Milestones:The Discovery of the Principle of Self-Complementarity in Antennas and the Mushiake Relationship, 1948

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Tohoku University
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Prof. Yasuto Mushiake with milestone plaque


Discovery of the Principle of Self-Complementarity in Antennas and the Mushiake Relationship, 1948


In 1948, Prof. Yasuto Mushiake of Tohoku University discovered that antennas with self-complementary geometries are frequency independent, presenting a constant impedance, and often a constant radiation pattern over very wide frequency ranges. This principle is the basis for many very-wide-bandwidth antenna designs, with applications that include television reception, wireless broadband, radio astronomy, and cellular telephony.

Street address(es) and GPS coordinates of the Milestone Plaque Sites

38.253517, 140.873299 The location of the plaque will be Tohoku University Archives in the Katahira Campus. The address of the Katahira Campus is 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan and GPS coordinates of Tohoku University Archives are 38.253517, 140.873299

Details of the physical location of the plaque

The mounting will be on the wall of the Tohoku University Archives.

How the intended plaque site is protected/secured

The plaque will be protected with other valuable historical materials by the office of Tohoku University Archives. Public visitors can visit Tohoku University Archives during daytime hours on weekdays. Visitors have to show an ID card issued by the office in the building.

Historical significance of the work

The frequency bandwidth of an antenna is critical to its useful operation. Prior to 1948, almost all antennas were designed based on resonant structures. As such, they operated over relatively narrow bandwidths, often over a range of less than 2:1 in frequency. The pattern, gain, and impedance of the antenna would vary over the operating frequency range. In fact, the operating bandwidth was typically defined in terms of the maximum allowable variation in one of these parameters over the frequency range.

In 1948, Prof. Yasuto Mushiake discovered the Principle of Self-Complementarity in antennas. A self-complementary antenna has a geometry such that its complement (where air is replaced by metal and metal replaced by air) can exactly overlay the original structure through translation and/or rotation. Figures 1-3 show truncated examples of such antennas (in the ideal case, the structures extend to infinity). The Principle of Self-Complementarity states that such self-complementary antennas have a constant impedance independent of frequency. Furthermore, Prof. Mushiake also derived what is referred to as the Mushiake Relationship, which is an expression for the impedance of such an antenna. In the case for such an antennas with two terminals, the Mushiake Relationship takes the form given in Equation (1) of Figure 4, where Z is the impedance of the antenna and Z0 is the impedance of the surrounding medium. If the surrounding medium is free space, then the Mushiake Relationship gives a value of 188.4 ohms as the impedance of the antenna.


Part of the Principle of Self-Complementarity is that not only the impedance, but in cases where the current decreases rapidly as the distance from the driving point increases, the radiation pattern and gain of such antennas remains constant with frequency. If the structure is truncated so as to be finite (as must obviously be the case in all practical antennas), the upper and lower bounds of the operating frequency bandwidth of the antenna over which these parameters remain essentially constant are respectively determined by the dimensions of the feed region and the overall size of the antenna.

The Principle of Self-Complementarity was originally derived for two-terminal antennas. Prof. Mushiake soon showed that it also applied to multi-terminal antennas, such as that shown in Figure 5. The general Mushiake Relationship for an n-terminal self-complementary antenna excited by n-phase electric sources in a star configuration with mth-order rotation is given by Equation (2) in Figure 4. For the case shown in Figure 5, where n = 4 and the excitations are in phase, the antenna’s impedance is approximately given by the value in Equation (3) of Figure 4, again independently of frequency.


In subsequent years, the Principle of Self-Complementarity was extended from two-dimensional to three-dimensional antennas (e.g., see Figure 6), and the Mushiake Relationship was similarly extended to such cases.


The historical significance of this principle and the associated relationship was huge. From the standpoint of antenna design, for the first time there was a whole category of antenna designs that were inherently wideband in nature, and the physics underlying this property was well understood. Furthermore, the most important practical design parameter – the antenna impedance – could be accurately predicted based on simple geometrical knowledge of the antenna. This principle led directly to the invention of the class of antennas called log-periodic antennas, which had high gains and relatively constant radiation patterns and impedances over bandwidths of 10:1 and more. From a societal standpoint, the timing of the discovery of the Principle of Self-Complementarity and the resultant development of log-periodic antennas was tremendously important. This came at the dawn of the development of television broadcasting. The log-periodic antenna was one of the most widely used antennas for home TV reception from the 1950s through widespread use of the delivery of television by cable.

The Principle of Self-Complementarity is finding many new uses in modern antenna design. Ultra-wideband antennas have become tremendously important in several areas of modern telecommunications, particularly as a part of efforts to optimize spectrum usage by spreading signals over very wide ranges of frequencies with very low energy at any one frequency. The 5G wireless communication standards call for ultra-wideband MIMO (multiple-input, multiple-output) antennas, often with omnidirectional patterns. Such antennas are designed using the Principle of Self-Complementarity.

Features that set this work apart from similar achievements

This work was the first discovery of the frequency-independent properties of self-complementary antennas. Rumsey and Deschamps used this work to develop the class of frequency-independent antennas known as log-periodic antennas. It was subsequently shown that it is the self-complementary aspect of the log-periodic antenna that is primarily responsible for its frequency-independent properties. Two major aspects of the significance of this work and that set it apart are the simplicity and universality of the Principle of Self-Complementarity and the Mushiake Relationship. It is easy to understand and simple to implement. Once a shape has been drawn in two or three dimensions, forming its complement to produce a self-complementary antenna is a simple, straightforward geometric construction. Given the self-complementary shape, the Mushiake Relationship immediately permits the impedance of the antenna to be calculated. Furthermore, because of the Principle of Self-Complementarity, the impedance, pattern, and gain of the antenna are independent of frequency, except for truncation effects.

The significance of this work can also be measured by the honors that have been given to Prof. Mushiake directly in recognition of the importance of the development of the Principle of Self-Complementarity and the Mushiake Relationship. In chronological order, Prof. Mushiake was elected an IEEE Fellow “For contributions to linear antennas and self-complementary antennas” (January 1976). This work was recognized with the Medal of Honor of the IEICE (Japan) in 1982. He received a Commendation for his distinguished achievements from the Minister of Science and Technology, Japan, in 1982. The work received the Medal of Honor with Purple Ribbon from the Emperor of Japan in 1985. (The Medal of Honor with Purple Ribbon is awarded by the Emperor of Japan for academic or artistic developments or accomplishments. As just one example, it is given to those who win medals in the Olympic games.) He received the Second Order of Merit with the Sacred Treasure from the Emperor of Japan in 1991. (The Order of Merit with the Sacred Treasure is awarded in eight classes, with the first being the highest. It is given by the Emperor of Japan for exceptionally distinguished civilian or military service.) His work was recognized with a Commendation from the Minister of Posts and Telecommunication, Japan in 1992, and he was made an Honorary Member of the IEICE, Japan.

Significant references

The following references have been uploaded as PDFs. The reference number is in square brackets at the front of the file name. If the original reference was in Japanese, the Japanese version has “J” appended to the end of the file name, and the English translation, provided by Prof. Sawaya of Tohoku University, has “E” appended to the end of the file name.

1. Y. Mushiake, “The Input Impedance of a Slit Antenna,” Joint Convention Record of Tohoku Sections of IEE and IECE of Japan, June 1948, pp. 25-26.

2. Y. Mushiake, “The Input Impedances of Slit Antennas,” J. IEE Japan, 69, 3, March 1949, pp. 87-88.

3. S. Uda and Y. Mushiake, “The Input Impedances of Slit Antennas,” Technical Report of Tohoku University, 14, 1, September 1949, pp. 46-59.

4. Y. Mushiake, “Multi-Terminal Constant Impedance Antenna,” 1959 National Convention Record of IECE of Japan, October 1959, p. 89.

5. V. H. Rumsey, “Frequency Independent Antennas,” 1957 IRE National Convention Record, Pt. 1, March 1957, pp. 114-118.

6. G. A. Deschamps, “Impedance Properties of Complementary Multiterminal Planar Structures,” IRE Transactions on Antennas and Propagation, AP-7, (special supplement), December 1959, pp. S371-S378.

7. Y. Mushiake and H. Saito, “Three-Dimensional Self-Complementary Antenna,” Joint Convention Record of Four Japanese Institutes Related to Electrical Engineering, Pt. 15, No. 1212, April 1963.

8. Y. Mushiake, “Constant-Impedance Antennas,” J. IECE Japan, 48, 4, April 1965, pp. 580-584.

9. Y. Mushiake, “A Report on Japanese Development of Antennas: From the Yagi-Uda Antenna to Self-Complementary Antennas,” IEEE Antennas and Propagation Magazine, 46, 4, August 2004, pp. 47-60.

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