### The History of the DQ Transformation

Park’s transformation can be considered to be the single theoretical contribution that triggered the development of advanced design, control, and analysis of electrical machines (motors and generators). To most practicing engineers and researchers in this field, the story goes like this:

After Nikola Tesla invented the AC machine in the 1880s, it took the electrical engineers over 3 decades of struggling with AC circuits analysis before Robert H. Park (1902–1994) published his seminal paper, in 1929, “*Two Reaction Theory of Synchronous Machines*”. In that paper, the brilliant young engineer solved the problem mathemagically by introducing the *dq0*-transformation that has been called after him, *the Park’s transformation, *which transforms the natural 3-phase AC reference frame into a fictitious 2-circuit rotating reference frame.

The Park’s transformation is a brilliant idea indeed, except that it was not invented by Park… Have you ever heard of André-Eugène Blondel (1863–1938)? Here is the complete story.

### Park’s Transformation in the Literature of AC Machines

The above story is told repeatedly in the literature of AC machines, like a Greek myth. Take for example Thomas A. Lipo, an esteemed author in the field of AC machines, in his book “Analysis of Synchronous Machines”, 2nd Edition, 2012, Chapter 2, page 77, he writes:

“

Unfortunately, the analysis of AC machines becomes involved at an early stage due to the nature of the numerous coupled magnetic circuits. Historically, this difficulty was overcome only in 1929 by R. H. Park, who formulated equations of transformation (Park’s transformation) from actual stator currents and voltages to different, equivalent currents and voltages.”

The author mentions other contributors that extended Park’s work, like Stanley, Kron, and so on, but with no mention of earlier contributors like Blondel, Behn-Eschenburg, Potier and Doherty the pioneers of the analysis of synchronous machines (the title of Lipo’s book!).

Another esteemed author in the field, Bimal Bose, in his book “Modern Power Electronics and AC Drives”, 2002, chapter 2, page 56, when talking about the time-varying inductances in salient-pole machines, wrote:

“… Although it is somewhat simple, the problem of time-varying parameters still remains. R. H. Park, in the 1920s, proposed a new theory of electric machine analysis to solve this problem. He formulated a change of variables which, in effect, replaced the variables (voltage, current, and flux linkages) associated with the stator windings of a synchronous machine with variables associated with fictitious windings rotating with the rotor at synchronous speed. […] With such a transformation (called Park’s transformation), he showed that all the time-varying inductances that occur due to an electric circuit in relative motion and electric circuits with varying magnetic reluctances can be eliminated…”

The author talks about the problem of *time-varying inductances, *which is the principal contribution of Blondel with his two-reaction method, as you will discover in the sequel. Park never claimed this contribution to himself.

Many other textbooks tell the same story and call the dq-transformation Park’s (Krause, Novotny, Leonhard, Boldea, etc.). Surprisingly, the same applies to the French literature (Philippe Barret, J-P. Louis, Pierre Mayé, etc.), where you find the name of Blondel associated only with the Blondel diagram, referred to as “the phasor diagram” in the English literature.

Very few references mention Blondel when talking about the dq-transformation. The only *recent *book I found is “Electric Machinery” by Fitzgerald et al., 6th edition, 2003, Appendix C, page 657, where the authors state that:

“The idea behind the [dq] transformation in an old one, stemming from the work of André Blondel in France, and the technique is sometimes referred to as the Blondel two-reaction method…”.

You may also find some scientific papers calling the dq-transformation “the Blondel-Park transformation”… Good luck to find them.

Let’s dive into the contributions of Both Blondel and Park to better understand them. And let’s try to understand why it is called Park’s, instead of Blondel’s, transformation.

### The contribution of Blondel

To understand the contribution of Blondel, Let’s start with the famous 1929 paper by Park.

#### Park’s 1929 Paper

When you read the *seminal* paper on the “two-reaction theory”, you’ll find that it starts with: *“This paper presents a generalization and extension of the work of Blondel, Dreyfus, and Doherty and Nickle…” *In the bibliography section, Park cites the work of Dreyfus, and Doherty and Nickle, but not of Blondel. The papers by Dreyfus are in German and start from 1911. Unfortunately, I could not find any reference that helps me in my search. So let’s check Doherty and Nickle.

#### Doherty and Nickle’s 1926 Paper

Doherty and Nickle also explicitly stated that their contribution extended Blondel’s two-reaction theory that resolves the m.m.f (magnetomotive force) along two axes direct and quadrature. The work of Blondel included the spatial harmonics of the m.m.f., which Park neglected in his paper to simplify the equations. The authors cite Blondel’s book “Synchronous Motors and Converters”, translated by C.D. Mailloux in 1913. So let’s check it.

#### Back to the Source: Blondel’s 1913 Book

Blondel’s book “Synchronous motors and converters” was translated from French and published in 1913, that is 16 years before Park’s paper. Here are some extracts from that book.

In the book, Blondel clearly states the essence of the dq-transformation; to exploit the similarities between polyphase (alternating-current) machines and direct-current machines. The author mentions that he had published his original work in 1899 (Park was not born yet).

The irony is: in 1902, when Park was born in Strasbourg, today in Eastern France, the quadragenarian Blondel was an accomplished professor and engineer teaching electricity in the “Ecole Nationale des Ponts et Chaussées” in France. In his course on synchronous machines, Blondel explains about “direct flux” and “transversal flux”.

### The contribution of Park

The goal of this writing is not to undermine the contribution of Park. Considering his publications, and according to his contemporaries, R.H. Park made substantial contributions to the field of AC machines, and his work was marked by a high degree of analytical rigor, physical insight, and pedagogy.

Park never claimed that he invented the dq-transformation. To his colleagues, like Clarke, Concordia, and Kron, his contribution was clear: using Blondel’s two-reaction theory, Park took some assumptions to simplify the equations (neglecting saturation and harmonics), and stated the differential equation of the *fundamental *harmonic of (ideal) synchronous machines in the dq reference frame. Yet, who said that his contemporaries were even aware of the work of Blondel? Here are the details.

#### According to Gabriel Kron (1901–1968)

In his discussion on Park’s paper, Kron articulates the contribution of Park as follows:

#### According to Edith Clarke (1893–1959)

Edith Clarke, in her book “Circuit Analysis of A-C Power System: Vol II”, mentions “Park’s equations” when referring to the differential equations of an *ideal synchronous machine* in the dq reference frame, but did not attribute the transformation to Park.

#### According to Charles Concordia (1908–2003)

In the biography of R. H. Park published in Memorial Tributes, Vol 8, 1996, Charles Concordia wrote:

Before Park’s work, several papers had been written on electric generator equations. However, they were so complex as to be of little practical use. David M. Jones, for whom Park then worked at General Electric, recognized this and also recognized that Park was the person who could bring order out of chaos. So he assigned the job to Park, with world-shaking results.

### Why Park not Blondel?

Back to the original question of this post; why is it called Park’s transformation? Here are some attempts to answer the question.

**The language**: The work of Blondel was widely known in France, but little known among English-speaking engineers and researchers. This is how the translator of Blondel’s book articulates it.

The same argument was made by Vladimir Karapetoff in his discussion on Doherty and Nickel’s 1918 paper:

**The medium of publication:**R. H. Park, like Clarke, Concordia, and Fortescue, published his work in the transactions of the AIEE (the predecessor of the IEEE), which made his work accessible to a wider audience.**The ecosystem of General Electric**in the first half of the 20th century: Park, Clarke, Concordia, Kron, and others were with the General Electric (GE) company when they published their work. At the time, GE had an army of talented engineers and researchers who contributed significantly to the theory and practice of synchronous machines. They were able to present complex theories in a comprehensible way, accessible to a wider audience. This may give the impression that they actually invented the entire theory of synchronous machines.

#### How does it make sense?

If you’re not convinced yet, think about it! Synchronous generators were invented in the 19th century when Blondel was a young talented engineer. With these machines, the rotating dq-reference frame is more intuitive than with other alternating-current motors, since the causality chain starts from the excitation (rotor) field that generates an electromotive force at the stator terminal. The similarity with the direct-current machines is easier to understand. In such a framework, the decomposition of stator flux into direct and quadrature flux components would not escape the mind of a smart engineer living in the golden age of electricity, just before the first world war. Why wait until baby Robert Park turns 27 and the beginning of the great depression?

As a bonus, here is a paper published in 1919, in France, that shows the use of the dq-transformation with a reference to Blondel. In the paper, the d-axis is called the p-axis.

### References

- Blondel, André.
*Synchronous motors and converters: theory and methods of calculation and testing*. McGraw-Hill Book Company, 1913. - Doherty, R. E., and C. A. Nickle. “Synchronous machines I-an extension of Blondel’s two-reaction theory.”
*Transactions of the American Institute of Electrical Engineers*45 (1926): 912–947. - Park, Robert H. “Two-reaction theory of synchronous machines generalized method of analysis-part I.”
*Transactions of the American Institute of Electrical Engineers*48.3 (1929): 716–727. - Concordia, Charles.
*Synchronous machines: theory and performance*. Wiley, 1951. - Clarke, Edith.
*Circuit analysis of AC power systems; symmetrical and related components*. Vol. 1. Wiley, 1943. - Clarke, Edith.
*Circuit analysis of AC power systems; symmetrical and related components*. Vol. 2. Wiley, 1950. - Fitzgerald, Arthur Eugene, et al.
*Electric machinery*. New York: McGraw-Hill, 2003. - Lipo, Thomas A.
*Analysis of synchronous machines*. CRC Press, 2017. - Bose, Bimal K.
*Modern Power Electronics and AC Drives.*Prentice-Hall, 2001. - Brylinskin E.
*Sur la réaction d’induit des alternateurs*, Revue Générale de l’Electricité, 23 Aout 1919.