## Opinion

### Science

#### Hoashi & Millman's Theorem

— Common knowledge for those who have studied electrical engineering

Yoshimichi Ohki

Professor, Faculty of Science and Engineering, Waseda University

The Institute of Electrical Engineers of Japan (IEEJ), one of Japan’s oldest science academies, honors important products, inventions, and other Japanese electrical technologies every year with its “One Step on Electro-Technology” awards. This year in 2015, Takeji Hoashi was honored for his discovery of a combination theorem of electrical circuits known as Hoashi & Millman's Theorem. Four other technological achievements received awards at the Eighth Awards Ceremony on March 15. A magazine published by the world's largest academy, the Institute of Electrical and Electronics Engineers (IEEE) in USA, covered the IEEJ awards and featured then Waseda University Assistant Professor Takeji Hoashi’s discovery of the Hoashi & Millman's Theorem in 1927.

Figure 1 Takeji Hoashi

Hoashi & Millman's Theorem is an important theory that every electrical engineering student has to learn. Although a young Assistant Professor (later Professor, and Professor Emeritus) of Waseda University discovered this theorem, it is named after a foreign scholar who discovered it long after Hoashi and introduced it to the world as Millman's Theorem. Even in Japan, many textbooks refer to it as “Millman's Theorem” even though it should be “Hoashi's Theorem.” I wanted to address this situation and asked the IEEJ to recognize the historical significance of this issue, which was then featured in the IEEE published journal.

Ohm, Faraday, and Henry studied the individual properties of resistors, inductors, and capacitors as circuit components. Meanwhile, the study of circuit networks, which are made of these components connected together, began when Kirchhoff discovered a law on connected components. Network theory, an important field in electrical engineering, has been thoroughly researched and is an essential technology even today in the 21st century. Its influence is evident in every electrical device ranging from small smartphones to large power supply facilities and electric power networks.

Before numerical analysis of electric currents across components and voltages in complex networks was possible, there was a much greater need for general theories on electric currents. Hoashi's Theorem is one of these general principles. This theorem has the following three characteristics:

- (i) It incorporates a circuit network consisting of arbitrarily connected arbitrary components.
- (ii) Calculation amounts are significantly decreased.
- (iii) The derived formula is clear and comprehensible.

In general, to obtain voltages across components and currents that flow through components, it is necessary to solve simultaneous equations. However, according to Hoashi's Theorem, a single formula can reveal the electric potential at any node.

Figure 2 An example of network that demonstrates the effect of Hoashi & Millman's Theorem

Look at the example in Figure 2. Assume you want potential *E _{A}* at node

*A*. Admittance

^{(1)}

*Y*and voltage source

_{AB}*E*are connected in series between nodes

_{AB}*A*and

*B*. Current

*I*that flows from node

_{AB}*B*to node

*A*is given by:

^{(2)}

Currents that flow from nodes *C*, *D*, ... to node *A* are also expressed in the same manner. Here, let us apply Kirchhoff's Current Law expressed by Formula ^{(2)} to node *A*.

This gives

The following can be obtained by solving this equation for *E _{A}*:

This is Hoashi's Theorem.

The denominator of Formula (4) adds up admittances connected to node *A*. The numerator adds up the products of each admittance and the voltage source connected to the admittance. Therefore, this formula is clear and comprehensible.

Although this formula is referred to as Millman's Theorem in the United States, as mentioned earlier, Millman published this theorem in 1940 and Hoashi's paper appeared thirteen years earlier in 1927 in Japan’s IEEJ's magazine and in an issue of the Waseda Electrotechnical Society^{(3)} (EWE) Journal. I hope many people will acknowledge this historical fact.

**^**(1) Admittance: the inverse of impedance; in high-school physics, impedance equals resistance, and admittance is the reverse of resistance.**^**(2) As mentioned in note (1), in high-school physics, this corresponds to Ohm's Law, Current = Voltage (or Difference in potential)/Resistance.**^**(3) The Waseda Electrotechnical Society was established in 1912 as an alumni association of departments related to electrical engineering at Waseda University. It functioned as an academy of electrical science and engineering and its journal (the EWE Journal) published electrical engineering reports.

#### Yoshimichi Ohki

Professor, Faculty of Science and Engineering, Waseda University

Profile:

Yoshimichi Ohki was born on December 21, 1950 and obtained a Doctoral degree in engineering after completing the Doctoral Course of electrical engineering at Waseda University. He joined the teaching staff of Waseda University as an Assistant and later worked as a full-time Lecturer and then Assistant Professor before becoming a Professor in April 1985. He currently works at the Department of Electrical Engineering and Bioscience at the Faculty of Science and Engineering, the Kagami Memorial Research Institute for Materials Science and Technology, and the Cooperative Major in Nuclear Energy. He is a Guest Professor at Shibaura Institute of Technology, and Professor Emeritus at Xi'an Jiaotong University.

Awards and honors: Minister of Education Award for Science and Technology, Waseda Research Award, IEEJ Fellow, IEEJ Outstanding Achievement Award, IEEJ Book of the Year Award, IEEJ Distinguished Paper Award, IEEE Fellow, IEEE Whitehead Memorial Award, and others

Academic posts: IEEJ ex-Vice President, ex-President of the Institute of Engineers on Electrical Discharges in Japan, Chairman of an IEEE Local Section

Around 400 Peer-reviewed papers