There is a widely spread error that it was introduced by Nicola Tesla. Nicola Tesla was mostly interested in 2-phase system that is usually referred as "poly-phase" system in the USA. He also created 2 phase induction motor. You can find a picture of that design from Internet. (I should mention, the first induction motor was invented and built by Ferraries from Italy, it was single-phase one and had slipping of 0,5!!! As result, efficiency was very low).
The 3-phase system was developed in the former "German Edison Company" (later AEG Company) by Mikhail Dolivo-Dobrovolsky (he was technical director of AEG until 1914) in 1889. I have read that he also, like Tesla, offered his idea of three phase system to Edisson and was rejected. Dolivo-Dobrovlski tried to find the optimal number of phase to provide the most cheap way to transmit power and to provide good rotating magnetic field in induction motor. He created an induction motor in the form like we know it now - with squirrel cage. (Later, Dolivo-Dobrvolsky found that squirrel cage suits only for relatively low power rates. For big powers, he proposed an induction motor with a wound rotor.) Three phase induction motor in contrary to 2-phase system, has very good starting torque. He realized that Ferraries' induction motor had too high slipping and made own with low one, providing good energy efficiency. (I should remark, that Dolivo-Dobrovlsky also created the first scientific methodology of electrical machines designing, he was able to find parameters of a machine before it was built. It was really new in that time.).
Besides that, Dolivo-Dobrovolsky discovered, comparing different systems, that the three phase system requires less cross-section of wire to transmit power. Imagine that we have 3 similar single-phase generators and 3 similar single-phase loads. Total cross-section area of wires will be 6A (3 direct and 3 back-going lines), where A is the cross-section area of a line. If we unify going back lines into one, there will be total current 3I, where I is the current of single direct line. If loads are the same, then providing shift in 120 degree between voltages we get the same shifts between equal currents. The sum of returning currents will be equal to zero and we theoretically have no need in going back lines!!! Thus, in the case of induction motor Dolivo-Dobrovlsky got AC power supply via cross-section area of 3A instead of 6A. Of, course later he added neutral wire, so the economical effect due to save in Cu material required for transmission system was 4/6 (even 3, 5/6, because in XX century it was common to use neutral wire with cross-section of 0.5A).
Those are 2 main reasons that made 3-phase system popular (cheap and reliable 3 phase induction motor requiring no mechanical contacts, and cheaper three-phase power supply system). Dolivo-Dobrovlsky also created the first 3-phase generator and 3-phase transformer. The whole system providing AC power supply through 170 km long transmission lines was presented at the Frankfurt mess in 1891.
Due to robust and cheap design, and good operational characteristics of the 3-phase induction motor, the industry got excellent source of mechanical power. We can say, that starting from this mess real large-scale electrification started because of huge interest towards induction motors. The fast-growing industry in 1890-s and 1900-s required huge number of these and three phase system became the most popular one. In early stages of electrification, a cheaper ac power supply through 3 phases was rather an additional bonus. Of course, if one had electricity for creation of mechanical power, he started use it also for lighting and other purposes. Power companies implemented the same three phase power supply systems for both industry and domestic sector.
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