B. Formulas for the solenoid force and current

Measuring the force exerted by the solenoid

To be able to accurately control the solenoid’s pulling force on a payload attached to its plunger, it is desired to have a deterministic equation that relates the following 3 quantities:

The mechanical force exerted by the solenoid on the payload,
the current in the wire that is wound around the solenoid, and
the distance between the solenoid and the payload.

Such an equation would allow the computer to easily control the force exerted on the plant by correctly calculating the current to supply to the solenoids. Because of the nonideal behavior of the solenoids, it was decided that the best way to formulate these expressions was to measure the force the solenoid exerts on an object at a range of distances within a range of currents.

Figure III.B.1: Apparatus used to measure the mechanical force exerted by the solenoid. A thin beam load cell is a strain gauge that functions as a transducer. It can measure the normal force acting on it as an electrical signal. A linear force pressing on the beam would deform it, which changes the electrical resistance of the cell. This image was taken from Alex Rokhvarg’s original thesis.

When the apparatus was first constructed in 1994, the setup in Figure III.B.1 was built to measure the solenoid force. It consisted of a thin beam load cell attached to a load and to the solenoid. A thin beam load cell is a strain gauge that functions as a transducer. It can measure the normal force acting on it as an electrical signal. A linear force pressing on the beam would deform it, changing the electrical resistance of the cell.

The solenoids are model D-Frame Intermittent Duty Part No. 53719-87. They were manufactured by Deltrol Controls. This apparatus measured the force the solenoid exerted on the load at nine displacement settings (0.1”, 0.15”, 0.2” , 0.3” , 0.4” , 0.5” , 0.6” , 0.7” , 0.8”) and at fifteen quantities of current (0.2 A to 3 A in 0.2 A increments).

Deriving the Solenoid Force Equations

The collected data was processed with Matlab using the least squares error methodology. The data was used to derive these expressions for the solenoid current and force. The controller calculates this current i (Amperes) to energize a solenoid with to have it exert a desired force f (Newtons) on an attached payload that is d (Meters) distance from it.

When the solenoid is energized with a given current i, the magnetic force f that it exerts on a payload that is d distance from it is

Functions G(f,d) and F(i,d) are inverses of one another. For more details on the collection of this data and the derivation of these equations, please see Alex Rokhvarg’s original thesis. [2]

[2] A. Rokhvarg, "Single-axis agonistic motion control system with electro-magnetic actuators," New Jersey Institute of Technology, Newark, NJ, 1995.

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