The electrical diagram on the right used in every electrical industry involving industrial process automation control technology shows a similar power circuit identical to the 3 phase DOL motor controller power circuit except for the distinguishable difference on the wiring connection of the circuit specifically intended to accommodate a 1 phase motor instead of a 3 phase motor.
In the absence of single (1) phase motor controller devices, an available three (3) phase magnetic contactor and thermal overload relay can be used as well in a 1 phase circuit. You will notice from the presented diagram that the wiring configuration to the terminal of the three pole main contactor and thermal overload relay utilized all three terminals despite the common knowledge of a two line electricity flow path inherent with single phase circuits. This type of connection is done to achieve a complete current flow measurement to maintain a balance load current detection from the 1 phase motor concentrated on the three poles of the thermal overload relay.Why? Because leaving out one terminal open for a 1 phase application would result in an imbalance distribution of load current detection on the 3 terminals of a 3 pole thermal overload relay. Since a three pole thermal overload relay is specifically designed to detect an equal three phase load current, then the application of an unequal load concentrated only on two terminals with one terminal open would eventually render an inaccurate current measurement resulting in an intermittent unnecessary tripping OFF of the thermal overload relay.