Analysing and Computing the Impact of Geometric Asymmetric Coils on Transformer Stray Losses

Abstract

Designing and manufacturing transformers often involves variations in heights and thicknesses of windings. However, such geometric asymmetry introduces a significant impact on the magnitude of stray transformer losses. This study examines the effects of asymmetric coils on the generation of stray losses within core clamps and transformer tank walls. A model has been introduced to ascertain the dispersion magnetic field’s value at a specific distance from the coil. The analysis extends to characterising the dispersion magnetic field reaching the tank walls by using electromagnetic simulation by a finite element method. It explores strategies to diminish stray losses, including the placement of magnetic shunts as protective shields for the tank walls. It delves into the efficacy of employing a transformer shell-type configuration to mitigate the magnetic dispersion field. The findings revealed that achieving greater symmetry in transformer coils can minimise stray losses. Specifically, the incorporation of magnetic shunts has the potential to reduce additional losses by 40%, while the adoption of a shell-type configuration alone can lead to a 14% reduction. This work provides valuable insights into optimising transformer designs, contributes a user-friendly tool for estimating additional tank losses, thereby enhancing the knowledge base for transformer manufacturers.

1. Introduction

It is now widely recognised that some of the dispersion magnetic field generates stray losses within the structural components of the transformer and its associated tank. It is also recognised that as transformer power increases, this stray flux can cause substantial losses, resulting in heating effects. Recent advances in research focuses on various aspects of stray losses within transformer tanks. These studies employ a variety of methodologies, including numerical techniques such as the finite element method, as well as analytical and experimental approaches [1,2,3,4,5,6]. In this work, stray magnetic field refers to the dispersion magnetic field that reaches the tank and core clamp, and that produces stray losses in them. Research has demonstrated that stray losses resulting from induced current by eddy currents on tanks and other structural elements of distribution transformers can amount more than 15% of the corresponding load losses [7]. In the transformer industry (especially in power transformers), this knowledge has led to adapting the transformer with magnetic shunts and other accessories to reduce load losses in their equipment. The challenge of identifying and mitigating parasitic losses in metallic components and transformer tanks is not a recent problem. It has been widely addressed in technical literature [8,9,10,11]. Some transformer manufacturers continue to use the empirical analytical formulas from this literature or from their experience in their current practices. Calculating stray losses requires an understanding of the stray magnetic field. To address this problem, efficient methods have emerged to quickly and approximately estimate the equivalent leakage reluctance, in particular by using the Reluctance Network Modelling (RNM) method [12,13]. The adoption of numerical methods, such as the Finite Element Method (FEM), has greatly facilitated the evaluation of the stray magnetic field and its associated additional losses. These methods make the simulation of stray magnetic fields in two- and three-dimensional scenarios more accessible and less complex. However, it is imperative to note that the utilisation of such methods comes at the expense of high computational requirements during the transformer design stage. These considerations are essential to effectively reduce stray losses and prevent hot spots. Furthermore, the application of the Finite Element Method (FEM) has allowed us to examine the impact of additional factors, such as geomagnetically induced currents, that contribute to increased stray losses in the transformer tank [14]. Minimising stray losses in transformers represents an important competitive advantage in the industry. Consequently, there has been an increase in research dedicated to identifying the optimal location of magnetic shunts, with the objective of decreasing stray losses within the transformer tank [15,16]. The research is also focused on finding the optimal geometry of the magnetic shunt [17]. The orientation of the shunt also plays an important role in reducing losses, which is why works such as [18] clearly and conclusively confirm the effectiveness of orienting the shunt horizontally. In [19] non-magnetic stainless steel, an insert was utilized to reduce the stray losses in a significant way in 80% of the region of the Tertiary Voltage Bushings of the transformer. The effect of magnetic shunts on shell-type transformers’ characteristics was recently analysed and determined. This helped in determining the magnetic flux density distributions, and the points with high values in the shell-type core and movable shunt sub-areas [20]. Despite the important technical literature on this topic, in the design stage there are still some technical doubts about how the stray magnetic field is affected when there are different widths and heights of the windings. In the design, it is common and very normal for the thicknesses of the windings to be different and the height of the high voltage winding to be similar to the height of the low voltage winding, but most of the time it is not possible to achieve this, so this work contributes to the analysis and quantification of the effect of these differences in dimensions in the windings on the stray magnetic flux that reaches the tank. Therefore, it helps to have an overview of its magnetic field density, which assists in controlling additional losses. In this work, the characterisation of the magnetic field that reaches the tank walls was added along with a quick-to-apply model to approximately estimate the stray magnetic field from the coil to the limits of the tank walls and the impact of reducing stray losses by placing magnetic shunts and using a shell-type configuration transformer. This work also considers three different materials (Mill Anneal Steel, Grain Oriented Silicon Steel and Amorphous Silicon Steel) for the shunts in order to analyse their effectiveness and cost. Therefore, the objective of this research is to serve as a valuable tool in the design and manufacturing processes of transformers. Therefore, this work has the following main contributions: (1) the development of a technique to perform a quick and accurate estimation of the stray magnetic field and losses of the tank when there is asymmetry in transformer windings; (2) characterisation of the stray magnetic field on the tank walls for a better visualization of shielding placement; and (3) study of the potential reduction in stray losses on tank walls and core clamps, through the adoption of shell-type configurations.

2. Modelling and Considerations

2.1. Geometric Model of Column-Type Transformers

Additional losses in all structural components in the transformer sometimes represent more than 20% of the load losses if they are not controlled [21]. Most of the additional losses occur in the tank, so this work lays more emphasis on this part and its control through magnetic shunts. Additional losses in core clamps or yoke beams are also considered, but the analysis of these losses in structural parts such as guides and flitch plates was discarded. Figure 1a shows the actual arrangement of the transformer tank being analysed, equipped with magnetic shunts. Figure 1b shows the geometric model used for the analysis, covering the structural components and their positioning. In particular, tank details are omitted to improve clarity of magnetic shunts and visualisation of core clamps. It is important to highlight that in this model the core is represented as a solid piece, without considering laminations. This simplification arises from the fact that the main focus of the research is the analysis and calculation of losses in the magnetic shunts and the tank. Figure 1c represents an overhead view of a cross section that includes the tank walls: front wall (Fw), left wall (Lw), rear or back wall (Bw), and right wall (Rw). The top (Tw) and the bottom (Gw) tank walls are also considered. The separation of tank walls allowed us to analyse in greater detail the amount of stray magnetic flux in each of the walls. Other parameters include primary winding height (hp) and thickness (wp), secondary winding height (hs) and thickness (ws), and magnetic shunt height (hsh).

2.2. Modelling Additional Losses

The measurement of stray losses in the transformer tank walls does not differentiate between hysteresis, eddy, and excess losses. We isolated each of the six walls, treating them as individual mild steel sheets. This simplification facilitates a more detailed analysis of stray losses. The value of |B_d| is derived through electromagnetic simulations using the Finite Element Method (FEM). Specifically, for calculating the loss components within the laminations of the ferromagnetic material, the methodology of this study aligns with the conceptual framework introduced in [22]. This strategic separation and focused analysis enhances the precision in evaluating the distinct contributions of hysteresis, eddy and excess losses in the transformer tank walls. The hysteresis loss density (Watts/m³) in each of the tank walls will be given by the Steinmetz equation [23]:

$$P_h=k_h{B_d}^{\alpha }f,$$

(1)

where k_h is a hysteresis factor that depends on the properties of the material with which the tank wall is made, k_h = 63 for mild steel commonly used for transformer tanks, α is the Steinmetz coefficient (the value of Steinmetz coefficient α is approximately 2 for all modern magnetic materials) and f is the operating frequency. The additional loss density (Watts/m³) due to eddy losses can be determined by [22] the following:

$$P_e=\frac{1}{6}\sigma {\pi }^2{B_d}^2{f}^2{q}^2,$$

(2)

where σ is the conductivity of the tank wall (In this case, σ = 6.67 × 10⁶ S/m for mild steel) and q is the wall thickness. The additional loss density (Watts/m³) due to excess losses can be determined by [22] the following:

$$P_x=8.76\sqrt{\sigma G{V}_{o}S}{B_d}^{1.5}{f}^{1.5},$$

(3)

where S is the cross section of the tank wall and G and Vo are dimensionless constants related to the quality of the ferromagnetic material with which the tank wall is made. For example, for M4 (0.27 mm), grain-oriented steel G = 0.1356 and Vo = 0.0110 at 1.5 T and 60 Hz, the excess losses in the tank walls were not considered in this work; however, it is possible to calculate them with this proposed technique by finding the |B_d| that reaches the tank wall. Moreover, determining the values of G and Vo of the tank wall would be outside the scope and technical feasibility of this work.

Equations (1)–(3) represent the density of losses per unit volume, so they need to be multiplied by the volume V of each wall to obtain the losses. The additional plate losses of the magnetic shunts can be calculated in the same way using (1)–(3), taking into account that the values of the coefficients k_h, σ, G and Vo change depending on the material used. Mild steel is used for the tank and yoke beams, while grain-oriented silicon steel (GOES) and amorphous silicon steel were analysed to be used for the magnetic shunts.

2.3. Stray Losses Modelling with FEM

Analysing the stray magnetic flux ϕd and its density B_d along with the extra losses in the tank and structural components using the Finite Element Method (FEM) requires the computation of induced losses attributed to eddy current (P_e), excluding hysteresis. To achieve this, a model founded on the solution of the magnetic vector potential A in the frequency domain [24,25] was used. The expressions for this model are as follows:

$$\mathrm{j}\mathsf{\omega }\mathsf{\sigma }\mathrm{A}+\nabla \times \left(\frac{1}{\mu }\nabla \times A\right)-J_e=0,$$

(4)

$$P_e=\frac{1}{2}Re\left[\sum _{i=1}^n\left({\rho }_i{J_{ei}}^{*}{J}_{ei})V_i\right)\right],$$

(5)

where variable n represents the number of elements. Each component contributing to the tank, including the tank walls and magnetic shunts, is characterised by a volume denoted as V_i. Furthermore, ρ_i stands for the diagonal matrix incorporating the resistivity values specific to the materials constituting each element. Additionally, J_ei represents the induced current density vector resulting from the stray or stray magnetic field.

3. Assessment and Model of the Stray Magnetic Field

For determining the stray magnetic field and the associated losses within the tank walls and other ferromagnetic structural components of the transformer, it is necessary to solve Equations (4) and (5) in a three-dimensional (3D) space within the frequency domain. To accomplish this, the Magnetic Fields module of COMSOL Multiphysics was used as the chosen simulation tool for this study. Complementary information of the simulation included the following: around 1,915,500 mesh elements were used (shunts included), the solution time was 88 min 27 s, the virtual memory used was 11.3 GB, Intel64 family computer equipment was used, and there was 16.32 GB RAM available, 6 cores, 3.2 GHz.

3.1. Assessment of the Stray Magnetic Field Density on Tank Walls

The magnitudes of |B_d| on the tank wall can be examined in Figure 2a, excluding the influence of magnetic derivations. For clarity, wall dimensions (Lw and Tw) have been omitted for better visualisation. In particular, the maximum values of |B_d|, which reach 0.9 T, are concentrated in the centre of the Rw side wall. This concentration is attributed to its proximity to the coil. Walls Fw and Bw are shown with a low value of |B_d| = 0.4 T at the centre of the wall. Figure 2b shows the characterised curve of |B_d| on the tank walls. Figure 3a shows the values of |B_d|, considering vertically placed magnetic leads. In this configuration, |B_d| on the tank walls can be reduced to 0.2 T, while the walls of the magnetic shunts have higher values of 0.9 T.

The characterisation of |B_d| on the tank walls protected by magnetic shunts is shown in Figure 3b; magnetic shunts thickness and gap separation with the tank were the same 9 mm (3/8″). In particular, wall Lw keeps |B_d| at elevated values, since it remains unprotected by derivations. On the contrary, the values of |B_d| on the wall Rw decreases to 0.4 T. This reduction depends on the type of material and the thickness of the shunts. In this case, magnetic shunts of the same core material, specifically GOES Steel, were used and simulated. The root mean square (rms) value of |B_d| in the absence of shunts was 0.68 T. After adding shunts, the rms values of |B_d| decreased to levels as low as 0.4 T, representing a substantial 41.1% reduction in |B_d| on the walls of the tank.

3.2. Impact of Winding Asymmetry

The dimensions of the transformer windings, including height and width, play an important role in setting the magnitudes of |B_d| that reach the tank wall and other ferromagnetic structural elements. These dimensions, in turn, influence additional losses. Consequently, this study evaluates the effects of winding geometric asymmetry on the values of |B_d|, examining the balance of the magnetic field intensity |H| that arises from the geometry of the windings. For the designers, it is evident that they will seek to centre the windings; however, it is not evident what effect would be caused to the “magnetic centre” by this geometric asymmetry of the windings, so this impact must be analysed and shown so that the designer can take precautions when they estimate the stray magnetic flux that reaches the tank walls and core clamps. In an ideal two-winding transformer, the magnetic field intensity |H| exhibits an isosceles trapezoidal characteristic, as shown in Figure 4. This ideal condition is achieved when the heights and thicknesses of both windings are equal (hs = hp and ws = wp), which is often observed in transformers with a 1:1 ratio. During the transformer design process, efforts are made to balance the ampere-turns of both windings. However, due to geometric complexities, variations in conductor dimensions and manufacturing adjustments, winding heights and thicknesses may differ. This deviation results in a scalene trapezoidal feature in |H|.

The variation in |H| serves as a key parameter to determine the strength of the scattered magnetic field with |H_d| reaching the core walls and other components. To explore this effect, several cases were analysed to determine the corresponding values of |H_d| and |B_d| that impact the walls of the tank. The results of these analyses are presented in Figure 5, which provides a complete visual representation of the various scenarios considered in this study. Case 1 is the ideal design (hp = hs, wp = ws). The parameters of other cases are mentioned as follows: Case 2, (hp = 0.85·hs, wp = ws), Case 3 (hp = 0.85·hs, wp = 1.15·ws) and Case 4 (hp = 0.85·hs, wp = 1.30·ws).

These cases were considered in a very general way to visualise the impact of the geometric asymmetry of the windings. Analysing and addressing a very specific case could be the content of another work; an example of a specific case could be when tap changers are added. Taps with a wide voltage regulation range, sometimes outside the standards (NFPA-70, IEC-60364-1) ± 5%, implies including more turns and having ampere-turns causes unbalances in the coil, which produces a magnetic unbalance between the coils and increases the stray magnetic field. Case 4 is a common scenario encountered in practical transformer design, where efforts are made to achieve a balance in winding dimensions. Despite these attempts, there is a disparity in both the heights and thicknesses of the windings. Regardless of the specific case, this variation in dimensions directly influences the magnitudes of |H_d|, |B_d|, consequently contributing to additional losses experienced in the tank walls.

3.3. Model to Determine Stray Magnetic Field on Tank Walls

In order to calculate the additional losses that occur in the tank walls, Equations (1)–(3) are used. However, it is necessary to first determine the effective value of |B_d|. Through the analysis performed to characterise |H| and the large amount of information obtained through finite element method (FEM) simulations and with the intention of helping the designer to estimate the stray magnetic field due to the magnetic imbalance as well as the geometric asymmetry of the windings, a model has been successfully derived. This model facilitates the estimation of the magnitude of |H| (expressed in A/m) and is expressed as follows:

$$H_e=\frac{1}{K_R}\left|\left(\frac{N_s{I}_s}{h_s}-\frac{N_p{I}_p}{h_p}\right)\right|,$$

(6)

where Np, Ns, Ip and Is are turns and currents of the primary and secondary windings, respectively, hp, wp, hs and ws are heights and widths of the primary and secondary windings, respectively and K_R is calculated by the Rogowski factor given by [26] the following:

$$K_R=1-\left(\frac{\mathrm{w}\mathrm{p}+\mathrm{w}\mathrm{s}\mathrm{p}+\mathrm{w}\mathrm{p}}{\pi \mathrm{h}\mathrm{w}}\right)\left(1-e^{\frac{-\pi \mathrm{h}\mathrm{w}}{\left(\mathrm{w}\mathrm{p}+\mathrm{w}\mathrm{s}\mathrm{p}+\mathrm{w}\mathrm{p}\right)}}\right),$$

(7)

where hw is the height of the core window. It is proposed that the estimation of the magnitude of |H_d| at a distance d from the external winding (high voltage winding) of the coil is given by the following:

$$H_d=H_e\left(e^{\frac{-d}{\left(\mathrm{w}\mathrm{p}+\mathrm{w}\mathrm{s}\mathrm{p}+\mathrm{w}\mathrm{p}\right)}}\right),$$

(8)

Afterwards, the density of the magnetic field that reaches the walls of the tank would be given by the following:

$$B_d=\left(e^{\frac{-d}{\left(\mathrm{w}\mathrm{p}+\mathrm{w}\mathrm{s}\mathrm{p}+\mathrm{w}\mathrm{p}\right)}}\right)H_e{\mu }_r{\mu }_0,$$

(9)

where µr is the relative permeability and µ₀ = 4π × 10⁻⁷. From (6)–(9), it is possible to show that the proposed technique seeks to determine the stray magnetic field based on the difference in the magnetomotive force (ampere-turns) produced by each winding and thereby later determine the magnetic field density that will reach the tank walls at a distance d from the coil.

4. Design Parameters and Loss Values

4.1. Estimation of Stray Losses with FEM and with the Proposed Model

To compare the results obtained from the Finite Element Method (FEM) simulations with those derived from the proposed model, load tests were carried out on a 25 MVA power transformer operating at 115 kV–34.5 kV, with Delta-connection. The complementary values of the design and manufacturing parameters of the transformer are presented in

Table 1. Design and manufacturing parameters of 25 MVA transformer.

Design Parameter	Value
Core window high (mm)	1482
LV—Margins (mm)	89
HV—Margins (mm)	89
Core Diameter (mm)	552
LV—HV Space (mm)	51
Tank Wall Thickness (mm)	9.5
Core-LV winding distance (mm)	19
Rw wall-HV winding distance (mm)	381
Fw wall-HV winding distance (mm)	330
LV width (mm)	69.8
LV height (mm)	1219.2
HV width (mm)	123.8
HV height (mm)	960
Volts/Turns	23.00

. It should be noted that the losses per phase in the low voltage winding amounted to approximately 43.75 kW, while those in the high voltage winding were around 57 kW. Core losses were measured at approximately 21.7 kW, resulting in an overall transformer efficiency of approximately 98.5% at full load. From these measured loss values and considering efficiency, it was determined that the estimated additional losses would be approximately 35.6 kW. It is worth noting that this value of additional losses cannot be measured directly, much less separated between those attributed to the tank and those associated with the support structures. However, these losses can be estimated based on the total losses measured.

Table 2. Stray or additional losses and |B_d| estimation.

Losses	By FEM	|Bd| by FEM	By (9 & 2)
Lw, left side wall	6455 W	0.520 T	6853 W
Rw right side wall	6853 W	0.528 T	7262 W
Fw, front wall	8322 W	0.534 T	7694 W
Bw, rear wall	7694 W	0.520 T	7091 W
Bottom wall	2517 W	0.493 T	2271 W
Top wall	906 W	0.420 T	1063 W
Top yoke beams	1252 W	0.600 T	1252 W
Bottom yoke beams	1634 W	0.680 T	1634 W

provides the losses obtained by electromagnetic simulations using finite element analysis for the tank walls and yoke beams.

The additional losses in the tank walls amounted to 32.23 kW and the losses in the yoke beams were calculated to be 1.84 kW, resulting in an estimated total additional loss of 34.0 kW using Equation (2). In contrast, finite element method (FEM) simulations yielded a total additional loss of 34.40 kW. The discrepancy between these values and those obtained by measurement is approximately 3.3% to 3.5%.

The third column of Table 2 shows the effective values of the magnetic field density estimated by FEM, as illustrated by the characteristic curve on the walls in Figure 2b. In particular, employing both FEM and the proposed model (Equation (9)) allows detailed quantification of losses in each tank wall and facilitates the determination of losses in the metallic elements that support the core, such as the yoke beams. The introduction of magnetic shunts is seen in the reduction of losses. The obtained values have been summarised in

Table 3. Stray losses and |B_d| estimation with magnetics shunts.

Losses	By FEM	|Bd| by FEM	By (9 & 2)
Lw, left side wall	6853 W	0.520 T	6455 W
Rw right side wall	3457 W	0.428 T	2720 W
Fw, front wall	3151 W	0.322 T	3199 W
Bw, rear wall	2814 W	0.320 T	3151 W
Bottom wall	1565 W	0.324 T	1651 W
Top wall	624 W	0.211 T	697 W
Top yoke beams	530 W	0.266 T	499 W
Bottom yoke beams	704 W	0.299 T	632 W

. According to FEM estimates, the total additional losses with magnetic shunts are 19.70 kW, while the proposed model (Equation (9)) indicates the losses of 19.0 kW, which means a notable loss reduction of approximately 42%.

4.2. Estimation of the Impact of Winding Asymmetry

The study used Finite Element Method (FEM) simulation and the proposed simple model (2) to assess the impact of asymmetry in transformer windings as well as to estimate losses associated with different heights and widths. Figure 6 illustrates magnetic field values obtained through FEM without magnetic shunts around the tank walls, showcasing variations in winding heights and widths. Case 4 characterised by larger dimensions of the outer winding (high-voltage winding) compared to the low-voltage winding yielded the highest values for stray magnetic fields. Conversely, Case 1, representing the ideal scenario with identical heights and widths for all windings, demonstrated the lowest values for stray magnetic fields. In Figure 7, a comparison for Case 1 reveals field values with and without magnetic shunts. Notably, the presence of magnetic shunt protection along certain walls led to a decrease in stray magnetic fields. However, in areas without protection, such as the left wall (Lw), the stray magnetic field increased. This observation underscores the effectiveness of magnetic shunts in reducing losses and highlights their practical utility in transformer design.

Table 4. Stray losses (W) without magnetics shunts.

	Lw	Rw	Fw	Bw	Gw	Tw	Tyb	Byb
Case 1	6171	6579	7972	7348	2416	868	1202	1569
Case 2	6352	6757	8181	7532	2472	891	1226	1603
Real	6455	6853	8322	7694	2517	906	1252	1634
Case 3	6958	7388	8996	8325	2711	978	1350	1763
Case 4	7165	7600	9287	8540	2814	1013	1396	1812

presents a compilation of results derived from varying the heights and thicknesses of the windings for the purpose of visualising their impact. Parameters Tyb and Byb were added, denoting Top yoke beams and Bottom yoke beams, respectively. The findings indicate that when the design aims for equal dimensions in both height and width, there is a noticeable reduction in the stray magnetic field. Consequently, this reduction contributes to a decrease in additional losses, amounting to approximately 4.2%. On the contrary, during the design phase, when the high voltage winding is configured with greater height and width than the low voltage winding, it results in an increase in the stray magnetic field. As a consequence, additional losses also increase, reaching approximately 16.1% in comparison to the ideal case (Case 1) and around 11.2% in the real case. The results, detailed in

Table 5. Stray losses (W) with magnetics shunts.

	Lw	Rw	Fw	Bw	Gw	Tw	Tyb	Byb
Case 1	6565	3388	3025	2699	1534	603	524	682
Case 2	6750	3419	3107	2761	1552	617	529	692
Real	6853	3457	3151	2814	1565	624	537	704
Case 3	7394	3734	3416	3050	1690	680	581	760
Case 4	7627	3837	3523	3129	1751	699	599	781

, demonstrate a consistent trend of magnetic shunts effectively contributing to a reduction in losses, achieving a significant decrease of around 44.2%. Figure 8 depicts the values of |B_d| obtained through the proposed model (6)–(9) for both the real case and case 2. Notably, the values of He and |H_d| were determined as 7.69 × 10⁴ A/m and 1.63 × 10⁴ A/m, respectively. This implies that this magnetic field strength would be present at a distance d—specifically at the edge of the right wall Rw of the tank (d = 381 mm). It is essential to clarify that the proposed model does not account for the ideal case due to the low probability of achieving symmetry in winding dimensions during manufacturing.

4.3. Materials for Magnetic Shunts

Three materials, namely Mill anneal electrical steel, Electrical Silicon Steel Grain Oriented (CRGO) and Amorphous Steel, were reviewed and analysed using finite element method (FEM) simulation. Mill Steel served as a reference point, with an approximate cost of $1200 per ton. It possesses a relative permeability of 2000 and a resistivity of 0.143 µΩm, for CRGO 0.48 µΩm and Amorphous Steel 1.3 µΩm, remembering that the relative permeability of these latter materials depends on their B–H curve. It is important to clarify that the manufacturer of these magnetic shunts uses scrap or residual material from the manufacturer of the core; the material was not purchased for this purpose. Also, the amorphous material is very fragile and difficult to work with, so it is rarely used as a shield for the tank, but it was considered to be analysed in this work since it is also in the scrap material from the cores.

To standardise the sizing of the magnetic shunt plates and facilitate an objective comparison among the considered materials, the widely recognised magnetic field penetration formula [27] was employed. This approach ensures a systematic evaluation based on the thickness of the shunt plates, allowing for a comprehensive assessment of their performance characteristics.

$$\delta =\sqrt{\frac{\rho }{\mu \pi f}},$$

(10)

where ρ is the material resistivity, µ is the permeability of the material and f is the operating frequency of the transformer. Taking into account a magnetic shunt plate with a thickness of 1 mm, Figure 9a illustrates the magnetic field distribution across a 1 mm thick steel sheet for the various materials under analysis, accompanied by visible magnetic field lines. To enhance the clarity of the magnetic field distribution values within the magnetic shunts, a more detailed presentation has been provided in Figure 9b. Observing Figure 9b, it is evident that CRGO Electrical Steel exhibits a higher magnetic field absorption compared to other materials. However, its cost is higher than that of Mill anneal steel, which has garnered more popularity as a magnetic shunt material owing to its cost-effectiveness and satisfactory absorption capacity. On the other hand, the amorphous material sheet is recognised for its excellence in transformer core development due to its high permeability and resistivity, resulting in minimised electrical losses. Nevertheless, creating magnetic shunts using amorphous material is not technically or economically feasible due to its limited magnetic field penetration capacity as the format is incorrect.

5. Impact of Shell-Type Configuration on Additional Losses

Transformer manufacturers widely recognise the shell-type configuration as an effective method to mitigate the stray of the magnetic field and, consequently, reduce additional losses. Despite its significance, the quantification of these benefits has not received widespread promotion. In Figure 10, the model of the shell-type configuration and its analysis path for a pad-mounted-type distribution transformer are depicted. The design and manufacturer parameters for the analysed 1500 kVA shell-pad-mounted transformer are detailed in

Table 6. Core- and shell-type pad-mounted transformer design and manufacturing parameters.

Design Parameter	Shell Type	Core Type
Core Window High (mm)	274	274
LV—Margins (mm)	20	20
HV—Margins (mm)	40	40
Core Length (mm)	237	237
LV—HV Space (mm)	10	10
Tank Wall Thickness (mm)	4.7	4.7
Core-LV Winding Distance (mm)	4	4
Rw wall-HV Winding Distance (mm)	−	38.1
Rw wall-Core leg Distance (mm)	25.4	−
Fw wall-HV Winding Distance (mm)	40	40
LV Width (mm)	55	55
LV High (mm)	274	274
HV Width (mm)	42	42
HV High (mm)	234	234
Volts/Turns	22.8	22.8
Tank Length (mm)	2121	1964
Tank Width (mm)	1370	1370
Tank Hight (mm)	1760	1960

. Noteworthy characteristics include a Delta connection at 22.5 kV in the primary winding, a 220/127 V square core cross section, an average magnetic field of 1.6 T in the core and an efficiency of 99% for the shell type compared to 98.8% for the core type. Figure 11 shows the magnetic field distribution in the active elements (core and coils) as well as on the tank walls of the shell-type transformer. In this representation, the average magnetic field in the core is approximately 1.6 T, while on the tank walls, the average field is around 300 mT, with darker spots reaching 350 mT in areas where the active elements are close to the walls.

Figure 12 shows a comparison of the magnetic field distribution along an analysis path for a 1500 kVA core-type pad mounted l transformer (represented by the solid blue line) and a shell-type transformer (represented by the dotted green line). Notably, the core-type transformer exhibits a shorter analysis path due to its shorter front and rear walls in comparison to the shell type. Furthermore, the magnetic field values for the core type are higher on the tank walls, reaching approximately 410 mT. In contrast, the shell type registers its highest magnetic field value around 350 mT, signifying a 14% reduction in the stray magnetic field reaching the magnetic tank. While this reduction would ideally correspond to a proportional decrease in additional losses, it is essential to consider that the tank for the shell-type transformer is longer to accommodate the active elements. As a result, the anticipated reduction in losses is more modest, estimated to be in the range of 6% to 8%.

Table 7. Stray losses for shell and core type.

	Shell Type		Core Type
	By FEM	|Bd| by (2)	By FEM	|Bd| by (2)
Lw, left side wall	218.0 W	231.4 W	295.9 W	314.8 W
Rw right side wall	217.9 W	230.9 W	289.2 W	312.9 W
Fw, front wall	422.2 W	390.3 W	412.9 W	451.3 W
Bw, rear wall	388.6 W	358.1 W	405.4 W	442.6 W
Bottom wall	308.9 W	278.7 W	286.4 W	315.4 W
Top wall	181.7 W	213.2 W	220.8 W	242.6 W

provides results obtained through the finite element method (FEM) and the proposed model (9), facilitating a comparison and confirming the effectiveness of reducing stray losses in the transformer tank with the adoption of a shell-type configuration. It must be clarified that although it is observed that stray losses are lower in the shell-type configuration, there are other elements that must be taken into account while designing and estimating total losses in the transformer. In a shell-type configuration, the core grows, and the no-load losses will be greater. In terms of manufacturing, its cost increases along with the increase in manufacturing time and materials volume; however, this work focuses on highlighting the decrease in the stray magnetic field. Consequently, a decrease in stray losses in the tank walls and core clamps is needed for the designer to have an idea of the impact of the stray loss reduction with the shell configuration and to make a decision for its manufacturing.

6. Conclusions

Reducing additional losses in transformers implies having a competitive advantage in the business of designing and manufacturing transformers. Some important technical literature is available on this topic; however, even in the design stage there are disadvantages such as computational resources and time for performing simulations with FEM to determine the stray magnetic field. This study has addressed this challenge, highlighting the impact that designing and manufacturing asymmetric coils has on the stray magnetic field. It has also addressed the challenge of showing an easy-to-use and low computational cost technique in the design stage to determine the magnetic field that would reach the tank walls and consequently the stray losses that are induced. The proposed technique was compared with the simulations carried out in a commercial FEM programme. From Section 4.1, with the measured values, the difference in the results was 3%, so this value can help with trusting its utilisation.

Also in this work, in addition to estimating the magnetic field on the tank walls, the magnetic field that reaches these walls was characterised, allowing for the visualisation and understanding of its behaviour with the purpose of estimating the thickness of the magnetic shunt plates and the best position to place them. It was demonstrated that the placement of the magnetic shunts allows the tank stray losses to be reduced by up to 40%. It was also evident that if the thickness and heights of the windings, in the design stage, are sought to be more equal, stray magnetic fields can be avoided, and consequently, stray losses on the tank walls can be reduced by 11%. It is possible to further reduce losses by using other accessories for the tank such as non-magnetic stainless steel; however, this work focused on determining the impact of using magnetic shunts to reduce stray losses. This work also demonstrates the advantage of using a shell-type configuration to reduce the magnetic field that hits the tank walls, demonstrating that this configuration by itself can reduce the stray losses of the tank walls up to 14%. However, it should also be considered that the shell configuration can increase the size of the tank, which means that losses may only be reduced between 6% and 8%. In general, with the results and evidence shown in this work, useful information regarding the design and manufacturing of transformers has been presented.