EGR 325, ELECTROMECHANICS

TECHNICAL PAPER #1

by

David W. Johnson

FORCE IN A SINGLY EXCITED MAGNETIC FIELD SYSTEM

from

EXPERIMENT #4

FEBRUARY 22, 2000

**0.0 INTRODUCTION**

It is often necessary in today's computer controlled industrial setting to convert an electrical signal into a mechanical action. To accomplish this, the energy in the electrical signal must be converted to mechanical energy. A variety of devices exist that can convert electrical energy into mechanical energy using a magnetic field. One such device, often referred to as a reluctance machine, produces a translational force whenever the electrical signal is applied. There are several variations of the reluctance machine but all operate on the same basic electromechanical principles.

The principles of electromechanical energy conversion are investigated. The motivation for this investigation is to show how the governing equations of an electromechanical device can be derived from a magnetic circuit analysis. An expression for the mechanical force will be derived in terms of the magnetic system parameters. This expression is tested in the lab.

**1.0 ANALYSIS**

**1.1 MODEL**

The conversion of electrical energy to mechanical energy follows the
law of conservation of energy. In general, the law of conservation of energy
states that energy is neither created nor destroyed. Equation (1) describes
the process of electromechanical energy conversion for a differential time
interval dt, where dW_{e} is the change in electrical energy, dW_{m}
is the change in mechanical energy, and dW_{f} is the change in
magnetic field energy. Energy losses in the form of heat are neglected.

dW_{e} = dW_{m} + dW_{f} (1)

If the electrical energy is held constant, the dWe term is zero for Equation (1). The differential mechanical energy, in the form of work, is the force multiplied by the differential distance moved. The force due to the magnetic field energy is shown in Equation (2). The negative sign implies that the force is in a direction to decrease the reluctance by making the air gap smaller.

(2)

An expression for the energy stored in the magnetic field can be found
in terms of the magnetic system parameters. This expression is then substituted
into Equation (2) for W_{f} to get an expression for the force.
This derivation is shown in Appendix A. The result is Equation (3), in
terms of the current, i, the constant for the permeability of free space,
m_{0},
the cross-sectional area of the air gap, A_{g}, the number of turns,
N, and the air gap distance, x .

(3)

To verify this relationship in the lab, it is convenient to have an expression for the current necessary to hold some constant force. In a design, the dimensions and force are often known. So, the user of the reluctance machine needs to know how much current to supply. Rearranging terms in Equation (3) yields Equation (4).

(4)

**1.2 SAMPLE CALCULATIONS**

For the simple magnetic system of Figure 1, the current necessary to suspend the armature can be calculated using Equation (4).

**Figure 1. **Electromechanical system.

For an air gap length of 0.12 mm, an air gap cross sectional area of
1092 mm^{2}, and a 230 turn coil the current required to just suspend
the 12.5 newton armature is

(5)

**2.0 EXPERIMENT**

**2.1 Description of THE EXPERIMENT AND SETUP**

The model for the electromechanical energy conversion process was tested in the laboratory using an apparatus similar to Figure 1. For this setup the measured weight of the armature is the mechanical force. A shim was placed in between the armature and the core to give the system a known air gap length. Care was taken to align the armature so that the area of the gap was equal to the cross-sectional area of the armature. The core was then energized with enough current to firmly hold the armature in place. The entire assembly was turned upside down so that the weight of the armature was supported by the force of the magnetic field. The current was slowly decreased until the armature dropped. At this point the force of the magnetic field was exactly equal to the weight of the armature.

**2.2 PRESENTATION OF MEASURED DATA**

The experimental steps described above were repeated for several different
shim sizes. Figure 2 is a Mathcad plot of the measured currents with the
data points shown as a stem plot and the calculated values shown as a line.

**Figure 2. **Plot of Measured and Calculated Values for Current
versus Air Gap

**3.0 RESULTS AND COMPARISON**

The experimental results for the current necessary to hold the armature in place match the calculations. It is difficult to precisely control the current supplied to the coil in the lab. That is, the resolution of the current steps effect the accuracy of the measured current. Even slow, careful measurements cannot completely elimate this error.

**4.0 CONCLUSIONS**

From the results of the experiment and the derived model, a designer
can pick a current value for a given force and known magnetic system parameters.
The derivation of the mathematical model follows the intuition of the system
operation. In other words, the idea of converting an electrical signal
to a mechanical action is modeled by analyzing the energy conversion process.

**Appendix A: Derivation of Magnetic Field Energy and Magnetic Force**

Let W_{f} be the energy stored in a magnetic field.

where l is flux linkages,

L(x) is the inductance as a function of the air gap length, x.

where A_{g} is the area of the air gap.

The magnetic force is