PERMANENT MAGNET TYPE ROTARY ELECTRIC MACHINE AND ELECTRIC POWER STEERING APPARATUS USING THE SAME TECHNICAL FIELD [0001] The present invention relates to a permanent magnet type rotary electric machine which uses permanent magnets for field system and an electric power steering apparatus using the same. BACKGROUND ART [0002] In recent years, a motor with small cogging torque has been required for various applications such as industrial servo motors and hoists for elevators. In ' focusing attention on such applications for vehicles, an electric power steering apparatus has become widespread for achieving an improvement in fuel consumption and an improvement in steering performance. Cogging torque of a motor for use in the electric power steering apparatus is transmitted •to a driver via gears; and therefore, reduction in cogging torque of the motor is strongly desired in order to obtain a smooth steering feeling. In response, one possible method to reduce the cogging torque is to provide supplemental grooves in a core of a stator. Such a method is disclosed in Patent Document 1, Patent Document 2, and Patent Document 3. RELATED ART DOCUMENT PATENT DOCUMENT [0003] Patent Document 1: Japanese Unexamined Patent Publication No. 2001-25182 Patent Document 2: Japanese Unexamined Patent Publication No. 2006-230116 Patent Document 3: Pamphlet of International Patent Unexamined Publication No. WO2009/084151 DISCLOSURE OF THE INVENTION PROBLEMS TO BE SOLVED BY THE INVENTION [0004] The supplemental grooves are provided over the whole in a rotational axis direction of the motor in a permanent magnet type rotary electric machine of . Patent Document 1; and therefore, a problem exists in that the equivalent length of air gap becomes longer and accordingly torque is reduced.' Furthermore, Patent Document 1, Patent Document 2, and Patent Document 3 exert an effect to reduce cogging torque of the number of pulsations and an integral multiple thereof of the least common multiple of the number of poles and the number of slots; however, a problem exists in that it is not possible to sufficiently suppress a cogging torque .component (a component which pulsates the number of times corresponding to the number of slots by one rotation of a rotor), the cogging torque component being generated by variations on the rotor side, for example, an attachment position error, a shape error, and/or variations in magnetic characteristics of permanent magnets. This invention has been made to solve the problem as described above, and an object of the present invention is to provide a permanent magnet type rotary electric machine which reduces cogging torque and an electric power steering apparatus using the same. MEANS FOR SOLVING THE PROBLEMS [0005] According to the present invention, there is provided a permanent magnet type rotary electric machine including: a rotor having a plurality of magnetic poles Composed of permanent magnets and a rotor core; and a stator including armature windings and a stator core which is provided with slots for incorporating the armature, windings and has a plurality of teeth facing the rotor. Each of the teeth of the stator core is ' provided with supplemental grooves at portions facing the rotor. In the permanent magnet type rotary electric machine, not less than two regions different in magnetic circuit design are provided in a rotational axis direction of the rotor, the regions being different by changing a cross-sectional shape' in the rotational axis direction in a cross-section perpendicular to a rotational shaft of the rotor having the permanent magnets and the rotor core; the supplemental grooves are provided in axial partial regions of the tooth of the stator core; and the region in which the supplemental groove is provided is each partial region for each region facing a region same in magnetic circuit design of the rotor. ADVANTAGEOUS EFFECT OF TEE INVENTION [0006] The permanent magnet type rotary electric machine according to the present invention can reduce cogging torque (component in which the number ,. of pulsations per one rotation of a rotor corresponds to the number of slots) , the cogging torque being generated by variations on the rotor side, for example, an attachment position error, a shape' error, and/or variations in magnetic characteristics of the permanent magnets. Objects, features, aspects, and advantageous effects other than the foregoing of the present invention will become more apparent from the following detailed description of the present invention for referring to the accompanying drawings.' BRIEF DESCRIPTION OF THE DRAWINGS [0007] Fig. 1 is a cross-sectional view exemplarily showing a permanent magnet type rotary electric machine according to Embodiment 1 of the present invention; Fig. 2 is a cross-sectional view taken along the line A1-A2 shown in Fig. 1; Fig. 3 is a cross-sectional view taken along the line B1-B2 shown in Fig. 1; Fig. 4 is a cross-sectional view taken along the line C1-C2 shown in Fig. 1; Fig. 5 is a cross-sectional view taken along the line D1-D2 shown in Fig. 1; Fig. 6 is a partial perspective view showing a stator core according to Embodiment 1; Fig. 7 is a perspective view showing a rotor according to Embodiment 1; and Fig. 8 is a partial perspective view showing a different stator core according to Embodiment 1. [0008] Fig. 9 is a perspective view showing a different rotor according to Embodiment 1; Fig. 10 is a partial cross-sectional ' view exemplarily showing a permanent magnet type rotary electric machine including the stator core of Fig. 6 and the rotor of Fig. 7; Fig. 11 is . a partial cross-sectional view exemplarily showing a permanent magnet type rotary electric machine including a stator core and the rotor of Fig.- 7; Fig. 12 is a partial cross-sectional view exemplarily showing a permanent magnet type rotary electric machine including the stator core of Fig. 8 and the rotor of Fig. 9; Fig. 13 is a partial cross-sectional view exemplarily showing a permanent magnet type rotary electric machine including a stator core and the rotor of Fig. 9; Fig.. 14 is a cross-sectional view showing a rotor in the case where an attachment position of a permanent magnet is out of position from an ideally equally-spaced position; Fig. 15 is an explanation view showing waveforms of a cogging - torque for a rotational angle of 30 degrees (mechanical angle) ; and Fig. 16 is an explanation view showing waveforms of a cogging torque for a rotational angle of 30 degrees (mechanical angle). [0009] Fig. 17 is an explanation view for comparing waveforms of cogging torque of the present invention with that of a known example (an supplemental groove is absent); Fig. 18 is an explanation view showing frequency analysis results of the cogging torque waveforms of Fig. 17; Fig. 19 is an explanation view showing a histogram of cogging torque of the known example; Fig. 20 is an explanation view, showing a histogram of cogging torque of .the present invention; Fig. 21 is an explanation view showing an supplemental groove and a slot opening portion; Fig. 22 is an explanation view showing the relationship between the width of an supplemental groove and cogging torque; Fig. 23 is a cross-sectional view showing a permanent magnet type rotary electric machine according to Embodiment 2; and Fig. 24 is a cross-sectional view showing a different example of a permanent magnet type rotary electric machine according to' Embodiment 2. [0010] Fig. 25 is a cross-sectional view showing a further different example of a permanent magnet type rotary electric machine according to Embodiment 2; Fig. 26 is a perspective view showing a rotor according to Embodiment 3; Fig, 27 is a partial perspective view showing a stator core according to Embodiment 3; Fig. 28 is a partial perspective view showing other example of a stator core according to Embodiment 3; Fig. 29 is a cross-sectional view showing a rotor according to Embodiment 3; Fig. 30 is a perspective view showing a rotor according to Embodiment 4; Fig. 31 is a partial perspective view showing a stator core according to Embodiment 4; Fig. 32 is a perspective view showing a different example of a rotor according to Embodiment 4; and Fig. 33 is a perspective view showing an. electric power steering apparatus according to Embodiment 5. MODE FOR CARRYING OUT THE INVENTION [0011] Embodiment 1. Fig. 1 shows a typical cross-sectional view in a plane parallel to a rotational shaft. And passing through the rotational shaft on a permanent magnet type rotary electric machine of Embodiment 1 of the present invention. Permanent magnets 1 are mounted on the surface of a rotor core 2. Protrusion portions 8 are provided near circumferential end portions of the permanent magnet 1 and the protrusion portions 8 are arranged so as to sandwich the permanent magnets 1. In the case where each of the protrusion portions 8 is arranged between the adjacent permanent magnets .1, the protrusion portions 8 and the permanent magnets 1 are not simultaneously illustrated in a cross-sectional view in a plane parallel to a rotational shaft 10 and passing through the rotational shaft 10;- however, the protrusion portions 8 and the permanent magnets 1 are simultaneously illustrated in Fig. 1 for ease of understanding. [0012] The rotational shaft 10 is press-fitted to the rotor core 2 and a rotor 30 is configured to be rotatable by bearings 11a, lib. A rotational angle sensor 14 that detects a- rotational angle is provided on the rotor 30. The rotational angle sensor 14 is formed of, for example, a resolver or a Hall sensor and a magnet or an encoder. A stator core 3 is provided so as to face the permanent magnets 1; and, for example, the stator core 3 can be formed by laminating magnetic steel sheets or formed of a dust core. Armature windings 4 are wound around the stator core 3. A stator 40 is fixed to a frame 13 by press-fitting, shrink-fitting, or the like and the frame 13 is fixed to a housing 12. [0013] Supplemental grooves 5 are provided at portions facing the permanent magnets 1 of the stator core 3. Further, the supplemental grooves 5 are provided at portions in a rotational axis direction. Fig. 1 exemplifies the supplemental grooves 5 arranged at three positions in the axis direction. Although description will be made in detail later, axial positional relationship in which the supplemental grooves 5 are provided is set according to the protrusion portions 8. [0014] Cross-sectional views in a plane perpendicular to the rotational shaft 10 of Fig. 1 are shown in Figs. 2, 3, 4, and 5. Figs. 2, 3, 4, and 5 are cross-' sections taken along the lines A1-A2, B1-B.2, C1-C2, and D1-D2 shown in Fig. 1, respectively. In these cross-^sectional views, the permanent magnets 1 are attached on the surface of the rotor core 2; and in these examples, the number of poles (the number of magnetic poles) is 10; Further, the permanent magnet 1 is semi cylindrical in cross-sectional shape and reduces harmonic components of magnetic flux to form induced voltage into a sine wave shape; and accordingly, torque pulsations are reduced. The rotor core 2 is provided with the protrusion portions 8, each being formed by a part of the rotor core 2 and made of the same material. The protrusion portions 8 serve to fix and retain the permanent magnets 1 so as not to slip in a circumferential direction. [0015] Meanwhile, the stator core 3 of the stator 40 is provided with slots 6/ each for winding the armature. Winding 4. In an example of Fig. 2, each of the armature windings 4 is intensively wound around a tooth 7 extending in the radial direction of the stator core 3; and the number of the slots is 12. . The armature winding is wound around all of the 12 teeth. Further, ] the number of phases' of the permanent magnet type rotary electric machine is 3; and if they are expressed by U phase, V phase, and W phase, the windings are arranged in the arrangement of D1+, U1-, VI-, V1+, W1+, W1-, 02-, U2+, V2+, V2-, W2-, and W2+ as shown in Fig. 2. In this case, signs + and - denote winding directions; and the winding directions of + and - are opposite to each other. Further, U1+ and Dl- are connected in series and U2- and U2+ are also connected in series. These two series circuits can be connected -in parallel or connected in series- The same applies to V phase and W phase. Moreover, three phases can be connected in star connection or in delta connection. [0016] Fig. 2 is the cross-section taken along the line A1-A2 shown in Fig. 1; and in ' the cross-section thereof, the supplemental groove is not provided in .the stator core 3 but the protrusion portion 8 is provided in the rotor core 2. Fig. 3 is the cross-section taken along the line B1-B2 shown in Fig. 1; and in the cross section thereof, the supplemental groove 5 is provided in the stator core 3 and the protrusion portion 8 is provided in the rotor core 2. Fig. 4 is the cross-section taken along the line C1--C2 shown in Fig. 1; and in the cross-section, thereof, the supplemental groove 5 is not provided in the stator core 3 and the protrusion portion 8 is not provided in the rotor core 2. Fig. 5 is the cross-section taken along the line D1-D2 shown in Fig. 1; and in the cross-section thereof, the supplemental groove 5 is provided in the stator core 3 but the protrusion portion 8 is not provided in the rotor core 2. [0017] Fig. 6 is a partial perspective view showing the' stator core of the permanent magnet type rotary electric machine of Fig- 1. Fig. 6 shows only one half, that is, six teeth out of the 12 teeth for ease of understanding the present invention. The stator core 3 is provided- with the teeth 7 radially extending and facing the permanent magnets; and the supplemental grooves 5 are provided on the surface of the end of each tooth 7, the surface being faced to the rotor. The supplemental grooves 5 are arranged at three positions in the axis direction and the supplemental groove 5 near the axial center is longer in axial length than other two supplemental grooves 5. [0018] Fig. 7 is a perspective view showing the rotor of the permanent magnet type rotary electric machine of Fig. 1. Portions in which the rotational shaft 10 is protruded from the end faces of the rotor core 2 are omitted for simplicity. Furthermore, a protective cover of the permanent magnet 1 is omitted. The permanent magnets 1 are provided on the surface of the rotor core 2 and each of the protrusion portions 8 is provided between the adjacent permanent magnets 1. Further, the protrusion portions 8 are provided on both axial end portions. In the case where the protrusion portion 8 is formed of a magnetic material, the configuration of magnetic circuit in a cross-section of a plane perpendicular to the rotational shaft 10 is different between a cross-section of a portion in which the protrusion portion 8 is present and a cross-section of a portion in which the protrusion portion 8 is not present. [0019] If the protrusion portions 8 are present, effects exist in that the protrusion portions 8 prevent the permanent magnets 1 from being out of position in the -circumferential direction and make the permanent magnets 1' position easily; however, in the case where the presence or absence of the protrusion portions 8 changes in the axis direction or the size of the protrusion portion 8 changes in the axis direction, a problem exists in that cogging torque increases because the magnetic circuit is not uniform in the axis direction. More particularly, cogging torque caused by variations in attachment position, shape, and/or characteristics of the permanent magnet 1 may increase. The present invention has an object to provide the arrangement of the supplemental groove 5 capable of effectively reducing the cogging torque in the permanent magnet type rotary electric machine including the rotor 30 that has the configuration of at least two types of magnetic circuits. [0020] Hereinafter,, the arrangement of the supplemental groove 5 intended for reduction -in cogging torque will, be described in detail. Fig. 10 is a cross-sectional view of the permanent magnet type rotary electric machine of Fig. 1, that is, the permanent magnet type rotary electric machine including the stator core of Fig. 6 and the rotor of the Fig. 7 and exemplarily shows the cross-sectional view in a plane passing through the rotational shaft. In Fig. 10, symbols A and B are used to discriminate the configuration of the magnetic circuit on the rotor 30 side. A region in which the protrusion portion 8 is present is expressed by A; and a region in which the protrusion portion is not present is expressed by B. In this example, the protrusion portions 8 are provided on both axial end portions; and therefore, arrangement is made in the order of the region A, the region B, and the region A from above toward the page space in Fig. 10. Further, the axial length of these regions is illustrated by symbols Lrl, Lr2, and Lr3, respectively. Meanwhile, in the stator 40 side, a region in which the supplemental groove 5 is absent is expressed by X; and a region in which the supplemental groove 5 is present is expressed' by Y. Arrangement is made in the order of X, Y, X, Y, X, Y, and X from above toward the page space in Fig. 10. Further, the axial length of these regions is illustrated by symbols Lsl, Ls2, Ls3, Ls4, Ls5, Ls6, and Ls7, respectively. [0021] A dashed line in- Fig. - 10 is a line showing a plane perpendicular to the shaft. The same applies to Fig. 11, Fig. 12, and Fig. 13. The dashed line in Fig. 10 shows that the position of an. axial end face of the protrusion portion 8 is the same as that of an axial end face of the supplemental groove 5. For example, the lower end face of the upper protrusion portion 8 toward the page space corresponds to the lower end face of the upper supplemental groove 5 toward the page space. Incidentally, Fig. 10, Fig. 11, Fig. 12, and Fig. 13 show that the axial position on the stator side corresponding to the axial position on the rotor side is the same at the positions of the dashed line. In Fig.10, a relation equation of Lrl=Lsl+Ls2 is established. Similarly, relation equations of Lr2=Ls3+Ls4+Ls5 and Lr3=Ls6+Ls7 are established. [0022] Next, description will be made on a mechanism in which cogging torque can be considerably reduced if the supplemental grooves 5 are arranged as shown in Fig. 10. Ten permanent magnets 1 in the rotor in Figs. 2 to 5 and Fig. 7 show examples in which the attachment positions of the permanent magnets are equally spaced and the cross-sectional shapes thereof are also the same. However, in fact, variations in manufacture are possible. For example, even when the permanent magnets 1 are accurately attached, the attachment position of the permanent magnets 1 is not equally spaced and may be out of position by approximately several urn to 100 um in the circumferential direction. Meanwhile, there may be assumed a case where a cross-sectional shape is not also an ideally bilaterally symmetrical shape, one of the right and left thicknesses increases, and the other thickness decreases. Such an example is shown in Fig. 14. [0023] Fig. 14 shows that the attachment position of the permanent magnets is out of position from an ideally equally-spaced position in a direction shown by an arrow. Further, a cross-sectional shape is not also bilaterally symmetrical and a portion of a semi cylindrical apex moves in the arrow direction, which shows the cross-sectional shape is not bilaterally symmetrical. As described above, if variations occur in the' rotor, cogging torque increases and a component” which pulsates the number of times same as the number of slots per one rotation- of the rotor appears. In the example of Embodiment 1, the number of slots is 12; and therefore, cogging torque appears 12 times by one rotation of the rotor, that is, cogging torque appears at a period of 30 degrees (360 degrees/12=30 degrees) in mechanical angle. [0024] Figs. 15, 16 are views showing waveforms of a cogging torque for a rotational angle of 30 degrees {mechanical angle). A waveform shown by the curve A-X in Fig. 15 shows a cogging torque waveform at the time when variations in the permanent magnets 1 are in a state of Fig. 14 in the case where the stator 40 and the rotor 30 of the permanent magnet type rotary electric machine are equally configured in "the. presence of the protrusion portion and the absence of the supplemental groove." Meanwhile, a waveform shown by the- curve A-Y shows a. cogging torque waveform at the time when variations in the permanent magnets 1 are in the state of Fig. 14 in the case where, the' stator 40 and the rotor 30 are equally configured in "the presence of the protrusion portion and the presence of the supplemental groove." It is observed that these waveforms 'are large in component with a period of 30 degrees in mechanical angle, which shows that the cogging torque is large. [0025] Further, a waveform shown by an explanatory note "invention" in Fig. 15 is a waveform of cogging torque when variations in the permanent magnets 1 are in the state of Fig. 14 in the case where "the • presence of the protrusion portion and the absence of the supplemental groove" and "the presence of the protrusion portion and the presence of the supplemental groove" are simultaneously provided, so that the waveform becomes an average waveform of two waveforms. This waveform is extremely small in component with a period of 30 degrees in mechanical angle and the cogging torque can be considerably reduced. This makes, phases of components with a period of 30 degrees in mechanical angle invert by providing the supplemental grooves 5 to cancel out cogging torque at the portion in which the supplemental groove 5 is provided and cogging torque at the portion in which the supplemental groove 5 is 'absent; and accordingly, the cogging, torque 'is considerably reduced. [0026] Similarly, Fig. 16 shows cogging torque wave forms at the time when variations in the permanent magnets 1 are in the state of Fig. 14 in the case where a waveform shown by the curve B-X is in "the absence of the protrusion portion and the absence of the supplemental groove" and a waveform shown by the curve B-Y is in "the absence of the protrusion portion and the presence of the supplemental groove." Meanwhile, a waveform of the invention is a waveform of cogging torque when variations in the permanent magnets 1 are in the state of Fig. 14 in the case where "the absence of the protrusion portion and the absence of the supplemental groove" and "the absence of the protrusion portion and the presence - of the supplemental groove" are simultaneously provided, so that the waveform becomes an average waveform, of two -waveforms. This case also shows that the waveform is extremely small in component with a period of 30 degrees in mechanical angle, and therefore, the cogging torque can be considerably reduced. [0027] Further, attention needs to be paid that the A-X \ waveform of Fig. 15 differs from the B-X waveform of Fig. 16 and the A-Y waveform of Fig. 15 differs from the B-Y waveform of Fig. 16. This shows that cogging torque cannot be sufficiently cancelled out in the A-X waveform and the B-Y waveform and cogging torque cannot also be sufficiently cancelled out in the B-X waveform and the A-Y waveform. Therefore, in order to sufficiently reduce a component with a period of 30 degrees in mechanical angle, the arrangement of the supplemental groove 5 needs to consider the arrangement of the protrusion portions 8, that is, magnetic circuit design on the rotor 30 side needs to be considered. Consequently, if the arrangement of the supplemental grooves is made as shown in Fig. 10, the A-X waveform and the A-Y waveform can be cancelled out in the region of the axial length Lrl and the region of the axial length Lr3; and the B-X waveform and the B-Y waveform can be cancelled out in the region of the axial length Lr2; and therefore,, the cogging torque can be considerably reduced. The axial length of providing • the supplemental groove 5 should be approximately 1/2 (for example, 1/2 ±10%) of the axial length of the stator core, and more preferably 1/2 (for example, l/2±5%) . [0028] Further, in Fig. 10, if the following is given: Lsl=Ls2=Lrl/2, Ls4=Ls3+Ls5=Lr2/2, and Ls6=Ls7=Lr3/2, the axial length of the regions of. A-X is equal to that of the regions of A-Y, and the axial length of the regions of .B-X is equal to that of the regions of B-Y; and therefore, it becomes a configuration in which the effect of reduction in cogging torque can be more exerted. [0029] Fig. 17 is a view for comparing waveforms of cogging torque of the configuration of Fig.10 with cogging torque in the case where the supplemental grooves 5 are not provided as a known example. Standardization is made so that the maximum value of the- cogging torque of the known example is 100%. Dashed lines depict the waveform of cogging torque of the known example and components which pulsate 12 times by one rotation of the rotor are noticeably observed. Meanwhile, the invention shown by a solid line is ' the cogging torque waveform of the configuration of Fig. 10 and confirmation can be made that cogging torque can be considerably reduced. Fig. 18 is frequency analysis results of the cogging torque waveforms of Fig. 17. The invention can considerably reduce a 12th-order component of • cogging torque caused by variations on the rotor 30 side as compared to the known example. [0030] The above has described the case where a pattern of variations is as shown in Fig. 14. In order to verify the advantageous effect of the invention, a histogram is created for the 12th-order component of cogging torque (component in which the number of pulsations corresponds to the number of slots) in a total of 10,000 rotary electric machines on the assumption that random variations occur in each of the ten permanent magnets. The results are shown in Fig. 19 and Fig. 20. Fig. 19 is a histogram of the known example and Fig. 20 is a histogram of the configuration of “Fig. 10 (the invention). A vertical axis is frequency; and a horizontal axis is the ' amplitude of cogging torgue and is shown in percent value as in a vertical axis of Fig. 17. Comparison is made on the assumption that variation conditions on the rotor 30 side are the same. Cogging torque is widely distributed in Fig. 19 and therefore a problem exists in- that the cogging torque becomes large according to the pattern of variations. [0031] However, in the configuration of the invention, it shows that the cogging torgue is small regardless of the pattern of variations in the permanent magnets 1 of the rotor 30. That is, the permanent magnet type rotary electric machine with high robustness against variations in manufacture on the rotor 30 side can be obtained. The configuration of the known supplemental grooves is targeted at the effect of reduction in component in which the number of pulsations per one rotation of the rotor corresponds to the least common multiple of the number of poles and the number of slots; and therefore, the effect of reduction cannot be sufficiently obtained for a component (component in which the number of pulsations corresponds to the number of slots) generated by the variations on the rotor 30 side. [0032] The configuration of Fig. 11 can also obtain an effect similar to that of Fig. 10. Fig. 11 is an example in which the arrangement of the protrusion portions 8 of the rotor core 2 is the same as that of Fig. 1.0, but the arrangement of the supplemental grooves 5 of the statdr core 3 is different. The example is such that the supplemental groove 5 provided in the region of Ls4 shown in Fig. 10 is provided in regions of Ls3 and Ls5; and accordingly, the supplemental grooves. 5 are arranged at two places in the axis direction.- The arrangement of such' supplemental grooves 5 can also obtain the effect of reduction in cogging torque by a mechanism described in the example of Fig. 10. Furthermore, as in Fig. 10, the axial length of providing the supplemental groove 5 should be. approximately ½ (for example, 1/2 ±10%) of the axial length of the stator core, and more preferably 1/2 (for example, l/2±5%) . [0033] Further, it goes without saying that, in Fig. 11, if the following is given: Lsl=Ls2=Lrl/2, Ls4-Ls3+Ls5=Lr2'/2, and Ls6=Ls7=Lr3/2, the axial . length of the regions of A-X is equal to .that of the'regions of A-Y, and the axial length of the regions of B-X is equal to that of the regions of B-Y; and therefore, it becomes a configuration in ' which the effect of reduction.in cogging torque can be more exerted. [0034] Although the examples in which the protrusion portions 8 are arranged on both axial end portions are described in Figs. 6, 7, 10, and 11, an application range of the present invention is not limited to these examples. Other examples are shown in Figs. 8 and 9. Fig. 8 is a partial perspective view showing a different stator core of a permanent magnet type rotary electric machine according to Embodiment 1. Fig. 8 shows one half, that is, six teeth out of the 12 teeth for ease of understanding. Fig, .9 is a perspective view showing a different -rotor of a permanent magnet type rotary electric machine according to Embodiment 1, Portions in which the rotational shaft 10 is protruded from the' end faces of the rotor core 2 are omitted for simplicity. Furthermore, a protective cover of the permanent magnet 1 is omitted. The permanent magnets 1 are provided ■ on the surface of the rotor core 2. Protrusion portions 8, each being formed by • a part of the rotor core 2 and made of the same pnaterial, ..are provided between 'the adjacent permanent magnets 1 to fix and retain the permanent magnets 1. Further,, the protrusion portions 8 are provided at positions apart from both axial end portions by a predetermined distance. [0035] Cross-sectional views corresponding to these •examples are shown in Fig. 12 and Fig. 13. In Fig. 12 and Fig. 13, the protrusion portions 8 of the rotor core are provided at two places, in the axis direction and the axial positions thereof are apart from end portions by Lrl or Lr5. "With respect to the configuration of such a rotor, Fig. 12 shows an example in which three supplemental grooves 5 are arranged in the axis direction of the stator core 3 and Fig. 13 shows an example in which two supplemental grooves 5 are . arranged. If the arrangement of the supplemental grooves 5 is made as shown in Fig. 12, the A-X waveform and the A-Y waveform can be cancelled out in a region of an axial length Lr2 and a region of an axial length Lr4; and the B-X waveform and the- B-Y waveform can . be cancelled out in a region of axial lengths Lrl, Lr3, and Lr.5; and therefore, the cogging torque can be considerably reduced. The axial length of providing the supplemental groove '5 should be approximately 1/2 (for example, 1/2 ± 10%) of the axial length of the ' stator core, and more preferably 1/2 (for example, l/2±5%) . [0036] Further, in Fig. 12, if the following is given: Lsl=Ls2=Lrl/2, Ls-3=Ls4=Lr2/2, Ls5+Ls7=Ls6=Lr3/2, Ls8=Ls9=Lr4/2, and LslO=Lsll=Lr5/2, the axial length of the regions of A-X is equal to that of the regions of A-Y, and the axial length of the regions of B-X is equal to that of the regions of B-Y; and therefore, it becomes a configuration in . which the effect of reduction in cogging torque can be more exerted. [0037] Meanwhile, if the arrangement of the supplemental grooves 5 is made as shown in Fig. 13, the A-X waveform and the A-Y waveform can be cancelled out in a•region of an axial length Lr2 and,a region of . an axial length Lr4; and the B-X waveform and the B- Y waveform can be cancelled out in a region of axial lengths Lrl, Lr3, and Lr5; and therefore, the cogging torque can be considerably reduced. The axial length of providing the supplemental groove 5 should be approximately 1/2 {for example, 1/2 ±10%) of the axial length of the stator core, and more preferably 1/2 (for example, l/2±5%) . [0038] Further, in Fig. 13, if the following is given: Lsl=Lrl, Ls2=Ls3=Lr2/2, ~Ls7=Ls8=Lr4/2, Ls4+Ls6=Lsl+Ls5+Ls9=(Lrl+Lr3+Lr5)/2, and Ls9=Lr5, the axial length of the' regions of A-X is equal to that of the regions of A-Y, and the axial length of the regions of B-X is equal to that of the regions of B-Y; and therefore, it becomes a configuration in which the effect of reduction in cogging torque can be more exerted. [0039] Although the examples shown in this time are those in which the presence or absence of the protrusion portion of the • rotor core changes in the axis direction, the application of the present invention is not limited to these examples. In the case where not less than two types of regions different .in magnetic circuit design of the rotor are provided in the axis direction, 'for example, in the case where the protrusion portions are not present but concave portions are present between the adjacent permanent magnets, in the case where the cross-sectional shape of the permanent magnet changes, and the like, the supplemental grooves are provided in a similar manner; and accordingly, cogging torque can be considerably reduced. [0040] As described above, in the rotor, if the configuration is made such that not less than two regions different in magnetic circuit design are provided in the axis direction, the regions being different by changing at least one of a cross-sectional shape in the rotational axis direction in a cross-section perpendicular' to the. rotational shaft of the rotor core and a cross-sectional shape in a cross-section perpendicular to the rotational shaft of the permanent magnets; the supplemental grooves are provided -in the axial partial regions of-the stator core; and the region in' which the supplemental groove is provided is each partial region for each region facing a region same in magnetic circuit design of the rotor, it becomes possible to considerably reduce cogging torque (component in which the number of pulsations by one rotation of the rotor corresponds to the number of slots), the cogging torque being generated by variations on the rotor side, for example, an attachment position error, a shape error, and/or variations in magnetic characteristics of the permanent magnets. Incidentally, the above mentioned region facing a region same in magnetic circuit design of the rotor represents a stator side region corresponding to .a region same in' magnetic circuit design of the rotor at the time when the rotor side and the stator side are delimited in a plane perpendicular to the shaft.. [0041] Further, if the configuration is made such that the region in which the supplemental groove is provided is a half of each axial length for each region facing a region same in magnetic circuit design of the rotor, an effect is sufficiently exerted in that cogging torque generated at the region in which the-supplemental groove is provided and cogging torque generated at the region in which the supplemental groove is not provided are cancelled out. Therefore, it" becomes possible to more effectively reduce cogging torque (component in which the number of pulsations by one rotation of the rotor corresponds to the number of slots) , the cogging torque being generated by variations on the rotor side, for example, an attachment position error, a shape error, and/or variations in magnetic characteristics of the permanent magnets. [0042] If the configuration is made such that a region provided with the protrusion portion in an axial portion of the rotor core is present, positioning of the permanent magnets can be made because of the presence of the protrusion portion; and therefore, an effect exists in that the accuracy of the attachment position improves. At the same time, the configuration is made such that the supplemental groove is provided; and accordingly, this allows to prevent cogging torque generated by the difference in magnetic circuit design due to the protrusion portion from increasing, and to effectively, reduce cogging torque (component in. which the number of pulsations by one rotation' of the rotor corresponds to the number of slots) generated by variations on the rotor side. Furthermore, the configuration is made such that the supplemental grooves are not provided at axial end portions as shown in Figs. 6 and 8; and accordingly, a portion having a large gap formed with respect to the rotor at each axial end portion can be reduced as compared to the case where the supplemental grooves are provided at the axial end portions. This increases an effect that prevents foreign particles from entering into the gap portion between the stator and the rotor. [0043] Next, the shape of the supplemental groove will be described. Fig. 21 is a view for explaining the width of the supplemental groove Wd, the depth of the supplemental groove Hd, the width of a. slot opening Ws, and the thickness of a tooth end portion Hs. In the structure of the present invention, phases of components in which the number of- cogging torque pulsations corresponds to the number of slots. are inverted-at a portion in which the supplemental groove is provided and a portion in which the supplemental groove is not provided to obtain a canceling out effect; and therefore, the selection of the shape of the supplemental groove has an influence on the size of the effect. Fig. 22 is a graph in which a horizontal axis is the ratio of the width of the supplemental groove Wd to the width of the slot opening Ws and a vertical axis is overall values of the cogging torque at the time when variations occur in the rotor. [0044] As compared to the case where Wd/Ws=0, in other-words, the supplemental groove is not provided; if Wd/Ws^l.O, the cogging torque is not more than 1/2 of the case where the supplemental groove is not provided. Further, if Wd/Ws^l.25, the cogging torque is 0.00.1 Nm that is an extremely small value. If the cogging torque due to variations of the rotor is suppressed to this extent, for example, in the case where a rotary electric machine is incorporated in an electric power steering apparatus (to be described later), an effect is obtained in that a driver can obtain good steering feeling without feeling cogging torque. [0045] If Wd/Ws^l.0, phases of components in which the . number Of cogging ' torque pulsations corresponds, to . the number of slots can be inverted by changing components of permeance pulsations due to the slots of the stator core; and therefore, cogging torque at the portion in which the supplemental groove 5 is provided and cogging torque, at the portion in which the supplemental groove is absent can be cancelled out each other. In the case where one supplemental groove is provided, Wd/Ws^l.O can be used; and in the case where not less than two supplemental grooves are provided, Wd/Ws^l.O can be used by defining the total of the widths of all the supplemental grooves provided in the tooth as Wd; and accordingly, similar effects can be obtained. Further, the depth of the supplemental groove fid is preferably larger than the thickness of the tooth end portion Hs. Also, phases of components in which the number of pulsations corresponds to the number of slots can be inverted by changing the component of permeance pulsations due to the slots of the st'ator core. It becomes possible to sufficiently exert an effect that cancels out the above mentioned: cogging torque at the. portion in which the supplemental groove is provided and cogging torque at the portion in which the supplemental groove is absent. [0046] Patent Document 1, Patent Document 2, and Patent Document 3 disclose examples in which two or not less than two supplemental grooves are provided in each tooth in the circumferential direction; however, in Embodiment 1, one supplemental groove is provided at -a circumferential center portion in each tooth. A cogging torque Sth-order component (S is the number" of slots) generated by variations on the rotor side is largely involved with an Sth-order component of permeance pulsations due to the slots of the stator; however, an effects exists in that it is easy to change an amplitude and a phase of the Sth-order component of the permeance pulsations by providing one supplemental groove. When the number of the supplemental grooves is smaller, the average length of gap becomes shorter. Therefore, one supplemental groove is provided at the circumferential center portion and only at an. axial portion; and accordingly, degradation of torque • during load application can be minimized. [0047] In. Embodiment 1, the supplemental groove has a shape formed by cutting out the core in a rectangular shape but the supplemental groove is not limited to this shape. For example, it goes without saying that similar effects can be' obtained by a shape formed by cutting out the core in an arc like shape, cutting out in a triangle shape, and the like. Furthermore, the permanent magnet type rotary electric machine with 10 poles and 12 slots shown in Embodiment 1 is larger in the winding factor of a fundamental wave" than that with the number of poles : the number of slots=2 : 3, which has been conventionally and widely used; and therefore, the rotary electric machine in Embodiment 1 is suitable for use in small size and high output machines. In addition,, as compared to a rotary electric machine with the same number of slots, the least common multiple of the number of poles and the number of slots is. 60 in the case of 10 poles and 12 slots, and 24 in the case of 8 poles and 12 slots; and accordingly, the rotary electric machine with 10 poles and 12 slots is smaller in - cogging torque of order of the least common multiple. However/ a . problem exists in that cogging torque caused by variations .on the rotor side of the rotary electric machine with 10 poles and 12 slots is larger than that with 8 poles and 12 slots and robustness against variations in manufacture is low. However, the problem can be solved by Embodiment 1; and therefore, there can be obtained a permanent magnet type rotary electric 'machine which achieves small size and high output, and reduction in cogging torque caused by variations in manufacture at the same time. [0048] ' Embodiment 1 shows the example in which the protrusion portions are provided on the rotor core. If the protrusion portions are present, effects exist in that positioning of the permanent magnets can be made and the permanent magnets can . be prevented from being out of position in the circumferential direction;-whereas, a problem exists in that a cogging torque Sth-oxder component (S is the number of slots) becomes large due to the difference of magnetic circuit design. However, Embodiment 1 can solve the problem and can achieve the effects of positioning the permanent magnets and preventing the permanent magnets from being out of position in the circumferential direction, and the reduction in the cogging torque Sth-order component at the same time. [0049] Embodiment 2. Embodiment 1 describes the example in which the number of poles (the number of magnetic poles) is 10 and the number of slots is 12; however, the present invention is not limited to this example. In the case of the combination of the following relationship 0.75