Modeling of a coil system considering frequency-dependent inductances and losses. Determination of the equivalent circuit impedances

Abstract

A methodology for determining the equivalent circuit impedances of a system of magnetically coupled coils is presented in this paper. The structure of the equivalent circuit has been proposed in previous works [Mombello (2000) Electr Eng 84:3–10; 84:11–19]. Starting from of a series of impedance measurements on the coil system for an appropriate frequency range, the necessary information is acquired to begin a process of identification of the circuit parameters. The parameter identification procedure is detailed, and consists of two successive minimization processes. Fundamental requirementts for preserving the precision of the results, which are highly sensitive to small variations in the parameters, are given.

Appendix: Initial values

1.1 First considerations

First, the case of fitting an impedance Z(ω) using functions as given in Eqs. (13) and (15) will be analyzed. It should be noted that Z(ω) is a complex function and hence has real and imaginary parts. Suppose initially that the impedance is to be fitted considering that r is equal to 1, so the series in Eq. (9) has the term Z ₀ and a single additional frequency-dependent term. The parameters to be calculated are the initial conditions R ₀, L ₀, γ, and λ.

Any term in the summation of Eq. (10) or (13) can be expressed as:

$$ y(\omega ) = {{\beta \omega ^2 } \over {\lambda _i ^2 + \omega ^2 }} $$

(A1)

where the parameter β has a different meaning in the case of either Eq. (10) (β=γλ) or Eq. (13) (β=γ).

This function increases monotonically for ω>0. Furthermore, it also holds that:

$$ \omega = 0\;\;\;\;y = 0;\;\;\;\;\omega \to \infty \;\;\;\;y = \beta $$

(A2)

It can be derived that the abscissa at the point of maximum slope is:

$$ \omega _{{\rm ms}} = {\lambda \over {\sqrt 3 }} $$

(A3)

and the corresponding ordinate

$$ y_{{\rm ms}} = {\beta \over 4} $$

(A4)

1.2 Estimation of the initial values

From the analysis of Eq. (10) and the frequency dependence of the resistance (Fig. 3a), it can be concluded that a reasonable initial value for R ₀ is:

$$ R_{0} = 0 $$

(A5)

Since the slope of the real part of Z still grows at a frequency f _m=500 kHz (f _m=2πω _m), the maximum slope point will be at a frequency higher than or near to f _m, therefore the proposal is

$$ \omega _{{\rm ms}} = \omega _{\rm m} $$

(A6)

where ω _m is the maximum measured frequency in this case. In other cases, a lower frequency could be chosen. It would be very difficult if a much higher frequency ω _m were chosen for the measurements, since at frequencies close to 1 MHz the capacitive effect is important and the impedance behavior very distorted.

Starting from Eqs. (A6) and (A3), the initial value of λ can be found:

$$ \lambda = \sqrt 3 \;\omega _{\rm m} $$

(A7)

By analyzing the resistance function and considering Eq. (A5), the resistance value R _m measured at the frequency f _m is

$$ R(\omega _{\rm m} ) = R_{\rm m} \cong {{\gamma \lambda \omega _{\rm m} ^2 } \over {\lambda ^2 + \omega _{\rm m} ^2 }} $$

(A8)

Combining Eqs. (A7) and (A8) and solving for γ

$$ \gamma = {4 \over {\sqrt 3 }}{{R_{\rm m} } \over {\omega _{\rm m} }} $$

(A9)

Furthermore, the measured inductance at f _m is

$$ L(\omega _{\rm m} ) = L_{\rm m} \cong L_{0} - {{\gamma \omega _{\rm m} ^2 } \over {\lambda ^2 + \omega _{\rm m} ^2 }} $$

(A10)

hence

$$ L_{0} \cong L_{\rm m} + {{\gamma \omega _{\rm m} ^2 } \over {\lambda ^2 + \omega _{\rm m} ^2 }} $$

(A11)

Starting from impedance Z _m measured at frequency f _m, the parameters R ₀, L ₀, γ and λ can be calculated by means of Eqs. (A5), (A7), (A9), and (A11).

This calculation has been carried out on the assumption that r=1 (note that r and n _2L have an identical meaning). This condition is not desirable, since it will give a very bad fitting as a result. After some computation, it can be seen that a suitable value for r is 5 or 6, which lead to very good fittings. In this last case, there are several terms present, and the same initial value of Eq. (A7) can be set for all terms λ _i (the values will change during minimization process and will no longer be equal), but the initial values of γ _i must be different for each term. A choice that works well is:

$$ \gamma _i = {\gamma \over r} + \Delta \gamma _i $$

(A12)

This means that all terms are similar (but not equal) at the beginning of the minimization process. However, small values Δγ _i have been introduced so that the minimization algorithm does not start with identical initial values for the parameters γ _i .

In optimizing the parameters of Eq. (21), it should be considered that parameters $K_{i_{11} }$ should be used instead of parameters γ _i in the equations in this Appendix and in Eq. (15).

1.3 Accuracy improvement in calculating the off-diagonal elements of Z

It is a well-known fact that the transient and resonant behavior of circuits containing self- and mutual inductances are strongly dependent on the leakage flux [3]. The impedances related to the leakage flux are the subtractive series connection impedances defined as:

$$ Z_{{\rm g}_{ij} } = Z_{ii} + Z_{jj} - 2Z_{ij} $$

(A13)

It can be seen that they have small values in comparison with the self- and mutual impedances, since all impedances present on the right-hand side of Eq. (A13) have the same order of magnitude. For this reason, a calculation of a leakage impedance $Z_{{\rm g}_{ij} }$ from the self- and mutual impedances Z _ii , Z _jj and Z _ij would give results containing severe errors on the assumption that Z _ij has been measured independently. However, this problem can be drastically overcome by measuring $Z_{{\rm g}_{ij} }$ directly and then calculating the mutual impedance Z _ij by means of Eq. (4). In this way, impedance $Z_{{\rm g}_{ij} }$ keeps its original value if it is recovered from the calculated value of Z _ij from Eq. (A13). This means that the leakage flux information contained in impedances $Z_{{\rm g}_{ij} }$ remains intact in matrix Z. This strategy makes unnecessary the application of a mathematical model that only contains leakage inductances at the expense of making some assumptions in order to avoid the inclusion of self- and mutual inductances.

After the impedances Z _ii , Z _jj , and $Z_{{\rm g}_{ij} }$ have been measured, it is necessary to obtain a fitting of them so as to express the impedance matrix in the form given in Eq. (17), in which each impedance of the matrix has the form given in Eq. (18). Two possible methods for this exist. The first method is to calculate impedance Z _ij first using Eq. (4) and the measured data of Z _ii , Z _jj , and $Z_{{\rm g}_{ij} }$ and then to carry out the fitting of Z _ij according to Eq. (22). By doing this, the total error $\varepsilon _{{\rm g}I_{ij} }$ (measurement error plus fitting error) contained in impedance $Z_{{\rm g}_{ij} }$ obtained from matrix Z is:

$$ \varepsilon _{{\rm g}I_{ij} } = \varepsilon _{ii} + \varepsilon _{jj} - 2\varepsilon _{I_{ij} } $$

(A14)

where ε _ii , ε _jj , and $\varepsilon _{I_{ij} }$ are the total errors of impedances Z _ii , Z _jj , and Z _ij respectively. The total error $\varepsilon _{I_{ij} }$ also includes the fitting and measurement errors (the last one is obtained composing the measurement errors of Z _ii , Z _jj , and $Z_{{\rm g}_{ij} }$ in Eq. (4). This error can be extremely large, since each error ε can have either positive or negative values for its real and imaginary parts, and they have a component that is proportional to the corresponding magnitude, and all of them (right-hand side of Eq. A13) correspond to magnitudes with the same order as a self-impedance (much larger than $Z_{{\rm g}_{ij} }$).

Fortunately, a second procedure exists. Starting from the measurements of impedances Z _ii , Z _jj , and $Z_{{\rm g}_{ij} }$ the fitting of each of them is fulfilled and then Eq. (4) is applied. The sum indicated in Eq. (4) is performed using Eqs. (22) and (29). This way the error in Z _ij is:

$$ \varepsilon _{II_{ij} } = {{\varepsilon _{ii} + \varepsilon _{jj} - \varepsilon _{{\rm g}_{ij} } } \over 2} $$

(A15)

In this case, the total error $\varepsilon _{{\rm g}II_{ij} }$ contained in impedance $Z_{{\rm g}_{ij} }$ obtained from matrix Z is

$$ \varepsilon _{{\rm g}II_{ij} } = \varepsilon _{ii} + \varepsilon _{jj} - 2\varepsilon _{II_{ij} } $$

(A16)

Substitution of Eq. (A15) in (A16) yields

$$ \varepsilon _{{\rm g}II_{ij} } = \varepsilon _{{\rm g}_{ij} } $$

(A17)

This reveals that the error of impedance $Z_{{\rm g}_{ij} }$ obtained from matrix Z is only affected by its measurement errors and fitting errors, but it does not contain errors coming from other impedances.

As a consequence, the convenience of carrying out measurements of subtractive series connection impedances $Z_{{\rm g}_{ij} }$ in order to obtain the mutual impedances Z _ij has been demonstrated, since impedances $Z_{{\rm g}_{ij} }$ can first be fitted and then their fittings can easily be added to the fittings of the self-impedances (Eq. 29). This would not be possible if short-circuit impedances $Z_{{\rm sh}_{ij} }$ were used for the calculation of the mutual impedances Z _ij .