Clarke and Park Transformation on Vector Motors

Clarke and Park Transformation are "simply" matrix of transformation to convert a system from one base to another one:
- Clarke transform a three phase system into a two phase system in a stationary frame.
- Then Park transforms a two phase system from a stationary frame to a rotating frame.

The Beauty of these transformations is after doing it, it simplifies a lot the math and calculation power to control the three phase motor (torque control, speed control, field weakening).
The transformations afford better mathematical tractability and provide more intuitive insight into the operation of rotating electrical machines. Clarke's transformation reduces a multi-phase system of an arbitrary number of phases to an equivalent two-phase system (in quadrature) plus a zero-sequence component that, in the balanced case remains zero and may be neglected. Currents, voltages and fluxes may be described then in a two-axis orthogonal coordinate system as vectors (such as in an X-Y plane) whose lengths describe the magnitude of that quantity. The vectors can be viewed as rotating at fundamental frequency, but may have relative phase displacement among them. Park's transformation may most easily be described as a change in perspective: rather than describing the aforementioned vectors as rotating in a fixed coordinate system, such as one that may be viewed as fixed to the stator (e.g. of a synchronous machine) where the electrical quantities are time-varying even in the steady-state, we can instead rotate the coordinate axes at synchronous or slip speed such that the coordinate axes are instead affixed to the rotor. In this way, the vectors are seen as stationary in the steady-state with some arbitrary phase displacement among them. This is the intent of the Park transformation.
Under certain conditions (such as balanced voltages or currents, and no harmonics) you can get away with describing a three-phase system using only two pieces of information. You can us your knowledge of two phases to infer the third, provided the power system is clean and balanced. This is similar to when you have a power meter, and you only need two CT's or VT's to infer the behavior of the third phase.

It then follows that you can choose which two vectors to describe the three phase behaviour, which can be represented by a Y that rotates 360 degrees for every cycle of mains power (each line is a phase). The two axis representation will look like an x-y axis superimposed on this rotating Y in the chart. One are to have an x-y axis where the x is stuck to phase A (Park) or another one is to have all three phases move around while the x-y axis remains stationary (Clarke), as described above.
Valuable responses have been provided but the word DC has not been mentioned, which provides opportunity to present a historical slant especially on the Park transformation which is arguably by far the single most important concept needed for an understanding of high-performance vector-controlled variable frequency drives. The Park transformation was first conceptualized in a 1929 paper authored by Robert Park. Park's paper was recently ranked 2nd most important paper ever written in 20th century in terms of power engineering impact. The novelty of Park's work was in his ability to transform any related machine's linear differential equation set from one with time varying coefficients to another set with time invariant coefficients. Most importantly, through the Park transformation, a vector-controlled variable speed drive allows a three-phase induction motor to behavior like a separated-excited DC motor with torque control decoupled from flux control.