Torque equation - Sizing equation of FRPMM 1 Magnetic circuit model

Chapter 7 137 Torque Ripple Reduction in DTC Induction Motor Drive

A. IM motor parameters

3. Sizing equation of FRPMM 1 Magnetic circuit model

3.4 Torque equation

Once the stator winding pole pair is selected, the stator flux linkage can be deduced using winding function theory, just as mentioned in Eq. (2). The winding function N(θs) in Eq. (2) can be written as:

Nð Þ ¼θ_s X^∞

i¼1, 3, 5

Nicos iPθð sÞ (18)

Ni¼ 2 iπ

P kwi (19)

Flux Reversal Machine Design

DOI: http://dx.doi.org/10.5772/intechopen.92428

where FCis:

FC ¼Brhm

μ_rμ₀ (6)

Then, it can be written in Fourier series as follows:

FPMð Þ ¼θ_s X^∞

i¼1, 3, 5

Fisin iZs

2 θ_s

� �

(7)

where the magnitude Fiis

F_i¼4 π 1 i

B_rh_m

μ₀μ_r 1þ �1ð Þ^iþ1² sin iπ 2SO

� �

(8) Then, the next step is to derive the specific airgap permeanceΛ(θs,θ) in Eq. (1).

Since the stator slotting effect has already been considered in Eqs. (5–8), the specific airgap permeanceΛ(θs,θ) can be replaced by the airgap permeance with smoothed stator and slotted rotorΛr(θs,θ). The model of smoothed stator and slotted rotor is shown in Figure 8. Then, theΛr(θs,θ) can be expressed by:

Λrðθ_s,θÞ≈ Λ0rþΛ1rcos Z½ rðθ_s�θÞ� (9) The coefficients of the airgap permeance functionΛ0randΛ1rin Eq. (9) can be obtained using the conformal mapping method [28, 29]:

Λ0r¼μ₀

g⁰ 1�1:6βbo

� �

(10)

g⁰¼gþh_m=μ_r (11)

Λ1r¼μ₀ g⁰ 4

πβ 0:5þ ðbo=tÞ² 0:78125�2 bð o=tÞ²

" #

sin 1:6πbo

� �

(12)

β¼0:5� 1 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ ^b_2tô_g^t0

� �2

r (13)

Figure 8.

Schematic of single-side salient structure on rotor.

Direct Torque Control Strategies of Electrical Machines

where bois the rotor slot opening width and t is the rotor slot pitch, as shown in Figure 8. Combining Eq. (1), Eqs. (5–13), the no-load airgap flux density B(θs,θ) can be finally calculated as:

Bðθ_s,θÞ ¼ X^∞

i¼1, 3

B_isin iZ_s 2 �Z_r

� �

θ_s�Z_rθ

� �

(14)

where the magnitude Biis Bi¼1

2FiΛ1r, i¼1, 3, 5 … (15) 3.3 Slot-pole combinations

As can be seen in Eq. (14), the number of pole pairs in the air gap flux density is iZs/2�Zr, i = 1,3,5 … Then, in order to make the flux density induce EMF in the armature windings, the pole pair number of the armature windings P should be equal to iZs/2�Zr, i = 1,3,5 … Besides, for three phase symmetry, the winding pole pair number must also meet the following requirement:

GCD Zð _s, PÞ¼3k, k¼1, 2, 3 … (16) All in all, the slot-pole combination of three-phase FRPMMs is ruled by the following equation:

P¼ min P¼iZs

2 �Zr; Zs

GCD Zð _s, PÞ¼3k

� �

i¼1, 3, 5 … k¼1, 2, 3 …

(17)

where min means to select the minimum number of these qualified harmonic orders so as to obtain a maximal pole ratio of FRPMMs. Therefore, the feasible slot- pole combinations can be summarized as Table 2. Non-overlapping windings (i.e., concentrated windings) are usually used in FRPMMs because of the higher fault tolerance and easier manufacture than regular overlapping windings. However, some FRPMMs are suggested to employ overlapping windings in order to have a larger winding factor and thus a higher torque density. Therefore, both winding factors, that is, kwn(using non-overlapping winding) and kwr(using overlapping winding) are calculated for each FRPMM so as to see the difference of using different winding types.

3.4 Torque equation

Nð Þ ¼θ_s X^∞

i¼1, 3, 5

Nicos iPθð sÞ (18)

Ni¼ 2 iπ

P kwi (19)

Flux Reversal Machine Design

DOI: http://dx.doi.org/10.5772/intechopen.92428

ZsZr234567810111213141516 6P1122112211 SPP110.50.5110.50.511 PR242.53.58105.56.51416 kwn0.50.50.8660.8660.50.50.8660.8660.50.5 kwr110.8660.866110.8660.86611 12P4211245542 SPP0.512210.50.40.40.51 PR0.525742.52.22.63.58 kwn0.8660.50.250.250.50.8660.9330.9330.8660.5 kwr0.86610.9660.96610.8660.9330.9330.8661 18P76543211234567 SPP3/70.50.60.7511.5331.510.750.60.53/7 PR2/70.50.81.2523.58105.543.252.82.516/7 kwn0.9020.8660.7350.6170.50.4920.1670.1670.4920.50.6170.7350.8660.902 kwr0.9020.8660.9450.94510.9450.960.960.94510.9450.9450.8660.902 PS:Non-overlappingwindingisrecommended.Other:Overlappingwindingisrecommended. kwnandkwrarefundamentalwindingfactorscalculatedbasedonnon-overlappingwindingtypeandrecommendedwindingtypes,respectively. Table2. Slot-polecombinationsofthree-phaseFRPMM.

Direct Torque Control Strategies of Electrical Machines

where Niis the ith harmonics of the winding function and kwiis the winding factor of the ith harmonics. As can be seen in Eq. (17), the pole pair number is iZs/2�Zr(i = 1,3,5 … ). So, the sum or difference of any two pole pair harmonics Pi1

and Pi2is a multiple of stator slot number, that is, Pi1¼i1Zs=2�Zr

P_i2¼i₂Z_s=2�Z_r Pi1�Pi2

j j ¼kZs, k¼1, 2, 3 … 8>

>: (20)

Therefore, all the flux density harmonics are tooth harmonics of each other, that is, they have the same absolute values of winding factors, and their absolute winding factor equals the fundamental winding factor kw1:

kwPi1

j j ¼jkwPi2j ¼kw1 (21) Then, combining Eq. (2), Eq. (3), Eqs. (18–21), the back-EMF can be finally obtained as:

Eph¼2ωmrglstkNsZrkw1

X^∞

i¼1

sgn∗ Bi iZs

2 �Z_r

� �

=P (22)

where

sgn ¼ 1, winding factor of iZð s=2�ZrÞ^thharmonic equals kw1

�1, winding factor of iZð s=2�ZrÞ^thharmonic equals�kw1

(

(23) Since the reluctance torque of FRPMM is negligible, the electromagnetic torque under id= 0 control can be expressed as Eq. (4). Then, combining Eq. (4) and Eq. (22), the average torque Teis able to be calculated as:

Te¼3I_phrgl_stkNsZrkw1

X^∞

i¼1

sgn∗ Bi iZs

2 �Zr

� �

=P (24)

So far, the general torque equation has been obtained as Eq. (24), but in this equation, some parameters such as Bi, Iphcannot be determined in the initial design stage of FRPMMs, so it is desirable that Eq. (24) can be transformed to a combination of several basic parameters, such as electric loading, magnetic loading, which can be easily determined in the initial design stage.

As known for electrical machines, the electric loading Aecan be written as:

A_e¼ 6N_sI_ph 2 ffiffiffi

πr_g (25)

Then, the equivalent magnetic loading of three-phase FRPMM Bmis defined as:

B_m ¼X^∞

i¼1

sgn∗ B_i

iZs

2 �Z_r

� �

=P (26)

So, the torque expression in Eq. (24) can be rewritten as:

Te¼ ffiffiffi p2

πr²_glstkkwZrAeBm (27) Flux Reversal Machine Design

DOI: http://dx.doi.org/10.5772/intechopen.92428

Direct Torque Control Strategies of Electrical Machines

and Pi2is a multiple of stator slot number, that is, Pi1¼i1Zs=2�Zr

P_i2¼i₂Z_s=2�Z_r Pi1�Pi2

j j ¼kZs, k¼1, 2, 3 … 8>

>: (20)

kwPi1

j j ¼jkwPi2j ¼kw1 (21) Then, combining Eq. (2), Eq. (3), Eqs. (18–21), the back-EMF can be finally obtained as:

Eph¼2ωmrglstkNsZrkw1

X^∞

i¼1

sgn∗ Bi iZs

2 �Z_r

� �

=P (22)

where

sgn ¼ 1, winding factor of iZð s=2�ZrÞ^thharmonic equals kw1

�1, winding factor of iZð s=2�ZrÞ^thharmonic equals�kw1

(

Te¼3I_phrgl_stkNsZrkw1

X^∞

i¼1

sgn∗ Bi iZs

2 �Zr

� �

=P (24)

As known for electrical machines, the electric loading Aecan be written as:

A_e¼ 6N_sI_ph 2 ffiffiffi

πr_g (25)

Then, the equivalent magnetic loading of three-phase FRPMM Bmis defined as:

B_m¼X^∞

i¼1

sgn ∗ B_i

iZs

2 �Z_r

� �

=P (26)

So, the torque expression in Eq. (24) can be rewritten as:

Te¼ ffiffiffi p2

πr²_glstkkwZrAeBm (27) Flux Reversal Machine Design

DOI: http://dx.doi.org/10.5772/intechopen.92428

Thus, the rotor volume Vr, which equalsπlstkr²g, can be obtained:

Vr¼ Te

ffiffiffi2

p k_wZ_rA_eB_m (28) and then the airgap radius rgand the stack length lstkcan be derived as:

r_g ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V_r=ðπk_lrÞ p3

(29) l_stk¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Vrk²_lr=π q3

(30) where klris the aspect ratio, equals to the ratio of rgto lstk. It can be found in Eq. (27) that the key parameters affecting the torque density are the airgap radius rg, stack length lstk, winding factor kw, rotor slot number Zr, electric loading Ae,and equivalent magnetic loading Bm, among which the stack length lstkcan be determined by the volume requirement, and winding factor kwis approximate to 1. So, the remaining parameters rg, Zr, Ae, Bmshould be determined at the initial stage of the design process. Thus, the influences of the above key parameters on important performances, such as average torque, pulsating torque, power factor, PM demag- netization performance, will be investigated in the following parts.

4. Influence of design parameters on key performances

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