Clarke Transform Clarification

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In summary, the standard Clarke Transform is a method for converting three-phase currents (I_a, I_b, I_c) into two-axis currents (I_alpha, I_beta). It involves resolving the currents along the x-axis and y-axis, but there is a missing factor of 2/3 that needs to be included in order to accurately compare the derived equations with the original ones. The Clarke Transformation is also known as the Power Invariant Transformation, which means that the transformation preserves the power of the system.
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The Clarke transform equation does not match with my calculations
The standard Clarke Transform is
##
i_{alpha} = i_a; -> 1
i_{beta} = \frac {(i_a + 2i_b)} {\sqrt3} ->2
##
1605953917984.png


I am trying to derive it, but missing a factor. Basically converting the 3 phase currents ## I_a, I_b, I_c ## into the 2 axis ##I_{\alpha}, I_{\beta} ##
resolving along the x-axis
##I_{\alpha} = I_a - I_b \sin(30) - I_c\sin(30) = \frac {3I_a} 2; ## since ##I_a+I_b+I_c=0## -->3
resolving along y-axis
##
I_{\beta} = I_b\cos30 - I_c\cos30; = \frac{\sqrt3(I_a+ 2I_b)} 2 --> 4
##
Now the derived 3 and 4 if i compare with original equations 1, 2, there is a factor ##\frac 2 3## is missing. What is this factor? Why should i include it?
 
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Clarke Transformation i found the explanation in the wiki, the Power invariant transformation. What does it mean?
 

1. What is Clarke Transform Clarification?

Clarke Transform Clarification is a mathematical technique used to convert a three-phase system into two orthogonal components, known as the alpha and beta components. This transformation simplifies the analysis of three-phase systems and is commonly used in power systems and motor control applications.

2. How does Clarke Transform Clarification work?

The Clarke Transform takes three-phase quantities, such as voltage or current, and transforms them into two components: the alpha component, which represents the average value, and the beta component, which represents the difference between the three phases. This transformation is achieved through a series of mathematical equations.

3. What are the benefits of using Clarke Transform Clarification?

Clarke Transform Clarification allows for easier analysis and control of three-phase systems. It simplifies the equations and calculations involved in analyzing and controlling these systems, making them more efficient and accurate. It also makes it easier to detect and correct imbalances in the three-phase system.

4. What are some applications of Clarke Transform Clarification?

Clarke Transform Clarification is commonly used in power systems, motor control, and other three-phase systems. It is also used in renewable energy systems, such as wind and solar power, as well as in electric vehicles and industrial automation.

5. Are there any limitations to Clarke Transform Clarification?

While Clarke Transform Clarification is a useful tool for simplifying the analysis of three-phase systems, it does have some limitations. It assumes a balanced three-phase system and may not accurately represent unbalanced or distorted systems. It also requires a good understanding of three-phase systems and mathematical calculations to properly implement.

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