Cauer ladder network representation of eddy-current fields using the partial element equivalent circuit method

Emiliani, Matteo; Di Rienzo, Luca; Hiruma, Shingo; Figoni, Isacco; de Falco, Carlo; Matsuo, Tetsuji

doi:10.1108/COMPEL-10-2025-0504

Purpose

This study aims to extract a reduced order model based on the Cauer ladder network (CLN) method from a partial element equivalent circuit (PEEC) formulation of the eddy current problem in parallel conductors. The obtained equivalent circuit can then be effectively used for time domain simulations.

Design/methodology/approach

The PEEC formulation has been written as a first-order linear dynamic system, and the CLN algorithm has been applied to extract the reduced-order model. The numerical results of the reduced order model have been then compared with the reference results obtained both with the finite element method (FEM) and the PEEC method for a canonical test case of two rectangular conductors.

Findings

The Cauer ladder network representation of the eddy current fields is proved to be accurate enough in frequency and in time domains, provided that a sufficient number of the equivalent circuit stages is considered.

Originality/value

The CLN method has already been applied in the past to FEM for eddy current problems. In this work, the CLN has been applied to the PEEC method for the first time.

1. Introduction

Among the existing computational methods for solving eddy-current problems, the partial element equivalent circuit (PEEC) method is gaining importance, due to the possibility of coupling it with circuit models (Ruehli et al., 2017; Nguyen et al., 2016).

The Cauer ladder network (CLN) method is a model order reduction approach that allows obtaining an equivalent Cauer circuit (Shindo et al., 2017). This method has been introduced and developed for finite element formulations (Kameari et al., 2017). Many applications of the CLN have been proposed in recent years. In particular, the CLN equivalent representation can be used to compute the earth return impedance of an high voltage direct current (HVDC) (Tang et al., 2025); to model the electromagnetic behavior of a printed circuit board to efficiently design the power converter operating at high switching frequencies (Henneron et al., 2025); to present the connection between CLN and the proper generalized decomposition with a novel double vector potential formulation (Köster et al., 2021).

In this paper, the application of the CLN method to a PEEC formulation is proposed for the first time. It is shown that the structure of PEEC matrices exhibits important advantages over FEM matrices when applying the CLN method.

Magneto-quasistatic conditions are assumed when deriving the PEEC formulation, neglecting wave propagation effects. This hypothesis can be considered valid up to 500 MHz so that the corresponding wavelength is approximately equal to ten times the distance between the centers of the two conductors, that is a characteristic size of the problem. In these conditions, capacitive and wave propagation effects can be safely neglected.

To explain the application of the CLN method to a magneto-quasistatic PEEC formulation, we refer here to a very simple test case of two parallel rectangular conductors (Morisco et al., 2019). The case of more complex geometries does not require conceptual modifications.

Once the CLN circuit has been obtained, the results both in time and frequency domains are compared with PEEC and finite element method (FEM) simulations (COMSOL AB, 2024), where the latter are taken as reference.

2. PEEC formulation

Consider two parallel rectangular conductors of width w and thickness t shown in Figure 1 carrying two opposite time-harmonic excitation currents $i_{1, 2} (t) = \pm \sqrt{2} \cdot R (I \cdot e^{j ω t})$ ⁠.

Figure 1.

A schematic shows two blocks of width w separated by distance D, with thickness t indicated on the side.

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The diagram shows two rectangular blocks placed horizontally. Each block has width labelled w. A small dot appears at the centre of each block. The vertical dimension at the left side is labelled t. A double-headed arrow between the two blocks indicates the separation distance labelled D. The blocks are aligned along the same horizontal line with space between them representing the distance D.

Geometry of the study case of two parallel rectangular conductors, where $t = 3.5 mm, w = 40 mm   and   D = 60 mm$ are, respectively, the thickness and the width of the two conductors and the distance between their centers

The conductors are subdivided into n smaller rectangular sub-conductors $Λ_{1} \dots Λ_{n}$ (Morisco et al., 2019). The dimensions of each sub-conductor, $w_{Λ}, t_{Λ}$ ⁠, are chosen smaller than the skin depth (⁠ $w_{Λ}, t_{Λ} ≪ δ$ ⁠) to assume the current density inside it as uniform (Ruehli et al., 2017).

Each sub-conductor is modeled as a series connection of its DC resistance and its partial inductance (Morisco et al., 2019). The conductors can be represented as a parallel network of sub-conductors as reported in Figure 2.

Figure 2.

A schematic shows multiple parallel branches with resistors and inductors, each branch labelled with current i sub A and voltage v sub A.

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The schematic presents several parallel electrical branches connected between two vertical nodes. Each branch contains a resistor labelled R with a subscript number followed by an inductor labelled L p with a matching subscript. Current arrows labelled i sub A with subscripts indicate flow from left to right in each branch. Voltage arrows labelled v sub A with subscripts are drawn above each branch in the opposite direction. The top branches are labelled R 1 and L p 1, and R 2 and L p 2. The bottom branch is labelled R n and L p n, with intermediate branches implied by dotted continuation.

Equivalent circuit of the conductors subdivided into n parallel sub-conductors. Their currents and voltages are denoted by $i_{Λ_{h}}$ and $v_{Λ_{h}}$ ⁠, respectively, where h = 1, …, n

The resistance and inductance matrices are given by the following:

\begin{array}{l} R_{Λ} = [\begin{array}{l} R_{11} & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ 0 & \dots & R_{n n} \end{array}] \end{array}

(1)

\begin{array}{l} L_{Λ} = [\begin{array}{l} L_{11} & \dots & M_{1 n} \\ ⋮ & ⋱ & ⋮ \\ M_{n 1} & \dots & L_{n n} \end{array}] \end{array}

(2)

The following complex symmetric impedance matrix thus describes the complete electromagnetic coupling of the system:

\begin{array}{l} Z_{Λ} = [\begin{matrix} Z_{11} & \dots & Z_{1 n} \\ ⋮ & ⋱ & ⋮ \\ Z_{n 1} & \dots & Z_{n n} \end{matrix}] = R_{Λ} + j ω L_{Λ} \end{array}

(3)

The relationship between sub-conductor voltages and currents reads:

v_{Λ} = Z_{Λ} \cdot i_{Λ}

(4)

where $v_{Λ} \in C^{n}$ is the vector of sub-conductor voltages and $i_{Λ} \in C^{n}$ is the vector of sub-conductor currents.

A connectivity matrix $C \in N^{n \times 2}$ is introduced, in which each row corresponds to a sub-conductor and each column to a conductor. Its elements are given by the following:

C_{i j} = {\begin{array}{l} 1, & if 1 \leq i \leq \frac{n}{2} and j = 1, \\ 1, & if \frac{n}{2} < i \leq n and j = 2, \\ 0, & otherwise . \end{array}

(5)

Each row of matrix $C$ contains exactly one element equal to 1, while the other is zero.

The symmetric terminal admittance matrix $Y \in C^{2 \times 2}$ can be then written as follows:

\begin{array}{l} Y = C^{T} \cdot Z_{Λ}^{- 1} \cdot C = [\begin{array}{l} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{array}] \end{array}

(6)

where, referring to Figure 3, its elements are defined as follows:

Y_{i i} = {\frac{i_{i}}{v_{i}} |}_{v_{j} = 0} and Y_{i j} = {\frac{i_{i}}{v_{j}} |}_{v_{i} = 0} for i, j = 1, 2 and i \neq j

(7)

Figure 3.

A diagram shows two rectangular volumes with currents i sub 1 and i sub 2 and corresponding voltages v sub 1 and v sub 2 along edges.

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The diagram shows two separate rectangular blocks placed side by side. Each block has a shaded front face and dashed lines indicating internal depth. In the first block, an arrow labelled i sub 1 points diagonally upward inside the volume. Along the right edge, a line labelled v sub 1 runs downward. In the second block, an arrow labelled i sub 2 appears in the same orientation. A line labelled v sub 2 is drawn along the right edge. Both blocks have identical shape and orientation with similar internal dashed outlines.

Geometry of the study case of two parallel rectangular conductors, with the two terminal currents (⁠ $i_{1} and i_{2}$ ⁠) and terminal voltages (⁠ $v_{1} and v_{2}$ ⁠)

From equation (6), the terminal impedance matrix of the conductors is obtained as follows:

\begin{array}{l} Z = Y^{- 1} = [\begin{matrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{matrix}] \end{array}

(8)

The terminal impedance of each of the two conductors is then defined as follows:

Z_{1} = Z_{11} - Z_{12} = R_{1} + j X_{1}

(9)

Z_{2} = Z_{22} - Z_{21} = R_{2} + j X_{2}

(10)

Equation (6) represents the transfer function of the following linear time-invariant system:

(R_{Λ} + s L_{Λ}) x = Cu,

(11)

y = C^{T} x

(12)

where $u$ and $y$ are, respectively, the input and output vectors of the system.

3. CLN formulation

Equations (11) and (12) describe the full-order model obtained from the PEEC formulation. In the following, a reduced-order model is obtained by applying the CLN method to this PEEC formulation.

Transfer function (6) can be rewritten as follows:

\begin{array}{l} H (s) & = C^{T} {(R_{Λ} + s L_{Λ})}^{- 1} C \\ = M^{T} {(I - s A)}^{- 1} N \end{array}

(13)

where $M = C$ ⁠, $N = R_{Λ}^{- 1} C$ and $A = - R_{Λ}^{- 1} L_{Λ}$ ⁠.

Algorithm 1, applied to the full-order transfer function (13), allows writing the reduced-order system by computing matrices $U_{2 i} \in R^{n \times 2}$ and $V_{2 i + 1} \in R^{n \times 2}$ ⁠, which are orthogonal with respect to the $R_{Λ}$ and $L_{Λ}$ inner products, respectively (Hiruma and Igarashi, 2020). The coefficient matrices $K_{i} \in R^{2 \times 2}$ (⁠ $i = 0, 1, \dots$ ⁠) are defined as follows:

{(U_{2 i}, U_{2 j})}_{R_{Λ}} = K_{2 i} δ_{i j}

(14)

{(V_{2 i + 1}, V_{2 j + 1})}_{L_{Λ}} = K_{2 i + 1} δ_{i j}

(15)

where $δ_{i j}$ is the Kronecker delta.

Since matrices $R_{Λ}, L_{Λ}$ are also positive-definite, the obtained coefficient matrices $K_{i} \in R^{2 \times 2}$ are positive-definite so that the obtained reduced-order Cauer circuit is passive.

Coefficient matrices $K_{i}$ correspond to the coefficients of the continued fraction expansion of transfer function (13). In fact, its Taylor series expansion around $s = 0$ is given by the following:

H (s) = \sum_{k = 0}^{\infty} M_{k} s^{k}

(16)

where $M_{k} = M^{T} A^{k} N$ ⁠, are referred to as the moments.

\begin{array}{l} Algorithm 1 Cauer circuit extraction via self-adjoint Lanczos algorithm \\ Input : symmetric (positive definite) matrices R_{Λ}, L_{Λ} \in ℝ^{n \times n}, matrix C \in ℕ^{n \times 2}, A = - R_{Λ}^{- 1} L_{Λ} . ​ \\ Let q be the total number of stages of the Cauer circuit . \\ Output : R_{Λ} and L_{Λ} orthogonal vectors U_{0}, U_{2}, U_{2 q - 2}, V_{1}, V_{3}, \dots, V_{2 q - 1} and parameters in the continued fraction expansion K_{1}, \dots, K_{2 q - 1} . \\ Initialisation : U_{0} = R_{Λ}^{- 1} C, {(U_{0}, U_{0})}_{R_{Λ}} = K_{0}, V_{- 1} = 0 . \\ for i = 1 to i = q do \\ V_{2 i - 1} = V_{2 i - 3} + U_{2 i - 2} K_{2 i - 2}^{- 1}, \\ K_{2 i - 1} = {(V_{2 i - 1}, V_{2 i - 1})}_{L_{Λ}}, \\ U_{2 i} = U_{2 i - 2} + A V_{2 i - 1} K_{2 i - 1}^{- 1}, \\ K_{2 i} = {(U_{2 i}, U_{2 i})}_{R_{Λ}}, \\ end  for \\ return U_{0}, U_{2}, \dots, U_{2 q - 2}; V_{1}, V_{3}, \dots, V_{2 q - 1}; K_{1}, \dots, K_{2 q - 1} . \end{array}

Setting in Algorithm 1 the initial matrix $U_{0} = R_{Λ}^{- 1} C$ ⁠, it is possible to construct the set of matrices $U_{0}, U_{2}, \dots, U_{2 q - 2}$ orthogonal with respect to $R_{Λ}$ inner product, i.e. the orthogonal matrices are $R_{Λ}$ -orthogonal to each other (Hiruma, 2021). A three-term recurrence relation is satisfied:

A U_{2 i} = U_{2 i + 2} K_{2 i} K_{2 i + 1} - U_{2 i} K_{2 i} (K_{2 i + 1} + K_{2 i - 1}) + U_{2 i - 2} K_{2 i} K_{2 i - 1}

(17)

where $i = 1, 2, \dots,$ ⁠. Equation (17) can be written in a compact matrix form as follows:

AU = UT + U_{2 q} K_{2 q - 2} K_{2 q - 1} E_{q}^{T}

(18)

where $E_{q} = {[0   0   \dots   1]}^{T} \in R^{2 q \times 2}, U = [U_{0}   U_{2}   \dots   U_{2 q - 2}] \in R^{n \times 2 q}$ and $T \in R^{2 q \times 2 q}$ is a tridiagonal matrix defined as follows:

\begin{array}{l} T = [\begin{matrix} - K_{0} K_{1} & K_{1} K_{2} & \dots & 0 \\ K_{0} K_{1} & - K_{2} (K_{1} + K_{3}) & ⋱ \\ ⋮ & ⋱ & ⋱ \\ - K_{2 q - 4} (K_{2 q - 5} + K_{2 q - 3}) & K_{2 q - 3} K_{2 q - 2} \\ 0 & K_{2 q - 4} K_{2 q - 3} & - K_{2 q - 2} (K_{2 q - 3} + K_{2 q - 1}) \end{matrix}] \end{array}

Using (14, 15) and (18), it can be shown that:

D_{e} T = T^{T} D_{e}

(19)

where $D_{e} = U^{T} R_{Λ} U = diag [K_{0}   K_{2}   \dots   K_{2 q - 2}] \in R^{2 q \times 2 q} .$

The moments $M_{k}$ (⁠ $k = 0, 1, \dots, 2 q - 2$ ⁠) are expressed as follows:

\begin{array}{l} M_{k} & = M^{T} A^{2 k} N \\ = K_{0} E_{1}^{T} T^{k} D_{e}^{- 1} U^{T} K T^{k} E_{1} \\ = K_{0} E_{1}^{T} T^{k} D_{e}^{- 1} D_{e} T^{k} E_{1} \\ = K_{0} E_{1}^{T} T^{2 k} E_{1} \end{array}

(20)

where $E_{1} = {[1   0   \dots   0]}^{T} \in R^{2 q \times 2} .$

Therefore, the partial realization of the transfer function becomes:

\begin{array}{l} H (s) & \approx H_{q} (s) \\ = \sum_{k = 0}^{2 q} M_{k} s^{k} \\ = K_{0} \sum_{k = 0}^{2 q} E_{1}^{T} T^{k} E_{1} s^{k} \\ = K_{0} E_{1}^{T} {(I - s T)}^{- 1} E_{1} \end{array}

(21)

In the CLN method, the approximation $H_{q} (s)$ of the original transfer function is represented by a continued fraction. It can be shown that:

T = - S^{T} D_{0} S D_{e}

(22)

where $D_{0} = V^{T} NV = diag [K_{1}   K_{3}   \dots   K_{2 q - 1}] \in R^{2 q \times 2 q}, V = [V_{1}   V_{3}   \dots   V_{2 q - 1}] \in R^{n \times 2 q},$ and $S \in R^{2 q \times 2 q}$ denotes:

\begin{array}{l} S = [\begin{matrix} 1 & - 1 & 0 & \dots & 0 \\ 0 & 1 & - 1 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ ⋮ & ⋱ \\ 0 & 0 & 0 & \dots & 1 \end{matrix}] . \end{array}

(23)

Substituting (22) in (21) results in:

\begin{array}{l} H_{q} (s) & = K_{0} E_{1}^{T} {(I + s S^{T} D_{0} S D_{e})}^{- 1} E_{1} \\ = K_{0} E_{1}^{T} D_{e}^{- 1} {(D_{e}^{- 1} + s S^{T} D_{0} S)}^{- 1} E_{1} \\ = E_{1}^{T} {(D_{e}^{- 1} + s S^{T} D_{0} S)}^{- 1} E_{1} \end{array}

(24)

Equation (24) is the transfer function of the reduced-order system written as follows:

(D_{e}^{- 1} + s S^{T} D_{0} S) x = E_{1} v_{i n}

(25)

i_{o u t} = E_{1}^{T} x

(26)

where:

\begin{array}{l} S^{T} D_{0} S = [\begin{matrix} K_{1} & - K_{1} & \dots & 0 \\ - K_{1} & K_{1} + K_{3} & ⋱ \\ ⋱ & ⋱ \\ K_{2 q - 5} + K_{2 q - 3} & - K_{2 q - 3} \\ 0 & - K_{2 q - 3} & K_{2 q - 3} + K_{2 q - 1} \end{matrix}] \end{array}

(27)

The system given by equations (25) and (26) describes the reduced-order model. In particular, equation (25) corresponds to the KVL of the equivalent Cauer circuit, and the transfer function of the reduced-order system corresponds to the admittance of the Cauer circuit. It can be written as follows:

H_{q} (s) = {K_{0}^{- 1} + {\frac{1}{s} K_{1}^{- 1} + {\dots + {[K_{2 q - 2}^{- 1} + {(\frac{1}{s} K_{2 q - 1}^{- 1})}^{- 1}]}^{- 1}}^{- 1}}^{- 1}}^{- 1}

(28)

where the inverse of the even coefficient matrices $K_{2 i}$ represent the resistive component of the Cauer circuit, $R_{2 i}$ and the odd coefficient matrices, $K_{2 i + 1}$ represent the inductive component of the Cauer circuit and $L_{2 i + 1}$ ⁠, for $i = 0, \dots, q - 1$ ⁠:

\begin{array}{l} R_{2 i} = [\begin{matrix} R_{2 i_{11}} & R_{2 i_{12}} \\ R_{2 i_{21}} & R_{2 i_{22}} \end{matrix}] = K_{2 i}^{- 1}, L_{2 i + 1} = [\begin{matrix} L_{2 i + 1_{11}} & L_{2 i + 1_{12}} \\ L_{2 i + 1_{21}} & L_{2 i + 1_{22}} \end{matrix}] = K_{2 i + 1} \end{array}

(29)

It can be concluded that the continued fraction expansion of the reduced-order transfer function can be directly obtained from Algorithm 1. The corresponding equivalent Cauer circuit is shown in Figure 4 (Matsuo et al., 2020).

Figure 4.

A circuit diagram shows two input currents i 1 and i 2 with resistors R 0 and R 2 and inductors L 1 and L 3 in branches.

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The diagram presents a multi-branch electrical circuit. Two input currents labelled i 1 and i 2 enter from the left along separate horizontal lines. Each line includes resistors labelled R 0 and R 2 enclosed in highlighted blocks. Vertical connections link nodes between the lines. Inductor pairs labelled L 1 and L 3 appear in lower branches connected between nodes and the bottom reference line. Voltages v 1 and v 2 are indicated vertically on the left side. The network extends horizontally with repeated structure towards the right.

Equivalent Cauer ladder network of the two parallel rectangular conductors

4. Numerical results

Considering $q = 5$ ⁠, the values of resistances and inductances of the reduced order Cauer circuit of Figure 4 are reported, respectively, in Tables 1 and 2.

Table 1.

Resistances of the Cauer ladder network $R_{2 i_{j k}}  [Ω]$ ⁠, for $i = 0, \dots, 4$ and $j, k = 1, 2$


Self-resistances
$R_{0_{11}} = R_{0_{22}}$	$R_{2_{11}} = R_{2_{22}}$	$R_{4_{11}} = R_{4_{22}}$	$R_{6_{11}} = R_{6_{22}}$	$R_{8_{11}} = R_{8_{22}}$
$1.19 \cdot 10^{- 4}$	$4.75 \cdot 10^{- 2}$	$2.75 \cdot 10^{- 1}$	1.01	2.58
Mutual resistances
$R_{0_{12}} = R_{0_{21}}$	$R_{2_{12}} = R_{2_{21}}$	$R_{4_{12}} = R_{4_{21}}$	$R_{6_{12}} = R_{6_{21}}$	$R_{8_{12}} = R_{8_{21}}$
0	$4.29 \cdot 10^{- 2}$	$2.52 \cdot 10^{- 1}$	$9.42 \cdot 10^{- 1}$	2.39

Table 2.

Inductances of the Cauer ladder network $L_{2 i + 1_{j k}}  [μH]$ ⁠, for $i = 0, \dots, 4$ and $j, k = 1, 2$


Self-inductances
$L_{1_{11}} = L_{1_{22}}$	$L_{3_{11}} = L_{3_{22}}$	$L_{5_{11}} = L_{5_{22}}$	$L_{7_{11}} = L_{7_{22}}$	$L_{9_{11}} = L_{9_{22}}$
$8.68 \cdot 10^{- 1}$	$4.21 \cdot 10^{1}$	$1.52 \cdot 10^{2}$	$2.39 \cdot 10^{2}$	$3.38 \cdot 10^{2}$
Mutual inductances
$L_{1_{12}} = L_{1_{21}}$	$L_{3_{12}} = L_{3_{21}}$	$L_{5_{12}} = L_{5_{21}}$	$L_{7_{12}} = L_{7_{21}}$	$L_{9_{12}} = L_{9_{21}}$
$5.21 \cdot 10^{- 1}$	$3.89 \cdot 10^{1}$	$1.39 \cdot 10^{2}$	$2.22 \cdot 10^{2}$	$3.13 \cdot 10^{2}$

Per-unit-length power losses of the system are computed for both the PEEC and the CLN formulation with $q = 20$ ⁠, for frequencies up to 10 kHz, considering three different meshings of the conductors reported in Table 3.

Table 3.

Meshes for the two conductors where $n_{x}$ and $n_{y}$ are, respectively, the number of subdivisions along the width and thickness of each conductor and n is the total number of sub-conductors

	Mesh parameters
Mesh	n_x	n_y	n
Mesh 1	31	3	120
Mesh 2	62	6	610
Mesh 3	124	12	2706

The resulting values are reported in Figures 5 and 6, where they are also compared with the FEM results obtained using COMSOL.

Figure 5.

A two-panel graph compares losses and relative error versus frequency for F E M and P E E C meshes.

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The two panels show frequency in hertz on the horizontal axis from 1 to 10000. The top panel plots losses in watt per metre. Curves for F E M and P E E C meshes increase gradually and then rise more sharply at higher frequency. The F E M curve is highest, while P E E C mesh 3 is closest to it, followed by mesh 2, with mesh 1 lowest. The bottom panel shows relative error. Error remains low at lower frequency and increases with frequency. Mesh 1 shows the largest increase, mesh 2 moderate, and mesh 3 the smallest.

Comparison between per-unit-length power losses of the two conductors computed by PEEC with respect to FEM reference results, considering the three different PEEC meshes

Figure 6.

A two-panel graph compares losses and relative error versus frequency for F E M and C L N meshes.

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The two panels show frequency in hertz on the horizontal axis from 1 to 10000. The top panel plots losses in watts per metre. Curves for F E M and C L N meshes increase gradually and then rise more sharply at higher frequency. The F E M curve is highest. C L N mesh 3 is closest to it, followed by mesh 2, with mesh 1 lowest. The bottom panel shows relative error. Error remains low at lower frequency and increases with frequency. Mesh 1 shows the largest increase, mesh 2 moderate, and mesh 3 the smallest.

Comparison with respect to FEM reference results of the per-unit-length power losses of the two conductors, computed by CLN with $q = 20$ ⁠, considering three different PEEC meshes

To correctly apply the CLN to the PEEC formulation, the PEEC must provide accurate results. In fact, if the CLN method is applied to PEEC without using a sufficiently refined mesh of the conductors, it cannot guarantee an accurate representation of the reference solution, as shown in Figures 5 and 6. For this reason, in the following computations, mesh 3 will be adopted.

The ladder elements of the CLN circuit represent the frequency resistive and inductive effects as can be noted in Figure 7. Here, the per-unit length terminal resistance and reactance of the conductors are shown.

Figure 7.

A two-panel graph shows resistance and reactance versus frequency, comparing F E M and C L N meshes.

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The two panels show frequency in hertz on the horizontal axis from 1 to 100000. The top panel shows resistance in ohms per metre. All curves increase slowly at low frequency and rise sharply after about 10000 hertz. The F E M curve is highest. C L N with 20 segments closely follows it, then 15, then 10, while 5 remains lowest. The bottom panel shows reactance in ohms per metre. All curves remain near zero at low frequency and increase rapidly at high frequency. All methods closely overlap, showing very small differences across mesh sizes.

Per-unit length terminal resistance and reactance computed by PEEC and by the CLN formulation, with different numbers of stages

Once a good meshing of the conductors is provided, the relationship between the number of stages of the reduced equivalent circuit and the accuracy of the CLN formulation is analyzed. In particular, the per-unit-length power losses computed with PEEC are compared with those obtained using the CLN formulation, considering different numbers of stages (Figure 8).

Figure 8.

A two-panel graph shows losses and relative error versus frequency, comparing F E M and C L N meshes.

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The two panels show frequency in hertz on the horizontal axis from 1 to 100000. The top panel shows losses in watt per metre. All curves increase gradually at low frequency and rise sharply beyond about 10000 hertz. The F E M and C L N 20 curves overlap closely and give the highest values. C L N 15 is slightly lower, C L N 10 is moderate, and C L N 5 is lowest. The bottom panel shows relative error. Error remains near zero at low frequency and increases with frequency. C L N 5 has the largest error, followed by 10, while 15 and 20 remain very small.

Per-unit length power losses computed by PEEC and by the CLN formulation, with different numbers of stages and the corresponding relative error

Computational times versus number of stages q of the reduced equivalent circuit are reported in Table 4. Note that the CLN method with 20 stages is almost 20 times faster than the full order PEEC formulation when computing the terminal impedance.

Table 4.

Computational time of the terminal impedance matrix of the conductors using the full-order PEEC formulation and applying CLN with different numbers of equivalent circuit stages

Method	$R_{Λ}, L_{Λ}  (s)$	Algorithm (s)	$Z  (s)$	Total (s)
PEEC	$8.07 \cdot 10^{- 2}$	0	4.16	4.24
5-CLN	$8.07 \cdot 10^{- 2}$	$8.29 \cdot 10^{- 2}$	$2.59 \cdot 10^{- 3}$	$1.66 \cdot 10^{- 1}$
10-CLN	$8.07 \cdot 10^{- 2}$	$9.11 \cdot 10^{- 2}$	$8.43 \cdot 10^{- 4}$	$1.73 \cdot 10^{- 1}$
15-CLN	$8.07 \cdot 10^{- 2}$	$1.23 \cdot 10^{- 1}$	$6.73 \cdot 10^{- 4}$	$2.04 \cdot 10^{- 1}$
20-CLN	$8.07 \cdot 10^{- 2}$	$1.36 \cdot 10^{- 1}$	$2.07 \cdot 10^{- 3}$	$2.19 \cdot 10^{- 1}$

Method	$R_{Λ}, L_{Λ}  (s)$	Algorithm (s)	$Z  (s)$	Total (s)
PEEC	$8.07 \cdot 10^{- 2}$	0	4.16	4.24
5-CLN	$8.07 \cdot 10^{- 2}$	$8.29 \cdot 10^{- 2}$	$2.59 \cdot 10^{- 3}$	$1.66 \cdot 10^{- 1}$
10-CLN	$8.07 \cdot 10^{- 2}$	$9.11 \cdot 10^{- 2}$	$8.43 \cdot 10^{- 4}$	$1.73 \cdot 10^{- 1}$
15-CLN	$8.07 \cdot 10^{- 2}$	$1.23 \cdot 10^{- 1}$	$6.73 \cdot 10^{- 4}$	$2.04 \cdot 10^{- 1}$
20-CLN	$8.07 \cdot 10^{- 2}$	$1.36 \cdot 10^{- 1}$	$2.07 \cdot 10^{- 3}$	$2.19 \cdot 10^{- 1}$

As an example of a time-domain simulation, the Joule losses are computed when the two conductors are excited by the following two opposite currents:

i_{1, 2} (t) = \pm I_{0} \cdot (1 - e^{- t / τ})

(30)

where $I_{0} = 10  A$ and $τ = 10^{- 2} s$ ⁠.

In time-domain analysis, the stage number is fixed to $q = 10$ to accurately represent the full order model in the frequency range 0–10 kHz (Figure 8), which contains the bandwidth of the input current waveform. The corresponding Joule losses are reported in Figure 9.

Figure 9.

A two-panel graph shows current response and losses over time, comparing F E M, P E E C, and C L N methods.

View large Download slide

The two panels show time in seconds from 0 to 0.05 on the horizontal axis. The top panel shows current in amperes. Current i 1 increases from 0 to about 9, while current i 2 decreases from 0 to about negative 9, both approaching steady values smoothly. The bottom panel shows losses in watts per metre. Losses start near 0 and rise steadily to about 2.4. The F E M, P E E C, and C L N 10 curves closely overlap, indicating very similar behaviour across methods.

Time-domain analysis of two parallel rectangular conductors, with the two current sources and the total Joule losses, considering $q = 10$ stages of the equivalent reduced Cauer circuit

5. Conclusion

The CLN method has been applied for the first time to a PEEC formulation in an efficient way, also thanks to the diagonal structure of the $R_{Λ}$ matrix that can be easily inverted, as required by the algorithm. The obtained reduced order model maintains the same advantages of the PEEC method, i.e. only active conductors must be discretized and open boundary problems are well treated.

For wide-band frequency domain analysis, the CLN representation can reduce computational times compared to the full-order PEEC and FEM methods. Furthermore, good accuracy can also be obtained in a time domain analysis.

The CLN representation of the eddy current fields extracted from a proper PEEC formulation has been shown to be accurate enough, provided that a sufficient number of equivalent circuit stages are considered. As expected, the accuracy of the CLN reduced-order model extracted from PEEC decreases as the working frequency increases.

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Matteo Emiliani, Luca Di Rienzo, Shingo Hiruma, Isacco Figoni, Carlo de Falco and Tetsuji Matsuo

Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at Link to the terms of the CC BY 4.0 licenceLink to the terms of the CC BY 4.0 licence.

Cauer ladder network representation of eddy-current fields using the partial element equivalent circuit method

1. Introduction

2. PEEC formulation

3. CLN formulation

4. Numerical results

5. Conclusion

References

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Cauer ladder network representation of eddy-current fields using the partial element equivalent circuit method

1. Introduction

2. PEEC formulation

3. CLN formulation

4. Numerical results

5. Conclusion

References

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