JP4683591B2 - Magnetization distribution calculation apparatus and calculation method - Google Patents

Description

本微分方程式は、Ｒと２√（Ｌ／Ｃ）に応じて以下に示す３つの場合に分けられる。
【００１５】
【数式２】

【００１６】
【数式３】

【００１７】
【数式４】

なお上式において、実際には着磁器のインダクタンスＬは、着磁器の磁化飽和の程度により変化するが、ここでは一定（ユーザが指定した一定の電流を流したときの値）として扱うことにする。
【００１８】
図３は式（１−２）から式（１−４）の基本式を用いて、着磁電流波形を計算する手順を示したフローチャートである。以下その内容を説明する。
【００１９】
ステップＳ３０１では、着磁器の構成等を示すデータとして着磁器本体のデータおよび電源回路のデータをメモリに読み込む。具体的には、着磁器本体についての着磁器分割モデル、着磁ヨークのＢＨ特性及び導電率、コイルの巻き数、着磁電流の目安（一定値）と、電源回路についてのコイルの抵抗Ｒ、コンデンサの容量Ｃ、着磁電流のピーク値Ｉである。
【００２０】
ステップＳ３０２では、ステップＳ３０１で読み込んだ着磁器本体のデータに基づいて、有限要素法により磁場計算を行い、着磁器のインダクタンスＬを計算する。なお、着磁電流の目安としては、着磁ヨークが飽和領域に達しない程度の微弱な電流でよい。
【００２１】
ステップＳ３０３では、電源回路のコンデンサの両端子電圧Ｖの初期値を設定する。本初期値は１Ｖ 程度でよい。
【００２２】
ステップＳ３０４では、ステップＳ３０１で読み込んだ電源回路のコンデンサの容量ＣおよびステップＳ３０３で設定した電源回路のコンデンサの両端子電圧Ｖの値とから電源回路のコンデンサの電荷量Ｑを求める。
【００２３】
ステップＳ３０５では、着磁電流波形を、上式（１−２）〜（１−４）により求める。
【００２４】
ステップＳ３０６ではステップＳ３０５で求めた着磁電流波形のピーク値をＪ_ｐｅａｋとして設定する。
【００２５】
ステップＳ３０７では、ステップＳ３０６で設定されたＪ_ｐｅａｋを、ユーザが指定した着磁電流のピーク値（ステップＳ３０１で読み込んだ着磁電流のピーク値Ｉ）と比較する。十分一致すればステップＳ３０６で得られた着磁電流波形を最終結果とする（ステップＳ３０７）。
【００２６】
一方、ステップＳ３０８でＪ_ｐｅａｋが、ユーザが指定した着磁電流のピーク値と一致しない場合には、電源回路のコンデンサ両端子電圧Ｖを変更して（ステップＳ３０８）、一致するまで繰り返す。
【００２７】
なお、ここではステップＳ３０２の着磁器のインダクタンスＬの計算に有限要素法を使用したが、境界要素法、積分法、差分法等他の方法を使用してもよい。
【００２８】
２．完全着磁曲線推定部
完全着磁曲線推定部は、磁化分布の算出に必要な磁化曲線（磁石に磁界Ｈを印加して着磁した場合の磁化Ｍの状態を示す曲線）をもとめるもので、磁化曲線（ＢＨ曲線）の測定において、飽和磁化に達するまでの十分な磁界（高磁界）がかけられない場合に、すでに測定された磁石の磁化曲線（低磁界の磁化曲線）を外挿することによって完全着磁部分までを推定する処理手段である。
【００２９】
以下にその方法とその方法の妥当性を示したうえで、処理の流れについて説明する。
【００３０】
１）完全着磁部分の推定方法
完全着磁部分の推定にあたっては、ＢＨ曲線が次のＳｔｏｎｅｒ−Ｗｏｈｌｆａｒｔｈ方程式で表されるものとして行う。なおここで、Ｈ_Ａは磁石の異方性磁界である。以下にその方法を説明する。
【００３１】
【数式５】

Ｓｔｏｎｅｒ−Ｗｏｈｌｆａｒｔｈ方程式を導入するにあたって、まず飽和磁化Ｍ_ｓが必要である。ここでは次の２つの方法のいずれかで決定する。
【００３２】
ａ．等方性磁石を想定し、単純に残留磁化Ｍ_ｒの２倍とする。
ｂ．図４に示すように、初磁化曲線（（ａ）図）から１／Ｈ^２外挿を行う（完全着磁曲線を２次曲線で近似する）ことにより得る（（ｂ）図）。
【００３３】
ここで、図４（ａ）は横軸に磁界Ｈを、縦軸に磁化Ｍをとった場合の磁化曲線であり、図４（ｂ）は横軸に１／Ｈ^２を、縦軸に磁化Ｍをとり、図４（ａ）の磁化曲線に基づいて各点をプロットし、２次曲線で近似することで１／Ｈ^２＝０のときの磁化Ｍの値、すなわち飽和磁化Ｍ_ｓをもとめるための図である。
【００３４】
本発明は基本的には等方性磁石に対して適用できるものであるが、僅かに異方性がある場合にも上記ｂの方法を適用することが可能であり、ａ、ｂの両方をユーザが選択できるようにすることで、利用の範囲を拡げている。
【００３５】
次に得られた飽和磁化Ｍ_ｓ及び測定値の残留磁化Ｍ_ｒを用いて角型比Ｒ_ｓを求め（上記ａを使用した場合０．５）、以下の手順で磁化曲線の測定値がない部分の磁界Ｈに対する磁化Ｍを求める。
【００３６】
（ｉ）可逆磁化回転領域（磁化曲線において、着磁時の磁化曲線と減磁時の磁化曲線のヒステリシスループが一本になった領域。すなわち図５に示す（Ｈ_ｂ、Ｍ_ｂ）以上の領域）の任意の点（Ｈ_ｆ、Ｍ_ｆ）に対して、まず次式（２−５）により任意の点（Ｈ_ｆ、Ｍ_ｆ）における磁化回転角φ_ｆを求め、次に（２−６）式により異方性磁界Ｈ_Ａを求める。
【００３７】
ここで、図５は、横軸に磁界Ｈを、縦軸に磁化Ｍをとった場合の磁化曲線であり、着磁時の磁化曲線と、可逆磁化回転領域以上（Ｈ_ｂ）の磁界を印加後に減磁した場合の減磁時の磁化曲線（減磁曲線）とを重ねて記載したものでＭ_ｓは飽和磁化、Ｍ_ｒは残留磁化を示す。
【００３８】
なお、（２−５）式は上記（２−２）式にＭ＝Ｍ_ｆを代入し磁化回転角φについて解いたもので、（２−６）は（２−１）式にＨ＝Ｈ_ｆを代入し異方性磁界Ｈ_Ａについて解いたものである。
【００３９】
また、可逆磁化回転領域の点が複数ある場合は、それらの異方性磁界Ｈ_Ａの平均値を使用する（すなわち、入力データ中の磁界Ｈ_ｂ以上のすべての点を使用する）。
【００４０】
【数式６】

（ｉｉ）上記（ｉ）で得た異方性磁界Ｈ_Ａを用いて、磁化曲線の測定値がない部分の磁界Ｈに対する磁化回転角φを（２−１）式によって求め（求めかたは次述）、（２−２）式により磁化Ｍを求める。なお（ｉｉ）において、次式（Ａ−１）のＳｔｏｎｅｒ−Ｗｏｈｌｆａｒｔｈ方程式に、磁界Ｈ、異方性磁界Ｈ_Ａが与えられたときの磁化回転角φを求める必要がある。
【００４１】
【数式７】

ここでは（Ａ−１）式を（Ａ−２）式、（Ａ−３）式のように置き、ニュートン法を用いて、φ＝０から出発し、（Ａ−４）式を繰り返すことにより、磁化回転角φを求める。
【００４２】
【数式８】

上記方法により、磁界Ｈに対する磁化Ｍが求まる。一般に磁化曲線は磁界Ｈが大きくなるにつれて飽和磁化Ｍ_ｓに近づき、その変化の度合いは小さくなる。従って磁化曲線を有限ないくつかの点で表現する場合には、磁界Ｈが大きな部分ほど点の間隔を大きくした方が少数の点で表現することができる。このとき磁化Ｍを与えて磁界Ｈを求める方法では、磁化Ｍの与え方が難しいが（飽和磁化Ｍ_ｓに近づくほど、間隔を小さくする必要がある）、本方法によれば、磁界Ｈを与えることから、点の間隔を容易に大きく設定できるので非常に便利であるという効果が得られる。
【００４３】
２）完全着磁部分の推定方法の妥当性
次に上記１）のＳｔｏｎｅｒ−Ｗｏｈｌｆａｒｔｈ方程式により得られたＢＨ曲線の外挿部分の妥当性を確認する方法を説明する。
【００４４】
【数式９】

（２−６）式で求めた異方性磁界Ｈ_Ａは、減磁曲線のＭ_ｒ点での接線の勾配をχ_ｒとして（図６は減磁時の磁化曲線（減磁曲線）で、磁界Ｈ＝０の時の磁化ＭをＭ_ｒ、Ｍ_ｒ点における減磁曲線の傾きをχ_ｒとして）、次式で表すことができることがわかっている。なお本式中＜＞は空間平均値を意味し、＜＞の値は完全異方性磁石ではゼロ、完全等方性磁石では２／３である。
【００４５】
また＜＞の値は、後述の代表粒子モデルに対しては、次式のようになる。
【００４６】
【数式１０】

そこで、上記（２−６）式で求めた異方性磁界Ｈ_Ａを（２−７）式に代入して得られるχ_ｒ（次式（２−１０））と、測定した磁化曲線のχ_ｒとを比較することで、上記で推定された磁化曲線の完全磁化部分の妥当性を確認することができる。
【００４７】
なお等方性磁石に対して、（２−１０）式の＜＞を計算するにあたっては、２／３とする方法と（２−９）を使用する方法の２つをあげたが、両者の差は小さく、どちらを採用してもよい。
【００４８】
【数式１１】

【００４９】
３）完全着磁部分の推定処理の流れ
次に完全着磁部分の推定処理の具体的な手順を図７を用いて以下に説明する。
【００５０】
ステップＳ７０１では、永久磁石の特性である、初磁化曲線、可逆回転磁化領域の最小磁界（Ｈ_ｂ）、残留磁化Ｍ_ｒをメモリに読み込む。なお、これらはいずれも測定によって得られるデータである。
【００５１】
ステップＳ７０２ではステップＳ７０１で読み込んだ磁化曲線を構成する各点のデータに関して、磁化Ｍを求める。
【００５２】
ステップＳ７０３では上記ａまたはｂの方法により、飽和磁化Ｍ_ｓを求め、ステップＳ７０４では、角型比Ｒ_ｓを求める。
【００５３】
ステップＳ７０５では（２−３）式により、平均的な磁化回転角ωを求める。ステップＳ７０６では（２−５）式および（２−６）式により、可逆磁化回転領域における異方性磁界Ｈ_Ａを求める。
【００５４】
ステップＳ７０７では（２−１）式、（２−２）式により、測定点のない領域の磁界Ｈに対する磁化Ｍを求める。
【００５５】
ステップＳ７０８では（２−１０）式より磁化曲線における残留磁化Ｍ_ｒ点での減磁曲線の傾きχ_ｒを求める。
【００５６】
３．初磁化過程計算部
初磁化過程計算部は、磁石を有限個の要素に分割し、磁化０の状態から各要素に対して着磁器より磁界Ｈを印加した場合の印加磁界の過渡解析を有限要素法等の一般的な磁界計算を用いて算出し、各要素ごとの最大磁界を抽出し、該最大磁界と磁石の特性とに基づいて各要素ごとの磁化（最大磁化点）を算出する処理である。以下に、有限要素法等の一般的な磁界計算を利用できる根拠を示した上で、初磁化過程計算部における処理手順を説明する。
【００５７】
１）有限要素法等の一般的な磁界計算を利用できる根拠
図８のように、単位体積の磁性体片に外部磁界Ｈ_０が印加され、平均磁化Ｍが誘起されたときの系の自由エネルギー変化は次式で表される（ここでは熱的、力学的効果等は考えない）。
【００５８】
【数式１２】

本式の積分項をここでは、「反磁界Ｈ_ｄによるエネルギー」と「有効磁界Ｈ_ｅｆｆによるエネルギー」とに便宜的に分けて取り扱うことにすると、（３−２）式のように表すことができる。
【００５９】
【数式１３】

次に、（３−２）式の右辺のそれぞれの項について考える。
【００６０】
まず反磁界係数テンソルの主軸をａ軸、ｂ軸方向、主値をそれぞれＮ_ａ、Ｎ_ｂ（Ｎ_ｂ≧Ｎ_ａ）とすると、反磁界エネルギーは、重ねあわせの原理に従って、ａ軸方向にＭｃｏｓ（θ−φ）を持つ場合とｂ軸方向に磁化Ｍｓｉｎ（θ−φ）をもつ場合のエネルギーの和として次式で表すことができる。ここで各ベクトルの方向は、図８に示すとおりで斜線領域で示した単位体積の磁性体片に対して、磁化容易軸方向と磁化困難軸方向とをそれぞれ破線矢印のようにとった場合、反磁界係数テンソルの主軸のうち、ａ軸と磁化容易軸方向とのなす角度をφ、磁化Ｍの方向と磁化容易軸方向とのなす角度をθ、外部磁界Ｈ_Ｏと磁化容易軸方向とのなす角度をΩとおく。また、μ_０は真空の透磁率である。
【００６１】
【数式１４】

次に磁化エネルギーは、有効磁界Ｈ_ｅｆｆの磁化容易軸方向と磁化困難軸方向の成分をそれぞれＨ_ｅ、Ｈ_ｈとし、またそれぞれの方向の磁化成分Ｍ_ｅ、Ｍ_ｈをＭｃｏｓθ、Ｍｓｉｎθで置き換えると、次のように表される。
【００６２】
【数式１５】

一方、外部磁界Ｈ_Ｏとの相互エネルギーは次の通りである。
【００６３】
【数式１６】

以上より、次の自由エネルギーＵの具体式（３−７）が得られる。
【００６４】
【数式１７】

（３−７）式に変分原理を適用する。すなわち∂Ｕ／∂Ｍ＝０、∂Ｕ／∂θ＝０を求めることにより、次の方程式（３−８）および（３−９）が導かれる。
【００６５】
【数式１８】

異方性磁性体片の磁化Ｍを求めるためには、本連立方程式を解けばよい。ここで、等方性材料からなる、磁性体の塊の場合について考える。このとき、磁性体内部の微小部分においては形状異方性に対して、
【００６６】
【数式１９】

としてよい。 また等方性であることから、
【００６７】
【数式２０】

である。これより（３−９）式は、次のように表される。
【００６８】
【数式２１】

よって、（３−１３）式が導かれる。
【００６９】
【数式２２】

（３−１３）より、磁化Ｍ、外部磁界Ｈ_０、反磁界Ｈ_ｄ、有効磁界Ｈ_ｅｆｆはすべて方向が一致することがわかる。一方このとき、（３−８）式は、次のようになる。
【００７０】
【数式２３】

ここで、反磁界Ｈ_ｄ、有効磁界Ｈ_ｅｆｆは（３−１５）、（３−１６）式の通りであることから、（３−１４）式は、（３−１７）式のようになる。
【００７１】
【数式２４】

磁化Ｍ、外部磁界Ｈ_０、反磁界Ｈ_ｄ、有効磁界Ｈ_ｅｆｆはすべて方向が一致すること及び（３−１７）式から、等方性磁石の塊の場合、（３−８）、（３−９）式における磁化Ｍと有効磁界Ｈ_ｅｆｆの関係は、等方性軟磁性体の磁化現象と同じになることがわかる。
【００７２】
従って、等方性磁石の初磁化過程は、磁石部分を、磁石の初磁化曲線をもった軟磁性体として扱うことで、有限要素法等の一般的な磁界計算で求めることができる。
【００７３】
なお本初磁化過程を、着磁電流の時間変化、渦電流考慮等過渡的に計算する場合において、どの時点での磁化を採用すべきかが問題となる。そこでここでは有限要素法で磁界計算する際の各要素に対して、時間経過中における最大値を採用することにする。
【００７４】
また、本初磁化過程によって、各粒子の磁化のうち、所定の分布の粒子で磁化反転が起こり、その結果、粒子磁化のベクトル総和は有限値Ｍ_ｍになるとして今後の説明を行う。
【００７５】
２）初磁化過程計算部における処理手順
次に上記１）をふまえて初磁化過程計算部における具体的な処理手順を図９を用いて以下に説明する。
【００７６】
ステップＳ９０１では、着磁器の構成等を示すデータとして着磁器本体のデータをメモリに読み込む。具体的には着磁器本体における着磁器分割モデル、着磁ヨークのＢＨ特性及び導電率、コイルの巻き数、着磁電流波形として着磁電流波形計算部で得た結果である。
【００７７】
ステップＳ９０２では磁石の特性データとして、上記完全着磁曲線推定部で得た完全着磁曲線、及び磁石の導電率をメモリに読み込む。
【００７８】
ステップＳ９０３では有限要素法等により磁界計算を行う。計算はヨーク、磁石の渦電流、飽和を考慮した非線型過渡解析となる。
【００７９】
ステップＳ９０４では各要素の磁束密度経時変化のピーク値を最大磁束密度点Ｂ_ｍとし、最大磁束密度点Ｂ_ｍに基づいてそれより最大磁化点Ｍ_ｍを求めて外部記憶装置に保存する。
【００８０】
４．残留磁化計算部
残留磁化計算部は、所定の磁界を印加して磁石を着磁後に磁界を０とした場合の磁石の残留磁化をもとめる処理手段であり、上記初磁化過程計算部によりもとめた最大磁化点Ｍ_ｍをとおる磁化曲線をもとめ、該磁化曲線に基づいて算出する。以下にその基本式と、最大磁化点Ｍ_ｍをとおる磁化曲線をもとめる手順を示した上で、残留磁化計算部における処理の流れについて説明する。
【００８１】
１）残留磁化計算部における基本式
磁石の減磁曲線の第一象限の最大磁化点（Ｈ_ｍ、Ｍ_ｍ）から第二象限の急降下部の寸前までの範囲は、次の単磁区粒子の一斉磁化回転モデルすなわちＳｔｏｎｅｒ−Ｗｏｈｌｆａｒｔｈ方程式で再現できる。なお式（４−１）において、Ｈは外部磁界である。
【００８２】
【数式２５】

ここでＨ_Ａ（Ｍ_ｍ）は上記初磁化過程計算部でもとめた最大磁化点（Ｈ_ｍ、Ｍ_ｍ）を通る減磁曲線における異方性磁界である。また、Ｍ_ｓ（Ｍ_ｍ）は図１０に示す磁化曲線において、最大磁化点（Ｈ_ｍ、Ｍ_ｍ）をとおる減磁曲線の飽和磁化であり、Ｒ_ｓ（Ｍ_ｍ）は該減磁曲線の角型比である。なおこれらの値は、初磁化過程における磁化反転の状態に依存するものであり、最大磁化点（Ｈ_ｍ、Ｍ_ｍ）によって決定されるものである。
【００８３】
ここで、各最大磁化点ごとの減磁曲線の角型比は、最大磁化点に依存せず、次のように一定と仮定する（（２−４）式の値と同じ）。
【００８４】
【数式２６】

減磁曲線が、メジャーループの磁化曲線の場合（メジャーループの磁化曲線とは、可逆磁化回転領域以上の磁界以上（すなわちＨ_ｂ以上）印加した後に減磁した場合の着磁および減磁曲線をいう）を考えると、可逆回転磁化領域の最小磁化点（Ｈ_ｂ、Ｍ_ｂ）において、磁化回転角φ_ｂは、（４−２）式より（４−６）式のようになり、このとき本状態での異方性磁界Ｈ_Ａ（Ｍ_ｂ）は（４−１）式より、（４−７）式となる。
【００８５】
【数式２７】

ここで、異方性磁界Ｈ_Ａ（Ｍ_ｂ）は次式のように、最大磁化点に比例すると仮定する。
【００８６】
【数式２８】

以上により、次の２）に示す手順で、任意の最大磁化点（Ｈ_ｍ、Ｍ_ｍ）に対して、減磁曲線を再現することができるようになった。
【００８７】
２）最大磁化点Ｍ_ｍをとおる磁化曲線をもとめる手順
最大磁化点Ｍ_ｍをとおる磁化曲線をもとめる手順を以下示す。
（ｉ）（４−３）、（４−５）式より磁化回転角ωを求め、（４−６）式より可逆磁化回転領域における磁化回転角φ_ｂを求める。そして（４−７）、（４−８）式により異方性磁界Ｈ_Ａ（Ｍ_ｍ）を求める。
【００８８】
（ｉｉ）（４−１）式を用いて、最大磁化点（Ｈ_ｍ、Ｍ_ｍ）における（Ｈ＝Ｈ_ｍ）磁化回転角φ_ｍを求める。これは上述の（Ａ−１）〜（Ａ−４）式を用いて求めることができる。
【００８９】
（ｉｉｉ）（４−２）式において、φ＝φ_ｍ、Ｍ＝Ｍ_ｍとおいて、飽和磁化Ｍ_ｓ（Ｍ_ｍ）を求める。本値は次式（４−９）の通りである。
【００９０】
（ｉｖ）任意の磁界Ｈに対して（４−１）式により磁化回転角φを求めた後、（４−２）式を用いて磁化Ｍを求める。
【００９１】
【数式２９】

本減磁曲線におけるＨ＝０での磁化（残留磁化Ｍ_ｒ（Ｍ_ｍ））は、（４−２）式におけるφ＝０の場合に相当し（←（４−１）式は、Ｈ＝０のときφ＝０となる）、（４−２）式に（４−９）式を代入した次式（４−１０）で求めることができる。
【００９２】
【数式３０】

【００９３】
３）残留磁化計算部における処理の流れ
次に残留磁化計算部における具体的な手順を図１１を用いて以下に説明する。ステップＳ１１０１では永久磁石の磁化曲線データである、可逆回転磁化領域の最小磁界Ｈ_ｂ、残留磁化Ｍ_ｒ、飽和磁化Ｍ_ｓ、χ_ｒｅｃ及び完全着磁曲線推定部で得た完全着磁を含む磁化曲線をメモリに読み込む。
【００９４】
ステップＳ１１０２ではステップＳ１１０１で読み込んだ残留磁化Ｍ_ｒと飽和磁化Ｍ_ｓとに基づいて角型比Ｒ_ｓを求める。一般には０．５に近い値となる。
【００９５】
ステップＳ１１０３では上記初磁化過程計算部で得た各要素の最大磁化点の値Ｍ_ｍを読み込む。
【００９６】
ステップＳ１１０４ではステップＳ１１０３で読み込んだ最大磁化点Ｍ_ｍでの磁化ベクトルの方向を磁化容易軸の方向Θ_ｍとして計算する（本磁化容易軸の方向は後述の残留磁化過程に必要なためである）。
【００９７】
ステップＳ１１０５では次の手順で、各要素の異方性磁界Ｈ_Ａ（Ｍ_ｍ）、及び残留磁化Ｍ_ｒ（Ｍ_ｍ）を求める。
（ｉ）（４−３）式により、磁化回転角ωを求める。
（ｉｉ）（４−６）、（４−７）式により、Ｈ_Ａ（Ｍ_ｓ）を求める。
（ｉｉｉ）各要素について、次の手順で残留磁化Ｍ_ｒ（Ｍ_ｍ）を求める。
ａ．ステップＳ１１０３で読み込んだ最大磁化点Ｍ_ｍ値に対するＨ_ｍ値を完全着磁曲線から求める。
ｂ．（４−８）式により、Ｈ_Ａ（Ｍ_ｍ）を求める。
ｃ．（４−１）式により、磁化回転角φ_ｍを求める。
ｄ．（４−１０）式により、Ｍ_ｒ（Ｍ_ｍ）を求める。
【００９８】
ステップＳ１１０６では各要素の磁化容易軸方向Θ_ｍ、残留磁化Ｍ_ｒ（Ｍ_ｍ）及び異方性磁界Ｈ_Ａ（Ｍ_ｍ）を外部記憶装置に出力する。
【００９９】
５．減磁過程計算部
減磁過程計算部は、残留磁化計算部により算出された各要素の残留磁化と、該残留磁化を有する各要素に、該各要素の磁化が発生する磁界が印加された場合の前記各要素の磁化を算出する処理手段である。以下にかかる減磁過程の磁化分布をもとめる基本式を導く過程を示したうえで、減磁過程計算部における処理手順を説明する。
【０１００】
１）減磁過程の磁化分布をもとめる基本式
減磁過程の計算にあたっては、出発点を残留角θ_ｒを持った残留磁化Ｍ_ｒの状態にとり、磁化回転角θは磁化容易軸から測るものとする。そして、磁化が（Ｍ_ｒ、θ_ｒ）の状態（変化前の状態）から（Ｍ、θ）の状態（変化後の状態）に変化したときの自由エネルギー密度の変化をＵとする。また、反磁界のない状態での磁化に要する仕事は、磁化をＭ_ｒからＭまで磁化容易軸方向に沿って変化させるのに要する仕事Ｕ_ｍａｇと、Ｍをθ_ｒ方向からずらす仕事（異方性エネルギーＵ_{ａｎｉｓｏ}）との和で近似する。すると自由エネルギー密度の変化Ｕは次の４つの寄与の和であらわすことができる。
【０１０１】
【数式３１】

Ｕ_{ｄｅｍａｇ} はいわゆる反磁界エネルギーの変化であり、次式で表される。
【０１０２】
【数式３２】

ａ、ｂは反磁界テンソルの主値Ｎ_ａ、Ｎ_ｂの方向を表す（Ｎ_ａ≦Ｎ_ｂ）。変化後の状態における磁化Ｍ、θ（磁化Ｍと磁化容易軸方向とのなす角度）を未知数とするため、この式に含まれる各磁化は次のようにおきかえることができる。なおここで、φは磁化容易軸方向と反磁界テンソルａ方向とのなす角（＜ ９０ｄｅｇ）であり、Ｍ_ａ、Ｍ_ｂは変化後の状態における磁化の反磁界テンソルの各軸方向の成分、Ｍ_ａｒ、Ｍ_ｂｒは変化前の状態における磁化の反磁界テンソルの各軸方向の成分、θ_ｒは変化前の状態における磁化Ｍ_ｒと磁化容易軸方向とのなす角度をそれぞれ表す（図１２）。また、μ_０は真空の透磁率である。
【０１０３】
【数式３３】

（５−３）式により（５−２）式は次のように表される。
【０１０４】
【数式３４】

次にＵ_ｍａｇを具体的に書き表すため、磁化容易軸方向磁化曲線の主要部を直線で近似する。すなわち、
【０１０５】
【数式３５】

このχ_ｒｅｃは当直線の勾配であり、第２象限の屈曲部寸前の点と残留磁化Ｍ_ｒの点を結ぶ直線の勾配である（図１３）。かくして、Ｕ_ｍａｇは次式で表される。なお、図１４に本式の積分の内容を図で示したもので、Ｕ_ｍａｇは図１４の斜線部分に他ならない。
【０１０６】
【数式３６】

異方性エネルギーＵ_{ａｎｉｓｏ}について、ここでは減磁曲線の急降下部前までの区間では異方性磁界は一定であると仮定して次のように表す。
【０１０７】
【数式３７】

一方、外部磁界との相互エネルギーは次の通りである。
【０１０８】
【数式３８】

以上よりＵの具体的表現は次のようになる。
【０１０９】
【数式３９】

このＵに変分原理を適用する。すなわち∂Ｕ／∂Ｍ＝０、∂Ｕ／∂θ＝０を求めることにより、Ｍとθを決める次の連立方程式を得る。
【０１１０】
【数式４０】

異方性磁石の磁化を求めるためには、本連立方程式を解けばよい。
ここで、等方性材料からなる磁石を考える。このとき、磁石内部の微小部分においては形状異方性に対して、
【０１１１】
【数式４１】

としてよい。 また等方性であることから、
【０１１２】
【数式４２】

である。これより、（５−１０）、（５−１１）式は、次のようになる。
【０１１３】
【数式４３】

ここで磁石内部の微小部分における有効磁界をＨ_ｅｆｆとすると、Ｈ_ｅｆｆは外部磁界Ｈ_０と反磁界Ｈ_ｄの和として表され、その磁化Ｍに平行な方向成分をＨ_ｅｆｆθ、垂直な方向成分をＨ_ｅｆｆ⊥とすると、それぞれ次のように記述される。ただしここで、Ｈ_０θ、Ｈ_ｄθは外部磁界及び反磁界の磁化Ｍに平行な方向成分、Ｈ_０⊥、Ｈ_ｄ⊥は垂直な成分である。
【０１１４】
【数式４４】

これによって（５−１４）、（５−１５）式は次のように表される。
【０１１５】
【数式４５】

本方程式を解くことによってＭ、θが求まる。
次に基準座標軸（ここでは直交座標系のＸ軸の方向を基準軸とする）に対して表現された磁界分布に対して、本方程式（５−１８）、（５−１９）を適用する場合について説明する。
【０１１６】
図１６に示すように磁化容易軸方向が基準軸に対して、Θ_ｍだけ傾いた状態にあり、基準軸（Ｘ軸）から測ったＨ_ｅｆｆ、磁化Ｍの方向をそれぞれΘ_ｅｆｆ、Θとしたとき、磁界Ｈ＝（Ｈ_ｘ、Ｈ_ｙ）は次のように表現できる。つまり次式によって磁界Ｈ＝（Ｈ_ｘ、Ｈ_ｙ）に対する、Ｈ_ｅｆｆとΘ_ｅｆｆが決まる。
【０１１７】
【数式４６】

次に、ここで求まったＨ_ｅｆｆとΘ_ｅｆｆを用いて、磁界Ｈ＝（Ｈ_ｅｆｆθ、Ｈ_ｅｆｆ⊥）は次のように求まる。なおここで、θは磁化容易軸方向と磁化Ｍのなす角度であり、Ω_ｅｆｆは磁化容易軸方向と有効磁界Ｈ_ｅｆｆとのなす角度である。
【０１１８】
【数式４７】

以上により、Ｘ軸方向を基準とした直交座標系で表現された磁界Ｈに対して、（５−２０）〜（５−２５）式を用いて、Ｈ_ｅｆｆθ、Ｈ_ｅｆｆ⊥が求まった。
【０１１９】
そこで更に、式 （５−１９）を用いて磁化回転角θを求め、（５−１８）を用いて磁化Ｍを求めれば、次式により磁化Ｍ＝（Ｍ_ｘ、Ｍ_ｙ）が求まる。
【０１２０】
【数式４８】

減磁後の磁化としては、磁化Ｍ自身が作る磁界Ｈ＝（Ｈ_ｘ、Ｈ_ｙ）が、式（５−１８）、（５−１９）を満たすような、自己無撞着な磁化Ｍ（Ｍとθ）を求めればよい。
【０１２１】
なお、等方性磁石の場合における上記本減磁過程の磁化容易軸の方向Θ_ｍは、（５−１３）式にあるように、初磁化過程直後における磁界の方向をとればよい。
【０１２２】
上記自己無撞着な磁化Ｍ（Ｍとθ）を求めるために、ここでは以下のような手順で計算を行う。
（ｉ）残留磁化過程で求まった各要素の磁化Ｍ_０の向きを磁化容易軸の方向とし、Θ_ｍを求める。
（ｉｉ）磁化Ｍ_０を、これから求めようとする減磁後の磁化Ｍの初期値に設定する。
（ｉｉｉ）各要素に対して、磁化Ｍと（ｉ）で求めた磁化容易軸とのなす角度θを求める。本θは初回は０、（ｘ）の反復後は（ｖｉｉｉ）で求めたθの値となる。
（ｉｖ）磁化Ｍを用いて、有限要素法で磁石単体での磁界計算を行う。
（ｖ）必要に応じて、磁界計算後の後処理を行い磁界の結果を得る。具体的には例えば磁界計算に有限要素法を用いた場合、磁束密度Ｂが求まるので、下式（５−２９）を使用して、磁界Ｈに変換する（磁石部分の透磁率をμ_０+χ_ｒｅｃとした）。
（ｖｉ）磁界Ｈ＝（Ｈ_ｘ、Ｈ_ｙ）から、式（５−２０）〜（５−２２）を用いて、Ｈ_ｅｆｆとΘ_ｅｆｆを求める。
（ｖｉｉ）式（５−２５）からΩ_ｅｆｆを求め、式（５−２３）、（５−２４） からＨ_ｅｆｆθ、Ｈ_ｅｆｆ⊥を求める。
（ｖｉｉｉ）式（５−１９）からθを求め、（５−１８）より磁化Ｍを求める。なおここで異方性磁界Ｈ_Ａには、（４−８）で求めたＨ_Ａ（Ｍ_ｍ）を使用する。
（ｉｘ）式（５−２８）よりΘを求め、式（５−２６）、（５−２７）より、新たな磁化Ｍを求める。
（ｘ）（ｉｘ）で求まった新たな磁化Ｍを用いて、（ｉｉｉ）〜（ｖｉｉ）を、自己無撞着な磁化Ｍが得られるまで繰り返す。なお実際には、反復により（ｖｉｉｉ）で新たなθとＭを求める際に、旧θとＭをそのまま更新（置換）するのではなく、変動分の０．２〜０．８だけを修正する緩和係数を導入することで収束性が向上する。
【０１２３】
【数式４９】

【０１２４】
２）残留磁化計算部における処理の流れ
上記に基づいて残留磁化計算部では図１７に示す流れで処理を行う。
【０１２５】
ステップＳ１７０１では、残留磁化過程計算部で得た、磁化容易軸Θ_ｍの方向、残留磁化Ｍ_ｒ（Ｍ_ｍ）、異方性磁界Ｈ_Ａの結果をメモリに読み込む。
【０１２６】
ステップＳ１７０２では磁化の大きさＭ及び方向θの初期値を設定する。大きさはＭ_ｒ（Ｍ_ｍ）、方向は磁化容易軸と一致する方向（θ＝０）とする。
【０１２７】
ステップＳ１７０３では磁化ベクトルを直交座標のｘ、ｙ成分（Ｍ_ｘ、 Ｍ_ｙ）で表す。
【０１２８】
ステップＳ１７０４では磁化分布（Ｍ_ｘ、Ｍ_ｙ）を用いて、磁石単体での磁場計算を行う。有限要素法等の一般的な磁界計算ソルバーを使用する。
【０１２９】
ステップＳ１７０５では磁界計算の結果求まる磁石の部分の磁束密度（Ｂ_ｘ、Ｂ_ｙ） から磁界（Ｈ_ｘ、Ｈ_ｙ）を求める 。
【０１３０】
ステップＳ１７０６では磁界（Ｈ_ｘ、Ｈ_ｙ）を元に、（５−２０）〜（５−２２）式により、Ｈ_ｅｆｆ、Θ_ｅｆｆを求める。
【０１３１】
ステップＳ１７０７では（５−２３）〜（５−２５）式により、Ω_ｅｆｆ、Ｈ_ｅｆｆθ、Ｈ_ｅｆｆ⊥を求める。
【０１３２】
ステップＳ１７０８では（５−１８）式、（５−１９）式を用いて、新たな磁化の大きさＭ及び方向θを求める。
【０１３３】
ステップＳ１７０９では磁化の大きさＭ及び方向θが十分収束したか否かを判定し、収束が不十分の場合には、ステップＳ１７０３に戻り、ステップＳ１７０３からステップＳ１７０８までの処理を繰り返す。一方、ステップＳ１７０９で磁化Ｍ及び方向θが十分収束したと判定された場合には、ステップＳ１７１０に進み、磁化Ｍ及び方向θから、（５−２６）、（５−２７）式より磁化ベクトルを求め磁化分布の結果として、外部記憶装置に出力する。
【０１３４】
なおここでは、減磁過程を計算する際に、減磁曲線の第２象限の屈曲部寸前の点と残留磁化Ｍ_ｒの点を結ぶ直線の勾配χ_ｒｅｃを使用したが、一般に本勾配は完全磁化曲線推定部で得られるχ_ｒ（（２−１０）式）とほぼ同じ値をもつものであり、χ_ｒで代用してもよい。
【０１３５】
また（５−２０）〜（５−２８）式では、計算をｘｙ平面の２次元に限定して説明を行ってきたが、容易にわかるように、これを３次元の式に拡張して、上記Ｓ１７０４のステップでは３次元の磁界計算ソルバーを用いることで、３次元の磁化分布を求めることも容易である。従って上記計算手順中で使用した（５−２０）〜（５−２８）式は、表記のような２次元に限定されるものではなく、３次元に拡張した式も含むものである。
【０１３６】
上記着磁電流波形計算部、完全着磁曲線推定部、初磁化過程計算部、残留磁化計算部、減磁過程計算部を持つ着磁計算装置を用いて、円筒形状をした永久磁石の着磁計算を行った例を示す。
【０１３７】
図１８は着磁器の分割モデルの一例で、１８０１はコイル、１８０２は磁石である。本着磁器にコンデンサを接続して電流を流したときの電流波形の着磁電流波形計算部による計算結果を図１９に示す。ピーク電流（Ｊ_ｐｅａｋ）を１０ｋＡとしており、本電流波形が実測値に近いものとなっていることを確認している。
【０１３８】
また、測定によって得られた初磁化曲線を元に、完全飽和磁化曲線を作成した結果を図２０に示す。横軸に磁界Ｈ、縦軸に磁化Ｍをとり、２００１に示す磁界領域については測定結果を、２００２に示す磁界領域については完全着磁曲線推定部により算出された結果をプロットしており、妥当な完全飽和磁化曲線が得られている様子がわかる。
【０１３９】
図２１の（ａ）は初磁化過程計算部による初磁化過程計算後、（ｂ）は残留磁化計算部による残留磁化計算後、（ｃ）は減磁過程計算部による減磁過程計算後の磁石内の磁化分布を矢印で表示した図を示す。また図２１（ｄ）は減磁過程計算部における減磁過程計算で得られた磁化分布の結果を用いて磁石単体での磁束の様子を図にしたものである。
【０１４０】
このように、各計算部の結果を結果を図にして表示することで、着磁計算過程を非常に分かり易く知ることができる。
【０１４１】
図２２は磁石単体での磁石表面の磁束密度分布を実験結果と比較したものである。実測値に対して極めてよく一致する計算結果が得られていることがわかる。以上述べてきた本着磁計算方法に対して、計算を行うのに必要なデータをまとめると図２３の通りである。ここにあげたデータは比較的容易に測定、調査または決定できるものばかりである。
【０１４２】
【実施形態２】
上記実施形態１の着磁電流波形計算部では、ＬＣＲ回路のうち、着磁器のインダクタンスＬ、コンデンサの容量Ｃ、コイルの抵抗Ｒをユーザが指定し、コンデンサ両端子間の電圧をプログラム内部で調整することにより、ユーザの指定した着磁電流波形のピーク値を求めた。
【０１４３】
しかし、これは電源回路中のコンデンサ両端子間の電圧をユーザが指定し、コイルの抵抗（電源回路全体からみたときのコイルの抵抗）をプログラム内部で調整することにより、ユーザの指定した着磁電流波形のピーク値を求めてもよい。
なお本方法を実現するに当たっての着磁電流波形計算部における処理の流れを計算の流れを図２４に示す。図３との違いは、ステップＳ２４０３においてコイルの抵抗（Ｒ）の初期値を設定し、ステップＳ２４０８において コイルの抵抗Ｒを更新するように変更する点である。これによって、着磁計算を行うのに必要なデータは図２５に示す通りで、着磁器のコイル抵抗ではなく、着磁器のコンデンサ両端子間の電圧を読み込む。
【０１４４】
本例のようにすることで、着磁回路中のコイルの抵抗が不明な場合にも簡単に計算を行うことが可能になる。
【０１４５】
【実施形態３】
上記実施形態１では、（４−８）式に示したように、異方性磁界を最大磁化点に比例すると仮定した。本仮定は次式（４−８）’に示すように、異方性磁界を最大磁化点を通る減磁曲線の飽和磁束密度Ｍ_ｓ（Ｍ_ｍ）に比例するとした方がより正確である。
【０１４６】
【数式５０】

そして、本仮定を用いた減磁曲線は、次の手順で求めることができる。
（ｉ）（４−３）、（４−５）式より磁化回転角ωを求め、（４−６）式より可逆磁化回転領域における磁化回転角φ_ｂを求める。
（ｉｉ）飽和磁化Ｍ_ｓ（Ｍ_ｍ）を適当な値（Ｍ_ｓ程度）に設定する。
（ｉｉｉ）（４−７）、（４−８）’式により異方性磁界Ｈ_Ａ（Ｍ_ｍ）を求める。
（ｉｖ）（４−１）式を用いて、最大磁化点（Ｈ_ｍ、Ｍ_ｍ）における（Ｈ＝Ｈ_ｍ）磁化回転角φ_ｍを求める。これは上記（Ａ−１）〜（Ａ−４）式を用いて求めることができる。
（ｖ）（４−２）式において、φ＝φ_ｍ、Ｍ＝Ｍ_ｍとおいて、飽和磁化Ｍ_ｓ（Ｍ_ｍ）を求める。本値は式（４−９）の通りである。
（ｖｉ）求まった飽和磁化Ｍ_ｓ（Ｍ_ｍ）を用いて（ｉｉｉ）〜（ｖ）の計算を再び行う。本処理を飽和磁化Ｍ_ｓ（Ｍ_ｍ）が十分収束するまで繰り返す。
（ｖｉｉ）求まった異方性磁界Ｈ_Ａ（Ｍ_ｍ）、飽和磁化Ｍ_ｓ（Ｍ_ｍ）を用いて、任意の磁界Ｈに対して（４−１）式により磁化回転角φを求めた後、（４−２）式を用いて磁化Ｍを求める。
【０１４７】
また、（ｉｉｉ）〜（ｖ）の収束解として得られたφ_ｍを用いて、（４−１０）式により残留磁化Ｍ_ｒ（Ｍ_ｍ）を求めることができる。
【０１４８】
【実施形態４】
上記実施形態１乃至３では、着磁計算を行う方法及び手順等について、着磁計算装置を構成する着磁電流波形計算部、完全着磁曲線推定部、初磁化過程計算部、残留磁化計算部、減磁過程計算部の５つの部分について説明を行ってきた。
【０１４９】
本実施形態では、これらの構成部分を実際に組み込んだモジュール、及び装置全体の構成について説明する。
【０１５０】
図２６は装置を構成するモジュール、入力データファイル、結果データファイル及び着磁計算の流れを説明する図である。図中符号がＰから始まるものは実効形式のプログラムモジュールを示し、Ｑから始まるものはユーザが用意すべき入力データ、Ｒから始まるものはプログラムが吐き出すファイルである。
【０１５１】
Ｐ０２は上述の着磁電流波形計算部を組み込んだ着磁電流波形計算モジュールであり、Ｐ０３は完全着磁曲線推定部を組み込んだ完全着磁曲線推定モジュールであり、Ｐ０４は初磁化過程計算モジュールを組み込んだ初磁化過程計算モジュールであり、Ｐ０５は残留磁化計算モジュールと減磁過程計算モジュールを組み込んだ残留磁化・減磁過程計算モジュールである。またここでは新たにＰ０１に示す着磁分割モデル自動生成モジュールを付加している。本モジュールは着磁磁石の極数、直径、着磁器のコイルの太さ、スロット当たりのコイルの巻き数を入力データとして、それにあった形状の分割モデルを内部で自動的に生成するものである。モータ等に使用する磁石としては一般的に円筒形をしたものが多く、このようにあらかじめ決まった形状の磁石を着磁する場合に、それにあった着磁器の分割モデルを生成するものである。
【０１５２】
Ｑ０１は着磁器の形状を表すデータファイルであり、Ｑ０２は着磁器の電源回路のデータファイルであり、Ｑ０３は磁石の初磁化曲線を表す点列（磁界と磁化の組み合わせ、または磁界と磁束密度の組み合わせ）のデータである。Ｑ０４は磁石のメジャーループの形状を表すデータである。Ｑ０３とＱ０４は磁石の磁気特性をＢＨカーブトレーサー等で測定して得られるものである。
【０１５３】
Ｑ０２の着磁器の電源回路データの具体的な内容の一例を図２８（ａ）に示す。着磁電流の目安値２８０１、コイルの巻き数２８０２、着磁電流のピーク値２８０３、コンデンサの容量２８０４、コンデンサ充電時の両端子電圧２８０５のデータが入っている。
【０１５４】
Ｑ０３の磁石の初磁化曲線を表す点列のデータの例を図２８（ｂ）に示す。点列データとして、磁界Ｈと磁束密度Ｂの組が１０個設定されている。
【０１５５】
Ｑ０４の磁石のメジャーループの形状を表すデータの例を図２８（ｃ） に示す。可逆磁化回転領域の最小磁界Ｈ_ｂ（２８０６）、残留磁化Ｍ_ｒ（２８０７）、及びχ_ｒｅｃ（２８０８）が入っている。
【０１５６】
なおここでは、着磁ヨークの磁気特性と導電率及び磁石の導電率はプログラムモジュールＰ０２〜Ｐ０５の中に内蔵されているものとする。
【０１５７】
このようにデータを磁石の特性データと着磁器の特性データとで分けて管理することで、管理が行い易く、使い易いシステムとなる。
【０１５８】
次にこれらの入力データ及びプログラムモジュールを用いて実際に着磁計算を行う際の手順及びデータの流れについて説明する。
【０１５９】
ユーザはまず着磁器分割モデル自動生成モジュールＰ０１を起動する。すると本モジュールはデータＱ０１をファイルから読み込み、その内容に従って着磁器の分割モデルのファイルＲ０１を出力する。図２７（ａ）に生成した分割モデルの一例を示す。
【０１６０】
図２７（ｂ）は分割モデル中での材料分布を分かり易くするために表示した図である。なお、実際に分割モデルを表示する際には、材料ごとに異なる色で表示すると分かり易く、特に本着磁計算においては、永久磁石２７０１、着磁ヨーク２７０２、空気２７０３、＋コイル２７０４及び−コイル２７０５で色を変えて表示するとよい。
【０１６１】
ユーザは次に着磁電流波形計算モジュールＰ０２を起動する。本モジュールは着磁器の電源回路データＱ０２をファイルから読み込み、計算された着磁電流波形を初磁化過程計算モジュールの入力フォーマットに沿って初磁化計算用入力ファイルの雛形Ｒ０２として出力する。
【０１６２】
次にユーザは完全着磁曲線推定モジュールＰ０３を起動する。本モジュールは磁石の初磁化曲線データファイルＱ０３、磁石のメジャーループデータファイルＱ０４、及び初磁化計算用入力ファイルの雛形Ｒ０２を読み込み、ファイルＲ０２の内容に、ここで求めた完全着磁曲線を付加して、初磁化計算用入力データＲ０３として出力する。
【０１６３】
次にユーザは初磁化過程計算モジュールＰ０４を起動する。本モジュールは着磁電流波形データと完全着磁曲線のはいった初磁化計算用入力データＲ０３と磁石のメジャーループデータファイルＱ０４、及び着磁器の分割モデルファイルＲ０１を読み込んだ後、内蔵する
初磁化過程計算部により、初磁化を計算し、初磁化ファイルＲ０４を出力する。なお初磁化ファイルＲ０４には分割モデルデータも含まれている。
【０１６４】
最後にユーザは残留磁化・減磁過程計算モジュールを起動する。本モジュールは初磁化ファイルＲ０４を読み込み、内蔵された残留磁化計算部、減磁過程計算部により、残留磁化と減磁過程を経た磁化分布を計算し、夫々ファイルＲ０５とファイル０６に出力する。
【０１６５】
残留磁化計算部と減磁過程計算部が同一のモジュール内にあることで、両者間で引き渡されるべきデータである磁化容易軸の方向Θ_ｍ 、残留磁化Ｍ_ｒ（Ｍ_ｍ） 、異方性磁界Ｈ_Ａ（Ｍ_ｍ）は、ファイルを介してではなく、モジュール内部で引き渡される。
【０１６６】
ここには表示していないが、ファイルＲ０２を用いて図２０に示すような経過時間×電流値の着磁電流波形をグラフ表示するモジュール、ファイルＲ０３を用いて完全着磁曲線を図２０に示すような磁界×磁化の磁化曲線をグラフ表示するモジュール、ファイルＲ０４、Ｒ０５、Ｒ０６の磁化分布の様子をそれぞれベクトル表示するモジュールを用意しておく。
【０１６７】
このように図化モジュールを用意しておくことで、着磁計算過程を非常に分かり易く追跡することが可能となる。なおこれらの図化モジュールの機能は、上記５つの計算モジュール機能の一部として、中に含めてもよい。
【０１６８】
本実施形態で説明したように、着磁計算過程に沿って着磁計算を行うモジュールをいくつかに分割しておき、それぞれを独立なプログラムとすることによって、一部のデータだけを変更して計算する際の重複計算の無駄が削除できるとともに、計算過程を逐一追跡しながらの着磁計算が可能となる。
【０１６９】
【他の実施形態】
また、本発明の目的は、前述した実施形態の機能を実現するソフトウェアのプログラムコードを記録した記憶媒体を、システムあるいは装置に供給し、そのシステムあるいは装置のコンピュータ（またはＣＰＵやＭＰＵ）が記憶媒体に格納されたプログラムコードを読出し実行することによっても、達成されることは言うまでもない。
【０１７０】
この場合、記憶媒体から読出されたプログラムコード自体が前述した実施形態の機能を実現することになり、そのプログラムコードを記憶した記憶媒体は本発明を構成することになる。
【０１７１】
プログラムコードを供給するための記憶媒体としては、例えば、フロッピディスク、ハードディスク、光ディスク、光磁気ディスク、ＣＤ−ＲＯＭ、ＣＤ−Ｒ、磁気テープ、不揮発性のメモリカード、ＲＯＭなどを用いることができる。
【０１７２】
また、コンピュータが読出したプログラムコードを実行することにより、前述した実施形態の機能が実現されるだけでなく、そのプログラムコードの指示に基づき、コンピュータ上で稼働しているＯＳ（オペレーティングシステム）などが実際の処理の一部または全部を行い、その処理によって前述した実施形態の機能が実現される場合も含まれることは言うまでもない。
【０１７３】
さらに、記憶媒体から読出されたプログラムコードが、コンピュータに挿入された機能拡張ボードやコンピュータに接続された機能拡張ユニットに備わるメモリに書込まれた後、そのプログラムコードの指示に基づき、その機能拡張ボードや機能拡張ユニットに備わるＣＰＵなどが実際の処理の一部または全部を行い、その処理によって前述した実施形態の機能が実現される場合も含まれることは言うまでもない。
【０１７４】
【発明の効果】
以上説明したように、本発明によれば既存の汎用磁界計算ソルバーを用いて磁石の磁化分布を算出することが可能となる。
【図面の簡単な説明】
【図１】本発明の実施形態１にかかる磁化分布算出装置の構成を示すブロック図である。
【図２】本発明の実施形態１にかかる磁化分布算出装置の着磁電流波形計算部で解く回路図である。
【図３】本発明の実施形態１にかかる磁化分布算出装置の着磁電流波形計算部の行う処理フローを示す図である。
【図４】本発明の実施形態１にかかる磁化分布算出装置の着磁電流波形計算部において１／Ｈ^２外挿により飽和磁化を求める方法を説明する図である。
【図５】磁化曲線の可逆磁化回転領域を説明する図である。
【図６】χ_ｒを説明する図である。
【図７】本発明の実施形態１にかかる磁化分布算出装置の完全磁化曲線推定部の行う処理フローを示す図である。
【図８】本発明の実施形態１にかかる磁化分布算出装置の初磁化過程計算部の内容を説明する磁化回転モデルの説明図である。
【図９】本発明の実施形態１にかかる磁化分布算出装置の初磁化過程計算部の行う処理フローを示すである。
【図１０】本発明の実施形態１にかかる磁化分布算出装置の残留磁化の計算方法を説明する図である。
【図１１】本発明の実施形態１にかかる磁化分布算出装置の残留磁化計算部の行う処理フローを示す図である。
【図１２】異方性磁石の減磁過程を説明する図である。
【図１３】χ_ｒｅｃを説明する図である。
【図１４】Ｈ_ｅｆｆの積分の計算を説明する図である。
【図１５】等方性磁石の減磁過程を説明する図である。
【図１６】減磁過程の磁化の回転を基準軸から測った角度で説明する図である。
【図１７】本発明の実施形態１にかかる磁化分布算出装置の減磁過程計算部の行う処理フローを示す図である。
【図１８】本発明の実施形態１にかかる着磁器分割モデルの一例を示す図である。
【図１９】本発明の実施形態１にかかる磁化分布算出装置の電流波形の計算結果の一例を示す図である。
【図２０】本発明の実施形態１にかかる磁化分布算出装置の完全飽和磁化曲線推定部の結果の一例を示す図である。
【図２１】本発明の実施形態１にかかる磁化分布算出装置の初磁化過程計算部、残留磁化計算部、減磁過程計算部の結果の一例を示す図である。
【図２２】本発明の実施形態１にかかる磁化分布算出装置により得られた磁石の磁化分布を用いて磁石単体での表面磁束密度分布を求めた結果の一例を示す図である。
【図２３】本発明の実施形態１にかかる磁化分布算出装置が必要とする入力データの一覧を示す図である。
【図２４】本発明の実施形態２にかかる磁化分布算出装置の着磁電流波形計算部の行う処理フローを示す図である。
【図２５】本発明の実施形態２の磁化分布算出装置の着磁電流計算手段が必要とする入力データの一覧を示す図である。
【図２６】本発明の実施形態４にかかる磁化分布算出装置の構成する主要モジュールとデータの流れの一例を示す図である。
【図２７】本発明の実施形態４にかかる磁化分布算出装置の着磁器の分割モデルの一例を示す図である。
【図２８】本発明の実施形態４にかかる磁化分布算出装置の着磁計算を行う入力データファイルの一例を示す図である。
【符号の説明】
１０１ ＣＰＵ
１０２ 表示装置
１０３ 入力装置
１０４ 外部記憶装置
１０５ メモリ
１０６ バス[0001]
BACKGROUND OF THE INVENTION
The present invention relates to an apparatus for obtaining a magnetization distribution by applying a magnetic field to a magnetic material and performing magnetization.
[0002]
[Prior art]
In recent years, many attempts have been made to simulate the hysteresis phenomenon of magnets by numerical analysis for designing devices such as magnetic recording and electromagnetic actuators. In particular, there is a VMSW method as one of methods for obtaining a magnetization distribution in a magnet when a hard magnetic material is magnetized by applying a strong magnetic field.
[0003]
This method is based on a technique for expressing hysteresis characteristics as a set of SW models (Stoner and Wahlfarth model) representing magnetization rotation behavior of uniaxial anisotropic crystal grains, and each magnetization is variable. This is a technique that has been modified. The concrete contents of this method are as follows: “Mori: A new static magnetic field analysis method for anisotropic materials and its verification by magnetization vector measurement, Electrotechnical Society study material, Static / Rotating Machine Joint Study Group SA-91-6, RM-91-15, pp51-60 (1991) ", and" Mori: a new static magnetic field analysis method II of anisotropic materials, IEEJ Technical Committee Materials, Static / Rotating Machine Joint Technical Committee SA-91-25, RM-91-88, pp97-106 (1991) ", in which analysis of magnetization and demagnetization processes of permanent magnets is performed.
[0004]
In addition, “M. Enokizono et al .: Magnetic Field Analysis of Anisotropic Permanent Magnets Problems by Fine Element Method 16, M. Eng. : Magnetic field analysis on anisotropic permanent magnet problem, Magnetic Society of Japan MAG-96-251, pp21-30 (1996) "Calculation of magnet magnetization and demagnetization process using VMSW method" The result of having performed is reported.
[0005]
However, in any of the above techniques, it was necessary to newly develop a dedicated magnetic field analysis solver (device) using the VMSW method in order to determine the magnetization distribution, and the load was large. In addition, a magnetizing current waveform is required as input data, and therefore measurement is required in advance.
[0006]
[Problems to be solved by the invention]
The present invention has been made to solve the above-described problems, and an object thereof is to provide a magnetization distribution calculation processing device capable of calculating the magnetization distribution of a magnet using an existing general-purpose magnetic field calculation solver.
[0007]
[Means for Solving the Problems]
In order to solve the above problems, for example, a magnetization distribution calculation processing device of the present invention has the following configuration. That is,
A magnetization distribution calculating device for calculating a magnetization distribution in a magnetic material when the magnetic material is magnetized by applying a magnetic field by a magnetizer,
The magnetic material is divided into a finite number of elements, a transient analysis of the applied magnetic field is performed when a magnetic field is applied from the magnetizer to each element that is not magnetized, and the maximum magnetic field for each element is extracted, Initial magnetization process calculation means for calculating the maximum magnetization point for each element based on the maximum magnetic field and the characteristics of the magnetic material;
Residual magnetization calculation means for calculating the residual magnetization of each element based on the maximum magnetization point of each element calculated by the initial magnetization process calculation means and the characteristics of the magnetic material;
A demagnetization process calculation unit that calculates a change in magnetization of each element due to a magnetic field generated by the residual magnetization of each element calculated by the residual magnetization calculation unit.
[0008]
DETAILED DESCRIPTION OF THE INVENTION
Hereinafter, a preferred embodiment of the present invention will be described in detail with reference to the accompanying drawings.
[0009]
Embodiment 1
FIG. 1 is a block diagram showing a magnetization distribution calculating apparatus according to an embodiment of the present invention. A central processing unit (CPU) 101 that controls the overall operation is connected via a bus 106 to a display device 102 that displays analysis results and an input device 103 that is input by an analyst. Similarly, a magnetism calculation program is stored through the bus, various information at the time of control operation is stored and read out, memory 104, data that does not fit in the system, and external data that stores data even after the system is terminated A storage device 105 is connected.
[0010]
As shown in FIG. 1, the magnetization distribution calculation program is composed of five parts: a magnetization current waveform calculation unit, a complete magnetization curve estimation unit, an initial magnetization process calculation unit, a residual magnetization calculation unit, and a demagnetization process calculation unit. The As shown in FIG. 1, the data to be used are as follows: 1) Initial magnetization curve, 2) Minimum magnetic field H in the reversible magnetization rotation region._b3) Residual magnetization M_r4) χ_rec5) There are five types of magnet conductivity, and data indicating the configuration of the magnetizer, etc. 1) Divided model of the magnetizer, 2) BH characteristics and conductivity of the magnetized yoke, 3) Number of turns of the coil, 4 6) coil resistance, 5) capacitor capacity, and 6) peak value of magnetizing current.
[0011]
In the following, detailed description will be given individually for each of the five calculation units that calculate the magnetization distribution of the magnet using the data indicating the characteristics of the magnet and the configuration of the magnetizer.
[0012]
1. Magnetization current waveform calculator
The magnetizing current waveform calculating unit is a processing means for obtaining a magnetizing current waveform (magnetizing current waveform) when magnetizing the magnet based on the data on the configuration of the magnetizing unit. The basic formula used in the above and the flow of processing will be described.
[0013]
The power circuit of the magnetizer is expressed as an LCR circuit as shown in FIG. 2, and magnetization is performed by discharging a capacitor having a capacitance C having a charge Q through a coil having a resistance R and an inductance L. The equation (basic equation) of the power supply circuit at this time is as shown in equation (1-1).
[0014]
[Formula 1]

This differential equation is divided into the following three cases according to R and 2√ (L / C).
[0015]
[Formula 2]

[0016]
[Formula 3]

[0017]
[Formula 4]

In the above equation, the inductance L of the magnetizer actually varies depending on the degree of magnetization saturation of the magnetizer, but here it is treated as constant (value when a constant current specified by the user is passed). .
[0018]
FIG. 3 is a flowchart showing a procedure for calculating a magnetizing current waveform using the basic expressions (1-2) to (1-4). The contents will be described below.
[0019]
In step S301, magnetizer body data and power supply circuit data are read into the memory as data indicating the configuration of the magnetizer. Specifically, a magnetizer split model for the magnet body, a BH characteristic and conductivity of the magnetizing yoke, the number of turns of the coil, a guideline (a constant value) of the magnetizing current, and a coil resistance R for the power supply circuit, The capacitance C of the capacitor and the peak value I of the magnetizing current.
[0020]
In step S302, magnetic field calculation is performed by the finite element method based on the magnet body data read in step S301, and the inductance L of the magnetizer is calculated. As a guide for the magnetizing current, a weak current that does not allow the magnetizing yoke to reach the saturation region may be used.
[0021]
In step S303, an initial value of the both-terminal voltage V of the capacitor of the power supply circuit is set. The initial value may be about 1V.
[0022]
In step S304, the charge amount Q of the capacitor of the power supply circuit is obtained from the capacitance C of the capacitor of the power supply circuit read in step S301 and the value of the both-terminal voltage V of the capacitor of the power supply circuit set in step S303.
[0023]
In step S305, the magnetizing current waveform is obtained by the above equations (1-2) to (1-4).
[0024]
In step S306, the peak value of the magnetizing current waveform obtained in step S305 is determined as J._peakSet as.
[0025]
In step S307, the J set in step S306 is set._peakIs compared with the peak value of the magnetizing current designated by the user (the peak value I of the magnetizing current read in step S301). If there is a sufficient match, the magnetization current waveform obtained in step S306 is taken as the final result (step S307).
[0026]
On the other hand, in step S308, J_peakHowever, if it does not coincide with the peak value of the magnetizing current specified by the user, the capacitor both-terminal voltage V of the power supply circuit is changed (step S308), and the process is repeated until they coincide.
[0027]
Here, the finite element method is used to calculate the inductance L of the magnetizer in step S302, but other methods such as a boundary element method, an integration method, and a difference method may be used.
[0028]
2. Complete magnetization curve estimator
The complete magnetization curve estimator obtains a magnetization curve (curve indicating the state of magnetization M when the magnet is magnetized by applying a magnetic field H) necessary for calculating the magnetization distribution. The magnetization curve (BH curve) When a sufficient magnetic field (high magnetic field) is not applied to reach saturation magnetization in the measurement of, it is possible to extrapolate the magnetized curve (low magnetic field curve) of the magnet that has already been measured to the fully magnetized part Is a processing means for estimating.
[0029]
The flow of processing will be described below after showing the method and the validity of the method.
[0030]
1) Method for estimating the fully magnetized part
In estimating the fully magnetized portion, the BH curve is assumed to be expressed by the following Stoner-Wohlfart equation. Where H_AIs the anisotropic magnetic field of the magnet. The method will be described below.
[0031]
[Formula 5]

In introducing the Stoner-Wohlfart equation, first, the saturation magnetization M_sis required. Here, it is determined by one of the following two methods.
[0032]
a. Assuming an isotropic magnet, simply the residual magnetization M_r2 times.
b. As shown in FIG. 4, 1 / H from the initial magnetization curve ((a) diagram).²Obtained by extrapolation (approximate the complete magnetization curve with a quadratic curve) (FIG. 5B).
[0033]
Here, FIG. 4A is a magnetization curve when the magnetic field H is taken on the horizontal axis and the magnetization M is taken on the vertical axis, and FIG. 4B is 1 / H on the horizontal axis.²Is obtained by plotting each point based on the magnetization curve of FIG. 4A and approximating it with a quadratic curve.²Value of magnetization M when = 0, that is, saturation magnetization M_sIt is a figure for seeking.
[0034]
The present invention is basically applicable to isotropic magnets, but the method b can be applied even when there is a slight anisotropy. By allowing the user to select, the range of use is expanded.
[0035]
Next obtained saturation magnetization M_sAnd the remanent magnetization M of the measured value_rUsing the squareness ratio R_s(0.5 when the above a is used), and the magnetization M with respect to the magnetic field H of the portion where there is no measured value of the magnetization curve is obtained by the following procedure.
[0036]
(I) Reversible magnetization rotation region (region in which the hysteresis loop of the magnetization curve at the time of magnetization and the magnetization curve at the time of demagnetization becomes one in the magnetization curve. That is, as shown in FIG._b, M_b) Above point) any point (H_f, M_f) First, an arbitrary point (H_f, M_f) Magnetization rotation angle φ_fNext, the anisotropic magnetic field H is calculated by the equation (2-6)_AAsk for.
[0037]
Here, FIG. 5 is a magnetization curve when the horizontal axis represents the magnetic field H and the vertical axis represents the magnetization M. The magnetization curve at the time of magnetization and the reversible magnetization rotation region or more (H_b) And the magnetization curve (demagnetization curve) at the time of demagnetization when demagnetized after application of the magnetic field of_sIs saturation magnetization, M_rIndicates remanent magnetization.
[0038]
In addition, the equation (2-5) is changed from the equation (2-2) to M = M_fIs substituted for the magnetization rotation angle φ, and (2-6) is expressed as H = H in equation (2-1)._fIs substituted for anisotropic magnetic field H_AIt was solved.
[0039]
Further, when there are a plurality of points in the reversible magnetization rotation region, their anisotropic magnetic field H_A(Ie, the magnetic field H in the input data)_bUse all the points above).
[0040]
[Formula 6]

(Ii) Anisotropic magnetic field H obtained in (i) above_AIs used to determine the magnetization rotation angle φ with respect to the magnetic field H of the portion where there is no measured value of the magnetization curve (see the description below), and the magnetization M is determined from the equation (2-2). In (ii), the Stoner-Wohlfart equation of the following formula (A-1) is replaced with the magnetic field H and the anisotropic magnetic field H._AIt is necessary to determine the magnetization rotation angle φ when.
[0041]
[Formula 7]

Here, by placing (A-1) as shown in (A-2) and (A-3), using Newton's method, starting from φ = 0 and repeating (A-4) Then, the magnetization rotation angle φ is obtained.
[0042]
[Formula 8]

By the above method, the magnetization M with respect to the magnetic field H is obtained. In general, the magnetization curve has a saturation magnetization M as the magnetic field H increases._sThe degree of change becomes smaller. Therefore, when the magnetization curve is expressed by a finite number of points, the part where the magnetic field H is larger can be expressed by a smaller number of points by increasing the interval between the points. At this time, in the method of obtaining the magnetic field H by applying the magnetization M, it is difficult to apply the magnetization M (saturation magnetization M_sAccording to this method, since the magnetic field H is applied, the distance between the points can be easily set large, so that it is very convenient.
[0043]
2) Validity of the method for estimating the fully magnetized part
Next, a method for confirming the validity of the extrapolated portion of the BH curve obtained by the Stoner-Wohlfart equation of 1) above will be described.
[0044]
[Formula 9]

Anisotropic magnetic field H calculated by equation (2-6)_AIs the demagnetization curve M_rThe gradient of the tangent at the point χ_r(FIG. 6 shows a magnetization curve (demagnetization curve) at the time of demagnetization, and the magnetization M when the magnetic field H = 0 is represented by M._r, M_rThe slope of the demagnetization curve at the point χ_r)) And is known to be represented by the following equation: In the formula, <> means a spatial average value, and the value of <> is zero for a completely anisotropic magnet and 2/3 for a completely isotropic magnet.
[0045]
The value of <> is expressed by the following equation for the representative particle model described later.
[0046]
[Formula 10]

Therefore, the anisotropic magnetic field H obtained by the above equation (2-6)_AObtained by substituting into the equation (2-7)_r(The following formula (2-10)) and χ of the measured magnetization curve_r, The validity of the complete magnetization portion of the magnetization curve estimated above can be confirmed.
[0047]
For isotropic magnets, in the calculation of <> in the formula (2-10), there are two methods: 2/3 and (2-9). The difference is small and either may be adopted.
[0048]
[Formula 11]

[0049]
3) Flow of estimation process for fully magnetized parts
Next, a specific procedure of the process for estimating the fully magnetized portion will be described below with reference to FIG.
[0050]
In step S701, the initial magnetization curve and the minimum magnetic field (H_b), Remanent magnetization M_rIs loaded into memory. These are all data obtained by measurement.
[0051]
In step S702, the magnetization M is obtained for the data of each point constituting the magnetization curve read in step S701.
[0052]
In step S703, saturation magnetization M is performed by the method a or b described above._sIn step S704, the squareness ratio R_sAsk for.
[0053]
In step S705, an average magnetization rotation angle ω is obtained from the equation (2-3). In step S706, the anisotropy magnetic field H in the reversible magnetization rotation region is expressed by the equations (2-5) and (2-6)._AAsk for.
[0054]
In step S707, the magnetization M with respect to the magnetic field H of the area | region without a measurement point is calculated | required by (2-1) Formula and (2-2) Formula.
[0055]
In step S708, the remanent magnetization M in the magnetization curve from the equation (2-10)._rSlope χ of demagnetization curve at point_rAsk for.
[0056]
3. Initial magnetization process calculator
The initial magnetization process calculation unit divides the magnet into a finite number of elements, and performs a transient analysis of the applied magnetic field when a magnetic field H is applied to each element from a state of magnetization 0 by a general method such as a finite element method. In this process, the maximum magnetic field for each element is extracted by using a simple magnetic field calculation, and the magnetization (maximum magnetization point) for each element is calculated based on the maximum magnetic field and the characteristics of the magnet. Below, after showing the grounds that general magnetic field calculation such as the finite element method can be used, the processing procedure in the initial magnetization process calculation unit will be described.
[0057]
1) Grounds for using general magnetic field calculations such as the finite element method
As shown in FIG. 8, an external magnetic field H is applied to a magnetic piece of unit volume.₀Is applied and the change in free energy of the system when the average magnetization M is induced is expressed by the following equation (here, thermal and mechanical effects are not considered).
[0058]
[Formula 12]

Here, the integral term of this equation is expressed as “demagnetizing field H_dEnergy "and" effective magnetic field H "_effFor convenience, it can be expressed as equation (3-2).
[0059]
[Formula 13]

Next, consider each term on the right side of equation (3-2).
[0060]
First, the main axis of the demagnetizing factor tensor is the a-axis, b-axis direction, and the main value is N_a, N_b(N_b≧ N_a), The demagnetizing field energy is expressed as the sum of the energy when Mcos (θ−φ) is present in the a-axis direction and when magnetization Msin (θ−φ) is present in the b-axis direction according to the superposition principle. Can be expressed as Here, the direction of each vector is as shown in FIG. 8, when the magnetization easy axis direction and the magnetization difficult axis direction are taken as indicated by broken line arrows, respectively, with respect to the magnetic piece of the unit volume indicated by the hatched area, Of the principal axes of the demagnetizing factor tensor, the angle between the a axis and the easy magnetization axis direction is φ, the angle between the magnetization M direction and the easy magnetization axis direction is θ, and the external magnetic field H_OAnd Ω is the easy axis direction of magnetization. Also, μ₀Is the permeability of the vacuum.
[0061]
[Formula 14]

Next, the magnetization energy is the effective magnetic field H_effThe components of the easy magnetization axis direction and the hard magnetization axis direction of H are respectively H_e, H_hAnd the magnetization component M in each direction_e, M_hIs replaced by M cos θ and M sin θ, as follows.
[0062]
[Formula 15]

On the other hand, the external magnetic field H_OThe mutual energy is as follows.
[0063]
[Formula 16]

From the above, the following specific expression (3-7) of the free energy U is obtained.
[0064]
[Formula 17]

Apply the variational principle to equation (3-7). That is, by calculating ∂U / ∂M = 0 and ∂U / ∂θ = 0, the following equations (3-8) and (3-9) are derived.
[0065]
[Formula 18]

In order to obtain the magnetization M of the anisotropic magnetic piece, the simultaneous equations may be solved. Here, consider the case of a lump of magnetic material made of an isotropic material. At this time, with respect to the shape anisotropy in the minute part inside the magnetic body,
[0066]
[Formula 19]

As good as Because it is isotropic,
[0067]
[Formula 20]

It is. Accordingly, the expression (3-9) is expressed as follows.
[0068]
[Formula 21]

Therefore, the expression (3-13) is derived.
[0069]
[Formula 22]

From (3-13), magnetization M, external magnetic field H₀, Demagnetizing field H_d, Effective magnetic field H_effIt can be seen that the directions are all the same. On the other hand, at this time, the expression (3-8) is as follows.
[0070]
[Formula 23]

Where demagnetizing field H_d, Effective magnetic field H_effSince (3-15) and (3-16) are as shown above, (3-14) is changed to (3-17).
[0071]
[Formula 24]

Magnetization M, external magnetic field H₀, Demagnetizing field H_d, Effective magnetic field H_effIn the case of an isotropic magnet lump, the magnetization M and the effective magnetic field H in the equations (3-8) and (3-9)_effThis relationship is the same as the magnetization phenomenon of the isotropic soft magnetic material.
[0072]
Therefore, the initial magnetization process of the isotropic magnet can be obtained by a general magnetic field calculation such as a finite element method by treating the magnet portion as a soft magnetic material having the initial magnetization curve of the magnet.
[0073]
In addition, when the initial magnetization process is calculated transiently, such as a change in magnetization current with time and eddy current, a problem arises as to which magnetization should be used. Therefore, here, the maximum value during the passage of time is adopted for each element when the magnetic field calculation is performed by the finite element method.
[0074]
In addition, by this initial magnetization process, magnetization reversal occurs in the particles having a predetermined distribution among the magnetizations of the respective particles, and as a result, the vector sum of the particle magnetization is a finite value M._mThe future explanation will be given.
[0075]
2) Processing procedure in the initial magnetization process calculator
Next, a specific processing procedure in the initial magnetization process calculation unit will be described with reference to FIG.
[0076]
In step S901, the magnetizer body data is read into the memory as data indicating the configuration of the magnetizer. Specifically, it is the result obtained by the magnetizing current waveform calculation unit as the magnetizer division model in the magnet body, the BH characteristics and conductivity of the magnetizing yoke, the number of turns of the coil, and the magnetizing current waveform.
[0077]
In step S902, as the magnet characteristic data, the complete magnetization curve obtained by the complete magnetization curve estimation unit and the conductivity of the magnet are read into the memory.
[0078]
In step S903, magnetic field calculation is performed by a finite element method or the like. The calculation is a nonlinear transient analysis that takes into account the eddy current and saturation of the yoke and magnet.
[0079]
In step S904, the peak value of the magnetic flux density change over time of each element is determined as the maximum magnetic flux density point B_mAnd the maximum magnetic flux density point B_mBased on the maximum magnetization point M_mIs stored in an external storage device.
[0080]
4). Residual magnetization calculator
The residual magnetization calculator is a processing means for determining the residual magnetization of the magnet when the magnetic field is set to 0 after applying a predetermined magnetic field and magnetizing the magnet, and the maximum magnetization point M determined by the initial magnetization process calculator._mIs calculated based on the magnetization curve. The basic formula and the maximum magnetization point M are as follows:_mThe flow of processing in the residual magnetization calculation unit will be described after showing the procedure for obtaining the magnetization curve passing through.
[0081]
1) Basic equation in the remanent magnetization calculator
Maximum magnetization point in the first quadrant of the magnet's demagnetization curve (H_m, M_m) To just before the steeply descending portion in the second quadrant, it can be reproduced by the simultaneous magnetization rotation model of the next single domain particle, that is, the Stoner-Wohlfart equation. In Expression (4-1), H is an external magnetic field.
[0082]
[Formula 25]

Where H_A(M_m) Is the maximum magnetization point (H_m, M_m) Is an anisotropic magnetic field in a demagnetization curve passing through. M_s(M_m) In the magnetization curve shown in FIG._m, M_m) Is the saturation magnetization of the demagnetization curve passing through_s(M_m) Is the squareness ratio of the demagnetization curve. These values depend on the magnetization reversal state in the initial magnetization process, and the maximum magnetization point (H_m, M_m).
[0083]
Here, it is assumed that the squareness ratio of the demagnetization curve for each maximum magnetization point does not depend on the maximum magnetization point and is constant as follows (same as the value of the equation (2-4)).
[0084]
[Formula 26]

When the demagnetization curve is a major loop magnetization curve (the major loop magnetization curve is a magnetic field greater than or equal to the reversible magnetization rotation region (ie, H_bConsidering the magnetization and demagnetization curve when demagnetized after application), the minimum magnetization point (H_b, M_b), The magnetization rotation angle φ_b(4-2) from (4-2), and the anisotropic magnetic field H in this state at this time_A(M_b) Is represented by equation (4-7) from equation (4-1).
[0085]
[Formula 27]

Where anisotropic magnetic field H_A(M_b) Is assumed to be proportional to the maximum magnetization point as in the following equation.
[0086]
[Formula 28]

As described above, an arbitrary maximum magnetization point (H_m, M_m), The demagnetization curve can be reproduced.
[0087]
2) Maximum magnetization point M_mTo find the magnetization curve through
Maximum magnetization point M_mThe procedure for determining the magnetization curve through
(I) The magnetization rotation angle ω is obtained from the equations (4-3) and (4-5), and the magnetization rotation angle φ in the reversible magnetization rotation region is obtained from the equation (4-6)._bAsk for. And the anisotropic magnetic field H is obtained by the equations (4-7) and (4-8)._A(M_m)
[0088]
(Ii) Using the equation (4-1), the maximum magnetization point (H_m, M_m(H = H)_m) Magnetization rotation angle φ_mAsk for. This can be obtained using the above-described equations (A-1) to (A-4).
[0089]
(Iii) In the equation (4-2), φ = φ_m, M = M_mSaturation magnetization M_s(M_m) This value is as the following formula (4-9).
[0090]
(Iv) After obtaining the magnetization rotation angle φ by the equation (4-1) for an arbitrary magnetic field H, the magnetization M is obtained by using the equation (4-2).
[0091]
[Formula 29]

Magnetization at H = 0 in this demagnetization curve (residual magnetization M_r(M_m)) Corresponds to the case of φ = 0 in equation (4-2) (← Equation (4-1) becomes φ = 0 when H = 0), and equation (4-2) shows (4 It can be obtained by the following equation (4-10) substituting equation -9).
[0092]
[Formula 30]

[0093]
3) Process flow in the residual magnetization calculator
Next, a specific procedure in the residual magnetization calculation unit will be described below with reference to FIG. In step S1101, the minimum magnetic field H of the reversible rotational magnetization region, which is the magnetization curve data of the permanent magnet._b, Remanent magnetization M_r, Saturation magnetization M_s, Χ_recAnd the magnetization curve including the complete magnetization obtained by the complete magnetization curve estimation unit is read into the memory.
[0094]
In step S1102, the residual magnetization M read in step S1101._rAnd saturation magnetization M_sBased on the squareness ratio R_sAsk for. Generally, the value is close to 0.5.
[0095]
In step S1103, the maximum magnetization point value M of each element obtained by the initial magnetization process calculation unit._mIs read.
[0096]
In step S1104, the maximum magnetization point M read in step S1103._mThe direction of the magnetization vector at_m(This is because the direction of the easy axis of magnetization is necessary for the remanent magnetization process described later).
[0097]
In step S1105, the anisotropic magnetic field H of each element is as follows._A(M_m) And residual magnetization M_r(M_m)
(I) The magnetization rotation angle ω is obtained from the equation (4-3).
(Ii) According to equations (4-6) and (4-7), H_A(M_s)
(Iii) For each element, the residual magnetization M_r(M_m)
a. Maximum magnetization point M read in step S1103_mH for value_mThe value is obtained from the complete magnetization curve.
b. According to the formula (4-8), H_A(M_m)
c. According to equation (4-1), the magnetization rotation angle φ_mAsk for.
d. According to the formula (4-10), M_r(M_m)
[0098]
In step S1106, the easy axis direction Θ of each element_m, Remanent magnetization M_r(M_m) And anisotropic magnetic field H_A(M_m) To an external storage device.
[0099]
5. Demagnetization process calculator
The demagnetization process calculation unit calculates the residual magnetization of each element calculated by the residual magnetization calculation unit and each element having the residual magnetization when a magnetic field generated by the magnetization of each element is applied. It is a processing means for calculating the magnetization. In the following, a process for deriving a basic equation for obtaining the magnetization distribution of the demagnetization process will be described, and the processing procedure in the demagnetization process calculation unit will be described.
[0100]
1) Basic formula for obtaining the magnetization distribution in the demagnetization process
When calculating the demagnetization process, the starting point is the residual angle θ_rRemanent magnetization M with_rIn this state, the magnetization rotation angle θ is measured from the easy magnetization axis. And the magnetization is (M_r, Θ_r) Is the change in the free energy density when the state changes from the state (before change) to the state (M, θ) (state after change). The work required for magnetization in the absence of a demagnetizing field is_rWork U required to change from M to M along the easy axis of magnetization_magAnd M is θ_rWork shifted from direction (anisotropic energy U_aniso) And approximate. Then, the change U of free energy density can be expressed by the sum of the following four contributions.
[0101]
[Formula 31]

U_demag Is a so-called change in demagnetizing field energy and is expressed by the following equation.
[0102]
[Formula 32]

a and b are main values N of the demagnetizing field tensor._a, N_bRepresents the direction of (N_a≦ N_b). Since the magnetizations M and θ (angles between the magnetization M and the easy axis direction) in the changed state are unknowns, the magnetizations included in this equation can be replaced as follows. Here, φ is an angle (<90 deg) formed by the easy axis direction of magnetization and the demagnetizing field tensor a direction, and M_a, M_bIs the component in each axial direction of the demagnetizing field tensor of magnetization in the state after change, M_ar, M_brIs the axial component of the demagnetizing field tensor of the magnetization in the state before the change, θ_rIs the magnetization M in the state before the change._rAnd the angle formed by the axis of easy magnetization (FIG. 12). Also, μ₀Is the permeability of the vacuum.
[0103]
[Formula 33]

The expression (5-2) is expressed as follows by the expression (5-3).
[0104]
[Formula 34]

Then U_magIn order to express this specifically, the main part of the magnetization easy axis direction magnetization curve is approximated by a straight line. That is,
[0105]
[Formula 35]

This χ_recIs the slope of this straight line, the point just before the bent part in the second quadrant and the residual magnetization M_rThe slope of the straight line connecting the points (FIG. 13). Thus, U_magIs expressed by the following equation. FIG. 14 is a diagram showing the contents of integration of this equation._magIs nothing but the shaded portion in FIG.
[0106]
[Formula 36]

Anisotropic energy U_anisoHere, it is expressed as follows assuming that the anisotropy field is constant in the section before the steep descent part of the demagnetization curve.
[0107]
[Formula 37]

On the other hand, the mutual energy with the external magnetic field is as follows.
[0108]
[Formula 38]

From the above, the specific expression of U is as follows.
[0109]
[Formula 39]

The variational principle is applied to this U. That is, by obtaining 方程式 U / ∂M = 0 and ∂U / ∂θ = 0, the following simultaneous equations for determining M and θ are obtained.
[0110]
[Formula 40]

In order to obtain the magnetization of the anisotropic magnet, the simultaneous equations may be solved.
Here, a magnet made of an isotropic material is considered. At this time, for the shape anisotropy in the minute part inside the magnet,
[0111]
[Formula 41]

As good as Because it is isotropic,
[0112]
[Formula 42]

It is. From this, the expressions (5-10) and (5-11) are as follows.
[0113]
[Formula 43]

Here, the effective magnetic field in the minute part inside the magnet is expressed as H._effThen, H_effIs the external magnetic field H₀And demagnetizing field H_dThe direction component parallel to the magnetization M is expressed as H_effθ, The vertical direction component is H_eff⊥Then, each is described as follows. Where H_0θ, H_dθIs a directional component parallel to the magnetization M of the external magnetic field and the demagnetizing field, H_0⊥, H_d⊥Is a vertical component.
[0114]
[Formula 44]

As a result, the expressions (5-14) and (5-15) are expressed as follows.
[0115]
[Equation 45]

By solving this equation, M and θ are obtained.
Next, when applying the equations (5-18) and (5-19) to the magnetic field distribution expressed with respect to the reference coordinate axis (here, the direction of the X-axis of the orthogonal coordinate system is the reference axis) Will be described.
[0116]
As shown in FIG. 16, the easy axis direction is Θ with respect to the reference axis._mH measured from the reference axis (X axis)_eff, The direction of magnetization M_eff, Θ, the magnetic field H = (H_x, H_y) Can be expressed as follows: In other words, the magnetic field H = (H_x, H_yH)_effAnd Θ_effIs decided.
[0117]
[Formula 46]

Next, H found here_effAnd Θ_effTo use the magnetic field H = (H_effθ, H_eff⊥) Is obtained as follows. Here, θ is an angle formed by the magnetization easy axis direction and the magnetization M, and Ω_effIs the easy axis of magnetization and the effective magnetic field H_effThe angle between
[0118]
[Formula 47]

Thus, with respect to the magnetic field H expressed in an orthogonal coordinate system with the X-axis direction as a reference, using equations (5-20) to (5-25), H_effθ, H_eff⊥Wanted.
[0119]
Therefore, if the magnetization rotation angle θ is obtained using the equation (5-19) and the magnetization M is obtained using (5-18), the magnetization M = (M_x, M_y) Is obtained.
[0120]
[Formula 48]

As the magnetization after demagnetization, the magnetic field H = (H_x, H_y) May satisfy self-consistent magnetization M (M and θ) such that equations (5-18) and (5-19) are satisfied.
[0121]
In the case of an isotropic magnet, the direction of the easy axis of the main demagnetization process Θ_mMay take the direction of the magnetic field immediately after the initial magnetization process, as in equation (5-13).
[0122]
In order to obtain the self-consistent magnetization M (M and θ), calculation is performed in the following procedure.
(I) Magnetization M of each element obtained in the remanent magnetization process₀Is the direction of the easy axis of magnetization, and Θ_mAsk for.
(Ii) Magnetization M₀Is set to the initial value of the magnetization M after demagnetization to be obtained.
(Iii) For each element, an angle θ between the magnetization M and the easy magnetization axis obtained in (i) is obtained. This θ is 0 for the first time, and after repeating (x), is the value of θ obtained in (viii).
(Iv) Using the magnetization M, calculate the magnetic field of the magnet alone by the finite element method.
(V) If necessary, post-processing after magnetic field calculation is performed to obtain a magnetic field result. More specifically, for example, when the finite element method is used for the magnetic field calculation, the magnetic flux density B is obtained.₀+ χ_rec)
(Vi) Magnetic field H = (H_x, H_y) To H using the formulas (5-20) to (5-22)_effAnd Θ_effAsk for.
(Vii) From equation (5-25), Ω_effAnd H from the equations (5-23) and (5-24)_effθ, H_eff⊥Ask for.
(Viii) Find θ from equation (5-19), and find magnetization M from (5-18). Here, the anisotropic magnetic field H_AThe H calculated in (4-8)_A(M_m).
(Ix) Θ is obtained from the equation (5-28), and a new magnetization M is obtained from the equations (5-26) and (5-27).
(X) Using the new magnetization M obtained in (ix), (iii) to (vii) are repeated until a self-consistent magnetization M is obtained. Actually, when new θ and M are obtained by iteration (viii), the old θ and M are not updated (replaced) as they are, but only the fluctuation amount of 0.2 to 0.8 is corrected. Convergence is improved by introducing a relaxation coefficient.
[0123]
[Formula 49]

[0124]
2) Process flow in the remanent magnetization calculator
Based on the above, the residual magnetization calculation unit performs processing in the flow shown in FIG.
[0125]
In step S1701, the easy magnetization axis Θ obtained by the residual magnetization process calculator_mDirection, remanent magnetization M_r(M_m), Anisotropic magnetic field H_AThe result of is read into memory.
[0126]
In step S1702, initial values of magnetization magnitude M and direction θ are set. The size is M_r(M_m), The direction is the direction (θ = 0) coinciding with the easy axis of magnetization.
[0127]
In step S1703, the magnetization vector is converted into x, y components (M_x, M_y).
[0128]
In step S1704, the magnetization distribution (M_x, M_y) To calculate the magnetic field of a single magnet. A general magnetic field calculation solver such as the finite element method is used.
[0129]
In step S1705, the magnetic flux density (B_x, B_y) To magnetic field (H_x, H_y)
[0130]
In step S1706, the magnetic field (H_x, H_y) Based on equations (5-20) to (5-22)_eff, Θ_effAsk for.
[0131]
In step S1707, Ω is calculated according to equations (5-23) to (5-25)._eff, H_effθ, H_eff⊥Ask for.
[0132]
In step S 1708, a new magnetization magnitude M and direction θ are obtained using equations (5-18) and (5-19).
[0133]
In step S1709, it is determined whether or not the magnetization size M and the direction θ have sufficiently converged. If the convergence is insufficient, the process returns to step S1703, and the processing from step S1703 to step S1708 is repeated. On the other hand, if it is determined in step S1709 that the magnetization M and the direction θ have sufficiently converged, the process proceeds to step S1710, and the magnetization vector is calculated from the equations M5-26 and 5-27 from the magnetization M and the direction θ. The obtained magnetization distribution is output to an external storage device.
[0134]
Here, when calculating the demagnetization process, the point immediately before the bent portion of the second quadrant of the demagnetization curve and the residual magnetization M_rThe slope of the line connecting the points_recIn general, this gradient is χ obtained by the complete magnetization curve estimator._r(Equation (2-10)) has almost the same value, and χ_rMay be substituted.
[0135]
Further, in the formulas (5-20) to (5-28), the calculation has been limited to two dimensions on the xy plane, but for easy understanding, this is expanded to a three-dimensional formula, In the step of S1704, it is easy to obtain a three-dimensional magnetization distribution by using a three-dimensional magnetic field calculation solver. Therefore, the formulas (5-20) to (5-28) used in the above calculation procedure are not limited to the two-dimensional notation as described, but also include formulas expanded to three dimensions.
[0136]
Magnetization of a permanent magnet having a cylindrical shape using the magnetization calculation device having the above-described magnetization current waveform calculation unit, complete magnetization curve estimation unit, initial magnetization process calculation unit, residual magnetization calculation unit, and demagnetization process calculation unit An example of calculation is shown.
[0137]
FIG. 18 shows an example of a divided model of a magnetizer, in which 1801 is a coil and 1802 is a magnet. FIG. 19 shows the calculation result of the magnetizing current waveform calculation unit of the current waveform when a current is passed through a capacitor connected to the present magnetizer. Peak current (J_peak) Is 10 kA, and it is confirmed that the current waveform is close to the actual measurement value.
[0138]
FIG. 20 shows the result of creating a complete saturation magnetization curve based on the initial magnetization curve obtained by the measurement. Taking the magnetic field H on the horizontal axis and the magnetization M on the vertical axis, the measurement result is plotted for the magnetic field region shown in 2001, and the result calculated by the complete magnetization curve estimation unit is plotted for the magnetic field region shown in 2002. It can be seen that a complete saturation magnetization curve is obtained.
[0139]
21A shows the magnet after the initial magnetization process calculation by the initial magnetization process calculation unit, FIG. 21B shows the magnet after the demagnetization process calculation by the demagnetization process calculation unit, and FIG. 21C shows the magnet after the demagnetization process calculation by the demagnetization process calculation unit. The figure which displayed the magnetization distribution in the inside with the arrow is shown. FIG. 21D shows the state of magnetic flux in a single magnet using the result of magnetization distribution obtained by the demagnetization process calculation in the demagnetization process calculation unit.
[0140]
In this way, by displaying the result of each calculation unit in the form of a graph, it is possible to know the magnetization calculation process very easily.
[0141]
FIG. 22 compares the magnetic flux density distribution on the magnet surface with a single magnet with the experimental results. It can be seen that a calculation result that matches the measured value very well is obtained. FIG. 23 summarizes data necessary for the calculation for the magnetization calculation method described above. The data listed here is relatively easy to measure, investigate or determine.
[0142]
Embodiment 2
In the magnetization current waveform calculation unit of the first embodiment, the user specifies the inductance L of the magnetizer, the capacitance C of the capacitor, and the resistance R of the coil in the LCR circuit, and adjusts the voltage between both terminals of the capacitor within the program. As a result, the peak value of the magnetizing current waveform designated by the user was obtained.
[0143]
However, this is because the user specifies the voltage between both terminals of the capacitor in the power supply circuit, and adjusts the coil resistance (coil resistance when viewed from the whole power supply circuit) within the program, so that the magnetization specified by the user The peak value of the current waveform may be obtained.
Note that FIG. 24 shows the flow of processing in the magnetizing current waveform calculation unit for realizing this method. The difference from FIG. 3 is that the initial value of the resistance (R) of the coil is set in step S2403 and the coil resistance R is updated in step S2408. As a result, the data necessary for performing the magnetization calculation is as shown in FIG. 25, and the voltage between the capacitor terminals of the magnetizer is read instead of the coil resistance of the magnetizer.
[0144]
By making it like this example, even if the resistance of the coil in a magnetizing circuit is unknown, it becomes possible to calculate easily.
[0145]
Embodiment 3
In the first embodiment, it is assumed that the anisotropic magnetic field is proportional to the maximum magnetization point as shown in the equation (4-8). In this assumption, as shown in the following equation (4-8) ', the saturation magnetic flux density M of the demagnetization curve passing through the anisotropic magnetic field through the maximum magnetization point._s(M_m) Is more accurate.
[0146]
[Formula 50]

And the demagnetization curve using this assumption can be calculated | required in the following procedure.
(I) The magnetization rotation angle ω is obtained from the equations (4-3) and (4-5), and the magnetization rotation angle φ in the reversible magnetization rotation region is obtained from the equation (4-6)._bAsk for.
(Ii) Saturation magnetization M_s(M_m) To an appropriate value (M_sDegree).
(Iii) An anisotropic magnetic field H according to equations (4-7) and (4-8) '_A(M_m)
(Iv) Using the equation (4-1), the maximum magnetization point (H_m, M_m(H = H)_m) Magnetization rotation angle φ_mAsk for. This can be obtained using the above equations (A-1) to (A-4).
(V) In the equation (4-2), φ = φ_m, M = M_mSaturation magnetization M_s(M_m) This value is as shown in Formula (4-9).
(Vi) Obtained saturation magnetization M_s(M_m) To (iii) to (v) again. Saturation magnetization M_s(M_m) Repeat until it converges sufficiently.
(Vii) The obtained anisotropic magnetic field H_A(M_m), Saturation magnetization M_s(M_m) Is used to determine the magnetization rotation angle φ with respect to an arbitrary magnetic field H using the equation (4-1), and then the magnetization M is determined using the equation (4-2).
[0147]
In addition, φ obtained as a convergence solution of (iii) to (v)_mAnd the remanent magnetization M according to the equation (4-10)_r(M_m).
[0148]
Embodiment 4
In the first to third embodiments, the magnetization current waveform calculation unit, the complete magnetization curve estimation unit, the initial magnetization process calculation unit, and the remanent magnetization calculation unit that constitute the magnetization calculation device for the method and procedure for performing the magnetization calculation The five parts of the demagnetization process calculation unit have been described.
[0149]
In the present embodiment, a configuration in which these components are actually incorporated and a configuration of the entire apparatus will be described.
[0150]
FIG. 26 is a diagram for explaining a module, an input data file, a result data file, and a flow of magnetization calculation constituting the apparatus. In the figure, the code starting with P indicates an effective format program module, the one starting with Q is input data to be prepared by the user, and the one starting with R is a file discharged by the program.
[0151]
P02 is a magnetization current waveform calculation module incorporating the above-described magnetization current waveform calculation unit, P03 is a complete magnetization curve estimation module incorporating a complete magnetization curve estimation unit, and P04 is an initial magnetization process calculation module. P05 is a remanent magnetization / demagnetization process calculation module incorporating a remanence magnetization calculation module and a demagnetization process calculation module. In addition, a magnetized division model automatic generation module shown in P01 is newly added here. This module automatically generates a segmented model of the appropriate shape using the number of magnetized magnet poles, diameter, magnetizer coil thickness, and number of coil turns per slot as input data. . Many magnets used for motors and the like are generally cylindrical, and when magnets having such a predetermined shape are magnetized, a divided model of the magnetizer corresponding to the magnet is generated.
[0152]
Q01 is a data file representing the shape of the magnetizer, Q02 is a data file of the power circuit of the magnetizer, and Q03 is a point sequence representing the initial magnetization curve of the magnet (combination of magnetic field and magnetization, or magnetic field and magnetic flux density). Data). Q04 is data representing the shape of the major loop of the magnet. Q03 and Q04 are obtained by measuring the magnetic characteristics of the magnet with a BH curve tracer or the like.
[0153]
An example of specific contents of the power supply circuit data of the Q02 magnetizer is shown in FIG. The data includes a guide value 2801 of the magnetizing current, the number of turns 2802 of the coil, a peak value 2803 of the magnetizing current, a capacity 2804 of the capacitor, and a voltage 2805 at the time of charging the capacitor.
[0154]
FIG. 28B shows an example of point sequence data representing the initial magnetization curve of the Q03 magnet. Ten sets of magnetic field H and magnetic flux density B are set as the point sequence data.
[0155]
An example of data representing the shape of the major loop of the Q04 magnet is shown in FIG. Minimum magnetic field H in the reversible magnetization rotation region_b(2806), residual magnetization M_r(2807), and χ_rec(2808) is included.
[0156]
Here, it is assumed that the magnetic characteristics and conductivity of the magnetized yoke and the conductivity of the magnet are built in the program modules P02 to P05.
[0157]
Thus, by managing the data separately for the magnet characteristic data and the magnetizer characteristic data, the system is easy to manage and easy to use.
[0158]
Next, a description will be given of a procedure and a data flow when actually performing magnetization calculation using these input data and program modules.
[0159]
The user first activates the magnetizer split model automatic generation module P01. Then, this module reads the data Q01 from the file, and outputs the file R01 of the magnetizer division model according to the contents. FIG. 27A shows an example of the generated division model.
[0160]
FIG. 27B is a diagram displayed for easy understanding of the material distribution in the divided model. When the divided model is actually displayed, it is easy to understand if it is displayed in a different color for each material. In particular, in this magnetization calculation, the permanent magnet 2701, the magnetization yoke 2702, the air 2703, the + coil 2704, and the −coil. In 2705, the color may be changed and displayed.
[0161]
Next, the user activates the magnetization current waveform calculation module P02. This module reads the power circuit data Q02 of the magnetizer from the file and outputs the calculated magnetization current waveform as a template R02 of the input file for initial magnetization calculation in accordance with the input format of the initial magnetization process calculation module.
[0162]
Next, the user activates the complete magnetization curve estimation module P03. This module reads the magnet initial magnetization curve data file Q03, the magnet major loop data file Q04, and the initial magnetization calculation input file template R02, and adds the complete magnetization curve obtained here to the contents of the file R02. And output as initial magnetization calculation input data R03.
[0163]
Next, the user activates the initial magnetization process calculation module P04. This module is loaded after reading the magnetizing current waveform data and the complete magnetization curve input data R03 for initial magnetization, the major loop data file Q04 of the magnet, and the divided model file R01 of the magnetizer.
The initial magnetization process calculation unit calculates the initial magnetization and outputs the initial magnetization file R04. The initial magnetization file R04 also includes division model data.
[0164]
Finally, the user activates the residual magnetization / demagnetization process calculation module. This module reads the initial magnetization file R04, calculates the magnetization distribution through the residual magnetization and the demagnetization process by the built-in residual magnetization calculation unit and demagnetization process calculation unit, and outputs them to the file R05 and the file 06, respectively.
[0165]
Since the remanent magnetization calculation unit and the demagnetization process calculation unit are in the same module, the direction Θ of the easy axis that is the data to be passed between them_m , Remanent magnetization M_r(M_m), Anisotropic magnetic field H_A(M_m) Is passed inside the module, not via a file.
[0166]
Although not displayed here, FIG. 20 shows a complete magnetization curve using a file R03, a module for displaying a graph of the magnetized current waveform of elapsed time × current value as shown in FIG. 20 using the file R02. A module for graphically displaying such a magnetic field × magnetization magnetization curve and a module for displaying the magnetization distribution states of the files R04, R05, and R06 as vectors are prepared.
[0167]
By preparing the plotting module in this way, the magnetization calculation process can be traced very easily. The functions of these plotting modules may be included as part of the above five calculation module functions.
[0168]
As explained in the present embodiment, the module for performing magnetization calculation is divided into several along the magnetization calculation process, and by making each an independent program, only a part of the data is changed. It is possible to eliminate the waste of duplicate calculation at the time of calculation, and it is possible to perform magnetization calculation while tracking the calculation process one by one.
[0169]
[Other Embodiments]
Another object of the present invention is to supply a storage medium storing software program codes for implementing the functions of the above-described embodiments to a system or apparatus, and the computer (or CPU or MPU) of the system or apparatus stores the storage medium. Needless to say, this can also be achieved by reading and executing the program code stored in the.
[0170]
In this case, the program code itself read from the storage medium realizes the functions of the above-described embodiments, and the storage medium storing the program code constitutes the present invention.
[0171]
As a storage medium for supplying the program code, for example, a floppy disk, a hard disk, an optical disk, a magneto-optical disk, a CD-ROM, a CD-R, a magnetic tape, a nonvolatile memory card, a ROM, or the like can be used.
[0172]
Further, by executing the program code read by the computer, not only the functions of the above-described embodiments are realized, but also an OS (operating system) operating on the computer based on the instruction of the program code. It goes without saying that a case where the function of the above-described embodiment is realized by performing part or all of the actual processing and the processing is included.
[0173]
Further, after the program code read from the storage medium is written into a memory provided in a function expansion board inserted into the computer or a function expansion unit connected to the computer, the function expansion is performed based on the instruction of the program code. It goes without saying that the CPU or the like provided in the board or the function expansion unit performs part or all of the actual processing, and the functions of the above-described embodiments are realized by the processing.
[0174]
【The invention's effect】
As described above, according to the present invention, it is possible to calculate the magnetization distribution of a magnet using an existing general-purpose magnetic field calculation solver.
[Brief description of the drawings]
FIG. 1 is a block diagram showing a configuration of a magnetization distribution calculating apparatus according to a first embodiment of the present invention.
FIG. 2 is a circuit diagram solved by a magnetization current waveform calculation unit of the magnetization distribution calculation apparatus according to the first embodiment of the present invention.
FIG. 3 is a diagram illustrating a processing flow performed by a magnetization current waveform calculation unit of the magnetization distribution calculation apparatus according to the first embodiment of the present invention.
FIG. 4 shows the 1 / H in the magnetization current waveform calculation unit of the magnetization distribution calculation apparatus according to the first embodiment of the present invention;²It is a figure explaining the method of calculating | requiring saturation magnetization by extrapolation.
FIG. 5 is a diagram illustrating a reversible magnetization rotation region of a magnetization curve.
FIG. 6_rFIG.
FIG. 7 is a diagram showing a processing flow performed by a complete magnetization curve estimation unit of the magnetization distribution calculating apparatus according to the first embodiment of the present invention.
FIG. 8 is an explanatory diagram of a magnetization rotation model for explaining the contents of an initial magnetization process calculation unit of the magnetization distribution calculation apparatus according to the first embodiment of the present invention.
FIG. 9 is a flowchart illustrating a process performed by an initial magnetization process calculation unit of the magnetization distribution calculation apparatus according to the first embodiment of the present invention.
FIG. 10 is a diagram for explaining a residual magnetization calculation method of the magnetization distribution calculation apparatus according to the first embodiment of the present invention.
FIG. 11 is a diagram showing a processing flow performed by a residual magnetization calculator of the magnetization distribution calculating apparatus according to the first embodiment of the present invention.
FIG. 12 is a diagram for explaining a demagnetization process of an anisotropic magnet.
FIG. 13_recFIG.
FIG. 14 H_effIt is a figure explaining calculation of integration.
FIG. 15 is a diagram illustrating a demagnetization process of an isotropic magnet.
FIG. 16 is a diagram for explaining rotation of magnetization in a demagnetization process at an angle measured from a reference axis.
FIG. 17 is a diagram illustrating a processing flow performed by a demagnetization process calculation unit of the magnetization distribution calculation device according to the first embodiment of the present invention;
FIG. 18 is a diagram illustrating an example of a magnetizer division model according to the first embodiment of the present invention.
FIG. 19 is a diagram illustrating an example of a calculation result of a current waveform of the magnetization distribution calculating apparatus according to the first embodiment of the present invention.
FIG. 20 is a diagram illustrating an example of a result of a complete saturation magnetization curve estimation unit of the magnetization distribution calculating apparatus according to the first embodiment of the present invention.
FIG. 21 is a diagram illustrating an example of results of an initial magnetization process calculation unit, a residual magnetization calculation unit, and a demagnetization process calculation unit of the magnetization distribution calculation apparatus according to the first embodiment of the present invention.
FIG. 22 is a diagram illustrating an example of a result of obtaining a surface magnetic flux density distribution of a single magnet using the magnetization distribution of the magnet obtained by the magnetization distribution calculating apparatus according to the first embodiment of the present invention.
FIG. 23 is a diagram showing a list of input data required by the magnetization distribution calculating apparatus according to the first embodiment of the present invention.
FIG. 24 is a diagram showing a process flow performed by a magnetization current waveform calculation unit of the magnetization distribution calculation apparatus according to the second embodiment of the present invention.
FIG. 25 is a diagram showing a list of input data required by a magnetization current calculation unit of the magnetization distribution calculation apparatus according to the second embodiment of the present invention.
FIG. 26 is a diagram illustrating an example of main modules and data flow included in the magnetization distribution calculating apparatus according to the fourth embodiment of the present invention.
FIG. 27 is a diagram illustrating an example of a divided model of a magnetizer of a magnetization distribution calculating apparatus according to a fourth embodiment of the present invention.
FIG. 28 is a diagram showing an example of an input data file for performing magnetization calculation of the magnetization distribution calculating apparatus according to the fourth embodiment of the present invention.
[Explanation of symbols]
101 CPU
102 Display device
103 Input device
104 External storage device
105 memory
106 Bus

Claims (26)

A magnetization distribution calculating device for calculating a magnetization distribution in a magnetic material when the magnetic material is magnetized by applying a magnetic field by a magnetizer,
The magnetic material is divided into a finite number of elements, a transient analysis of the applied magnetic field is performed when a magnetic field is applied from the magnetizer to each element that is not magnetized, and the maximum magnetic field for each element is extracted, Initial magnetization process calculation means for calculating the maximum magnetization point for each element based on the maximum magnetic field and the characteristics of the magnetic material;
Residual magnetization calculation means for calculating the residual magnetization of each element based on the maximum magnetization point of each element calculated by the initial magnetization process calculation means and the characteristics of the magnetic material;
A magnetization distribution calculating device comprising: Demagnetization for calculating the residual magnetization of each element calculated by the residual magnetization calculating means and the magnetization of each element when a magnetic field generated by the magnetization of each element is applied to each element having the residual magnetization The magnetization distribution calculation apparatus according to claim 1, further comprising a process calculation unit. The residual magnetization calculation means includes
The residual magnetization when the applied magnetic field is removed is calculated based on a maximum magnetization point calculated by the initial magnetization process calculation means and a demagnetization curve passing through the maximum magnetization point. 2. The magnetization distribution calculating device according to 2. In the residual magnetization calculating means,
The residual magnetization curve when the magnetic field at the maximum magnetization point is H _m and the magnetization is M _m is:
Based on equations (4-3) to (4-5), the magnetization rotation angle is based on the saturation magnetization M _s on the magnetic material and the residual magnetization M _r when a magnetic field of a reversible magnetization rotation region or more is applied to the magnetic material. Find ω,
Based on the magnetization M _b with respect to the magnetic field H _{b at} a predetermined magnetization point and the saturation magnetization M _s when a magnetic field of a reversible magnetization rotation region or more is applied to the magnetic material, the magnetization rotation angle φ _b from the equation (4-6) Seeking
From the equations (4-7) and (4-8) based on the magnetization rotation angle φ _b , the magnetization rotation angle ω, the magnetic field H _b , the saturation magnetization M _s, and the magnetization M _m. Find the anisotropic magnetic field H _A (M _m ) at the maximum magnetization point,
Based on the anisotropic magnetic field H _A (M _m ), the magnetization rotation angle φ _m at the maximum magnetization point is obtained by setting H = H _m from the equation (4-1),
In equation (4-2), when φ = φ _m and M = M _m , the saturation magnetization M _s (M _m ) of the demagnetization curve passing through the maximum magnetization point is obtained,
4. The magnetic field H is obtained by obtaining the magnetization M by using the equation (4-2) after obtaining the magnetization rotation angle φ by the equation (4-1) with respect to an arbitrary magnetic field H. Magnetization distribution calculation device.

In the residual magnetization calculating means,
The residual magnetization M _r (M _m ) at the maximum magnetization point having the magnetic field H _m and the magnetization M _m is:
On the basis of the magnetic and the magnetic material and the saturation magnetization M _s of the material and the residual magnetization M _r in the case of applying a magnetic field over the reversible magnetization rotation region (4-3) and (4-5) the magnetization rotation angle from the equation Find ω,
Based on the magnetization M _b with respect to the magnetic field H _{b at} a predetermined magnetization point and the saturation magnetization M _s when a magnetic field of a reversible magnetization rotation region or more is applied to the magnetic material, the magnetization rotation angle φ _b from the equation (4-6) Seeking
From the equations (4-7) and (4-8) based on the magnetization rotation angle φ _b , the magnetization rotation angle ω, the magnetic field H _b , the saturation magnetization M _s, and the magnetization M _m. Find the anisotropic magnetic field H _A (M _m ) at the maximum magnetization point,
Based on the anisotropic magnetic field H _A (M _m ), the magnetization rotation angle φ _m at the maximum magnetization point is obtained by setting H = H _m from the equation (4-1),
5. The magnetization distribution calculation apparatus according to claim 4, wherein the equation (4-10) is also used.

In the residual magnetization calculating means,
When the magnetic field at the maximum magnetization point is H _m and the magnetization is M _m , the demagnetization curve passing through the maximum magnetization point is
On the basis of the magnetic and the magnetic material and the saturation magnetization M _s of the material and the residual magnetization M _r in the case of applying a magnetic field over the reversible magnetization rotation region (4-3) and (4-5) the magnetization rotation angle from the equation Find ω,
Based on the magnetization M _b with respect to the magnetic field H _{b at} a predetermined magnetization point and the saturation magnetization M _s when a magnetic field of a reversible magnetization rotation region or more is applied to the magnetic material, the magnetization rotation angle φ _b from the equation (4-6) Seeking
Based on the magnetization rotation angle φ _b , the magnetization rotation angle ω, the magnetic field H _b , the saturation magnetization M _s, and M _s (M _m ) in which a predetermined value is set, formula (4-7) And the anisotropy magnetic field H _A (M _m ) at the maximum magnetization point from the equation (4-8) ′,
Based on the anisotropic magnetic field H _A (M _m ), the magnetization rotation angle φ _m at the maximum magnetization point is obtained by setting H = H _m from the equation (4-1),
In equation (4-2), when φ = φ _m and M = M _m , the saturation magnetization M _s (M _m ) of the demagnetization curve passing through the maximum magnetization point is obtained,
4. The magnetic field H is obtained by obtaining the magnetization M by using the equation (4-2) after obtaining the magnetization rotation angle φ by the equation (4-1) with respect to an arbitrary magnetic field H. Magnetization distribution calculation device.

In the residual magnetization calculating means,
The residual magnetization M _r (M _m ) for the maximum magnetization point having the magnetic field H _m and the magnetization M _m is:
On the basis of the magnetic and the magnetic material and the saturation magnetization M _s of the material and the residual magnetization M _r in the case of applying a magnetic field over the reversible magnetization rotation region (4-3) and (4-5) the magnetization rotation angle from the equation Find ω,
Based on the magnetization M _b with respect to the magnetic field H _{b at} a predetermined magnetization point and the saturation magnetization M _s when a magnetic field of a reversible magnetization rotation region or more is applied to the magnetic material, the magnetization rotation angle φ _b from the equation (4-6) Seeking
Based on the magnetization rotation angle φ _b , the magnetization rotation angle ω, the magnetic field H _b , the saturation magnetization M _s, and M _s (M _m ) in which a predetermined value is set, formula (4-7) And the anisotropy magnetic field H _A (M _m ) at the maximum magnetization point from the equation (4-8) ′,
Based on the anisotropic magnetic field H _A (M _m ), the magnetization rotation angle φ _m at the maximum magnetization point is obtained by setting H = H _m from the equation (4-1),
The magnetization distribution calculation apparatus according to claim 6, wherein the equation (4-10) is also used.

In the demagnetization process calculation means,
The x-direction component M _x and the y-direction component M when the easy axis of each element of the magnetic material has an angle of Θ _m with respect to a predetermined reference axis and the residual magnetization M _r is left as it is. a magnetization M having _y, and an angle Θ between the magnetization M and the reference axis,
1) A predetermined value is input to an angle θ formed by the magnetization M and the easy axis,
2) Calculate the magnetic field of each element of the magnetic material when a predetermined value is input to the magnetization M,
3) Find the x-direction component H _x and the y-direction component H _y of the magnetic field H of each element based on the magnetic field,
4) Based on the H _x and H _y , the effective magnetic field H _eff and the angle Θ _eff formed by the effective magnetic field H _eff and the reference angle are obtained from the equations (5-20) to (5-22),
5) on the basis of the effective magnetic field _{H eff} and the said angle theta _eff, (5-23) from the equation to (5-25) _below, Ω _{eff, H} effθ, determined the _{H eff⊥,}
The Omega _eff, the _{H Effshita,} based on the _{H Eff⊥,} determined to θ in terms of the assignment (5-19) from the equation, the anisotropic magnetic field at maximum magnetization point _H A a _{(M m)} to _{H A} further, (5-18) from equation obtains the magnetization M, (5-26) from the equation to (5-28) below, obtains the new _M x, _{M y,}
Claims wherein 5) Any new M _x which stopped _using the magnetization M with M _y, the 1) to 5), characterized in that the finding by repeating until self-consistent magnetization M will be obtained The magnetization distribution calculation apparatus according to 5 or 7.

The magnetization distribution calculating apparatus according to claim 8, wherein the magnetization distribution calculating apparatus is expressed in a Cartesian coordinate system. 10. The magnetization distribution calculation apparatus according to claim 9, wherein an expression obtained by extending the expressions (5-20) to (5-28) in three dimensions is used. 9. The magnetization distribution calculation apparatus according to claim 8, wherein the calculation of the magnetic field of each element of 2) uses any one of a finite element method, an integration method, a difference method, and a boundary element method. When repeating the above 1) to 5) until the self-consistent magnetization M is obtained by using the magnetization M having the new M _x and M _y stopped in the above 5), the newly acquired M _x and M The magnetization distribution calculation apparatus according to claim 8, wherein _y is multiplied by a predetermined coefficient. A magnetization distribution calculation method for calculating a magnetization distribution in a magnetic material when the magnetic material is magnetized by applying a magnetic field by a magnetizer,
The initial magnetization process calculation means divides the magnetic material into a finite number of elements, performs a transient analysis of the applied magnetic field when a magnetic field is applied from the magnetizer to each element that is not magnetized, and An initial magnetization process calculation step of extracting a maximum magnetic field and calculating a maximum magnetization point for each element based on the maximum magnetic field and the characteristics of the magnetic material;
A residual magnetization calculating means for calculating a residual magnetization of each element based on the maximum magnetization point of each element calculated by the initial magnetization process calculating step and the characteristics of the magnetic material;
A magnetization distribution calculation method comprising: A demagnetization process calculating step of calculating a residual magnetization of each element calculated by the residual magnetization calculating step and a magnetization of each element when a magnetic field is applied to each element having the residual magnetization from the magnetizer; The magnetization distribution calculation method according to claim 13. The residual magnetization calculation step includes:
The residual magnetization when the applied magnetic field is removed is calculated based on the maximum magnetization point calculated by the initial magnetization process calculation step and a demagnetization curve passing through the maximum magnetization point. 14. The magnetization distribution calculation method according to 14 . In the residual magnetization calculation step,
The residual magnetization curve when the magnetic field at the maximum magnetization point is H _m and the magnetization is M _m is:
On the basis of the magnetic and the magnetic material and the saturation magnetization M _s of the material and the residual magnetization M _r in the case of applying a magnetic field over the reversible magnetization rotation region (4-3) equation to (4-5) the magnetization rotation angle from the equation Find ω,
Based on the magnetization M _b with respect to the magnetic field H _{b at} a predetermined magnetization point and the saturation magnetization M _s when a magnetic field of a reversible magnetization rotation region or more is applied to the magnetic material, the magnetization rotation angle φ _b from the equation (4-6) Seeking
From the equations (4-7) and (4-8) based on the magnetization rotation angle φ _b , the magnetization rotation angle ω, the magnetic field H _b , the saturation magnetization M _s, and the magnetization M _m. Find the anisotropic magnetic field H _A (M _m ) at the maximum magnetization point,
Based on the anisotropic magnetic field H _A (M _m ), the magnetization rotation angle φ _m at the maximum magnetization point is obtained by setting H = H _m from the equation (4-1),
In equation (4-2), when φ = φ _m and M = M _m , the saturation magnetization M _s (M _m ) of the demagnetization curve passing through the maximum magnetization point is obtained,
16. The magnetic field H is obtained by obtaining the magnetization M using the equation (4-2) after obtaining the magnetization rotation angle φ by the equation (4-1) for an arbitrary magnetic field H. Magnetization distribution calculation method.

In the residual magnetization calculation step,
The residual magnetization at the maximum magnetization point having the magnetic field H _m and the magnetization M _m is:
On the basis of the magnetic and the magnetic material and the saturation magnetization M _s of the material and the residual magnetization M _r in the case of applying a magnetic field over the reversible magnetization rotation region (4-3) and (4-5) the magnetization rotation angle from the equation Find ω,
Based on the magnetization M _b with respect to the magnetic field H _{b at} a predetermined magnetization point and the saturation magnetization M _s when a magnetic field of a reversible magnetization rotation region or more is applied to the magnetic material, the magnetization rotation angle φ _b from the equation (4-6) Seeking
From the equations (4-7) and (4-8), based on the magnetization rotation angle φ _b , the magnetization rotation angle ω, the magnetic field H _b , the saturation magnetization M _s, and the magnetic field M _m. Find the anisotropic magnetic field H _A (M _m ) at the maximum magnetization point (H _m , M _m ),
Based on the anisotropic magnetic field H _A (M _m ), the magnetization rotation angle φ _m at the maximum magnetization point is obtained by setting H = H _m from the equation (4-1),
The method of calculating a magnetization distribution according to claim 16, wherein the residual magnetization M _r (M _m ) is obtained using the equation (4-10).

In the residual magnetization calculation step,
When the magnetic field at the maximum magnetization point is H _m and the magnetization is M _m , the demagnetization curve passing through the maximum magnetization point is
On the basis of the magnetic and the magnetic material and the saturation magnetization M _s of the material and the residual magnetization M _r in the case of applying a magnetic field over the reversible magnetization rotation region (4-3) and (4-5) the magnetization rotation angle from the equation Find ω,
Based on the magnetization M _b with respect to the magnetic field H _{b at} a predetermined magnetization point and the saturation magnetization M _s when a magnetic field of a reversible magnetization rotation region or more is applied to the magnetic material, the magnetization rotation angle φ _b from the equation (4-6) Seeking
Based on the magnetization rotation angle φ _b , the magnetization rotation angle ω, the magnetic field H _b , the saturation magnetization M _s, and M _s (M _m ) in which a predetermined value is set, formula (4-7) And the anisotropy magnetic field H _A (M _m ) at the maximum magnetization point from the equation (4-8) ′,
Based on the anisotropic magnetic field H _A (M _m ), the magnetization rotation angle φ _m at the maximum magnetization point is obtained from the equation (4-1),
In equation (4-2), when φ = φ _m and M = M _m , the saturation magnetization M _s (M _m ) of the demagnetization curve passing through the maximum magnetization point is obtained,
16. The magnetic field H is obtained by obtaining the magnetization M by using the equation (4-2) after obtaining the magnetization rotation angle φ by the equation (4-1) with respect to an arbitrary magnetic field H. Magnetization distribution calculation method.

In the residual magnetization calculation step,
The residual magnetization M _r (M _m ) for the maximum magnetization point having the magnetic field H _m and the magnetization M _m is:
On the basis of the magnetic and the magnetic material and the saturation magnetization M _s of the material and the residual magnetization M _r in the case of applying a magnetic field over the reversible magnetization rotation region (4-3) and (4-5) the magnetization rotation angle from the equation Find ω,
The stop than in a predetermined magnetization point when the reversible magnetization rotation region or more field of the magnetic material is applied to the magnetic field H _b a magnetization rotation angle phi _b in the magnetization M _b at that time (4-6) equation,
Based on the magnetization rotation angle φ _b , the magnetization rotation angle ω, the magnetic field H _b , the saturation magnetization M _s, and M _s (M _m ) in which a predetermined value is set, formula (4-7) And the anisotropy magnetic field H _A (M _m ) at the maximum magnetization point from the equation (4-8) ′,
Based on the anisotropic magnetic field H _A (M _m ), the magnetization rotation angle φ _m at the maximum magnetization point is obtained from the equation (4-1),
The method of calculating a magnetization distribution according to claim 18, wherein the equation (4-10) is also used.

In the demagnetization process calculation step,
The easy axis of each element of the magnetic material has an angle of theta _m with respect to a predetermined reference axis, and in a state of having a residual magnetization M _r, in the case of applying a predetermined magnetic field H from the magnetizer magnetization M with x direction component M _x and y-direction component M _y, and the angle Θ between the magnetization M and the reference axis,
1) A predetermined value is input to an angle θ formed by the magnetization M and the easy axis,
2) Calculate the magnetic field of each element of the magnetic material when a predetermined value is input to the magnetization M,
3) Find the x-direction component H _x and the y-direction component H _y of the magnetic field H of each element based on the magnetic field,
4) Based on the H _x and H _y , the effective magnetic field H _eff and the angle Θ _eff formed by the effective magnetic field H _eff and the reference angle are obtained from the equations (5-20) to (5-22),
5) on the basis of the effective magnetic field _{H eff} and the said angle theta _eff, (5-23) from the equation to (5-25) _below, Ω _{eff, H} effθ, determined the _{H eff⊥,}
The Omega _eff, the _{H Effshita,} based on the _{H Eff⊥,} determined to θ in terms of the assignment (5-19) from the equation, the anisotropic magnetic field at maximum magnetization point _H A a _{(M m)} to _{H A} further, (5-18) from equation obtains the magnetization M, (5-26) from the equation to (5-28) below, obtains the new _M x, _{M y,}
Claims wherein 5) Any new M _x which stopped _using the magnetization M with M _y, the 1) to 5), characterized in that the finding by repeating until self-consistent magnetization M will be obtained The magnetization distribution calculation method according to 17 or 19.

The magnetization distribution calculation method according to claim 20, wherein the magnetization distribution is expressed in a Cartesian coordinate system. The method of calculating a magnetization distribution according to claim 21, wherein an expression obtained by extending the expressions (5-20) to (5-28) in three dimensions is used. 21. The magnetization distribution calculation method according to claim 20, wherein the magnetic field of each element of 2) is calculated using any one of a finite element method, an integration method, a difference method, and a boundary element method. When repeating the above 1) to 5) until the self-consistent magnetization M is obtained by using the magnetization M having the new M _x and M _y stopped in the above 5), the newly acquired M _x and M 21. The magnetization distribution calculation method according to claim 20, wherein _y is multiplied by a predetermined coefficient. The computer, computer-readable storage medium storing a control program for functioning as the means of the magnetization distribution calculating apparatus according to any one of claims 1 to 12. The computer control program for functioning as the means of the magnetization distribution calculating apparatus according to any one of claims 1 to 12.

Images