same circumstances, the circuit analysis program provided a ratio of 3.3073. All of these calculations were performed with the 8 by 96 grid.

Table 1 contains a comparison between the integral equation solution method and the circuit analysis solution method. In this calculation each conductor is modeled as a 5 by 60 grid.
Table 1. Other Comparisons for Three Phase [R.sub.AC]/[R.sub.DC]
Predictions

Separation, meter  [R.sub.AC]/[R.sub.DC]--  [R.sub.AC]/[R.sub.DC]--
(inch)                Integral Equation        Circuit Analysis

0 (0)                     3.0956                   3.0781

0.01 (0.39)               3.1173                   3.0868

0.02 (0.79)               3.1529                   3.1158

0.04 (1.57)               3.2081                   3.1772

0.06 (2.36)               3.2413                   3.2201

0.1 (3.94)                3.2738                   3.2638

0.2 (7.87)                3.2965                   3.2937

[infinity]                3.3078                   3.3075


The AC to DC loss ratio for another three phase bus was determined. Each conductor consisted of a 3 inch (0.0762 m) by 1/4 inch (0.00635 m) piece of copper. The geometry of the conductors is illustrated in Figure 7.

Figure 8 shows the AC to DC heat loss ratio as a function of conductor separation. In the calculation, two sizes of grids were used. The 8 by 96 grid results (involving 2304 unknown current densities) are shown in the figure. The most striking detail of the results shown in Figure 8 is that it differs greatly from the results seen for the wide parallel faces shown in Figure 6.

[FIGURE 8 OMITTED]

For a single bar, the AC to DC resistance ratio was seen to be 1.1026 and for a separation of 4 m, the AC to DC loss ratio was 3.3078, three times the single-phase results.

It should be noticed that the geometry of Figure 5 is interesting in that the proximity effect here reduces the AC resistance losses whereas the geometry of Figure 7 is particularly prone to increasing the AC resistive losses. The Figure 7 conductor geometry is typical of switchgear construction.

Table 2 shows a comparison between the two calculations used in this work. The agreement is very good.
Table 2. Other Comparisons for Three Phase [R.sub.AC]/[R.sub.DC]
Predictions

Separation, meter  [R.sub.AC]/[R.sub.DC]--  [R.sub.AC]/[R.sub.DC]--
(inch)                Integral Equation        Circuit Analysis

0 (0)                     4.7850                      4.6168

0.01 (0.39)               4.1847                      4.0570

0.02 (0.79)               3.9206                      3.8247

0.04 (1.57)               3.6715                      3.6118

0.06 (2.36)               3.5538                      3.5129

0.1 (3.94)                3.4443                      3.4216

0.2 (7.87)                3.3596                      3.3511

[infinity]                3.3078                      3.3075


Enclosure Loss

Enclosure (or stray) loss occurs when current carrying conductors set up magnetic fields in the vicinity of conducting materials. While there are cases where bus bars are close to support beams for leads and cabinets in switchgear, the most common situation is that the leads are parallel to a conducting surface such as a sheet metal panel. The enclosure loss in the switchgear models will be modeled by the losses created in a conducting (possibly steel) plate of finite thickness.

The loss model of the conductors and plate arrangement is presented in Del Vecchio (2003). It is assumed that the magnetic field produced by the induced currents in the conducting plate do not interfere with the field set up by the conductors. Also the plate is sufficiently far from the conductors so that the external magnetic field is not influenced by either skin or proximity effect. The three phase geometry is illustrated in Figure 9. There are two cases where the first case involves the conductors being stacked vertically and the other where they are laid out parallel to the surface. The plate is assumed to be infinitely wide and long. The analysis determines the losses per unit length of conductor.

[FIGURE 9 OMITTED]

The analysis starts with a single current filament parallel to the plate surface. The magnetic field and resulting induced eddy currents in the plate are determined analytically. The rectangular conductor is modeled by a series of current filaments where each filament carries a current equal to the total current divided by the number of filaments used to model the conductor. In this investigation, five filaments were used to model each rectangular conductor. The details of the mathematics are covered in Del Vecchio (2003).

As a test of the technique, the results of every example presented by Del Vecchio were verified. This included both single bar and three phase examples.

Switchgear Losses

The main loss mechanisms of switchgear are the circuit breakers and the bus bars. The geometry of construction for several manufacturers was studied and several common geometrical similarities were identified. Using the conductor loss model based on eddy currents and proximity effect and the enclosure loss model based on induced currents in nearby sheet metal structures, loss models for common bar and cabinet configurations were developed. Those loss findings are reported in Piesciorovsky and White (2009).

CONCLUSION

Owing to a lack of reliable information, it is necessary to determine the losses in both medium and low voltage switchgear by the sum of the losses of the component parts. By identifying the major loss mechanisms for switchgear, a model which accounts for the individual losses can be developed. With the exclusion of circuit breakers which are discussed in Piesciorovsky and White (2009), the loss model is made up of two separate parts, the first of which treats both skin and proximity effect in rectangular conductors. The second part treats rectangular bus bars and conducting plates and is used to model stray or enclosure loss in conducting metallic structures. The commonality of materials, component sizes, and construction details in both medium and low voltage switchgear together with the convenient situation that only a handful of components account for the majority of power losses allows for this approach to be taken. Given the constraints of information availability, the approach taken in this effort represents good engineering judgment.

The models used in this study were drawn from previous effort. However, in two cases, errors and explanation difficulties in earlier presentations necessitated greater coverage of some of the model details. These points are explained in the text. Owing to these problems, the authors deemed it necessary to provide the correct details.

The ideal situation to improve on that presented here is to have the opportunity to visit an electrical equipment manufacturer so that component losses modeled in this effort can be closely investigated and overall equipment power loss can be compared to that produced by this investigation.

ACKNOWLEDGMENTS

The authors would like to thank the American Society of Heating Refrigeration and Air Conditioning Engineers (ASHRAE) for funding this work, especially TC 9.2-Industrial Air Conditioning and TC 9.1-Large Building Air Conditioning Systems.

REFERENCES

Arnold, A. H. M., "The Alternating-Current Resistance of Hollow, Square Conductors," Journal of the Institution of Electrical Engineers, vol. 82, pp. 537-545, May 1938.

Barr, A.W., "Calculation of Frequency-Dependent Impedance for Conductors of Rectangular Cross Section," AMP Journal of Technology, Vol. 1 November, 1991, pp.91-100.

Berleze, Sergio L. M. and Rene Robert, "Skin and Proximity Effects in Nonmagnetic Conductors," IEEE Transactions on Education, vol. 46, no. 3, pp. 368-372, Aug. 2003.

Cockcroft, J. D., "Skin Effect in Rectangular Conductors at High Frequencies," Proc. Roy. Soc. A, vol. 122, pp. 533-542, 1929.

Del Vecchio, Robert M., "Eddy-Current Losses in a Conducting Plate Due to a Collection of Bus Bars Carrying Currents of Different Magnitudes and Phases," IEEE Transactions on Magnetics, Vol. 39, No. 1, 2003, pp. 549-552.

Dwight, H. B., "Skin Effect in Tubular and Flat Conductors," Trans. AIEE, vol. 37, pt. II, pp. 1379-1403, 1918.

Dwight, H. B., "Effective Resistance of Isolated Nonmagnetic Rectangular Conductors," Trans. AIEE, vol. 66, pp. 549-552, 1947.

IEEE Std. C37.23-2003, IEEE Standard for Metal-Enclosed Bus, New York, The Institute of Electrical and Electronics Engineers, April, 2004.

IEC 60287-1-3:2002, "Electric Cables: Calculation of the Current Rating, Current Rating Equations (100% load factor) and Calculation of Losses. Current Sharing Between Parallel Single-Core Cables and Calculation of Circulating Current Losses," International Electrotechnical Commission, 2002.

Mocanu, C. I., "A.C. Current Resistance of Bars of Rectangular Cross-section at Low Frequencies," Revue Roumaine des Sciences Techniques, vol. 20, no. 3, pp. 339-354, 1975.

Piesciorovsky, Emilio and Warren N. White, "Building Heat Load Contribution from Medium and Low Voltage Switchgear, Part II: Component and Overall Switchgear Heat Gains," ASHRAE Transactions--to appear.

Rubin, I. M., "Heat Losses from Electrical Equipment in Generating Stations," IEEE Transactions on Power Apparatus and Systems, vol. PAS-98, no. 4, July-Aug. 1979, pp. 1149-1152.

Silvester, P., "AC Resistance and Reactance of Isolated Rectangular Conductors," IEEE Transactions on Power Apparatus and Systems, vol. PAS-86, no. 6 JUNE 1967.

Silvester, P., Modern Electromagnetic Fields, Englewood Cliffs, NJ: Prentice Hall, 1968.

White, Warren N., Anil Pahwa, and Chris Cruz, "Heat Loss from Electrical and Control Equipment in Industrial Plants: Part I--Methods and Scope," ASHRAE Transactions, vol. 110 (2) pp. 842-851, 2004.

White, Warren N., Anil Pahwa, and Chris Cruz, "Heat Loss from Electrical and Control Equipment in Industrial Plants: Part II--Results and Comparisons," ASHRAE Transactions, vol. 110 (2) pp. 852-870, 2004a.

DISCUSSION

Larry Spielvogel, Consulting Engineer, King of Prussia, PA: a) Were the current transformers you show split core or solid core? Were you able to determine the measurement accuracy of the current transformers? What was it? b) Is there any measured experimental data published to support the conclusions in the paper? c) In your presentation, it was mentioned that ASHRAE would publish a manual with this information. If all of the heat gain information necessary to determine cooling loads were in manuals, dozens of them would be required, Why is this information not going into the ASHRAE Handbooks along with all other heat gain information?

Warren N. White: The authors thank Mr. Spielvogel for his questions and interest in our work.In regard to his first question, the current transformers together with the Watt meter used in our measurements were tested by successfully replicating test measurements performed in RP-1104. The current transformers are marketed by Summit Technology, and the authors refer Mr. Spielvogel to this company regarding construction details of the current probes. The manufacturer-quoted uncertainty of the instrument is +/-1% for currents ranging from 10 to 3000 RMS AC amps. In regard to the second question, both low- and medium-voltage switchgear have been divided into three parts, which are bus bars, circuit breakers, and auxiliary components. The major losses in switchgear can be attributed to bus bars and circuit breakers. In regard to bus bars, the analytical model we used in our work, which consisted of a numerical solution of the integral equation representation of the magnetic field laws, showed very good agreement with published measurements of singlebar power losses. Proximity effect comes into play when there are multiple bus bars, as in a three-phase circuit. Searching the technical literature shows very little work in the area of measuring proximity effect, with the exception of round conductors. All of this work shows that the main influence of proximity effect is to increase the conductor heat losses. To remedy the lack of proximity effect measurement data, this investigation used two independent analytical models to predict the conductor losses. The analytical models consist of different approaches and assumptions. The results from the two models are different but very close, as demonstrated in the paper. Also, it was demonstrated in our work that proximity effect could both increase and decrease the AC losses. It would be a very interesting and useful endeavor to study proximity effect in rectangular conductors through both analytical and experimental means, and we urge Mr. Spielvogel to encourage ASHRAE to make such a study possible. Given the range of circuit breakers tested in RP-1104 and RP-1395 and the agreement with manufacturer data, we believe the manufacturer data for untested breakers to be useful. In regard to Mr. Spielvogel's last question regarding the venue for the publication of the heat load information from RP-1104 and RP-1395, the answer to this question rests with the ASHRAE Technical Committees sponsoring the work.

Warren N. White, PhD

Emilio C. Piesciorovsky

This paper is based on findings resulting from ASHRAE Research Project RP-1395.

Warren N. White is an associate professor in the Department of Mechanical and Nuclear Engineering and Emilio C. Piesciorovsky is a graduate student in the Department of Electrical and Computer Engineering, Kansas State University, Manhattan, KS.
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Author:White, Warren N.; Piesciorovsky, Emilio C.
Publication:ASHRAE Transactions
Article Type:Report
Geographic Code:1USA
Date:Jul 1, 2009
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