Modified method of identification of mutual fractional-order inductance

The paper presents a method for identifying the parameters Mγ, γ of a fractional-order transformer, which parameters Lβ1, β1, Lβ2, β2 have been previously determined. This method is based on the measurement of the phase resonance frequency in a few systems containing: the investigated fractionalorder transformer and two standard capacitors. The measurements need to be performed only for one series opposite-aiding connection of the fractional-order transformer. The dependencies allowing the determination of the fractional-order mutual inductance parameters have been given


Introduction
There are many works devoted to the analysis of systems with fractional-order elements L β , C α , their realization and parameter identification, e.g. [1][2][3].
For several years, there has been a rapid growth of interest in fractional differential-integral calculus application in describing fractional-order magneticallycoupled coils systems [4][5][6]. The work [4] describes the concept and properties of such fractional-order coupled inductances. In [5], the electromagnetic Maxwell equations of the fractional-order mutual inductance are analyzed. The wireless power transmission system has been modeled as a fractional-order coupled coils system in [6]. The existence of fractional-order coupled coils (fractional-order transformer) implies the need to determine the parameters of the fractional-order elements. In [7], a method has been proposed for parameters identification of the fractional-order coils with an iron core, which is based on the approximation of the transient response to the unit-step voltage using the least squares method.
The paper is an extension and continuation of [8], where the new method for the identification of all the parameters L β1 , β 1 , L β2 , β 2 , M γ , γ, of the fractional-order coupled inductances, has been proposed. The paper presents a proposal for a modified method of the identification of the fractional-order parameters M γ , γ of the mutual inductance, based on the phase resonance phenomenon in the series circuit of the class RL β C α , compared to [8], without the need of the input impedance measurement in the combination of series and opposite-aiding connection of the transformer system.

Modification proposal
The equivalent circuit of the system for the parameters γ, M γ determination of the fractional-order mutual inductance, is shown in Fig. 1. The circuit impedance, seen from the source terminals, is given by a formula: where: R -the equivalent resistance of the series connection of the coil resistances. Transforming, the real and imaginary part of the impedance is: and: Parameters L β1 , β 1 , L β2 , β 2 were determined previously, according to the procedure described in [8]. The circuit from Fig. 1 should be brought into phase resonance state, which will occur when the phase shift between the voltage measured on the series connection of the magnetically-coupled coil system as well as the capacitor C 1 and the current, will be equal to zero.
The measurement described above should be repeated twice, for two values of classic capacitances C 1 ,C 2 and the detection of the phase resonance frequency values, ω 1 , ω 2 respectively.
Next, using the general phase resonance condition Im{Z(jω)} = 0 and transforming, we get the relation: Then the value of the parameter γ can be determined as: However, the parameter M γ can be determined by substituting the obtained value of the coefficient γ with the formula (3) for one of the performed measurements, for example for: The described algorithm has been illustrated with a simulation example.

Example
The circuit from Fig. 1 has been supplied from a source with an adjustable frequency value, for which the input voltage value has been assumed U(jω) = 1 V.
Parameters of the primary and secondary side of the transformer have been determined according to the procedure diagram [8] and were respectively: β 1 = 0.503, L β1 = 8.813 mH•s (1-β) , β 2 = 0.502, L β2 = 3.113 mH•s (1-β) . For two capacitors with known capacitances C 1 = 10 mF, C 2 = 3,53 mF in the investigated circuit, as in Fig. 1, two values of resonance frequencies f 1 = 100 Hz, f 2 = 200 Hz have been recorded. From the dependences (4) and (5), the searched values of the fractional-order parameters of the mutual inductance have been determined: and:

Summary
The paper proposes a modified method, compared to the method presented in [8], for identifying M γ , γ parameters of a fractional-order transformer. This method is based on the measurement of the phase resonance frequency in a circuit containing the analyzed transformer and two switchable standard capacitors. The dependencies allowing the determination of the fractional-order mutual inductance parameters have been given, on the basis of the described measurements. The advantage of the modified method for determining the fractional-order parameters is the fact, that only one series opposite-aiding connection of the fractional-order coupled coils is enough to determine the searched parameters. The need to measure the input impedance of the circuit from Fig. 1 is also avoided.