_{1}

^{*}

In this paper is proposed a methodology to properly determine the leakage and magnetizing inductance of a single-phase high-frequency coaxial transformer. Both, leakage and magnetizing inductance equations are determined considering the transformer dimensions, the ferromagnetic characteristics, and the number of turners. Finite element methods (FEM) and a low scale prototype are also used to validate the equations . The results show that the leakage and magnetizing inductances can be precisely calculated with an error lower than 5% and 3%, respectively.

Coaxial transformers are normally used in high-frequency radio applications because it confines the flux leakage inside of the transformer windings [

In this section, the design of the leakage and the magnetizing inductances of a coaxial transformer with two windings are described.

To calculate the leakage inductance, a coaxial transformer described in [

Leakage inductances can be determined in different ways. In this paper, the inductances are obtained calculating first the magnetic field and energy in different parts of the transformer, and subsequently, obtaining the total leakage inductance. Thus, to carry out the calculations, the coaxial transformer can be divided in four zones, as shown in

Thus, applying this method for each zone, and summing the four resulting energies, the leakage inductance (per meter) in a coaxial transformer can be determined by

where the symbol μ_{o} is the permeability of free space, N is the transformer turn ratio, and R_{1}, R_{2}, and R_{3} are shown in

To validate the theoretical calculation obtained in (2), Finite Element Methods (FEM) were developed using Maxwell software and the results are shown in

A 2D FEM numerical model of the coaxial transformer in _{1} and R_{3}.

The next step is to calculate the coaxial transformer magnetizing inductance. By applying the same methodology described to obtain the leakage inductance, the magnetizing inductance can be theoretically determined by

where the symbol

With the leakage and the magnetizing inductances, calculated by (2) and (3), respectively, a two by two matrix inductance for this coaxial transformer can be obtained, and it can be compared with the matrix inductance generated by the Maxwell software, to validate the theoretical models.

Using Maxwell software it is also possible to generate a 3D model to evaluate the fringing flux effect in the external cables in a real application as shown in

For a three-winding transformer, the inductance matrix is defined as

To calculate these six parameters in a real transformer, six steps must be carried out: measure three reflected magnetizing inductances on each side, considering the other two windings open, then short-circuit two windings and measure the remaining two leakage inductances for all windings. For transformers with more than three windings a π-model is over determined.

To validate the theoretical and the finite element results, prototypes of the coaxial transformer, shown in

To determine the matrix inductance the T-model, described in [

The equation given by (6) is represented by the circuit shown in _{12} = M_{21} = M. The primary and secondary leakage inductance equations are defined based on

Theoretical, simulated, and experimental results were obtained at 10 kHz. Theory and simulation considered the skin effect.

Leakage [nH] | Magnetizing [μH] | |
---|---|---|

Theoretical (1 tube) | 65.13 | 28.22 |

Theoretical (2 tubes) | 65.13 | 28.22 |

3D Maxell software (2 tubes) | 96.61 | 28.34 |

Experimental (1 tube) | 68.54 | 27.75 |

Experimental (2 tubes) | 92.99 | 27.61 |

The theoretical, the simulated, and the experimental inductance matrix of the configuration described in

A methodology to determine the inductances of a coaxial transformer was developed in this paper. If the dimensions and material characteristics used to build the transformer are known, the leakage and magnetizing inductances can be precisely calculated. All results were obtained in three ways: theoretically, by Finite Element Methods, and with measurements on an experimental setup.

The magnetizing inductance of this transformer is quite low (~28 μH). To obtain a higher magnetizing inductance, three changes can be made: increasing the number of turns of the outer tube, increasing the toroidal core dimensions, or increasing both dimensions and number of turns.

Gierri Waltrich, (2016) Inductances Design of High-Frequency Coaxial Transformers. Open Access Library Journal,03,1-6. doi: 10.4236/oalib.1102820