transformer engineering design and practice 1_phần 6 doc

127 4 Eddy Currents and Winding Stray Losses The load loss of a transformer consists of losses due to ohmic resistance of windings (I 2 R losses) and some additional losses. These additional losses are generally known as stray losses, which occur due to leakage field of windings and field of high current carrying leads/bus-bars. The stray losses in the windings are further classified as eddy loss and circulating current loss. The other stray losses occur in structural steel parts. There is always some amount of leakage field in all types of transformers, and in large power transformers (limited in size due to transport and space restrictions) the stray field strength increases with growing rating much faster than in smaller transformers. The stray flux impinging on conducting parts (winding conductors and structural components) gives rise to eddy currents in them. The stray losses in windings can be substantially high in large transformers if conductor dimensions and transposition methods are not chosen properly. Today’s designer faces challenges like higher loss capitalization and optimum performance requirements. In addition, there could be constraints on dimensions and weight of the transformer which is to be designed. If the designer lowers current density to reduce the DC resistance copper loss (I 2 R loss), the eddy loss in windings increases due to increase in conductor dimensions. Hence, the winding conductor is usually subdivided with a proper transposition method to minimize the stray losses in windings. In order to accurately estimate and control the stray losses in windings and structural parts, in-depth understanding of the fundamentals of eddy currents starting from basics of electromagnetic fields is desirable. The fundamentals are described in first few sections of this chapter. The eddy loss and circulating current loss in windings are analyzed in subsequent sections. Methods for Copyright © 2004 by Marcel Dekker, Inc. Chapter 4128 evaluation and control of these two losses are also described. Remaining components of stray losses, mostly the losses in structural components, are dealt with in Chapter 5. 4.1 Field Equations The differential forms of Maxwell’s equations, valid for static as well as time dependent fields and also valid for free space as well as material bodies are: (4.1) (4.2) (4.3) (4.4) where H=magnetic field strength (A/m) E=electric field strength (V/m) B=flux density (wb/m 2 ) J=current density (A/m 2 ) D=electric flux density (C/m 2 ) ρ =volume charge density (C/m 3 ) There are three constitutive relations, J= σ E (4.5) B=µ H (4.6) D= ε E (4.7) where µ=permeability of material (henrys/m) ε =permittivity of material (farads/m) σ =conductivity (mhos/m) The ratio of the conduction current density (J) to the displacement current density (∂D/∂t) is given by the ratio σ /(j ωε ), which is very high even for a poor metallic conductor at very high frequencies (where ω is frequency in rad/sec). Since our analysis is for the (smaller) power frequency, the displacement current density is Copyright © 2004 by Marcel Dekker, Inc. Eddy Currents and Winding Stray Losses 129 neglected for the analysis of eddy currents in conducting parts in transformers (copper, aluminum, steel, etc.). Hence, equation 4.2 gets simplified to (4.8) The principle of conservation of charge gives the point form of the continuity equation, (4.9) In the absence of free electric charges in the present analysis of eddy currents in a conductor we get (4.10) To get the solution, the first-order differential equations 4.1 and 4.8 involving both H and E are combined to give a second-order equation in H or E as follows. Taking curl of both sides of equation 4.8 and using equation 4.5 we get For a constant value of conductivity (σ), using vector algebra the equation can be simplified as (4.11) Using equation 4.6, for linear magnetic characteristics (constant µ) equation 4.3 can be rewritten as (4.12) which gives (4.13) Using equations 4.1 and 4.13, equation 4.11 gets simplified to (4.14) or (4.15) Equation 4.15 is a well-known diffusion equation. Now, in the frequency domain, equation 4.1 can be written as follows: (4.16) Copyright © 2004 by Marcel Dekker, Inc. Chapter 4130 In above equation, term j ω appears because the partial derivative of a sinusoidal field quantity with respect to time is equivalent to multiplying the corresponding phasor by j ω . Using equation 4.6 we get (4.17) Taking curl of both sides of the equation, (4.18) Using equation 4.8 we get (4.19) Following the steps similar to those used for arriving at the diffusion equation 4.15 and using the fact that (since no free electric charges are present) we get (4.20) Substituting the value of J from equation 4.5, (4.21) Now, let us assume that the vector field E has component only along the x axis. (4.22) The expansion of the operator ∇ leads to the second-order partial differential equation, (4.23) Suppose, if we further assume that E x is a function of z only (does not vary with x and y), then equation 4.23 reduces to the ordinary differential equation (4.24) We can write the solution of equation 4.24 as (4.25) where E xp is the amplitude factor and γ is the propagation constant, which can be given in terms of the attenuation constant α and phase constant β as Copyright © 2004 by Marcel Dekker, Inc. Eddy Currents and Winding Stray Losses 131 γ = α +j β (4.26) Substituting the value of E x from equation 4.25 in equation 4.24 we get (4.27) which gives (4.28) (4.29) If the field E x is incident on a surface of a conductor at z=0 and gets attenuated inside the conductor (z>0), then only the plus sign has to be taken for γ (which is consistent for the case considered). (4.30) (4.31) Substituting ω =2 π f we get (4.32) Hence, (4.33) The electric field intensity (having a component only along the x axis and traveling/penetrating inside the conductor in +z direction) expressed in the complex exponential notation in equation 4.25 becomes E x =E xp e - γ z (4.34) which in time domain can be written as E x =E xp e - α z cos( ω t- β z) (4.35) Substituting the values of α and β from equation 4.33 we get (4.36) The conductor surface is represented by z=0. Let z>0 and z<0 represent the regions corresponding to the conductor and perfect loss-free dielectric medium Copyright © 2004 by Marcel Dekker, Inc. Chapter 4132 respectively. Thus, the source field at the surface which establishes fields within the conductor is given by (E x ) z=0 =E xp cos ω t Making use of equation 4.5, which says that the current density within a conductor is directly related to the electrical field intensity, we can write (4.37) Equations 4.36 and 4.37 tell us that away from the source at the surface and with penetration into the conductor there is an exponential decrease in the electric field intensity and (conduction) current density. At a distance of penetration the exponential factor becomes e -1 (=0.368), indicating that the value of field (at this distance) reduces to 36.8% of that at the surface. This distance is called as the skin depth or depth of penetration δ , (4.38) All the fields at the surface of a good conductor decay rapidly as they penetrate few skin depths into the conductor. Comparing equations 4.33 and 4.38, we getthe relationship, (4.39) The depth of penetration or skin depth is a very important parameter in describing the behavior of a conductor subjected to electromagnetic fields. The conductivity of copper conductor at 75°C (temperature at which load loss of a transformer is usually calculated and guaranteed) is 4.74×10 7 mhos/m. Copper being a non-magnetic material, its relative permeability is 1. Hence, the depth of penetration of copper at the power frequency of 50 Hz is or 10.3 mm. The corresponding value at 60 Hz is 9.4 mm. For aluminum, whose conductivity is approximately 61% of that of copper, the skin depth at 50 Hz is 13.2 mm. Most of the structural elements inside a transformer are made of either mild steel or stainless steel material. For a typical grade of mild steel (MS) material with relative permeability of 100 (assuming that it is saturated) and conductivity of 7×10 6 mho/m, the skin depth is δ MS =2.69 mm at 50 Hz. A non- magnetic stainless steel is commonly used for structural components in the Copyright © 2004 by Marcel Dekker, Inc. Eddy Currents and Winding Stray Losses 133 vicinity of the field due to high currents. For a typical grade of stainless steel (SS) material with relative permeability of 1 (non-magnetic) and conductivity of 1.136×10 6 mho/m, the skin depth is δ ss =66.78 mm at 50 Hz. 4.2 Poynting Vector Poynting’s theorem is the expression of the law of conservation of energy applied to electromagnetic fields. When the displacement current is neglected, as in the previous section, Poynting’s theorem can be mathematically expressed as [1,2] (4.40) where v is the volume enclosed by the surface s and n is the unit vector normal to the surface directed outwards. Using equation 4.5, the above equation can be modified as, (4.41) This is a simpler form of Poynting’s theorem which states that the net inflow of power is equal to the sum of the power absorbed by the magnetic field and the ohmic loss. The Poynting vector is given by the vector product, P=E×H (4.42) which expresses the instantaneous density of power flow at a point. Now, with E having only the x component which varies as a function of z only, equation 4.17 becomes (4.43) Substituting the value of E x from equation 4.34 and rearranging we get (4.44) The ratio of E x to H y is defined as the intrinsic impedance, (4.45) Substituting the value of γ from equation 4.30 we get Copyright © 2004 by Marcel Dekker, Inc. Chapter 4134 (4.46) Using equation 4.38, the above equation can be rewritten as (4.47) Now, equation 4.36 can be rewritten in terms of skin depth as E x =E xp e -z/δ cos( ω t-z/ δ ) (4.48) Using equations 4.45 and 4.47, H y can be expressed as (4.49) Since E is in the x direction and H is in the y direction, the Poynting vector, which is a cross product of E and H as per equation 4.42, is in the z direction. (4.50) Using the identity cosA cosB=1/2[cos(A+B)+cos(A-B)], the above equation simplifies to (4.51) The time average Poynting vector is then given by (4.52) Thus, it can be observed that at a distance of one skin depth (z= δ ), the power density is only 0.135 (=e -2 ) times its value that at the surface. This is very important fact for the analysis of eddy currents and losses in structural components of transformers. If the eddy losses in the tank of a transformer due to incident leakage field emanating from windings are being analyzed by using FEM analysis, then there should be at least two or three elements in one skin depth for getting accurate results. Let us now consider a conductor with field E xp and the corresponding current density J xp at the surface as shown in figure 4.1. The fields have the value of 1 p.u. Copyright © 2004 by Marcel Dekker, Inc. Eddy Currents and Winding Stray Losses 135 at the surface. The total power loss in height (length) h and width b is given by the value of power crossing the conductor surface [2] within the area (h ×b), (4.53) The total current in the conductor is found out by integrating the current density over the infinite depth of the conductor. Using equations 4.34 and 4.39 we get (4.54) If this total current is assumed to be uniformly distributed in one skin depth, the uniform current density can be expressed in the time domain as (4.55) Figure 4.1 Penetration of field inside a conductor Copyright © 2004 by Marcel Dekker, Inc. Chapter 4136 The total ohmic power loss is given by (4.56) The average value of power can be found out as (4.57) Since the average value of a cosine term over integral number of periods is zero we get (4.58) which is the same as equation 4.53. Hence, the average power loss in a conductor may be computed by assuming that the total current is uniformly distributed in one skin depth. This is a very important result, which is made use of in calculation of eddy current losses in conductors by numerical methods. When a numerical method such as Finite Element Method (FEM) is used for estimation of stray losses in the tank (made of mild steel) of a transformer, it is important to have element size less than the skin depth of the tank material as explained earlier. With the other transformer dimensions in meters, it is difficult to have very small elements inside the tank thickness. Hence, it is convenient to use analytical results to simplify the numerical analysis. For example in [3], equation 4.58 is used for estimation of tank losses by 3-D FEM analysis. The method assumes uniform current density in the skin depth allowing the use of relatively larger element sizes. The above-mentioned problem of modeling and analysis of skin depths can also be taken care by using the concept of surface impedance. The intrinsic impedance can be rewritten from equation 4.46 as (4.59) The real part of the impedance, termed as surface resistance, is given by (4.60) After calculating the r.m.s. value of the tangential component of the magnetic field intensity (H rms ) at the surface of the tank or any other structural component in the transformer by either numerical or analytical method, the specific loss per unit surface area can be calculated by the expression [4,5] Copyright © 2004 by Marcel Dekker, Inc. [...]... E and H vectors have components in only the x and y directions respectively, and that they are function of z only The diffusion equation 4.15 can be rewritten for this case with the complex permeability as (4 .63 ) A solution satisfying boundary conditions, Hy=H0 at z=0 and Hy=0 at z=∞ (4 .64 ) is given by Hy=H0e-kz (4 .65 ) where constant k is (4 .66 ) and α=cos(θ/2)+sin(θ/2) and β=cos(θ/2)-sin(θ/2) (4 .67 )... by Marcel Dekker, Inc  162 Chapter 4 Figure 4. 16 Details of 10 MVA transformer (4.121) Let us now calculate the circulating current loss for a transformer with LV winding having 12 parallel conductors The dimensions of windings are shown in figure 4. 16 The transformer specification and relevant design details are: 10 MVA, 33 /6. 9 kV, star/star, 50 Hz Transformer Volts/turn= 46. 87, Z=7.34% LV conductor:... values of By and Bx for each conductor can be obtained from the FEM solution, and then the axial and radial components of the eddy loss are calculated for each conductor by using equations 4.102 and 4.103 respectively The By and Bx values are assumed to be constant over a single conductor and equal to the value at the center of the conductor If the cylindrical coordinate system is used, By and Bx components... considered and the effect of the radial leakage field is neglected Due to this assumption the calculation is quite easy, and the extra loss on account of circulating currents can be found very quickly while the transformer Copyright © 2004 by Marcel Dekker, Inc Eddy Currents and Winding Stray Losses 161 design is in progress Although these methods may not give the accurate loss value, designers can... (4 .68 ) Using equation 4.8 and the fact that Hx=Hz=0 we get (4 .69 ) The time average density of eddy and hysteresis losses can be found by computing the real part of the complex Poynting vector evaluated at the surface [1], (4.70) Now, (4.71) and (4.72) (4.73) Copyright © 2004 by Marcel Dekker, Inc Eddy Currents and Winding Stray Losses 139 In the absence of hysteresis (θ=0), α=β=1 as per equation 4 .67 ... used [ 16, 17,18] Once the three-dimensional field solution is obtained, the three components of the flux density (Bx, By and Bz) are resolved into two components, viz the axial and radial components, which enables the use of equations 4.102 and 4.103 for the eddy loss evaluation For small distribution transformers with LV winding having crossmatic conductor (thick rectangular bar conductor), each and. .. two winding transformer in [20] For specific transformer dimensions given in the paper, the coefficient of additional loss is 1.0 46, i.e., the eddy loss is 4 .6% of the DC I2R loss In another approach [21], a boundary-value field problem is solved for magnetic vector potential in cylindrical coordinate system using modified Bessel and Struve functions For symmetrically placed LV (foil) and HV windings,... of the CTC conductor For transformer specifications with higher value of load loss capitalization (dollars per kW of load loss), the CTC conductor may give a lower total capitalized cost (material cost plus loss cost) particularly in large transformers The low voltage winding in large generator transformers is usually designed with the CTC conductor to minimize the eddy loss and improve productivity... conductor can be found out by using equations 4.104 and 4.105 as (J0 and Je are 90° apart, square of their sum is sum of their squares) (4.1 06) The power loss per unit length due to the exciting current alone is (4.107) Therefore, the ratio of effective AC resistance to DC resistance of a thin circular conductor can be deduced from equations 4.1 06 and 4.107 as (4.108) For thick circular conductors (R>>δ),... scheme may be used with caution in small transformers when the radial field at ends and within body of windings is insignificant In the discussion of paper [25], Rabins and Cogbill have given a very useful and quick method to check the correctness of any transposition scheme As per this method, for each conductor a sum is calculated where p is a particular position and Np is number of turns (out of total . µ) equation 4.3 can be rewritten as (4 .12 ) which gives (4 .13 ) Using equations 4 .1 and 4 .13 , equation 4 .11 gets simplified to (4 .14 ) or (4 .15 ) Equation 4 .15 is a well-known diffusion equation constant k is (4 .66 ) and α =cos( θ /2)+sin( θ /2) and β =cos( θ /2)-sin( θ /2) (4 .67 ) (4 .68 ) Using equation 4.8 and the fact that H x =H z =0 we get (4 .69 ) The time average density of eddy and hysteresis. the procedure similar to that given in Section 3 .1. 1 as (4 .10 0) and the mean eddy loss per unit volume in the section is (4 .10 1) Equation 4 .10 1 tells us that for a winding consisting of a number