transformer engineering design and practice 1_phần 6 doc

Eddy Currents and Winding Stray LossesThe load loss of a transformer consists of losses due to ohmic resistance of windings I2R losses and some additional losses.. The stray losses in th

Eddy Currents and Winding Stray Losses

The load loss of a transformer consists of losses due to ohmic resistance of

windings (I2R losses) and some additional losses These additional losses are

generally known as stray losses, which occur due to leakage field of windings andfield of high current carrying leads/bus-bars The stray losses in the windings arefurther classified as eddy loss and circulating current loss The other stray lossesoccur in structural steel parts There is always some amount of leakage field in alltypes of transformers, and in large power transformers (limited in size due totransport and space restrictions) the stray field strength increases with growingrating much faster than in smaller transformers The stray flux impinging onconducting parts (winding conductors and structural components) gives rise toeddy currents in them The stray losses in windings can be substantially high inlarge transformers if conductor dimensions and transposition methods are notchosen properly

Today’s designer faces challenges like higher loss capitalization and optimumperformance requirements In addition, there could be constraints on dimensionsand weight of the transformer which is to be designed If the designer lowers

current density to reduce the DC resistance copper loss (I2R loss), the eddy loss in

windings increases due to increase in conductor dimensions Hence, the windingconductor is usually subdivided with a proper transposition method to minimizethe stray losses in windings

In order to accurately estimate and control the stray losses in windings andstructural parts, in-depth understanding of the fundamentals of eddy currentsstarting from basics of electromagnetic fields is desirable The fundamentals aredescribed in first few sections of this chapter The eddy loss and circulatingcurrent loss in windings are analyzed in subsequent sections Methods for

evaluation and control of these two losses are also described Remainingcomponents of stray losses, mostly the losses in structural components, are dealtwith in Chapter 5.

E=electric field strength (V/m)

B=flux density (wb/m2)

J=current density (A/m2)

D=electric flux density (C/m2)

ρ=volume charge density (C/m3)

There are three constitutive relations,

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/(jωε), which is very high even for a poor metallic

conductor at very high frequencies (where ω is frequency in rad/sec) Since ouranalysis is for the (smaller) power frequency, the displacement current density is

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 continuityequation,

(4.9)

In the absence of free electric charges in the present analysis of eddy currents in aconductor 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 besimplified 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)

In above equation, term jω appears because the partial derivative of a sinusoidal

field quantity with respect to time is equivalent to multiplying the corresponding

phasor by jω 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 equation4.15 and using the fact that (since no freeelectric charges are present) we get

γ=α+jβ (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=2π f we get

(4.32)Hence,

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

respectively Thus, the source field at the surface which establishes fields withinthe conductor is given by

(E x ) z=0 =E xpcosωt

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 withpenetration into the conductor there is an exponential decrease in the electric fieldintensity and (conduction) current density At a distance of penetration

the exponential factor becomes e-1(=0.368), indicating that thevalue of field (at this distance) reduces to 36.8% of that at the surface Thisdistance 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 penetratefew skin depths into the conductor Comparing equations 4.33 and 4.38, we gettherelationship,

(4.39)

The depth of penetration or skin depth is a very important parameter indescribing the behavior of a conductor subjected to electromagnetic fields Theconductivity of copper conductor at 75°C (temperature at which load loss of atransformer is usually calculated and guaranteed) is 4.74×107 mhos/m Copperbeing a non-magnetic material, its relative permeability is 1 Hence, the depth ofpenetration 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, whoseconductivity is approximately 61% of that of copper, the skin depth at 50 Hz is13.2 mm Most of the structural elements inside a transformer are made of eithermild steel or stainless steel material For a typical grade of mild steel (MS)material with relative permeability of 100 (assuming that it is saturated) andconductivity of 7×106 mho/m, the skin depth is δMS=2.69 mm at 50 Hz A non-magnetic stainless steel is commonly used for structural components in the

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 of1.136×106 mho/m, the skin depth is δss=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 theprevious 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 bemodified as,

(4.41)

This is a simpler form of Poynting’s theorem which states that the net inflow ofpower is equal to the sum of the power absorbed by the magnetic field and theohmic loss The Poynting vector is given by the vector product,

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,

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 veryimportant fact for the analysis of eddy currents and losses in structuralcomponents of transformers If the eddy losses in the tank of a transformer due toincident leakage field emanating from windings are being analyzed by using FEManalysis, then there should be at least two or three elements in one skin depth forgetting 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

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 densityover the infinite depth of the conductor Using equations 4.34 and 4.39 we get

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 weget

(4.58)

which is the same as equation 4.53 Hence, the average power loss in a conductormay be computed by assuming that the total current is uniformly distributed in oneskin depth This is a very important result, which is made use of in calculation ofeddy current losses in conductors by numerical methods When a numericalmethod such as Finite Element Method (FEM) is used for estimation of straylosses in the tank (made of mild steel) of a transformer, it is important to haveelement size less than the skin depth of the tank material as explained earlier Withthe other transformer dimensions in meters, it is difficult to have very smallelements 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 forestimation of tank losses by 3-D FEM analysis The method assumes uniformcurrent density in the skin depth allowing the use of relatively larger element sizes.The above-mentioned problem of modeling and analysis of skin depths canalso be taken care by using the concept of surface impedance The intrinsicimpedance 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 (Hrms) at the surface of the tank or any other structural component in thetransformer by either numerical or analytical method, the specific loss per unitsurface area can be calculated by the expression [4,5]

Thus, the total losses in the transformer tank can be determined by integrating thespecific loss on its internal surface

4.3 Eddy Current and Hysteresis Losses

All the analysis done previously assumed linear material (B-H) characteristics meaning that the permeability (µ) is constant The material used for structural

components in transformers is usually magnetic steel (mild steel), which is a

ferromagnetic material having a much larger value of relative permeability (µ r ) as compared to the free space (for which µ r =1) The material has non-linear B-H characteristics and the permeability itself is a function of H Moreover, the

characteristics also exhibit hysteresis property Equation 4.6 (B=µH)has to be

suitably modified to reflect the non-linear characteristics and hysteresis behavior

Hysteresis introduces a time phase difference between B and H; B lags H by an

angle (θ) known as the hysteresis angle One of the ways in which the

characteristics can be mathematically expressed is by complex or ellipticalpermeability,

In this formulation, where harmonics introduced by saturation are ignored, thehysteresis loop becomes an ellipse with the major axis making an angle of θ with

the H axis as shown in figure 4.2 The significance of complex permeability is that

a functional relationship between B and H is now realized in which the permeability is made independent of H resulting into a linear system [6] Let us

now find an expression for the eddy current and hysteresis loss for an infinite space shown in figure 4.3

half-Figure 4.2 Elliptic hysteresis loop Figure 4.3 Infinite half space

The infinite half-space is an extension of the geometry shown in figure 4.1 in thesense that the region of the material under consideration extends from -∞ to +∞ in

the x and y directions, and from 0 to ∞ in the z direction Similar to Section 4.1, we

assume that 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

α=cos(θ/2)+sin(θ/2) and β=cos(θ/2)-sin(θ/2) (4.67)

(4.71)and

(4.72)

(4.73)

In the absence of hysteresis (θ=0), α=β=1 as per equation 4.67 Hence, the eddyloss per unit surface area is given by

(4.74)

Substituting the expression for skin depth from equation 4.38 and using the r.m.s

value of magnetic field intensity (Hrms) at the surface we get

an equation,

Figure 4.4 Step-magnetization

B=(sign of H)B s (4.76)

where B s is the saturation flux density The magnetic field intensity H at the

surface is sinusoidally varying with time (=H0 sinωt) The extreme depth to whichthe field penetrates and beyond which there is no field is called as depth ofpenetration δs This depth of penetration has a different connotation as compared

to that with constant or linear permeability In this case, the depth of penetration issimply the maximum depth the field will penetrate at the end of each half period.The depth of penetration for a thick plate (thickness much larger than the depth ofpenetration so that it can be considered as infinite half space) is given by [1,7,8]

Comparing this with equation 4.74, it can be noted that if we put µ=B s /H0, δs will

be equal to δ and in that case the loss in the saturated material is 70% higher than

the loss in the material having linear H characteristics Practically, the actual

B-H curve is in between the linear and step characteristics, as shown in figure 4.4 (b)

In [7], it is pointed out that as we penetrate inside into the material, eachsucceeding inner layer is magnetized by a progressively smaller number ofexciting ampere-turns because of shielding effect of eddy currents in the regionbetween the outermost surface and the layer under consideration In step-magnetization characteristics, the flux density has the same magnitudeirrespective of the magnitude of mmf Due to this departure of the step curve

response from the actual response, the value of B s is replaced by 0.75×B s Fromequations 4.77 and 4.78, it is clear that and hence the constant 1.7 inequation 4.78 would reduce to i.e., 1.47 As per Rosenberg’s theory,the constant is 1.33 [7] Hence, in the simplified analytical formulations, linearcharacteristics are assumed after taking into account the non-linearity by thelinearization coefficient in the range of 1.3 to 1.5 For example, a coefficient of 1.4

is used in [9] for the calculation of losses in tank and other structural components

in transformers

After having seen in details the fundamentals of eddy currents, we will nowanalyze eddy current and circulating current losses in windings in the followingsections Analysis of stray losses in structural components, viz tank, frames, flitchplates, high current terminations, etc., is covered in Chapter 5

4.5 Eddy Loss in Transformer Winding

4.5.1 Expression for eddy loss

Theory of eddy currents explained in the previous sections will be useful whilederiving the expression for the eddy loss in windings The losses in a transformerwinding due to an alternating current are usually more than that due to directcurrent of the same effective (r.m.s.) value There are two different approaches ofanalyzing this increase in losses In the first approach, we assume that the loadcurrent in the winding is uniformly distributed in the conductor cross section(similar to the direct current) and, in addition to the load current, there exist eddycurrents which produce extra losses Alternatively, one can calculate losses due tothe combined action of the load current and eddy currents The former method ismore suitable for the estimation of eddy loss in winding conductors, in whicheddy loss due to the leakage field (produced by the load current) is calculated

separately and then added to the DC I2R loss The latter method is preferred for

calculating circulating current losses, in which the resultant current in eachconductor is calculated first, followed by the calculation of losses (which give the

total of DC I2R loss and circulating current loss) We will first analyze eddy losses

in windings in this section; the circulating current losses are dealt with in the nextsection

Consider a winding conductor, as shown in figure 4.5, which is placed in an

alternating magnetic field along the y direction having the peak amplitude of H0

The conductor can be assumed to be infinitely long in the x direction The current density J x and magnetic field intensity H y are assumed as functions of z only.

Rewriting the (diffusion) equation 4.15 for the sinusoidal variation of the fieldquantity and noting that the winding conductor, either copper or aluminum, has

constant permeability (linear B-H characteristics),

(4.79)

Figure 4.5 Estimation of eddy loss in a winding conductor

A solution satisfying this equation is

where γ is defined by equation 4.32 In comparison with equation 4.65, equation

4.80 has two terms indicating waves traveling in both +z and -z directions (which

is consistent with figure 4.5) The incident fields on both the surfaces, having peak

amplitude of H0, penetrate inside the conductor along the z axis in opposite

directions (it should be noted that equation 4.80 is also a general solution of

equation 4.63, in which case C1=0 and C2=H0 for the boundary conditionsspecified by equation 4.64) For the present case, the boundary conditions are

H y =H 0 at z=+b and H y =H 0 at z=-b (4.81)Using these boundary conditions, we can get the expression for the constants as

The loss produced per unit surface area (of the x-y plane) of the conductor in terms

of the peak value of current density is given by

(4.85)

Now, using equation 4.39 we get

Substituting this magnitude of current density in equation 4.85 we get

(4.86)

Neglecting higher order terms and substituting the expression of δ from equation4.38 we get

dividing by t and finally substituting resistivity (ρ) in place of conductivity, we get

the expression for the eddy loss in the winding conductor per unit volume as

(4.94)

In case of thin conductors, the eddy currents are restricted by the lack of space orhigh resistivity and are said to be resistance limited In other words, since the field

of the eddy currents is negligible for thin conductors, the behavior is resistance

structural component made of magnetic steel having sufficiently large thickness,etc.), the resultant current distribution is greatly influenced and limited by theeffect its own field and the currents are said to be inductance limited (currents areconfined to the surface layers)

Now, let us analyze the case when dimension (thickness) of the conductor isquite small as compared to its depth of penetration, which is usually the case for

rectangular paper insulated conductors used in transformers For 2b<<δ, i.e.,

ξ<<1, equation 4.89 can be simplified to

(4.91)

limited Equation 4.94 matches exactly with that derived in [10] by ignoring themagnetic field produced by the eddy currents These currents are 90° out of phasewith the load current (uniformly distributed current which produces the leakage

field and is also responsible for DC I2R loss in windings) flowing in the conductor.

The eddy currents are shown to be lagging by 90° with respect to the load (source)current for a thin circular conductor in the later part of this section The totalcurrent flowing in the conductor can be visualized to be a vector sum of the eddy

current (I eddy ) and load current (I load ), having the magnitude of

because these two current components are 90° out of phase in a thin conductor

This is a very important and convenient result because it means that the I2R losses

due to load current and eddy current losses can be calculated separately and thenadded later for thin conductors

Equation 4.94 is very well-known and useful formula for calculation of eddylosses in windings If we assume that the leakage field in windings is in axialdirection only, then we can calculate the mean value of eddy loss in the wholewinding by using the equations of Section 3.1.1 The axial leakage field for an

inner winding (with a radial depth of R and height of H W) varies linearly frominside diameter to outside diameter as shown in figure 4.6 The thickness of theconductor, which is its dimension perpendicular to the axial field, is usually quite

small Hence, the same value of flux density (B0) can be assumed along both its

vertical surfaces (along width w) The position of the conductor changes along the

radial depth as the turns are wound Hence, in order to calculate the mean value ofthe eddy loss of the whole winding, we have to first calculate the mean value of The r.m.s value of ampere turns are linearly changing from 0 at the inside

diameter (ID) to NI at the outside diameter (OD) The peak value of flux density at

a distance x from the inside diameter is

where B gp is the peak value of flux density in the LV-HV gap,

Figure 4.6 Leakage flux density in winding

indicating that the mean eddy loss in the second layer close to the gap is 1.75times the mean eddy loss for the entire winding Similarly, for a 4-layer winding,

giving the mean eddy loss in the 4th layer as 2.31 times the mean eddy loss for the

entire winding Hence, it is always advisable to calculate the total loss (I2R+eddy)

in each layer separately and estimate the temperature rise of each layer Such acalculation procedure helps designers to take countermeasures to eliminate hightemperature rise in windings Also, the temperatures measured by fiber-opticsensors (if installed) will be closer to the calculated values when such a calculationprocedure is adopted

Eddy loss calculated by equation 4.99 is approximate since it assumes theleakage field entirely in the axial direction As seen in Chapter 3, there exists aradial component of the leakage field at winding ends and in winding zones whereampere-turns per unit height are different for LV and HV windings For smalldistribution transformers, the error introduced by neglecting the radial field maynot be appreciable, and equation 4.99 is generally used with some empiricalcorrection factor applied to the total calculated stray loss value Analytical/numerical methods, described in Chapter 3, need to be used for the correctestimation of the radial field The amount of efforts required for getting theaccurate eddy loss value may not get justified for very small distributiontransformers For medium and large power transformers, however, the eddy lossdue to the radial field has to be estimated and the same can be found out by usingequation 4.94, for which the dimension of the conductor perpendicular to the

radial field is its width w Hence, the eddy losses per unit volume due to axial (B y ) and radial (B x ) components of leakage field are

(4.102)

(4.103)

Thus, the leakage field incident on a winding conductor (see figure 4.7) is

resolved into two components, viz B y and B x , and losses due to these two