PROCEDIMIENTO DE PROCEDIMIENTOS

Next, three different air gaps of a salient-pole machine are investigated. The importance of these air gaps lies in the following: the first air gap is employed in the calculation when the machine is magnetized with a rotor field winding; the second is required for the calculation of the direct-axis armature reaction, and thereby the armature direct-axis inductance; and finally, the third air gap gives the armature quadrature-axis reaction and the quadrature-axis inductance. The air gaps met by the rotor field winding magnetizing are shaped with pole shoes such that as sinusoidal a flux density distribution as possible is obtained in the air gap of the machine.

The air-gap flux density created by the rotor field winding magnetizing of a salient-pole machine can be investigated with an orthogonal field diagram. The field diagram has to be constructed in an air gap of accurate shape. Figure 3.7 depicts the air gap of a salient-pole machine, into which the field winding wound around the salient-pole body creates a mag-netic flux. The path of the flux can be solved with a magmag-netic scalar potential. The magmag-netic field refracts on the iron surface. However, in the manual calculation of the air gaps of a syn-chronous machine, the permeability of iron is assumed to be so high that the flux lines leave the equipotential iron surface perpendicularly.

If the proportionΘδof the current linkageΘfof the rotor acts upon the air gap, each duct in the air gap takes, with the notation in Figure 3.7, a flux

Φ_δ = µ0Θ_δl x

nδ = Θ_δRm, (3.14)

τ_p

leakage flux main flux

main flux

leakage flux

x

δ

Figure 3.7 Field diagram of the rotor pole of an internal salient-pole synchronous machine with DC magnetizing in an area of half a pole pitchτp/2. The figure also indicates that the amount of leakage flux in this case is about 15%, which is a typical number for pole leakage. Typically, a designer should be prepared for about 20% leakage flux when designing the field winding

where l is the axial length of the pole shoe and n the number of square elements in the radial direction. If the origin of the reference frame is fixed to the middle of the pole shoe, we may write in the cosine form

Φ_δ

lx = µ0Θ_δ

nδ = ˆBδcosθ. (3.15)

In the field diagram consisting of small squares, the side of a square equals the average widthx. The magnetic flux density of the stator surface can thus be calculated by the average width of squares touching the surface. On the other hand, nδ is the length of the flux line from the pole shoe surface to the stator surface:

nδ = µ0Θδ

Bˆ_δcosθ = δ0e

cosθ, (3.16)

whereδ0eis the air gap in the middle of the pole, corrected with the Carter factor. Now the pole shoe has to be shaped such that the length of the flux density line of the field diagram is inversely proportional to the cosine of the electrical angleθ. A pole shoe shaped in this way creates a cosinusoidal magnetic flux density in the air gap, the peak value of which is ˆB_δ. The maximum value of flux penetrating thorough a full-pitch winding coil is called the peak value of flux, although it is not a question of amplitude here. The peak value of the flux is obtained

by calculating a surface integral over the pole pitch and the length of the machine. In practice, the flux of a single pole is calculated. Now, the peak value is denoted ˆΦm

Φˆm=

where l^is the equivalent core length l^≈ l + 2δ in a machine without ventilating ducts (see Section 3.2),τpis the pole pitch,τp= πD/(2p), x is the coordinate, the origin of which is in the middle of the pole, andθ = xπ/τp.

When the flux density is cosinusoidally distributed, we obtain by integration for the air-gap flux

Φˆm= 2

πBˆ_δτpl^. (3.18)

By reformulating the previous equation, we obtain Φˆm = Dl^

Next, the air gap experienced by the stator winding current linkage is investigated. The stator winding is constructed such that its current linkage is distributed fairly cosinusoidally on the stator surface. The stator current linkage creates an armature reaction in the magnetiz-ing inductances. As a result of the armature reaction, this cosinusoidally distributed current linkage creates a flux of its own in the air gap. Because the air gap is shaped so that the flux density created by the rotor pole is cosinusoidal, it is obvious that the flux density created by the stator is not cosinusoidal, see Figure 3.8. When the peak value of the fundamental current linkage of the stator is on the d-axis, we may write

Θs1(θ) = Θd^(θ) = ˆΘd^cosθ. (3.20) The amplitude of the current linkage is calculated by Equation (2.15). The permeance dΛ of the duct at the positionθ is

dΛ = µ0

The magnetic flux density at the positionθ is

Bd(θ) = dΦ dS = µ0

δ0e

Θˆd^cos²θ. (3.22)

The distribution of the air-gap flux density created by the stator current is proportional to the square of the cosine when the current linkage of the stator is on the d-axis. To be able to calculate the inductance of the fundamental, this density function has to be replaced by a cosine function with an equal flux. Thus, we calculate the factor of the fundamental of the

τp

Θs1

B_d(q ) (q ) (q )

(q ) d

q δ0 q

1d

Bˆ

τp

Θs1

B_q

d d

q ˆ1q

B

(a) (b)

Figure 3.8 (a) Cosine-squared flux density Bd(θ), created in the shaped air gap by a cosinusoidal stator current linkageΘs1occurring on a direct axis of the stator, where the peak value of the fundamental component of Bd(θ) is ˆB1d. (b) The cosinusoidal current linkage distribution on the quadrature axis creates a flux density curve Bq(θ). The peak value of the fundamental component of Bq(θ) is ˆB1q

Fourier series. The condition for keeping the flux unchanged is µ0

δ0e

Θˆd^ +π/2

−π/2

cos²θ dθ = ˆB1d +π/2

−π/2

cosθ dθ. (3.23)

The amplitude of the corresponding cosine function is thus Bˆ1d= π

4 µ0

δ0e

Θˆd^ = µ0

δde

Θˆd^. (3.24)

In the latter presentation of Equation (3.24), the air gapδdeis an equivalent d-axis air gap experiencing the current linkage of the stator. Its theoretical value is

δde= 4δ0e

π . (3.25)

Figure 3.8a depicts this situation. In reality, the distance from the stator to the rotor on the edge of the pole cannot be extended infinitely, and therefore the theoretical value of Equa-tion (3.25) is not realized as such (it is only an approximaEqua-tion). However, the error is only marginal, because when the peak value of the cosinusoidal current linkage distribution is at the direct axis, the current linkage close to the quadrature axis is very small. Equation (3.25) gives an interesting result: the current linkage of the stator has to be higher than the current linkage of the rotor, if the same peak value of the fundamental component of the flux density is desired with either the stator or rotor magnetization.

Figure 3.8b illustrates the definition of the quadrature air gap. The peak value of the stator current linkage distribution is assumed to be on the quadrature axis of the machine. Next, the flux density curve is plotted on the quadrature axis, and the fluxΦqis calculated similarly as in Equation (3.19). The flux density amplitude of the fundamental component corresponding to this flux is written as

Bˆ1q= pΦq

Dl = µ0

δqe

Θˆq^, (3.26)

whereδqeis the equivalent quadrature air gap. The current linkages are set equal: ˆΘf= ˆΘd^ = Θˆ_q^, the equivalent air gaps behaving inversely proportional to the flux density amplitudes

Bˆ_δ: ˆB1d: ˆB1q= 1 δ0e

: 1 δde

: 1 δqe

. (3.27)

Direct and quadrature equivalent air gaps are calculated from this (inverse) proportion. A direct-axis air gap is thus approximately 4δ0e/π. A quadrature-axis air gap is more problem-atic to solve without numerical methods, but it varies typically between (1.5–2–3) × δde. According to Schuisky (1950), in salient-pole synchronous machines, a quadrature air gap is typically 2.4-fold when compared with a direct air gap in salient-pole machines.

The physical air gap on the centre line of the magnetic pole is set toδ0. The slots in the stator create an apparent lengthening of the air gap when compared with a completely smooth stator. This lengthening is evaluated with the Carter factor. On the d-axis of the rotor pole, the length of an equivalent air gap is nowδ0ein respect of the pole magnetization. In this single air gap, the pole magnetization has to create a flux density ˆB_δ. The required current linkage of a single pole is

Θf= δ0eBˆ_δ

µ0 . (3.28)

The value for current linkage on a single rotor pole isΘf= NfIf, when the DC field wind-ing current on the pole is If and the number of turns in the coil is Nf. The flux linkage and the inductance of the rotor can now be easily calculated. When the pole shoes are shaped according to the above principles, the flux of the phase windings varies at no load as a si-nusoidal function of time,Φm(t )= ˆΦmsinωt, when the rotor rotates at an electric angular frequencyω. By applying Faraday’s induction law as presented in Equation (1.8), we can cal-culate the induced voltage. The applied form of the induction law, which takes the geometry of the machine into account, is written with the flux linkageΨ as

e1(t )= −dΨ (t)

dt = −kw1NdΦm(t )

dt , (3.29)

and the fundamental component of the voltage induced in a single pole pair of the stator is written as

e1p(t )= −Npkw1ω ˆΦmcosωt. (3.30) Here Npis the number of turns of one pole pair of the phase winding. The winding factor kw1of the fundamental component takes the spatial distribution of the winding into account.

The winding factor indicates that the peak value of the main flux ˆΦmdoes not penetrate all the coils simultaneously, and thus the main flux linkage of a pole pair is ˆΨm= Npkw1ω ˆΦm. By applying Equation (3.18) we obtain for the voltage of a pole pair

e1p(t )= −Npkw1ω2

πBˆ_δτpl^cosωt, (3.31) the effective value of which is

E_1p = 1

√2ωNpk_w12

πBˆ_δτpl^= 1

√2ω ˆΨmp. (3.32)

The maximum value of the air-gap flux linkage ˆΨmpof a pole pair is found at instants when the main flux best links the phase winding observed. In other words, the magnetic axis of the winding is parallel to the main flux in the air gap.

The voltage of the stator winding is found by connecting an appropriate amount of pole pair voltages in series and in parallel according to the winding construction.

Previously, the air gapsδdeandδqewere determined for the calculation of the direct and quadrature stator inductances. For the calculation of the inductance, we have also to define the current linkage required by the iron. The influence of the iron can easily be taken into account by correspondingly increasing the length of the air gap,δdef = ( ˆUm,δde/( ˆUm,δde+ ˆUm,Fe))δde. We now obtain the effective air gapsδdef andδqef. With these air gaps, the main inductances of the stator can be calculated in the direct and quadrature directions

L_pd = 2 πµ0

D_δl^ pδdef

k_w1N_p2

, Lpq= 2 πµ0

D_δl^ pδqef

k_w1N_p2

. (3.33)

N_p is the number of turns (N_s/p) of a pole pair. The main inductance is the inductance of a single stator phase. To obtain the single-phase equivalent circuit magnetizing inductance, for instance for a three-phase machine, the main inductance has to be multiplied by 3/2 to take the effects of all three windings into account. When deriving the equations, Equation (2.15) for the current linkage of a stator is required, and also the equation for a flux linkage of a single pole pair of the stator

Ψmp= −kw1Np

2 πBˆ_δτpl^

which is included in the previous voltage equations. The peak value for the air-gap flux density is calculated with an equivalent air gap and a stator current linkage, which leads to Equation (3.33). The inductances will be discussed in detail later in Section 3.9 and also in Chapter 7.