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Clarke and Park transforms
The Clarke and Park transforms are fundamental mathematical tools in the field of electrical and electronic engineering, used to simplify the analysis and control of three-phase systems. The Clarke transform converts a three-phase signal (a, b, c) into a two-phase system (α, β), facilitating the study of symmetrical and unbalanced components. The Park transform, on the other hand, brings these components into a rotating reference frame (d, q). This transformation is especially useful in the control of electrical machines and power electronic converters, since it allows precise regulation of electrical quantities.
Reference frame theory uses mathematical transformations to simplify the modeling, analysis and simulation of balanced three-phase circuits, power converters and electrical machines. Three main types of reference frames are widely used in electrical engineering. These reference frames are classified according to the speed of the reference frame and the nature of the variables involved.
Natural reference frame abc
The reference frame has an electric angular velocity (ω) equal to zero and is referred to as a three-phase stationary frame. The three-phase variables exhibit a time-varying nature (AC) with a phase shift of 120 degrees. This frame represents the actual correlation between the machine or power converter construction and the corresponding mathematical model.
The generic balanced three-phase variables are considered as follows:
Where 𝐹 is the peak amplitude (V, A or Wb), 𝜔 is the angular frequency (rad/s), θ0 is the initial phase and 𝑡 is the time.
The three-phase function is represented by a phasor in the equivalent natural reference frame space, as shown below:
Where a is the Fortescue operator:
Stationary reference frame αβ: Clarke transform
The Clarke transform is a mathematical tool used in electrical engineering, and in particular for vector control, to model a three-phase system using a two-phase model. It was proposed by Edith Clarke, one of the first female pioneers in the field of electrical engineering. She invented a graphing calculator that facilitated the solving of electrical equations. She was the first woman to present a scientific paper at the annual meeting of the American Institute of Electrical Engineers (AIEE) in 1926. Furthermore, she was also the first woman to earn a Master of Science degree in electrical engineering from MIT and the first professor of electrical engineering at the University of Texas at Austin.
The Clarke transform is also known as the αβ transform, and it is precisely these complex coordinates (α as the real component and β as the imaginary component of the two-phase system) onto which we project our three-phase system (a, b, c) of phase voltages or currents.
The phasor in the natural reference frame space, 𝑓𝑎𝑏𝑐(𝑡), is decomposed into real and imaginary components to form a phasor in the stationary reference frame space.
The angle between the α and β component is calculated as follows:
Clarke’s transformation matrix is applied as follows:
Synchronous reference frame dq: Park transform
The Park transform, also known as the dq-axis transform, is a mathematical tool used in electrical machine theory to simplify the analysis and control of alternating current (AC) machines, such as synchronous or induction motors and generators. Robert H. Park is remembered for developing the “Park equations”, presented in 1929, which greatly simplified the calculation of the dynamic performance of alternating current generators and motors, and which have been fundamental in the engineering of electric power systems. His work allowed a practical and efficient analysis of the stability and reliability of electrical systems in the face of disturbances. In addition, during World War II, he developed magnetic mines for the U.S. Navy, earning multiple patents. Throughout his life, Park was a prolific innovator, with 64 patents, and his contribution remains key in the field of electrical engineering. Park submitted his paper to the AIEE entitled “Two reactions theory of synchronous machines” in which he presented a generalization and extension of the work of Blondel, Dreyfus and Doherthy and Nickle. This paper established general methods for calculating current, power, and torque in salient-pole and smooth-break synchronous machines under both steady-state and transient conditions.
Park’s transformation converts the three-phase current and voltage variables (which vary sinusoidally in time) to a rotating reference coordinate system (direct “d” and quadrature “q” axes), where these variables become constant or of lower frequency. This facilitates the analysis of electrical machines and the design of controllers.
The phasor in natural reference frame space, 𝑓𝑎𝑏𝑐 (𝑡), is decomposed into real and imaginary components to form a phasor in synchronous reference frame space.
When the a-axis is aligned to the q-axis the equation is:
When the a-axis is aligned to the d-axis the equation is:
Where θ (In both cases, θ = ωt) is the angle between the a and q axes for the q-axis alignment or the angle between the a and d axes for the d-axis alignment, ω is the rotational speed of the dq reference frame, t is the time, from the initial alignment.
Example
A three-phase RMS voltage signal of 220 V and 60 Hz is shown for 0.45 seconds at reference frames abc, αβ0 and dq0 under three conditions. Between instants t=0 and t=0.15, the system is three-phase balanced. Between instants t=0.15 and t=0.3, phase “a” undergoes a 60% decrease in the RMS value of the voltage. Finally, between instants t=0.30 and t=0.45, harmonics of the 5th and 7th positive sequences are introduced, with an amplitude of 20% and 15% of the RMS voltage value, respectively.
Applications
Clarke and Park transformations were originally developed to facilitate the analysis and modeling of electrical machines. Today, dq0-based models are used in a wide range of applications, such as electrical control and drive, modeling of multiple machines and power converters, electric vehicles, microgrid simulation, phase-locked loops (PLLs) and active power filters. In addition, these transformations are essential in the field of renewable energy. In numerous instances, dq components are no longer limited to the direct and quadrature axes of an electrical machine, suggesting a broader interpretation of reference frames, decoupled from specific technologies or applications.
Conclusion
The Clarke and Park transforms are essential mathematical tools for the analysis and control of three-phase electrical systems, and they simplify the study of electrical signals. The Clarke transform converts three-phase signals into a two-phase system, allowing for the identification of symmetrical and unbalanced components. The Park transform, however, brings these components into a rotating reference frame, which is useful for precise control of electrical machines and power converters. These transformations are applied to various areas, such as motor control, converters, microgrids, and renewable energy, allowing for a more general interpretation of reference frames for complex electrical analysis.
Reference
[1] O’Rourke, C. J., Qasim, M. M., Overlin, M. R., & Kirtley, J. L. (2019). A geometric interpretation of reference frames and transformations: Dq0, Clarke, and park. IEEE transactions on energy conversion, 34(4), 2070–2083. https://doi.org/10.1109/tec.2019.2941175
