克拉克變換 （Clarke transformation）也稱為
α
β
γ
{\displaystyle \alpha \beta \gamma }
電機工程學 裏簡化三相電 分析的數學變換 逆變器 的控制上，可以將平衡的三相系統轉換為互相垂直的二相系統，方便信號的處理。
克拉克變換和派克變換 （park transform）都是簡化三相電分析用的數學變換，在電機控制時也常一起使用。
伊迪絲·克拉克 [ 1]
伊迪絲·克拉克用在三相電流的克拉克變換如下[ 2]
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{\displaystyle i_{\alpha \beta \gamma }(t)=Ti_{abc}(t)={\frac {2}{3}}{\begin{bmatrix}1&-{\frac {1}{2}}&-{\frac {1}{2}}\\0&{\frac {\sqrt {3}}{2}}&-{\frac {\sqrt {3}}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}&{\frac {1}{2}}\\\end{bmatrix}}{\begin{bmatrix}i_{a}(t)\\i_{b}(t)\\i_{c}(t)\end{bmatrix}}}
其中
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{\displaystyle i_{abc}(t)}
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{\displaystyle i_{\alpha \beta \gamma }(t)}
T
{\displaystyle T}
逆變換為：
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{\displaystyle i_{abc}(t)=T^{-1}i_{\alpha \beta \gamma }(t)={\begin{bmatrix}1&0&1\\-{\frac {1}{2}}&{\frac {\sqrt {3}}{2}}&1\\-{\frac {1}{2}}&-{\frac {\sqrt {3}}{2}}&1\end{bmatrix}}{\begin{bmatrix}i_{\alpha }(t)\\i_{\beta }(t)\\i_{\gamma }(t)\end{bmatrix}}.}
上述的克拉克變換保持了電機變數的量值。考慮以下對稱的三相電流信號
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{\displaystyle {\begin{aligned}i_{a}(t)=&{\sqrt {2}}I\cos \theta (t),\\i_{b}(t)=&{\sqrt {2}}I\cos \left(\theta (t)-{\frac {2}{3}}\pi \right),\\i_{c}(t)=&{\sqrt {2}}I\cos \left(\theta (t)+{\frac {2}{3}}\pi \right),\end{aligned}}}
其中
I
{\displaystyle I}
i
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{\displaystyle i_{a}(t)}
i
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{\displaystyle i_{b}(t)}
i
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{\displaystyle i_{c}(t)}
平方平均數
θ
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{\displaystyle \theta (t)}
ω
t
{\displaystyle \omega t}
。 將上述三相電流信號進行變換，可得
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cos
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sin
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{\displaystyle {\begin{aligned}i_{\alpha }=&{\sqrt {2}}I\cos \theta (t),\\i_{\beta }=&{\sqrt {2}}I\sin \theta (t),\\i_{\gamma }=&0,\end{aligned}}}
變換後的電流量值和變換前相同。
電機系統的電壓和電流，經過上述的變換後，實功和虛功會和原系統會差一個系數，原因是因為
T
{\displaystyle T}
酉矩陣 （unitary matrix）。若要讓實功和虛功的值在變換前後相同，需要用以下的變換
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{\displaystyle i_{\alpha \beta \gamma }(t)=Ti_{abc}(t)={\sqrt {\frac {2}{3}}}{\begin{bmatrix}1&-{\frac {1}{2}}&-{\frac {1}{2}}\\0&{\frac {\sqrt {3}}{2}}&-{\frac {\sqrt {3}}{2}}\\{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\\\end{bmatrix}}{\begin{bmatrix}i_{a}(t)\\i_{b}(t)\\i_{c}(t)\end{bmatrix}},}
變換矩陣是酉矩陣，而且其逆矩陣恰好為其轉置矩陣[ 3]
i
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{\displaystyle {\begin{aligned}i_{\alpha }=&{\sqrt {3}}I\cos \theta (t),\\i_{\beta }=&{\sqrt {3}}I\sin \theta (t),\\i_{\gamma }=&0.\end{aligned}}}
其逆變換為
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{\displaystyle i_{abc}(t)={\sqrt {\frac {2}{3}}}{\begin{bmatrix}1&0&{\frac {1}{\sqrt {2}}}\\-{\frac {1}{2}}&{\frac {\sqrt {3}}{2}}&{\frac {1}{\sqrt {2}}}\\-{\frac {1}{2}}&-{\frac {\sqrt {3}}{2}}&{\frac {1}{\sqrt {2}}}\\\end{bmatrix}}{\begin{bmatrix}i_{\alpha }(t)\\i_{\beta }(t)\\i_{\gamma }(t)\end{bmatrix}}.}
在平衡系統中，
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{\displaystyle i_{a}(t)+i_{b}(t)+i_{c}(t)=0}
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{\displaystyle i_{\gamma }(t)=0}
[ 4] [ 5]
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{\displaystyle {\begin{aligned}i_{\alpha \beta }(t)&={\frac {2}{3}}{\begin{bmatrix}1&-{\frac {1}{2}}&-{\frac {1}{2}}\\0&{\frac {\sqrt {3}}{2}}&-{\frac {\sqrt {3}}{2}}\end{bmatrix}}{\begin{bmatrix}i_{a}(t)\\i_{b}(t)\\i_{c}(t)\end{bmatrix}}\\&={\begin{bmatrix}1&0\\{\frac {1}{\sqrt {3}}}&{\frac {2}{\sqrt {3}}}\end{bmatrix}}{\begin{bmatrix}i_{a}(t)\\i_{b}(t)\end{bmatrix}}\end{aligned}}}
是只考慮前二個方程的克拉克變換，其逆變換如下
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[
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{\displaystyle i_{abc}(t)={\frac {3}{2}}{\begin{bmatrix}{\frac {2}{3}}&0\\-{\frac {1}{3}}&{\frac {\sqrt {3}}{3}}\\-{\frac {1}{3}}&-{\frac {\sqrt {3}}{3}}\end{bmatrix}}{\begin{bmatrix}i_{\alpha }(t)\\i_{\beta }(t)\end{bmatrix}}}
克拉克變換可以視為是將三個相量 （電壓或電流）投影到二個固定的座標軸（alpha軸和beta軸）上。若三相平衡的話，所有資訊都可以保留，因為方程
I
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0
{\displaystyle I_{a}+I_{b}+I_{c}=0}
I
γ
{\displaystyle I_{\gamma }}
I
γ
{\displaystyle I_{\gamma }}
I
γ
{\displaystyle I_{\gamma }}
另一種理解的方式是方程
I
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+
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+
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=
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{\displaystyle I_{a}+I_{b}+I_{c}=0}
I
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+
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+
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=
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{\displaystyle I_{a}+I_{b}+I_{c}=0}
[[Image:AlphaBeta geometric interpretation.gif|center|frame|上圖是
α
β
γ
{\displaystyle \alpha \beta \gamma }
δ
{\displaystyle \delta }
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
α
{\displaystyle \alpha }
I
α
β
γ
{\displaystyle I_{\alpha \beta \gamma }}
ω
{\displaystyle \omega }
γ
{\displaystyle \gamma }
^ O'Rourke, Colm J. A Geometric Interpretation of Reference Frames and Transformations: dq0, Clarke, and Park . IEEE Transactions on Energy Conversion. December 2019, 34, 4 (4): 2070–2083. Bibcode:2019ITEnC..34.2070O S2CID 203113468 doi:10.1109/TEC.2019.2941175 hdl:1721.1/123557 （英語） . ^ W. C. Duesterhoeft; Max W. Schulz; Edith Clarke. Determination of Instantaneous Currents and Voltages by Means of Alpha, Beta, and Zero Components. Transactions of the American Institute of Electrical Engineers. July 1951, 70 (2): 1248–1255. ISSN 0096-3860 S2CID 51636360 doi:10.1109/T-AIEE.1951.5060554 ^ S. CHATTOPADHYAY; M. MITRA; S. SENGUPTA. Area Based Approach for Three Phase Power Quality Assessment in Clarke Plane . Journal of Electrical Systems. 2008, 04 (1): 62 [2020-11-26 ] . ^ 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.
^ Clarke Transform . www.mathworks.com.
–透過MIT Open Access Articles （英語）.
