Principal Laws and Methods in Electrical Machine Design
1.4 Application of the Principle of Virtual Work in the Determination of Force and Torque
S
J·dS. (1.72)
Inside the area under observation, when a closed line integral according to Equation (1.71) is written only for the areaS(<S), the flux of a flux tube in a current-carrying area becomes
Φ=µ0l h
b
S
J·dS (1.73)
and thus, in that case, if (h/b) =(h/b), then in factΦ< Φ. If the current densityJ in the current-carrying area is constant,Φ=ΦS/S is valid. When crossing the bound-ary between a current-carrying slot and currentless iron, the flux of the flux tube cannot change. Therefore, the dimensions of the line grid have to be altered. WhenJis constant and Φ=Φ, Equations (1.72) and (1.73) yield for the dimensions in the current-carrying area
h b
= Sh
Sb. (1.74)
This means that near the indifference point d the ratio (h/b) increases. An orthogonal field dia-gram can be drawn for a current-carrying area by correcting the equivalent linear current den-sity by iterating the created diagram. For a current-carrying area, gradient lines are extended from potential lines up to the indifference point. Now, bearing in mind that the gradient lines have to divide the current-carrying area into sections of equal size, next the orthogonal flux lines are plotted by simultaneously paying attention to changing dimensions. The diagram is altered iteratively until Equation (1.74) is valid to the required accuracy.
1.4 Application of the Principle of Virtual Work in the Determination of Force and Torque
When investigating electrical equipment, the magnetic circuit of which changes form during operation, the easiest method is to apply the principle of virtual work in the estimation of force and torque. Examples of this kind of equipment are double-salient-pole reluctant machines, various relays and so on.
Faraday’s induction law presents the voltage induced in the winding, which creates a current that tends to resist the changes in flux. The voltage equation for the winding is written as
u =Ri+dΨ
dt =Ri+ d
dtLi, (1.75)
whereuis the voltage connected to the coil terminals,Ris the resistance of the winding and Ψ is the coil flux linkage, andLthe self-inductance of the coil consisting of its magnetizing inductance and leakage inductance:L=Ψ/i=NΦ/i=N2Λ=N2/Rm(see also Section 1.6).
If the number of turns in the winding isNand the flux isΦ, Equation (1.75) can be rewritten as
u =Ri+NdΦ
dt . (1.76)
The required power in the winding is written correspondingly as ui =Ri2+N idΦ
dt , (1.77)
and the energy
dW =Pdt =Ri2dt+N idΦ. (1.78)
The latter energy componentNidΦis reversible, whereasRi2dtturns into heat. Energy cannot be created or destroyed, but may only be converted to different forms. In isolated systems, the limits of the energy balance can be defined unambiguously, which simplifies the energy analysis. The net energy input is equal to the energy stored in the system. This result, the first law of thermodynamics, is applied to electromechanical systems, where electrical energy is stored mainly in magnetic fields. In these systems, the energy transfer can be represented by the equation
dWel=dWmec+dWΦ+dWR (1.79)
where
dWelis the differential electrical energy input, dWmecis the differential mechanical energy output, dWΦis the differential change of magnetic stored energy, dWRis the differential energy loss.
Here the energy input from the electric supply is written equal to the mechanical energy together with the stored magnetic field energy and heat loss. Electrical and mechanical energy have positive values in motoring action and negative values in generator action. In a magnetic system without losses, the change of electrical energy input is equal to the sum of the change of work done by the system and the change of stored magnetic field energy
dWel=dWmec+dWΦ, (1.80)
dWel=eidt. (1.81)
In the above, eis the instantaneous value of the induced voltage, created by changes in the energy in the magnetic circuit. Because of this electromotive force, the external electric circuit converts power into mechanical power by utilizing the magnetic field. This law of energy conversion combines a reaction and a counter-reaction in an electrical and mechanical
Φ R
F i
u
x
N turns FΦ
Ψ, e
Figure 1.14 Electromagnetic relay connected to an external voltage sourceu. The mass of a moving yoke is neglected and the resistance of the winding is assumed to be concentrated in the external resistor R. There areNturns in the winding and a fluxΦflowing in the magnetic circuit; a flux linkageΨ ≈ NΦis produced in the winding. The negative time derivative of the flux linkage is an emf,e. The force Fpulls the yoke open. The force is produced by a mechanical source. A magnetic forceFΦtries to close the air gap
system. The combination of Equations (1.80) and (1.81) yields dWel=eidt =dΨ
dt idt =idΨ =dWmec+dWΦ. (1.82) Equation (1.82) lays a foundation for the energy conversion principle. Next, its utilization in the analysis of electromagnetic energy converters is discussed.
As is known, a magnetic circuit (Figure 1.14) can be described by an inductanceL deter-mined from the number of turns of the winding, the geometry of the magnetic circuit and the permeability of the magnetic material. In electromagnetic energy converters, there are air gaps that separate the moving magnetic circuit parts from each other. In most cases – because of the high permeability of iron parts – the reluctanceRmof the magnetic circuit consists mainly of the reluctances of the air gaps. Thus, most of the energy is stored in the air gap. The wider the air gap, the more energy can be stored. For instance, in induction motors this can be seen from the fact that the wider is the gap, the higher is the magnetizing current needed.
According to Faraday’s induction law, Equation (1.82) yields
dWel=idΨ. (1.83)
The computation is simplified by neglecting for instance the magnetic nonlinearity and iron losses. The inductance of the device now depends only on the geometry and, in our example, on the distancexcreating an air gap in the magnetic circuit. The flux linkage is thus a product
of the varying inductance and the current
Ψ =L(x)i. (1.84)
A magnetic forceFΦis determined as
dWmec=FΦdx. (1.85)
From Equations (1.83) and (1.85), we may rewrite Equation (1.80) as
dWΦ=idΨ −FΦdx. (1.86)
Since it is assumed that there are no losses in the magnetic energy storage, dWΦ is deter-mined from the values ofΨ andx. dWΦis independent of the integration path A or B, and the energy equation can be written as
WΦ(Ψ0,x0)=
With no displacement allowed (dx=0), Equations (1.86) now (1.87) yield
WΦ(Ψ,x0)= Ψ 0
i(Ψ,x0)dΨ. (1.88)
In a linear system,Ψ is proportional to currenti, as in Equations (1.84) and (1.88). We there-fore obtain The magnetic field energy can also be represented by the energy densitywΦ=WΦ/V = B H/2 [J/m3] in a magnetic field integrated over the volumeV of the magnetic field. This gives
Assuming the permeability of the magnetic medium constant and substituting B =µH gives
This yields the relation between the stored energy in a magnetic circuit and the electrical and mechanical energy in a system with a lossless magnetic energy storage. The equation for differential magnetic energy is expressed in partial derivatives
dWΦ(Ψ,x)= ∂WΦ
∂Ψ dΨ +∂WΦ
∂x dx. (1.92)
SinceΨ andxare independent variables, Equations (1.86) and (1.92) have to be equal at all values of dΨ and dx, which yields
i= ∂WΦ(Ψ,x)
∂Ψ , (1.93)
where the partial derivative is calculated by keeping xconstant. The force created by the electromagnet at a certain flux linkage levelΨ can be calculated from the magnetic energy
FΦ = −∂WΦ(Ψ,x)
∂x . (1.94a)
The minus sign is due to the coordinate system in Figure 1.14. The corresponding equation is valid for torque as a function of angular displacementθwhile keeping flux linkageΨ constant
TΦ= −∂WΦ(Ψ, θ)
∂θ . (1.94b)
Alternatively, we may employ coenergy (see Figure 1.15a), which gives us the force directly as a function of current. The coenergyWΦ is determined as a function ofiandxas
WΦ (i,x)=iΨ −WΦ(Ψ,x). (1.95)
B
0 i 0 H
Ψ
energy density
coenergy density coenergy
(a) (b) (c)
0 i
Ψ
energy energy
coenergy
W' W'
W x, const. W x, co
nst.
Figure 1.15 Determination of energy and coenergy with current and flux linkage (a) in a linear case (Lis constant), (b) and (c) in a nonlinear case (Lsaturates as a function of current). If the figure is used to illustrate the behaviour of the relay in Figure 1.14, the distancexremains constant
In the conversion, it is possible to apply the differential ofiΨ
d(iΨ)=idΨ +Ψ di. (1.96)
Equation (1.95) now yields
dWΦ (i,x)=d (iΨ)−dWΦ(Ψ,x). (1.97) By substituting Equations (1.86) and (1.96) into Equation (1.97) we obtain
dWΦ (i,x)=Ψdi+FΦdx. (1.98)
The coenergyWΦ is a function of two independent variables,iandx. This can be represented by partial derivatives
dWΦ (i,x)= ∂WΦ
∂i di+∂WΦ
∂x dx. (1.99)
Equations (1.98) and (1.99) have to be equal at all values of diand dx. This gives us Ψ =∂WΦ (i,x)
∂i , (1.100)
FΦ= ∂WΦ(i,x)
∂x . (1.101a)
Correspondingly, when the currentiis kept constant, the torque is TΦ = ∂WΦ(i, θ)
∂θ . (1.101b)
Equation (1.101) gives a mechanical force or a torque directly from the currentiand dis-placement x, or from the angular displacement θ. The coenergy can be calculated with i andx
WΦ (i0,x0)= i 0
Ψ (i,x0)di. (1.102)
In a linear system,Ψ andiare proportional, and the flux linkage can be represented by the inductance depending on the distance, as in Equation (1.84). The coenergy is
WΦ (i,x)= i 0
L(x)idi= 1
2L(x)i2. (1.103)
Using Equation (1.91), the magnetic energy can be expressed also in the form WΦ =
V
1
2µH2dV. (1.104)
In linear systems, the energy and coenergy are numerically equal, for instance 0.5Li2 = 0.5Ψ2/L or (µ/2)H2 = (1/2µ)B2. In nonlinear systems, Ψ andi or B andH are not pro-portional. In a graphical representation, the energy and coenergy behave in a nonlinear way according to Figure 1.15.
The area between the curve and flux linkage axis can be obtained from the integralidΨ, and it represents the energy stored in the magnetic circuitWΦ. The area between the curve and the current axis can be obtained from the integralΨdi, and it represents the coenergyWΦ. The sum of these energies is, according to the definition,
WΦ+WΦ =iΨ. (1.105)
In the device in Figure 1.14, with certain values ofxandi(orΨ), the field strength has to be independent of the method of calculation; that is, whether it is calculated from energy or coenergy – graphical presentation illustrates the case. The moving yoke is assumed to be in a positionxso that the device is operating at the point a, Figure 1.16a. The partial derivative in Equation (1.92) can be interpreted as WΦ/x, the flux linkage Ψ being constant and x→0. If we allow a changexfrom position a to position b (the air gap becomes smaller), the stored energy change−WΦ will be as shown in Figure 1.16a by the shaded area, and the energy thus becomes smaller in this case. Thus, the forceFΦis the shaded area divided by xwhenx→0. Since the energy change is negative, the force will also act in the negative x-axis direction. Conversely, the partial derivative can be interpreted asWΦ
x,ibeing constant andx→0.
Ψ Ψ
0 i 0 i
a a
b b
c
initial state x, const.
after displ.
after displ.
initial state x, const.
x
x− x− x
Ψ
WΦ
− + W'Φ
i
(a)Ψ is constant, i decreases (b) i is constant, flux linkage increases
Figure 1.16 Influence of the changexon energy and coenergy: (a) the change of energy, whenΨ is constant; (b) the change of coenergy, wheniis constant
The shaded areas in Figures 1.16a and b differ from each other by the amount of the small triangle abc, the two sides of which areiandΨ. When calculating the limit,xis allowed to approach zero, and thereby the areas of the shaded sections also approach each other.
Equations (1.94) and (1.101) give the mechanical force or torque of electric origin as partial derivatives of the energy and coenergy functionsWΦ(Ψ,x) andWΦ (x,i).
Physically, the force depends on the magnetic field strengthHin the air gap; this will be studied in the next section. According to the study above, the effects of the field can be repre-sented by the flux linkageΨ and the currenti. The force or the torque caused by the magnetic field strength tends to act in all cases in the direction where the stored magnetic energy de-creases with a constant flux, or the coenergy inde-creases with a constant current. Furthermore, the magnetic force tends to increase the inductance and drive the moving parts so that the reluctance of the magnetic circuit finds its minimum value.
Using finite elements, torque can be calculated by differentiating the magnetic coenergyW with respect to movement, and by maintaining the current constant:
T =ldW
In numerical modelling, this differential is approximated by the difference between two suc-cessive calculations:
T = l
W(α+α)−W(α)
α . (1.107)
Here,lis the machine length andαrepresents the displacement between successive field solutions. The adverse effect of this solution is that it needs two successive calculations.
Coulomb’s virtual work method in FEM is also based on the principle of virtual work. It gives the following expression for the torque:
T =
where the integration is carried out over the finite elements situated between fixed and moving parts, having undergone a virtual deformation. In Equation (1.108),lis the length,Jdenotes the Jacobian matrix, dJ/dϕ is its differential representing element deformation during the displacement dϕ,|J|is the determinant ofJand d|J|/dϕis the differential of the determinant, representing the variation of the element volume during displacement dϕ. Coulomb’s virtual work method is regarded as one of the most reliable methods for calculating the torque and it is favoured by many important commercial suppliers of FEM programs. Its benefit compared with the previous virtual work method is that only one solution is needed to calculate the torque.
Figure 1.17 Flux solution of a loaded 30 kW, four-pole, 50 Hz induction motor, the machine rotating counterclockwise as a motor. The figure depicts a heavy overload. The tangential field strength in this case is very large and produces a high torque. The enlarged figure shows the tangential and normal components of the field strength in principle. Reproduced by permission of Janne Nerg