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New algorithm for optimizing technological parameters of soft magnetic composites has been derived on the base of topological structure of the power loss characteristics. In optimization magnitudes obeying scaling, it happens that one has to consider binary relations between the magnitudes having different dimensions. From mathematical point of view, in general case such a procedure is not permissible. However, in a case of the system obeying the scaling law it is so. It has been shown that in such systems, the binary relations of magnitudes of different dimensions is correct and has mathematical meaning which is important for practical use of scaling in optimization processes. The derived structure of the set of all power loss characteristics in soft magnetic composite enables us to derive a formal pseudo-state equation of Soft Magnetic Composites. This equation constitutes a relation of the hardening temperature, the compaction pressure and a parameter characterizing the power loss characteristic. Finally, the pseudo-state equation improves the algorithm for designing the best values of technological parameters.

Recently novel concept of technological parameters’ optimization has been applied in Soft Magnetic Composites (SMC) by Ślusarek et al., [

The scaling is very useful tool due to the three reasons:

• it reduces number of independent variables

• and determines general form of loss of power characteristic in a form of homogenous function in general sense (h.f.g.s.),

• as well as enables us to use binary relations between magnitudes of diﬀerent dimensions.

Reduction of independent variables is based on definition of the h.f.g.s., namely,

According to the assumption concerning

where

Choice for the

where

[˚C] | [MPa] | [−] | [−] | [m^{2}∙s^{−2}T^{α}^{−β}] | [m^{2}∙s^{−1}T^{2α−β}] | [m^{2}T^{3α−β}] | [m^{2}∙sT^{4α−β}] |

500 500 500 500 400 450 550 600 | 500 600 700 900 800 800 800 800 | −1.312 −1.383 −1.735 −1.395 −1.473 −1.596 −2.034 −1.608 | −0.011 −0.125 −0.517 −0.082 −0.28 −0.123 −1.326 −0.232 | 0.171 0.153 0.156 0.101 0.183 0.145 0.106 1.220 | 3.606 × 10^{−5} 3.328 × 10^{−5 } 2.393 × 10^{−5} 6.065 × 10^{−5} 1.347 × 10^{−5} 2.482 × 10^{−5} 1.407 × 10^{−4} 8.941 × 10^{−4 } | 1.953 × 10^{−8 } 9.254 × 10^{−8 } 2.309 × 10^{−8 } −8.031 × 10^{−8 } 3.689 × 10^{−9 } −1.218 × 10^{−9 } −1.066 × 10^{−8 } −5.302 × 10^{−8 } | −2.255 × 10^{−12} −1.177 × 10^{−12} −8.075 × 10^{−14} 7.877 × 10^{−13} 1.185 × 10^{−13 } 6.120 × 10^{−14} 4.541 × 10^{−13} 1.664 × 10^{−11} |

power loss characteristics presented in

In this paper we will prove that if the considered characteristics obey the scaling, then the binary relation between them is invariant with respect to this transformation and comparison of two magnitudes of different dimensions has mathematical meaning. Reach measurement data of power losses in Somaloy 500 have been transformed into parameters of (3) vs. hardening temperature and compaction pressure

Let the power loss characteristic has the form determined by the scaling (2). It is important to remain that

Let us concentrate our attention at the point on the

Let us take into account the two characteristics and let us assume that

Therefore, the considered binary relation is the strong inequality and corresponds to natural order presented in

Let

• Let us perform the scaling with respect to

where

• Substituting appropriate relations of (8) to (7) we derive:

• Collecting all powers of

Therefore (6) is invariant with respect to scaling. This binary relation has mathematical meaning and constitutes the total order in the set of characteristics.

The result derived in Section 2 can be supplemented with the following binary equivalence relation. Let

be the

Theorem:

Let

in

state equation we transform each power loss characteristic into a number

This function must satisfy the following condition. Let us concentrate our attention at the two following points:

Let us consider the two characteristics

While, the other technological parameters powder compositions and volume fraction are constant. Let us assume that for (14) the following relation holds:

It results from the derived structure of

Moreover,

where the integration domain is common for the all characteristics. We have selected the common domain of ^{−1}∙T^{−α}]. Using (3) we transform (17) to the working formula for the measure

where

T | p | V |
---|---|---|

[K] | [MPa] | [W∙kg^{−1}T^{−β}] |

723.15 | 800 | 40.60 |

773.15 | 900 | 43.75 |

773.15 | 700 | 47.25 |

673.15 | 800 | 50.30 |

773.15 | 600 | 57.12 |

823.15 | 800 | 81.50 |

773.15 | 500 | 89.28 |

742.15 | 764 | 492.3 |

753.15 | 780 | 509.2 |

804.15 | 764 | 528.5 |

711.15 | 764 | 547.0 |

873.15 | 800 | 720.0 |

where

In order to extent (19) to a realistic equation we apply again the scaling hypothesis (2) [

where

In order to extent (19) to a full state-equation we apply the Padé approximant by analogy to virial expansion derived by Ree and Hoover [

where

At the beginning we have to notice that the data collected in ^{ }for both phases have been performed by using MICROSOFT EXCEL 2010, where

where

Function

Formula (24) represents the minimal iso-power loss curve. All points satisfying (24) are solutions of the optimization problem for technical parameters of SMC.

By introducing the binary relations we have revealed twofold. The power loss characteristics do not cross each other which makes the topology’s set of this curves very useful and effective that we can perform all calculations in the one-dimension space spanned by the scaled frequency or here in the case of pseudo-statee quation in the scaled temperature. For general knowledge concerning such a topology we refer to the papers by Egenhofer [

T_{c } | p_{c } | G_{0 } | G_{1 } | G_{2 } | ||
---|---|---|---|---|---|---|

0.1715 | 1.2812 | 21.622 | 37.729 | 370,315,315 | −47,752,251 | 1,734,952 |

G_{3 } | G_{4 } | D_{1 } | D_{2 } | D_{3 } | D_{4 } | - |

−1.3764 | −678.26 | 170.80 | 6243.8 | 386.96 | −28.699 | - |

T_{c } | p_{c } | G_{0 } | G_{1 } | G_{2 } | ||
---|---|---|---|---|---|---|

0.1810 | 1.5550 | 22.949 | 30.197 | 365,210,688 | −47,714,207 | 1,762,773 |

G_{3 } | G_{4 } | D_{1 } | D_{2 } | D_{3 } | D_{4 } | - |

−1.3763 | −683.38 | 170.77 | 5739.9 | 387.81 | −22.514 | - |

tween magnitudes of different dimensions in the sense of different physical magnitudes. Also, this paper is the first one which presents an application of scaling in designing the technological parameters’ values by using the pseudo-state equation of SMC. The obtained result is the continuous set of points satisfying (24). All solutions of these equations are equivalent for the optimization of the power losses. Therefore, the remaining degree of freedom can be used for optimizing magnetic properties of the considered SMC. Ultimately, one must say that the degree of success achieved when applying the scaling depends on the property of the data. The data must obey the scaling.

The work has been supported by National Center of Science within the framework of research project Grant N N507 249940.