Ferromagnetic coil frequency response and dynamics modeling with fractional elements

Abstract

The modeling of coils with ferromagnetic cores is considered. An attempt is made at the application of fractional derivatives because of the success in the modeling of real phenomena in various other fields. Three coils with ferromagnetic cores have been selected for the analysis. In order to obtain a basis for the study, measurements have been taken on a specially designed setup including an arbitrary function generator and a data acquisition device. The measurement process has been controlled by an original algorithm written in C#. The studied model has been considered in several varieties (various numbers of fractional coil and capacitor branches) in a ladder-like structure. The estimation process has been performed in an original program written in C#, applying the COBYLA method for constrained optimization. The modeling and simulation results have been compared through their frequency characteristics. A more critical approach has also been applied, where the measurements and simulations are compared for both a non-sinusoidal waveform and a step function on the source output.

Appendices

Appendix A: Fractional DAE formulation

The equations for the considered circuit can be given in terms of a general formula [60] of fractional DAE (differential algebraic equations):

$$\begin{aligned} {\left\{ \begin{array}{ll} \varvec{M}_{\mathrm {I}} \varvec{y} (t) + \varvec{M}_{\mathrm {II}} \varvec{x} (t) = \varvec{T} \varvec{v} (t), \\ \mathbf {D} _t ^{ \varvec{\alpha } } \varvec{x}(t) + \varvec{M}_{\mathrm {III}} \varvec{y} (t) + \varvec{M}_{\mathrm {IV}} \varvec{x} (t) = \varvec{0}_{n_x}, \end{array}\right. } \end{aligned}$$

(12)

where

• $\varvec{x}(t)$ is the vector of $n_x$ state variables (in this case capacitor voltages and coil currents),

• $\varvec{y}(t)$ is a vector with $n_y$ elements, which can be any remaining variables deemed useful (e.g., node potentials, capacitor currents and coil voltages),

• $\varvec{v}(t)$ is the vector of source time functions, which will only have one entry in the case of the studied problem—the voltage source E(t),

• $\mathbf {D}_t^{ \varvec{\alpha } }\varvec{x}(t)$ is the vector of fractional derivatives:

  $$\begin{aligned} \mathbf {D} _t ^{ \varvec{\alpha } } \varvec{x}(t) = \left[ \begin{array}{llll} _{t_0} ^{} {\mathrm {D}} _t ^{ \alpha _1 } x_1(t)&_{t_0} ^{} {\mathrm {D}} _t ^{ \alpha _2 } x_2(t)&\ldots&_{t_0} ^{} {\mathrm {D}} _t ^{ \alpha _{n_x} } x_{n_x}(t) \end{array} \right] ^{\mathrm {T}}, \end{aligned}$$

  (13)

• $\varvec{M}_{\mathrm {I}}$ is an $n_y \times n_y$ matrix,

• $\varvec{M}_{\mathrm {II}}$ is an $n_y \times n_x$ matrix,

• $\varvec{M}_{\mathrm {III}}$ is an $n_x \times n_y$ matrix,

• $\varvec{M}_{\mathrm {IV}}$ is an $n_x \times n_x$ matrix,

• $\varvec{T}$ is an $n_y \times n_v$ matrix,

• the notation $\varvec{0}_k$ represents a vector of k zeros.

Now the appropriate entries in Eq. (12) can be given. The dependency on time is only given, when there has been a need to emphasize it (e.g., u(t) is given simply as u and $\varvec{y}(t)$ is given as $\varvec{y}$). The number of differential equations in the system is determined by the number of fractional coils and capacitors, where $n_x = n + m$. The derivative orders are given in:

$$\begin{aligned} \varvec{\alpha } = [\alpha _1\;\;\alpha _2\;\;\ldots \;\;\alpha _n\;\;\beta _1\;\;\beta _2\;\;\ldots \;\;\beta _m]. \end{aligned}$$

(14)

The vector $\varvec{x}$ is filled with the coil currents and capacitor voltages:

$$\begin{aligned} \varvec{x} = [i_{L_1}\;\;i_{L_2}\;\;\ldots \;\;i_{L_n}\;\;u_{C_1}\;\;u_{C_2}\;\;\ldots \;\;u_{C_m}]^{\mathrm {T}}. \end{aligned}$$

(15)

The source vector has only one entry, being the time function of the source:

$$\begin{aligned} \varvec{v}(t) = [E(t)]. \end{aligned}$$

(16)

The additional variables are the model output voltage u and current i (for easy access when obtaining the results), along with the coil voltages and the capacitor currents:

$$\begin{aligned} \varvec{y} = [u\;\;i\;\; u_{L_1}\;\;u_{L_2}\;\;\ldots \;\;u_{L_n}\;\;i_{C_1}\;\;i_{C_2}\;\;\ldots \;\;i_{C_m}]^{\mathrm {T}}, \end{aligned}$$

(17)

where one can notice that $n_y = 2 + n + m$.

A current balance at the node with the voltage u yields the following equation:

$$\begin{aligned} i - \sum _{k = 1}^{m} i_{C_k} - \frac{u}{R_1} + \frac{u_{L_1}}{R_1} = 0, \end{aligned}$$

(18)

which allows to fill the following entries for $\varvec{M}_{\mathrm {I}}$:

$$\begin{aligned} \begin{array}{c} \varvec{M}_{{\mathrm {I}}\;1,2} = 1, \\ \varvec{M}_{{\mathrm {I}}\;1,k} = -1, k = n+3, n+4, \ldots n+m+2, \\ \varvec{M}_{{\mathrm {I}}\;1,1} = -\frac{1}{R_1} \\ \varvec{M}_{{\mathrm {I}}\;1,3} = \frac{1}{R_1}, \end{array} \end{aligned}$$

(19)

Next, the following equation is considered:

$$\begin{aligned} (R_{\mathrm {gen}} + R_{\mathrm {meas}}) i + u = E, \end{aligned}$$

(20)

which yields the only nonzero entry in $\varvec{T}$:

$$\begin{aligned} \varvec{T}_{2,1} = 1 \end{aligned}$$

(21)

along with the entries:

$$\begin{aligned} \begin{array}{c} \varvec{M}_{{\mathrm {I}}\;2,1} = 1, \\ \varvec{M}_{{\mathrm {I}}\;2,2} = R_{\mathrm {gen}} + R_{\mathrm {meas}}. \end{array} \end{aligned}$$

(22)

For the node above the coil branches, one can derive a current balance, which yields the following equation:

$$\begin{aligned} \frac{u}{R_1} - \frac{u_{L_1}}{R_1} - \frac{u_{L_1}}{R_{\mathrm {F}}} - \sum _{k = 1}^{n} i_{L_k} = 0, \end{aligned}$$

(23)

which yields the following entries for $\varvec{M}_{\mathrm {I}}$:

$$\begin{aligned} \begin{array}{c} \varvec{M}_{{\mathrm {I}}\;3,1} = \frac{1}{R_1} \\ \varvec{M}_{{\mathrm {I}}\;3,3} = -\frac{1}{R_1} -\frac{1}{R_{\mathrm {F}}}, \end{array} \end{aligned}$$

(24)

and the following ones for $\varvec{M}_{\mathrm {II}}$:

$$\begin{aligned} \varvec{M}_{{\mathrm {II}}\;3,k} = -1, \;\; k = 1, 2, \ldots n. \end{aligned}$$

(25)

Each additional fractional coil (i.e., $n > 1$) voltage dependency:

$$\begin{aligned} u_{L_1} - u_{L_k} - R_k i_{L_k} = 0, \;\; k = 2, 3, \ldots n, \end{aligned}$$

(26)

yields the following entries:

$$\begin{aligned} \begin{array}{c} \varvec{M}_{{\mathrm {I}}\;3+k, 3} = 1, \;\; k = 1, 2, \ldots n - 1, \\ \varvec{M}_{{\mathrm {I}}\;3+k, 3+k} = -1, \;\; k = 1, 2, \ldots n - 1, \\ \varvec{M}_{{\mathrm {II}}\;3+k, 1+k} = -1, \;\; k = 1, 2, \ldots n - 1. \end{array} \end{aligned}$$

(27)

Each fractional capacitor voltage dependency yields an equation:

$$\begin{aligned} u - R_{Ck} i_{C_k} - u_{C_k} = 0, \;\; k = 1, 2, \ldots m. \end{aligned}$$

(28)

which gives the following entries:

$$\begin{aligned} \begin{array}{c} \varvec{M}_{{\mathrm {I}}\;2+n+k, 1} = 1, \;\; k = 1, 2, \ldots m. \\ \varvec{M}_{{\mathrm {I}}\;2+n+k, 2+n+k} = -R_{Ck}, \;\; k = 1, 2, \ldots m. \\ \varvec{M}_{{\mathrm {II}}\;2+n+k, n+k} = -1, \;\; k = 1, 2, \ldots m. \end{array} \end{aligned}$$

(29)

The differential equation for each fractional coil:

$$\begin{aligned} L_k \; _{t_0}^{} {\mathrm {D}}_t ^{\alpha _k} i_{L_k} = u_{L_k}, \;\; k = 1, 2, \ldots n, \end{aligned}$$

(30)

introduces entries in $\varvec{M}_{\mathrm {III}}$:

$$\begin{aligned} \begin{array}{c} \varvec{M}_{{\mathrm {III}}\;k, 2+k} = -\frac{1}{L_k} \;\; k = 1, 2, \ldots n, \end{array} \end{aligned}$$

(31)

The same is done for each fractional capacitor differential equation:

$$\begin{aligned} C_k \; _{t_0}^{} {\mathrm {D}}_t ^{\alpha _k} u_{C_k} = i_{C_k}, \;\; k = 1, 2, \ldots m, \end{aligned}$$

(32)

which introduces the following entries:

$$\begin{aligned} \begin{array}{c} \varvec{M}_{{\mathrm {III}}\;n+k, 2+n+k} = -\frac{1}{C_k} \;\; k = 1, 2, \ldots m. \end{array} \end{aligned}$$

(33)

Appendix B: Numerical solution of fractional DAE

The SubIval numerical method has been designed to be used in time-stepping solvers [60], where for each time step $t = t_{\mathrm {now}}$, following the method’s internal computations involving subinterval dynamics and polynomial differintegration, for each variable x(t) under a fractional time derivative one can obtain the approximation:

$$\begin{aligned} _{t_0} ^{} {\mathrm {D}} _{t_{\mathrm {now}}} ^{{\alpha }} x(t) \approx a x_{\mathrm {now}} + b . \end{aligned}$$

(34)

The details on this can be found i.a. in [60, 65]. A time-step size adaptive numerical solver basing on SubIval has been implemented in C# and the computations of the time dependent analyses of this paper have been performed with that solver.

There are a few options for the solver that need to be given. These are:

• the maximum polynomial order in the SubIval internal computations ($p = 4$ has been selected for the computations in this paper),

• the maximum allowed estimated error value ($e_{\mathrm {max}} = 10^{-1}\;\%$ in this paper) and the so-called control error, which is the value the estimated error should tend to ($e_{\mathrm {ctrl}} = 10^{-2}\;\%$ in this paper); a detailed description on this is given in [65],

• the minimum and maximum time-step size values, denoted by ${\Delta }t_{\mathrm {min}}$ and ${\Delta }t_{\mathrm {max}}$, respectively (for the step response computations ${\Delta }t_{\mathrm {min}} = t_{\mathrm {max}}/2000$ and ${\Delta }t_{\mathrm {max}} = t_{\mathrm {max}}/100$ have been selected, while for the non-sinusoidal response computations ${\Delta }t_{\mathrm {min}} = T/2000$ and ${\Delta }t_{\mathrm {max}} =T/20$).

It is also worthwhile to mention that widely available solvers for fractional differential equations or fractional DAE are a rarity. Unlike solvers for ordinary differential equations of DAE where first-order derivatives appear, there are many solvers available, e.g., allowing to solve even very complicated problems [66]; however, not ones where fractional derivatives appear. Only a few instances of solvers can be found [49, 62, 67]. A solver basing on SubIval (versions for Matlab and Octave) is also available. It can be found on the web page [61].