#
Modelling of the Electrically Excited Synchronous Machine with the Rotary Transformer Design Influence^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. The EESM Model with Vector Control System

#### 2.1.1. Modeling of the EESM

#### 2.1.2. Calculation of the Field Excitation Current

#### 2.1.3. Transformation Block

#### 2.1.4. Motor Voltage/Current Models

#### 2.2. Selection of the EESM for Modeling and Analysis

#### Parameters of the Real EESM Used for Modeling

#### 2.3. The Wireless Power Transfer System Applied for the EESM Excitation

#### 2.3.1. The Geometries of Wireless Power Transfer Systems Applied for EESM Excitation

#### 2.3.2. Analytical Approach for Modeling of Rotary Transformers

^{2}.

^{2}, where I is the conductor current in A and the ${S}_{w}$ is the conductor cross-section in mm

^{2}and the ${A}_{w}$ is the cross-section of the one side winding window area in mm

^{2}which can be calculated as ${A}_{w}=NI/{k}_{cu}J$.

_{c}area which occurs due to the fringing effect. The coefficient $k$ denotes the ratio between the extension of the magnetic path length due to the fringing flux and the actual air gap.

#### 2.3.3. The Time Constant Determination of the Axial and Radial Rotary Transformers

#### 2.4. Estimation of the Change in Dynamic Performance of the EESM Control System Due to the Wireless Power Transfer Excitation

#### The Method Used for Comparison of the Dynamic Performances between Direct and Electromagnetic Coupling Approaches

## 3. Results and Discussion

#### 3.1. Results of the Axial and Radial Rotary Transformer Modeling

#### 3.1.1. Design Methodology of the Axial and Radial Rotary Transformers with Different Supply Frequencies

^{2}) is kept constant in both topologies (i.e., radial and axial) in order to perform a precise comparison between the two topologies. Moreover, the second main cross-section ${A}_{c2}$ and auxiliary cross-section area ${A}_{c3}$ are kept equal to the first main cross-section ${A}_{c1}$ of the ferromagnetic core (i.e., ${A}_{c1}={A}_{c2}={A}_{c3}$) in all cases of the supply frequency.

#### 3.1.2. The Comparison of the Analytical Models with Numerical Modeling of the Axial and Radial Rotary Transformers Topologies

#### 3.2. Modeling Results of the EESM Control System with Rotary Transformer (i.e., WPT) and Comparison to the Conventional Excitation (i.e., Sliding Contacts)

## 4. Conclusions

- -
- The axial topology of the rotary transformer outperforms the radial type, since the minimum size with the best magnetic properties can be obtained with the axial topology of the rotary transformer for the same supply frequency. Namely, our results show that the best design in terms of size, time constant and inductance values can be obtained with the axial rotary transformer topology at the supply frequency of 400 Hz. In addition, the rotary transformer supplied with 400 Hz enables the usage of the classic H-bridge inverter which is an important advantage in terms of cost-effectiveness of the WPT system, while at higher frequencies (e.g., 1000 Hz), a special and more expensive high frequency inverter is required.
- -
- The comparison of the analytical and numerical results in 12% difference for both topologies (axial and radial). Thus, the developed analytical tool can be used as an efficient alternative for time-consuming numerical simulations.
- -
- The usage of the rotary transformer instead of direct contact degrades the dynamic performance of the EESM by less than 5.38% in the case of 400 Hz design of the rotary transformer. Further improvement of the dynamic response of the EESM with rotary transformer can be achieved with the proper adjustment of the PI controllers’ gains of the control system. The future work towards improvement of the EESM dynamic response can also include the advanced optimization procedure for tuning the PI gains of the model controllers including the automatic recalculation of the geometry of the rotary transformer based on the rotary transformers’ time constant as an optimization goal. These results may have an important contribution in the field of the development of the power electronics required for the excitation circuits of the EESM.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Modeling parameters of the three-phase synchronous generator ET16F-130/A from Mecc Alte S.p.A [26].

Parameters | Value (SI) | Value (pu) | |
---|---|---|---|

Apparent power | S | 5500 VA | 1.0011 |

Nominal active power | P_{n} | 4400 W | - |

Nominal voltage | U_{n} | 400 V | 1.2247 |

Nominal current | I_{n} | 7.93 A | 0.7071 |

Nominal field current | I_{f_no_load}I _{fn} | 1.11 A 4.11 A | 0.0711 0.2639 |

Nominal field voltage | U_{fn} | 44 V | 0.124 |

Stator winding resistance | R_{s} | 1.5 Ω | 0.0515 |

Field winding resistance without resistance of brushes | R_{f} | 7.25 Ω | 0.3378 |

Field winding time constant | τ_{f} | 206.8 ms | - |

Nominal frequency | f | 50 Hz | - |

Number of pole pairs | p | 1 | - |

Moment of inertia | J | 0.0129 kgm^{2} | - |

Power factor | cosφ | 0.8 | - |

Nominal speed | n | 3000 RPM | - |

**Figure A1.**Experimentally determined inductance of the excitation winding of the three-phase synchronous generator ET16F-130/A from Mecc Alte S.p.A.

## Appendix B

## Appendix C

- Direct excitation of the rotor winding via slip rings/brushes. In this case, the supplied AC voltage from the grid via variable transformer have been rectified with the diode rectifier and fed into the rotor windings.
- Indirect excitation of the rotor winding via transformer (with the following parameters obtained from the short circuit test: short circuit resistance R
_{k}= 0.8 Ω and short circuit reactance X_{k}= 0.5 Ω). In this case, the supplied AC voltage from the grid via variable transformer have been supplied to the primary winding of the conventional transformer. The voltage from the secondary transformer was rectified with the diode rectifier and fed into the rotor windings via slip rings/brushes.

**Figure A3.**Experimental setup for measurements of the three-phase synchronous generator ET16F-130/A from Mecc Alte S.p.A.

**Figure A4.**Experimentally measured excitation current response of the field winding of the three-phase synchronous generator with and without the presence of the transformer.

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**Figure 2.**The equivalent circuit of the EESM (

**a**) in the d-axis direction referred to the stator winding; (

**b**) in the q-axis direction referred to the stator winding.

**Figure 3.**The modeled EESM-ET16F-130/A (i.e., a real electrical machine available in the Laboratory of Electrical Machines, UNI LJ).

**Figure 4.**The modeled single-phase rotary transformer topologies: (

**a**) axial and (

**b**) radial. (The air gap width in the figures between the secondary and primary parts of the rotary transformers is intentionally enlarged for demonstrational purposes).

**Figure 5.**The modeled rotary transformers: (

**a**) single-phase axial rotary transformer; (

**b**) magnetic reluctance equivalent circuit of the single-phase axial rotary transformer topology; (

**c**) single-phase radial rotary transformer and (

**d**) magnetic reluctance equivalent circuit of the single-phase radial rotary transformer topology.

**Figure 6.**Equivalent electric circuit (

**a**) and simplified equivalent electric circuit (

**b**) of the single-phase rotary transformer [22].

**Figure 7.**Frequency dependence of the rotary transformer volume for axial topology calculated for three different magnetic flux densities (B = 1 T, 1.3 T and 1.5 T): (

**a**) whole frequency range with zoomed area at the frequency interval 0–500 Hz; (

**b**) zoomed area at the frequency interval 500–2500 Hz.

**Figure 8.**The calculated profile of the time constant vs. supply frequency for the axial (

**a**) and radial (

**b**) topology.

**Figure 9.**Magnetic flux density distribution (

**a**) and flux lines’ distribution (

**b**) in the core of axial rotary transformer with the supply frequency 400 Hz. With “cross” and “dots”, the direction of the current in the windings of the rotary transformer have been marked.

**Figure 10.**Comparison of speed (

**a**) and torque (

**b**) curves of the EESM operation with direct contact approach and electromagnetic coupling approach.

**Figure 11.**Comparison of excitation voltage curves of the EESM during transients with direct contact approach and electromagnetic coupling approach.

Parameters | Axial Rotary Transformer Topology | Radial Rotary Transformer Topology |
---|---|---|

Outer radius of the core (primary/secondary side) | ${r}_{3}$ | ${r}_{3}$ |

Outer radius of the winding window | ${r}_{2}$ | ${r}_{2}$ |

Inner radius of the winding window | ${r}_{1}$ | ${r}_{1}$ |

Shaft radius | ${r}_{sh}$ | ${r}_{sh}$ |

Height of the winding window | ${l}_{N}$ | b |

Width of the winding window | b | ${l}_{N}$ |

Width of the first main cross-section ${A}_{c1}$ | ${w}_{1}$ | ${w}_{1}$ |

Width of the second main cross-section ${A}_{c2}$ | ${w}_{2}$ | ${w}_{2}$ |

Width of the auxiliary cross-section ${A}_{c3}$ | ${w}_{3}$ | ${w}_{3}$ |

Air gap width | $\delta $ | $\delta $ |

Winding window cross-section area | ${A}_{w}$ | ${A}_{w}$ |

First main cross-section area of the ferromagnetic core | ${A}_{c1}$ | ${A}_{c1}$ |

Second main cross-section area of the ferromagnetic core | ${A}_{c2}$ | ${A}_{c2}$ |

Auxiliary cross-section area of the ferromagnetic core | ${A}_{c3}$ | ${A}_{c3}$ |

Core material | Cogent M250-35A | Cogent M250-35A |

Parameters | Axial Rotary Transformer Topology | Radial Rotary Transformer Topology | ||||
---|---|---|---|---|---|---|

Supply frequency | 50 Hz | 400 Hz | 1000 Hz | 50 Hz | 400 Hz | 1000 Hz |

${r}_{3}$ | 103 mm | 51 mm | 42 mm | 99 mm | 47 mm | 38 mm |

${r}_{2}$ | 84 mm | 47 mm | 40 mm | 69 mm | 32 mm | 25 mm |

${r}_{1}$ | 59 mm | 22 mm | 15 mm | 59 mm | 22 mm | 15 mm |

${r}_{sh}$ | 8 mm | 8 mm | 8 mm | 8 mm | 8 mm | 8 mm |

${l}_{N}$ | 25 mm | 25 mm | 25 mm | 25 mm | 25 mm | 25 mm |

$b$ | 10 mm | 10 mm | 10 mm | 10 mm | 10 mm | 10 mm |

${w}_{1}$ | 51 mm | 14 mm | 7 mm | 51 mm | 14 mm | 7 mm |

${w}_{2}$ | 19 mm | 4 mm | 2 mm | 19.7 mm | 4.7 mm | 2.7 mm |

${w}_{3}$ | 29 mm | 10 mm | 6 mm | 29 mm | 10 mm | 6 mm |

δ | 0.3 mm | 0.3 mm | 0.3 mm | 0.3 mm | 0.3 mm | 0.3 mm |

Total volume | 2.4 × 10^{−3} m^{3} | 2.1 × 10^{−4} m^{3} | 0.845 × 10^{−4} m^{3} | 2.4 × 10^{−3} m^{3} | 2.59 × 10^{−4} m^{3} | 1.28 × 10^{−4} m^{3} |

Turns (primary/secondary) | 33/33 | 33/33 | 33/33 | 33/33 | 33/33 | 33/33 |

Winding DC resistance (primary/secondary) | 0.1361 Ω/0.1361 Ω | 0.0657 Ω/0.0657 Ω | 0.0524 Ω/0.0524 Ω | 0.1414 Ω/0.1218 Ω | 0.071 Ω/0.0514 Ω | 0.0577 Ω/0.0381 Ω |

Parameters | Analytical Calculation | Numerical Calculation | Analytical Calculation | Numerical Calculation | Analytical Calculation | Numerical Calculation |
---|---|---|---|---|---|---|

Axial rotary transformer topology | ||||||

50 Hz | 400 Hz | 1000 Hz | ||||

${L}_{m}$ | 24.3 mH | 25.5 mH | 3.1 mH | 3.31 mH | 1.3 mH | 1.46 mH |

${L}_{\sigma 1}$ | 85.66 μH | 85.28 μH | 41.33 μH | 40.56 μH | 32.94 μH | 31.94 μH |

${L}_{\sigma 2}$ | 85.66 μH | 85.28 μH | 41.33 μH | 40.56 μH | 32.94 μH | 31.94 μH |

${L}_{\sigma}={L}_{\sigma 1}+{L}_{\sigma 2}$ | 171.32 μH | 170.56 μH | 82.66 μH | 81.12 μH | 65.88 μH | 63.88 μH |

τ | 89.9 ms | 94.3 ms | 23.9 ms | 25.8 ms | 13.3 ms | 14.54 ms |

Radial rotary transformer topology | ||||||

${L}_{m}$ | 29.3 mH | 29.15 mH | 5.1 mH | 4.8 mH | 2.6 mH | 1.89 mH |

${L}_{\sigma 1}$ | 89.01 μH | 86.08 μH | 44.69 μH | 41.71 μH | 36.3 μH | 40.57 μH |

${L}_{\sigma 2}$ | 76.67 μH | 79.4 μH | 32.35 μH | 34.87 μH | 23.96 μH | 15.19 μH |

${L}_{\sigma}={L}_{\sigma 1}+{L}_{\sigma 2}$ | 165.68 μH | 165.48 μH | 77.04 μH | 76.58 μH | 60.26 μH | 55.76 μH |

τ | 111.9 ms | 111.4 ms | 42.4 ms | 39.84 ms | 27.8 ms | 20.31 ms |

**Table 4.**The time constants of the rotary transformers at different supply frequencies and their RMSE error.

Supply Frequency of the Rotary Transformer | Rotary Transformer’s Parameters | |
---|---|---|

Axial rotary transformer | Time constant value (τ) | RMSE (Δ) |

50 Hz | 89.9 ms | 7.86% |

400 Hz | 23.9 ms | 5.38% |

1000 Hz | 13.3 ms | 6.06% |

Radial rotary transformer | Time constant value (τ) | RMSE (Δ) |

50 Hz | 111.9 ms | 8.76% |

400 Hz | 42.4 ms | 6.16% |

1000 Hz | 27.8 ms | 6.43% |

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## Share and Cite

**MDPI and ACS Style**

Manko, R.; Vukotić, M.; Makuc, D.; Vončina, D.; Miljavec, D.; Čorović, S.
Modelling of the Electrically Excited Synchronous Machine with the Rotary Transformer Design Influence. *Energies* **2022**, *15*, 2832.
https://doi.org/10.3390/en15082832

**AMA Style**

Manko R, Vukotić M, Makuc D, Vončina D, Miljavec D, Čorović S.
Modelling of the Electrically Excited Synchronous Machine with the Rotary Transformer Design Influence. *Energies*. 2022; 15(8):2832.
https://doi.org/10.3390/en15082832

**Chicago/Turabian Style**

Manko, Roman, Mario Vukotić, Danilo Makuc, Danijel Vončina, Damijan Miljavec, and Selma Čorović.
2022. "Modelling of the Electrically Excited Synchronous Machine with the Rotary Transformer Design Influence" *Energies* 15, no. 8: 2832.
https://doi.org/10.3390/en15082832