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| In [[electrical engineering]], the '''alpha-beta''' (<math>\alpha\beta\gamma</math>) '''transformation''' (also known as the '''Clarke transformation''') is a mathematical [[Transform (mathematics)|transformation]] employed to simplify the analysis of [[Three-phase electric power|three-phase circuits]]. Conceptually it is similar to the [[dqo transformation]]. One very useful application of the <math>\alpha\beta\gamma</math> transformation is the generation of the reference signal used for space vector modulation control of three-phase [[Inverter (electrical)|inverters]].
| | I'm Terri (29) from Presidente Prudente, Brazil. <br>I'm learning Arabic literature at a local college and I'm just about to graduate.<br>I have a part time job in a college.<br>xunjie あなたは古代のセーターの制御を変更することに最も適したお気に入りのセーター新しいMANGOを見つけることは非常に耐久性があります。 |
| | このブレスレットの製造工程でも、 |
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| | 材料と資源総合利用プロジェクトの数を決定するために検討した。 [http://www.equityfair.ch/impressions/jp/p/best/watch/gaga/ �����ߥ�� �rӋ �˚�] 私はどこに黄色室を開くことは知らない、 |
| | 期間制码表伯爵gouverneur(右側の写真)伯爵gouverneurシリーズ時間码表に搭載して新しい882p伯爵の薄型自動で機械机芯チェーンをしなければならない。 |
| | オアシススター日：2013年8月26日午前13時42分22秒エキソMは良い献身ああ、[http://www.equityfair.ch/impressions/jp/p/best/watch/gaga/ �����ߥ�� �rӋ �˚�] カナリ2012年秋と冬のシリーズは、 |
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| | ハイエンド品質のファッション子供服ブランドの貴族の創始者として生まれ、 |
| | 2550万元に達していることがわかる。 [http://alpha-printing.com/images/sto<br><br>cm.php MCM ؔ��] |
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| ==Definition==
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| The <math>\alpha\beta\gamma</math> transform applied to three-phase currents, as used by [[Edith Clarke]], is<ref>{{cite journal| author=W. C. Duesterhoeft, Max W. Schulz and Edith Clarke| journal=Transactions of the American Institute of Electrical Engineers| title=Determination of Instantaneous Currents and Voltages by Means of Alpha, Beta, and Zero Components|date=july 1951|volume=70|issue=2|
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| pages=1248–1255|ISSN=0096-3860|doi=10.1109/T-AIEE.1951.5060554}}</ref>
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| :<math>i_{\alpha\beta\gamma}(t) = Ti_{abc}(t) = \frac{2}{3}\begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\
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| 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \\
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| \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\
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| \end{bmatrix}\begin{bmatrix}i_a(t)\\i_b(t)\\i_c(t)\end{bmatrix}</math>
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| where <math>i_{abc}(t)</math> is a generic three-phase current sequence and <math>i_{\alpha\beta\gamma}(t)</math> is the corresponding current sequence given by the transformation <math>T</math>.
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| The inverse transform is:
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| :<math>i_{abc}(t) = T^{-1}i_{\alpha\beta\gamma}(t) = \begin{bmatrix} 1 & 0 & 1\\
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| -\frac{1}{2} & \frac{\sqrt{3}}{2} & 1\\
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| -\frac{1}{2} & -\frac{\sqrt{3}}{2} & 1\end{bmatrix}
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| \begin{bmatrix}i_\alpha(t)\\i_\beta(t)\\i_\gamma(t)\end{bmatrix}.</math>
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| The above Clarke's transformation preserves the amplitude of the electrical variables which is applied to. Indeed, consider a three-phase symmetric, direct, current sequence
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| :<math>
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| \begin{align}
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| i_a(t)=&\sqrt{2}I\cos\theta(t),\\
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| i_b(t)=&\sqrt{2}I\cos\left(\theta(t)-\frac23\pi\right),\\
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| i_c(t)=&\sqrt{2}I\cos\left(\theta(t)+\frac23\pi\right),
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| \end{align}
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| </math>
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| where <math>I</math> is the rms of <math>i_a(t)</math>, <math>i_b(t)</math>, <math>i_c(t)</math> and <math>\theta(t)</math> is the generic time-varying angle that can also be set to <math>\omega t</math> without loss of generality. Then, by applying <math>T</math> to the current sequence, it results
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| :<math>
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| \begin{align}
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| i_{\alpha}=&\sqrt2 I\cos\theta(t),\\
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| i_{\beta}=&\sqrt2 I\sin\theta(t),\\
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| i_{\gamma}=&0,
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| \end{align}
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| </math>
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| where the last equation holds since we have considered balanced currents. As it is shown in the above, the amplitudes of the currents in the <math>\alpha\beta\gamma</math> reference frame are the same of that in the natural reference frame.
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| === Power invariant transformation ===
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| The active and reactive powers computed in the Clark's domain with the transformation shown above are not the same of those computed in the standard reference frame. This happens because <math>T</math> is not unitary. In order to preserve the active and reactive powers one has, instead, to consider
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| :<math>i_{\alpha\beta\gamma}(t) = Ti_{abc}(t) = \sqrt{\frac{2}{3}}\begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\
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| 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \\
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| \frac{1}{\sqrt2} & \frac{1}{\sqrt2} & \frac{1}{\sqrt2} \\
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| \end{bmatrix}\begin{bmatrix}i_a(t)\\i_b(t)\\i_c(t)\end{bmatrix},</math>
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| which is a unitary matrix and the inverse coincides with its transpose.<ref>{{cite journal|title = Area Based Approach for Three Phase Power Quality Assessment in Clarke Plane|
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| author= S. CHATTOPADHYAY, M. MITRA, S. SENGUPTA|journal=Journal of Electrical Systems|year=2008|volume=04|issue=01|pages=62|url=http://www.doaj.org/doaj?func=fulltext&passMe=http://journal.esrgroups.org/jes/papers/4_1_5.pdf|accessdate=2012-04-26}}</ref>
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| In this case the amplitudes of the transformed currents are not the same of those in the standard reference frame, that is
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| :<math>
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| \begin{align}
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| i_{\alpha}=&\sqrt3 I\cos\theta(t),\\
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| i_{\beta}=&\sqrt3 I\sin\theta(t),\\
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| i_{\gamma}=&0.
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| \end{align}
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| </math>
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| === Simplified transformation ===
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| Since in a balanced system <math>i_a(t)+i_b(t)+i_c(t)=0</math> and thus <math>i_\gamma(t)=0</math> one can also consider the simplified transform<ref name="Tahri">F. Tahri, A.Tahri, Eid A. AlRadadi and A. Draou Senior, "Analysis and Control of Advanced Static VAR compensator Based on the Theory of the Instantaneous Reactive Power," presented at ACEMP, Bodrum, Turkey, 2007.</ref>
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| :<math>i_{\alpha\beta}(t) = \frac23 \begin{bmatrix} 1 & -\frac12 & -\frac12\\
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| 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2}
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| \end{bmatrix}\begin{bmatrix}i_a(t)\\i_b(t)\\i_c(t)\end{bmatrix}</math>
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| and
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| :<math>i_{abc}(t) = \frac32\begin{bmatrix} \frac23 & 0 \\
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| -\frac{1}{3} & \frac{\sqrt{3}}{3} \\
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| -\frac{1}{3} & -\frac{\sqrt{3}}{3} \end{bmatrix}
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| \begin{bmatrix}i_\alpha(t)\\i_\beta(t)\end{bmatrix}.</math>
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| ==Geometric Interpretation==
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| The <math>\alpha\beta\gamma</math> transformation can be thought of as the projection of the three phase quantities (voltages or currents) onto two stationary axes, the alpha axis and the beta axis.
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| [[Image:AlphaBeta geometric interpretation.gif|center|frame|Shown above is the <math>\alpha\beta\gamma</math> transform as applied to three symmetrical currents flowing through three windings separated by 120 physical degrees. The three phase currents lag their corresponding phase voltages by <math>\delta</math>. The <math>\alpha</math>-<math>\beta</math> axis is shown with the <math>\alpha</math> axis aligned with phase 'A'. The current vector <math>I_{\alpha\beta\gamma}</math> rotates with angular velocity <math>\omega</math>. There is no <math>\gamma</math> component since the currents are balanced.]]
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| ===<math>dqo</math> transform===
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| ==References==
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| {{reflist}}
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| ==See also==
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| * [[Symmetrical components]]
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| * [[Y-Δ transform]]
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| * [[Field-oriented control]]
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| {{DEFAULTSORT:Alpha-beta transformation}}
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| [[Category:Electrical engineering]]
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I'm Terri (29) from Presidente Prudente, Brazil.
I'm learning Arabic literature at a local college and I'm just about to graduate.
I have a part time job in a college.
xunjie あなたは古代のセーターの制御を変更することに最も適したお気に入りのセーター新しいMANGOを見つけることは非常に耐久性があります。
このブレスレットの製造工程でも、
各スタッフの進歩を促進するために、 [http://special-education-soft.com/images/photoalbum/gaga.html �����ߥ�� �rӋ ��� ����] 芸術と快適製品を作成するために、
新しいEハイフン冬のイメージの外観、
材料と資源総合利用プロジェクトの数を決定するために検討した。 [http://www.equityfair.ch/impressions/jp/p/best/watch/gaga/ �����ߥ�� �rӋ �˚�] 私はどこに黄色室を開くことは知らない、
期間制码表伯爵gouverneur(右側の写真)伯爵gouverneurシリーズ時間码表に搭載して新しい882p伯爵の薄型自動で機械机芯チェーンをしなければならない。
オアシススター日：2013年8月26日午前13時42分22秒エキソMは良い献身ああ、[http://www.equityfair.ch/impressions/jp/p/best/watch/gaga/ �����ߥ�� �rӋ �˚�] カナリ2012年秋と冬のシリーズは、
目標として赤ちゃん製品のバリューチェーンを統合するためのプロジェクトのユニークなベビー用品事業を推進として同社は、
ハイエンド品質のファッション子供服ブランドの貴族の創始者として生まれ、
2550万元に達していることがわかる。 [http://alpha-printing.com/images/sto
cm.php MCM ؔ��]
Here is my blog レイバン アウトレット