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The liquid starter is one that uses a washing soda solution and water in a tank. From what I've read, the dippers are attached to a shaft controlled by a small motor to lower the dippers into the solution and thus vary the resistance. I'd like to know what these dippers do and how they affect the change in resistance of the rotor circuit. Also, where in the system are the electrodes and what role do they play?

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Sounds like you're talking about a liquid rheostat, also called an electrolytic starter in the motor starter context.

Ancient technology, once used for stage lighting dimmers and such like. The resistance varies with the electrode area immersed, either by moving the electrode(s) or the electrolyte level (eg. via displacement), as well as the resistivity of the liquid and the spacing of the electrodes.

R = \$\rho L\over A\$ will apply if the configuration has a constant cross section, otherwise the calculations will be more complex.

Here is a video showing the internals of such a device, and another showing a crude hand-operating exciting device.

Frankly I'm not 100% sure which part is considered the dipper. I suspect it's the moving electrodes. That term does not seem to be used in US based patents.

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The "dippers" are indeed the electrodes and represent the resistance of the circuit depending on the amount of the electrode surface area that is immersed in the electrolyte (the "washing fluid" as you call it). Immersing them shorts them to ground through the electrolyte and the grounded conductive tank, relative to the surface area in the electrolyte, so the more they are lowered into the liquid, the less resistance there is in the rotor circuit and the more torque the motor will produce and that can indirectly change the speed on a given load with a wound rotor induction motor.

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