# Numerical and Experimental Evaluation and Heat Transfer Characteristics of a Soft Magnetic Transformer Built from Laminated Steel Plates

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Setup

_{max}used to design the transformer was 1.5 T. The number of primary turns is 170, distributed in 2 layers, and the number of secondary turns is 288, distributed in 4 layers. The main transformer dimensions are shown in Figure 1.

**2**13F), down height (2

**1**3F), primary winding (21

**3**F), and the front side of the limb (213

**F**). The detailed arrangement of these sensors and the nomenclature adopted for each measurement point is depicted in Figure 2.

_{floor}, just below the transformer during the experiments.

#### 2.2. Numerical Model

_{floor}.

^{−3}.

#### 2.2.1. Constitutive Equations and Boundary Conditions

_{p}represent the solid material density and specific heat, respectively, $\mathit{q}=-k\nabla T$ is the heat rate conducted inside the solid body, Q

_{gen}denotes the heat generated inside the solid bodies per unit volume, which corresponds to the electromagnetic heating of the components of the transformer, and Q

_{dis}refers to the heat dissipated at the boundaries of the system per unit volume. For simplicity, no contact thermal resistances are considered between the different solid bodies that conform the transformer provided that a sensitivity analysis of the thermal contact resistance between the core and the windings resulted in maximum differences of the maximum temperature of the system below 3 °C. Furthermore, due to the symmetry of the system and the boundary conditions, two symmetry planes are employed. The first is a vertical symmetry plane (plane XZ in Figure 6) and the second symmetry plane considers the symmetry of the system regarding the central limb of the core (plane YZ in Figure 6). Therefore, the volume of the model is reduced to one-fourth, largely decreasing the computational cost of the simulations.

_{a}, and supported by a wooden plate, resting on an insulator plate in contact with another wooden plate over the laboratory floor. The whole transformer is initially at room temperature. The boundary conditions considered in the numerical model are natural convection and radiation at the external walls of the transformer and a prescribed boundary temperature at the base of the insulation plate, T

_{floor}, which was measured experimentally. The initial (IC) and boundary (BC) conditions can be expressed as follows:

_{conv}refers to the natural convection coefficient and ε is an average surface emissivity used for the external surfaces of the transformer. The natural convection coefficient, h

_{conv}, was estimated using the widely known correlations of McAdams [29] and Churchill and Chu [30] as described below, whereas a nominal value of the average emissivity of the external surfaces of ε = 0.8 was selected based on the literature [16,21]. However, due to the incertitude associated with the actual emissivity of the boundaries of the transformer, different values of ε were used along this work to evaluate its effect on the steady-state temperature of the transformer. Furthermore, to simplify the numerical model and to avoid the use of surface-to-surface radiation conditions, the external boundary regions of the transformer that have a large view factor to other regions of the transformer, i.e., the regions of the core and windings in the internal part of each of the limbs, are not considered in ambient radiation. The duration of the experiments is above 10 h, which may cause significant variability in the laboratory conditions during the tests. Thus, in both Equations (3) and (4), the temperature of the ambient air T

_{a}and the bottom surface of the insulation plate T

_{floor}are time-dependent, and the values measured along the experiment are used as an input to the numerical model for validation of the transient heat transfer simulations.

_{p}, μ, and k are the thermal expansion coefficient, the density, the specific heat, the dynamic viscosity, and the thermal conductivity of air, respectively.

#### 2.2.2. Materials and Physical Properties

^{3}. Additionally, the density of the core was estimated by measuring the weight of the whole transformer, including the core and the fittings, and subtracting the winding mass, resulting in an equivalent density of the core of 9012 kg/m

^{3}.

#### 2.3. Heat Generation Due to Electromagnetic Losses

^{3}and the volume of the core and the windings should be considered.

_{rms}represents the root mean current of each of the R-S-T phases of the transformer, R

_{pr}and R

_{sec}are the resistances of the primary and secondary, respectively. Finally, P

_{loss_pr}and P

_{loss_sec}correspond, respectively, to the power losses in the primary and secondary, considering the primary is connected in delta and the secondary in star (see description of Figure 4).

## 3. Results and Discussion

#### 3.1. Experimental Results

_{a}and the floor temperature T

_{floor}. This set of thermocouples show the thermal behavior of the transformer.

- The thermal behavior of the left and right limbs is similar. For that reason, only temperatures of the left and central limbs, i.e., T1xxx and T2xxx, were considered.
- The thermal behavior of the front and rear sides is also similar. Thus, only temperatures of the front side were evaluated (TxxxF).
- The temperature value reached at the intermediate depth (Txx2x), i.e., in the secondary winding, must be a value between the core surface and the primary winding temperatures. Hence, only temperatures on the core surface and the primary winding were selected (Txx1x and Txx3x).
- The temperature distribution in the center limb is symmetrical. Therefore, only the temperatures of the front and left of the center limb were chosen (T1xxF and T1xxL).

_{a}and the temperature at the base of the insulator situated below the transformer, T

_{floor}. Thus, all the rest of the temperatures measured during the experiment remain between the red and blue curves of Figure 8. All the registered temperature signals present an inverse exponential increase, reaching steady-state conditions after 15 h since the transformer was switched on. The heating time and the temperature were, as expected, different depending on the position of the temperature sensors. For instance, the minimum measured temperature corresponds to sensor T133L, which is situated on the left-hand side of the first limb, below the center (down location) of the primary winding, and reaches a steady temperature after around 10 h. The maximum measured temperature in the system occurs at the plane of symmetry of the transformer, on the core surface (T221F), reaching steady conditions around 15 h after the transformer is switched on. From a practical point of view, it is of great interest to demonstrate the capabilities of the model to predict this maximum temperature, as it will be the limiting operative factor of the transformer. Thus, this point will be further analyzed and compared with the outcome of the simulations in the subsequent sections.

#### 3.2. Numerical Results

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Main dimensions of the transformer. Dimensions of steel plates (mm): 1.60 × 536; 2.70 × 65; 3.70 × 536, 4.70 × 55.

**Figure 7.**Schematic view of the core laminated sheet, including a detail of the sheet in the transversal direction and cross-section of the wiring in the primary and secondary. The scaling and the number of layers are schematic and do not correspond to reality.

**Figure 8.**Time evolution of the maximum, minimum, wood bottom, and ambient temperature measured by the thermocouples in the system.

**Figure 9.**Snapshots of the simulated temperature distribution, in °C, of the transformer under steady-state conditions. Cases 1 (

**left**) and 6 (

**right**).

**Figure 10.**Comparison of the maximum temperature measured in the prototype and the numerical model estimation for the cases listed in Table 1 for various surface emissivity values.

**Figure 11.**Temperature distribution, in °C, in a horizontal slice in the central section of the core and the windings for different time instants. Case 6.

**Figure 12.**Time evolution of the temperature at different points of the transformer. Comparison between experiments and simulations. Case 6.

**Table 1.**Cases simulated, including the total power input in the transformer and the power loss per unit volume in each of the components.

Case | Power Input, Transformer (W) | Total Power Losses (W) | Percentage of Power Losses in the Core (%) | Power Losses per Unit Volume—Core (kW/m^{3}) | Power Losses per Unit Volume—Winding (kW/m^{3}) |
---|---|---|---|---|---|

1 | 1205.5 | 71.8 | 94.3 | 8.8 | 0.96 |

2 | 4202.1 | 106.6 | 62.0 | 8.6 | 9.5 |

3 | 6547.3 | 165.7 | 39.4 | 8.5 | 23.6 |

4 | 7647.5 | 211.4 | 32.4 | 8.9 | 33.6 |

5 | 9399.7 | 304.8 | 27.0 | 10.7 | 52.3 |

6 | 11,160.3 | 437.5 | 24.3 | 13.9 | 77.9 |

7 | 12,534.2 | 578.7 | 23.1 | 17.4 | 104.7 |

Left Limb | Centre Limb | |||||
---|---|---|---|---|---|---|

Front | Left | Right | Front | Left | ||

Down | Core | T111F | T111L | T111R | T211F | T211L |

Centre2 | T113F | T113L | T113R | T213F | T213L | |

Centre | Core | T121F | T121L | T121R | T221F | T221L |

Centre2 | T123F | T123L | T123R | T223F | T223L | |

Up | Core | T131F | T131L | T131R | T231F | T231L |

Centre2 | T133F | T133L | T133R | T233F | T233L |

**Table 3.**Temperature values under steady-state conditions for different points in the transformer. Comparison between experimental and numerical results.

Position | T_{sim} (°C) | T_{exp} (°C) | Deviation (°C) |
---|---|---|---|

231F | 103.3 | 104.0 | 0.7 |

221F | 111.2 | 114.2 | 3.0 |

211F | 106.7 | 104.0 | 2.7 |

231L | 103.3 | 104.2 | 0.9 |

221L | 111.2 | 112.5 | 1.3 |

211L | 106.7 | 100.1 | 6.6 |

131F | 100.6 | 100.4 | 0.2 |

121F | 107.0 | 110.9 | 3.9 |

111F | 103.8 | 100.6 | 3.2 |

131L | 100.7 | 99.2 | 1.5 |

121L | 106.8 | 107.9 | 1.1 |

111L | 103.9 | 101.4 | 2.5 |

131R | 100.7 | 99.8 | 0.9 |

121R | 106.8 | 107.2 | 0.4 |

111R | 103.9 | 98.2 | 5.7 |

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**MDPI and ACS Style**

Cano-Pleite, E.; Barrado, A.; Garcia-Hernando, N.; Olías, E.; Soria-Verdugo, A.
Numerical and Experimental Evaluation and Heat Transfer Characteristics of a Soft Magnetic Transformer Built from Laminated Steel Plates. *Sensors* **2021**, *21*, 7939.
https://doi.org/10.3390/s21237939

**AMA Style**

Cano-Pleite E, Barrado A, Garcia-Hernando N, Olías E, Soria-Verdugo A.
Numerical and Experimental Evaluation and Heat Transfer Characteristics of a Soft Magnetic Transformer Built from Laminated Steel Plates. *Sensors*. 2021; 21(23):7939.
https://doi.org/10.3390/s21237939

**Chicago/Turabian Style**

Cano-Pleite, Eduardo, Andrés Barrado, Néstor Garcia-Hernando, Emilio Olías, and Antonio Soria-Verdugo.
2021. "Numerical and Experimental Evaluation and Heat Transfer Characteristics of a Soft Magnetic Transformer Built from Laminated Steel Plates" *Sensors* 21, no. 23: 7939.
https://doi.org/10.3390/s21237939