## Force In a Singly and Multiply Excited Magnetic Field System

Table of Contents

**Force In a Singly and Multiply Excited Magnetic Field System**

## Singly and Multiply Excited Magnetic Field System

**1. Model& Analysis**

The law of conservation of energy governs how electrical energy is transformed into mechanical energy. Energy is neither created nor destroyed, according to the law of conservation of energy. Equation (1), where dW e is the change in electrical energy, dWm is the change in mechanical energy, and dWf is the change in magnetic field energy, describes the process of electromechanical energy conversion for a differential time interval dt. Heat-related energy losses are disregarded.

The dWe term for Equation is zero if the electrical energy is maintained constant (1). The force times the differential distance travelled makes up the differential mechanical energy, or work. Equation depicts the force resulting from the energy of the magnetic field (2). The negative sign suggests that the force is acting to reduce resistance by narrowing the air gap.

In terms of the magnetic system parameters, one can find an expression for the energy held within the magnetic field. The force is then expressed by substituting this expression for Wf in Equation (2). In Appendix A, this derivation is illustrated. In terms of the current, I the permeability of free space constant, m0, the cross-sectional area of the air gap, Ag, the number of turns, N, and the air gap distance, x, the result is Equation (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).

**2. Sample Calculations**

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

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

**3. Derivation of Magnetic Field Energy and Magnetic Force**

**Force In A Multiply Excited Magnetic Field System**

For continuous energy conversion devices like Alternators, synchronous motors etc., multiply excited magnetic systems are used. In practice ,doubly excited systems are very much in use.

The Figure 3.8 shows doubly excited magnetic system. This system has two independent sources of excitations. One source is connected to coil on stator while other is connected to coil on rotor.

i1= Current due to source 1

i2 = Current due to source 2

l1 = Flux linkages due to i1

l2 = Flux linkages due to i2

q = Angular displacement of rotor

Tf = Torque developed

Due to two sources, there are two sets of three independent variables i.e., (l1, l2, q) or (i1, i2, q).

**Case 1 : **Independent variables l1, l2, q i.e., l1, l2 are constants.

From the earlier analysis it is known,

While the fields energy is,

Now let

L11 = Self inductance of stator

L22 = Self inductance of rotor

L12 = L21 = Mutual inductance between stator and rotor

l1 = L11 i1 + L12 i2

l2 = L12 i1 + L22 i2

So Solve equation (3) and equations (4) to express i1Â and i2Â in terms ofÂ l1Â andÂ l2Â asÂ l1Â andÂ l2Â are independent variables.

Multiply equation (3) by L12 and equation (4) by L11,

L12l1 = L11 L12 i1 + L212 i2

L11l2 = L11 L12 i1 + L11 L22 i2

Subtracting the two,

L12l1 – L11l2 = [L212 – L11 L12 ] i1

Note that negative sign is absorbed in defining Î².

Similarly i1 can be expressed in terms of l1 and l2 as,

The self and mutual inductances of the coils are dependent on the angular position q of the rotor.

**Case 2: **Independent variables i1, i2, q i.e. i1 and i2 are constants.

**Force in a doubly exited system**

so Where i1Â and i2Â are constants which are the stator and rotor current respectively.