JP2002354780A - Cylindrical field type linear motor - Google Patents

Abstract

(57) [Problem] To provide a low-loss cylindrical field linear motor that can provide a sufficiently large maximum thrust per unit volume, is not affected by a gap suction force, is easy to maintain, and has a low loss. SOLUTION: An armature core 12 having a plurality of slots for accommodating a group of ring-shaped coils 11 in a thrust direction is provided over an entire circumference of the ring-shaped coil.
A mover having a group of armature cores arranged such that the surface on the magnetic gap side thereof is a regular polygon, and a frame 17 having a groove for positioning the group of armature cores in the circumferential direction; A hollow magnetic rod 14 concentrically arranged on the center axis of the group of coiled coils, and a magnetic rod 14 arranged at an equal pitch in the thrust direction on the outer peripheral side of the hollow magnetic rod 14 and magnetically facing the inner surface of the regular polygon of the armature core. And a stator having a plurality of ring-shaped magnets 13 formed with a gap therebetween, thereby forming a cylindrical field type linear motor.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a movable armature type linear motor, and more particularly to a cylindrical field type linear motor having a cylindrical field structure and an associated armature.

[0002]

2. Description of the Related Art Conventionally, there has been a so-called linear slider capable of freely moving a table with respect to a fixed base by a linear motor as shown in FIGS. 5 is a plan view of a conventional linear slider viewed from above, FIG. 6 is a side view of FIG. 5 as viewed in the direction of arrow C, and FIG. 7 is a sectional view of FIG. 6 as viewed in the direction of arrow AA in FIG. . 5 to 7, reference numeral 40 denotes a mover, 50 denotes a stator, and 60 denotes a linear motor (linear slider). Reference numeral 59 denotes a fixed base, 41 denotes guide rails provided on both left and right ends of the fixed base 59, 42 denotes a slider which forms a linear guide in pairs with the guide rail 41, and 43 denotes a slider along the guide rail 41 (in the left-right direction in FIG. 7 (in the direction perpendicular to the plane of FIG. 6). Reference numeral 51 denotes a field yoke fixed in parallel on a fixed base 59, 52 denotes a plurality of permanent magnets disposed along the field yoke 51 (in the horizontal direction in FIG. 5 and in the direction perpendicular to the paper in FIG. 6), and 44 denotes a permanent magnet. An armature core formed by laminating electromagnetic steel plates provided to face the permanent magnet 52 with a magnetic gap therebetween, and an armature coil 45 (FIG. 6) wound around the winding accommodating portion of the armature core 44. , And is fixed by a resin mold 46 (FIG. 6). As described above, the linear motor 60 is
And the stator 50, and the mover 40 is
4 and the armature coil 45 constitute a main part.
Is composed of a field yoke 51 and a field permanent magnet 52,
This constitutes a linear motor 60 having a gap facing structure. The field part in which the field permanent magnets 52 (FIG. 7) are fixed to the field yoke 51 at a constant pitch is fixed so as to be sandwiched in parallel on the same surface on which the linear guide is mounted in parallel. The armature portion around which the armature coil 45 is wound is fixed to the sliding portion of the linear guide through a magnetic gap with the field portion, and when the armature coil 45 is energized, it is moved in the thrust direction. It generates thrust.

[0003] On the other hand, as a cylindrical linear motor, a linear motor described in Japanese Utility Model Laid-Open No. 6-62787 is known. FIG. 8 shows this cylindrical linear motor. FIG. 8A is a perspective view of a cylindrical linear motor.
(B) is a sectional view of the cylindrical linear motor. A cylindrical armature core 813 laminated with a ferromagnetic material (FIG. 8)
On the inner diameter side of (b), a cooling pipe 811 whose inside becomes a refrigerant passage 812 is embedded, and on the outer diameter side of the armature core 813,
A cylindrical armature winding 814 is fitted. The armature winding 814 electrically joins each phase element coil at a phase of 120 ° to each coil side to form a band-shaped smooth armature coil, and further joins the smooth armature coil to each phase side to form a cylinder. After spirally wrapping it around the core of the shape at an extreme pitch and molding, it is fixed with resin or the like. On the completed armature winding 814, a stator magnetic pole having U, W ', and V as a single pole pitch P is formed in a spiral shape in the axial direction. The stator 81 includes a cooling pipe 811, an armature core 813, an armature winding 814, and a can 815. Both ends of the armature core 813 are supported by a support portion 816,
816 (FIG. 8A). Support portion 816,
816 includes connecting portions 811 provided at both ends of the cooling pipe 811.
a, 811b (FIG. 8 (a)). The refrigerant is supplied to the connection portion 811a, flows through the cooling pipe 811 and is discharged from the connection portion 811b. The outer diameter of the can 815 of the stator 81 is fixed to the field pole 8 via a magnetic gap.
2a, 82b and 82c are opposed to each other. Field pole 82
Reference numerals a, 82b, and 82c denote cylindrical permanent magnets in the axial direction.
A field in which permanent magnets 821 having an axial length slightly shorter than the pole pitch P of the armature windings 814 that are divided and radially magnetized face each other so that their polarities are opposite to each other. The magnetic poles 82a, 82b, and 82c are arranged in series at an equal pole pitch in the axial direction, with the polarities of adjacent field magnetic poles being alternated. The field poles 82a, 82b, 82c are fixed to the inner diameter of a ferromagnetic cylindrical field yoke 822.
The mover 2 is composed of the field magnetic poles 82a, 82b, 82c and the field yoke 822. Therefore, when a balanced three-phase alternating current is supplied to the armature winding 814, the current of each phase changes in a sinusoidal manner with a phase of 120 ° within each pole pitch, and the magnetic field generated by the armature winding 814 changes to a pole. The direction is reversed by the pitch. In this magnetic field, the field poles 82a, 82 having the upper and lower polarities reversed.
When b and 82c are inserted, the lower half field pole and the upper half field pole receive an armature reaction in the same direction, so that a thrust is generated.

[0004]

However, the conventional linear motor has the following problems. First, FIG.
7 have the following problems. 1) A gap attraction force proportional to the square of the magnetic flux density B between the armature core 44 and the field permanent magnet 52 acts to cause distortion in the drive table or excessive load on the linear guide. The guide life was adversely affected. 2) The linear motor is basically a substitute for the “ball screw + rotary motor”, and the ideal linear motor has the best shape that can be stored in the space of the ball screw.
However, since the maximum thrust of this linear motor is determined by the field facing area (gap area) S of the armature, in order to secure a required thrust, it is determined by “width W × length L in the thrust direction” of the armature. It was decided.

On the other hand, the cylindrical linear motor shown in FIG. 8 has the following problems. 1) In FIG. 8B, since the permanent magnet 821 is divided into upper and lower parts, a gap is formed in the divided part, and no thrust can be obtained in this gap part. Therefore, the maximum thrust per unit volume is sufficiently large. I couldn't take it. 2) It is necessary to flow the refrigerant over the entire length on which the armature winding 814 is laid, which requires a large amount of the refrigerant, and
It was necessary to provide a liquid-tight structure over the entire length inside the stator. Further, for example, when the refrigerant leaks due to some trouble, it is necessary to remove the stator 81 from the support portion 816 and perform maintenance on the sealing portion (not shown) of the cooling pipe 811. Was expensive. 3) This cylindrical linear motor has no core (coreless)
Therefore, the energy density is low, and more current must be passed to obtain the same torque as that of the cored type linear motor, so that the linear motor has a large loss. The present invention has been made in order to solve the above-mentioned problem, and it is possible to obtain a sufficiently large maximum thrust per unit volume, and further, there is no influence of a gap suction force, maintenance is easy, and a low-loss cylindrical magnetic field is provided. It is an object of the present invention to provide a linear motor.

[0006]

In order to solve the above problems, the invention of a cylindrical field type linear motor according to claim 1 is provided.
In a cylindrical field type linear motor including a mover and a stator, the mover is configured by disposing a plurality of ring-shaped coils each having a winding wound in a ring shape at an equal pitch in an axial direction (thrust direction). A ring-shaped coil group, and a plurality of armature cores each including a plurality of slots for accommodating the ring-shaped coil group in a thrust direction. The armature core is provided on the magnetic gap side of the armature core over the entire circumference of the ring-shaped coil. Armature core group arranged so that the surface of the armature becomes a regular polygon, and a frame provided with a groove for positioning the armature core group in the circumferential direction, wherein the stator has the ring shape. A hollow magnetic rod concentrically arranged on the central axis of the coil group,
A plurality of ring-shaped magnets arranged at equal pitches in the thrust direction on the outer peripheral side of the hollow magnetic rod and provided with a magnetic gap with respect to the inner surface of the regular polygon of the armature core. It is characterized by. According to a second aspect of the present invention, in the cylindrical field-type linear motor according to the first aspect, the length of the ring-shaped magnet laid in the thrust direction is “the length of the cylindrical armature in the thrust direction (L)”. It is characterized by being the sum of "drive stroke". According to a third aspect of the present invention, in the cylindrical field-type linear motor according to the first or second aspect, the height of the teeth located at both ends of each armature core is increased by the height of the main armature core located between the both ends. It is characterized by being lower than the height of the teeth. According to a fourth aspect of the present invention, in the cylindrical field linear motor according to any one of the first to third aspects, an armature portion is formed by fitting an armature core into a ring-shaped coil. Is fitted into the groove of the frame, and the armature portion excluding the inner diameter portion is resin-molded.
According to a fifth aspect of the present invention, in the cylindrical linear motor of any one of the first to fourth aspects, a slot for inserting a nut is provided on an outer periphery of the frame. The invention according to claim 6 provides any one of claims 1 to 5.
In the cylindrical field linear motor described in the paragraph, the crossover and connection of the ring-shaped coil are performed in a fan-shaped space between the armature cores arranged in a regular polygon. The invention according to claim 7 provides any one of claims 1 to 6
In the cylindrical field type linear motor described in the above item, the ring-shaped magnet is constituted by a radially anisotropic ring magnet integrally formed. The invention according to claim 8 is the cylindrical field linear motor according to any one of claims 1 to 6, wherein the ring-shaped magnet is
A plurality of fan-shaped magnet pieces are combined to form a ring-shaped fan-shaped magnet. According to a ninth aspect of the present invention, in the cylindrical field-type linear motor according to any one of the first to eighth aspects, cooling jacket holes are formed at four corners of the frame, and the cooling jacket holes at the four corners are communicated with each other. The cooling jacket is thus constituted. The invention according to claim 10 is the invention according to claims 1 to
10. The cylindrical field type linear motor according to any one of items 9, wherein the shape of the regular polygonal armature is at least a regular decagon. An eleventh aspect of the present invention is the cylindrical field linear motor of the tenth aspect, wherein the ring-shaped coil has a regular polygon having the same angle as the regular polygonal armature core. According to a twelfth aspect of the present invention, in the cylindrical field linear motor according to any one of the first to eleventh aspects, the shape of the outer peripheral surface of the ring-shaped magnet is processed into an arc shape when viewed in a radial cross section. Features. With the above configuration, when the field facing area (gap area) is S and the outer diameter of the cylindrical field is Df (gap area), S = π × Df × length in the thrust direction. Is effectively used to obtain a cylindrical field type linear motor that can design the field facing area (gap area) S to the maximum. Accordingly, by doing so, a cylindrical field type linear motor having a large maximum thrust per unit volume can be obtained, and the gap suction force can be reduced by 3%.
Because it is 60 degrees, they cancel each other,
There will be no effect.

[0007]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS An embodiment of the present invention will be described below with reference to FIGS. FIG. 1 is a front view of a cylindrical magnetic field type linear motor of the present invention, showing a state in which a bracket and sub teeth of FIG. 2 are removed, and FIG.
FIG. 3 is a perspective view of the same, and FIG. 4 is a perspective view showing a specific example of the ring-shaped magnet. In FIG. 1, 11 is a ring-shaped coil, 12 is an armature core, 13 is a ring-shaped magnet, 14 is a magnetic rod, 17 is a frame, 18 is a resin, 22 is a cooling jacket hole, and 24 is a slot. The mover side mainly includes a ring-shaped coil 11, an armature core 12, a frame 17, and a bracket 25 (FIG. 2), and the stator side mainly includes a ring-shaped magnet 13,
It is composed of a magnetic rod 14. Hereinafter, each of these components will be described. The ring-shaped coil 11 is obtained by winding a plurality of windings in a ring shape, and as shown in FIG. 6) are arranged at equal pitches in the axial direction (thrust direction) of the ring-shaped coils. The armature core 12 includes a large number of armature cores.
The shape of the armature cores is, as shown by 12 in FIG. 2, punched out of a substantially rectangular electromagnetic steel sheet, and one side (the side facing the magnetic rod 14 in the figure) of the ring-shaped coil 1
In the thrust direction, there are eight slots in the thrust direction at equal intervals, each having two slots each having a size large enough to accommodate only one ring-shaped coil 11, It has a so-called comb shape. As shown in FIG. 1, one armature core 12 is formed by laminating a plurality of such armature cores to form one pole, which is arranged over the entire circumference of the ring-shaped coil (12 poles in FIG. 1). By doing
The surface on the magnetic gap side of the armature core is formed in a regular polygon (a regular dodecagon in FIG. 1).

The ring-shaped magnet 13 is disposed on the outer peripheral side of the magnetic rod 14 via a magnetic gap (mechanical gap) with respect to the inner surface of the regular polygonal armature core. The plurality of ring magnets 13 are arranged on the magnetic rod 14 in the thrust direction at equal pitches, and the length laid in the thrust direction is “the length of the cylindrical armature in the thrust direction (L). ”And“ drive stroke ”. The magnetic rod 14 is inserted into the central opening of the ring-shaped magnet 13 to support the ring-shaped magnet 13 and form a part of a magnetic circuit. The length in the thrust direction is the sum of the “length in the thrust direction (L) of the cylindrical armature” and the “drive stroke”. The frame 17 is for fixing the 12-pole armature core 12, and has a positioning groove for accommodating these armature cores 12 in the circumferential direction. Ring-shaped coil 1
After the armature portion is formed by fitting the armature core 12 into 1, the entire armature portion is fitted into the positioning groove of the frame 17. Then, the entire armature portion excluding the inner diameter portion is molded with the resin 18 so as to be mechanically rigid and to improve the heat radiation characteristics. In addition, slots 22 for inserting nuts are provided on four outer peripheral surfaces of the frame 17. This allows the user to fix the armature section to the drive table from any direction, and to attach a linear guide to the other surface, increasing the degree of freedom in supporting and fixing the armature on the movable side. Becomes
Regarding the crossover process and the connection process of the ring-shaped coil 11, the fan-shaped space G between the armature cores arranged in a regular polygon is used.
(FIG. 1), so that space can be saved.
In the above description, the ring-shaped coil 11 is formed in a ring shape. However, it is also possible to form a regular polygon having the same square number according to the regular polygon armature core 12.

FIG. 2 is a side sectional view of the cylindrical field type linear motor of the present invention. In FIG. 2 (a), 11 is a ring-shaped coil, 12 is an armature core, 13 is a ring-shaped magnet, 14 Is a magnetic rod, 16 is a sub tooth, 1
7 is a frame, 23 is a non-magnetic spacer, and 25 is a bracket. The two ring-shaped coils 11 are put together in one slot of the armature core 12, and three slits are used to constitute three phases for a total of six coils 11.
A plurality of ring-shaped magnets 13 are arranged on the magnetic rod 14 in the thrust direction at an equal pitch. The magnetic poles of the ring-shaped magnet 13 are polarized in N and S poles in the radial direction (radial direction). In FIG. 2, N, S
This is because only the side close to the ring-shaped coil 11 (the radially outer side) is shown to avoid complication. Also, a ring-shaped magnet 1
Ring magnet 13 adjacent to 3 in the thrust direction
Uses a magnetizer whose magnetization direction is different from that of the former. While repeating this in the thrust direction, and with the non-magnetic spacers 23 interposed between the adjacent ring-shaped magnets, they are alternately laid at a constant pitch. The sub teeth 16 are provided at both ends of the armature core 12 in the thrust direction (the left-right direction in the drawing). The sub-teeth 16 has an edge effect at this part, thereby offsetting the detent force, so that the height of the sub-teeth 16 is lower than the height of the teeth of the main armature core located between both ends. Further, as shown in FIG. 2 (b), when the shape of the outer peripheral surface of the ring-shaped magnet 13 'is processed into an arc shape when viewed in a radial cross section, the change in the gap permeance between the magnet and the armature core is minimized. Therefore, the dependent force can be reduced. By supplying AC power from the polyphase power supply to the ring-shaped coil 11, a moving magnetic field is generated in the cylindrical armature, and thrust can be generated in the thrust direction by interaction with the field magnetic flux.

FIG. 3 is a perspective view of a cylindrical linear motor of the present invention. In FIG. 3, reference numeral 11 denotes a ring-shaped coil, 12 denotes an armature core, 13 denotes a ring-shaped magnet,
14 is a magnetic rod, 16 is a sub tooth, 17 is a frame, and 21 is a cooling jacket hole. In FIG. 3, a part of the frame 17 of FIG. 2 is removed. The cooling jacket holes 21 are for allowing a coolant for cooling the armature to flow, and holes for cooling jackets are formed at four corners of the frame 17 as shown in the figure. Then, in order to allow the refrigerant to flow directly, a bypass pipe for connecting these four corner holes is provided, or a U-shaped bent pipe is symmetrically fitted, and the cooling jacket is connected by connecting the two U-shaped pipes. Make up.

FIG. 4 shows a specific example 2 of the ring-shaped magnet 13.
This is an example. FIG. 4A shows a fan-shaped magnet 1 in which a large number of fan-shaped magnet pieces are joined to form a ring.
FIG. 4B shows a radially anisotropic ring magnet 20 integrally formed in a ring shape.
The common point is that each of them has a ring shape and the arrangement direction of the magnetic poles is polarized to the outer diameter side and the inner diameter side in the radial direction. When a plurality of such ring-shaped magnets 19 (or 20) are arranged in the thrust direction, the ring-shaped magnets 1 adjacent in the thrust direction are arranged.
9 '(or 20') has a magnetization direction different from that of the adjacent one. While repeating this in the thrust direction and alternately interposing the non-magnetic spacer 20 between the adjacent ring-shaped magnets, the magnets are alternately laid at the same pitch. Therefore, since the embodiment of the present invention has the above configuration, the gap suction force between the armature core and the magnet is canceled because the gap shape is cylindrical, so that the drive table may be distorted or the linear guide may be excessively distorted. Since no heavy load is applied, the guide life is not adversely affected. The linear motor has a shape that can be stored in the space of the ball screw.
f × π × thrust length in the thrust direction L ”, so that the thrust per volume can be doubled as compared with the prior art.
Compared to the cylindrical linear motor of FIG. 8, the thrust per volume can be far exceeded. Also, since the armature has an armature winding with a core on the mover,
Compared to the coreless type, only a small current needs to be passed to obtain the same torque, and a low-loss linear motor can be obtained. Further, since the cooling jacket holes are provided at the four corners of the frame supporting the armature, even if the coolant leaks, it is easy to remove the armature composed of the armature, the frame, etc. without removing it. Maintenance of bypass pipes and cooling jacket holes.
Maintenance costs can be reduced.

[0012]

As described above, according to the present invention, since the gap shape is cylindrical, the gap attraction between the armature core and the magnet is canceled, so that the drive table may be distorted, Since no excessive load is applied to the guide, the guide life is not adversely affected. This linear motor has a shape that can be stored in the space of the ball screw. The maximum thrust is obtained by the gap area "gap diameter Df x π x length in the thrust direction L". 2 compared to
Can be doubled. Further, compared with the cylindrical linear motor of FIG. 8, the thrust per volume can be far surpassed. Since the armature has an armature having an armature winding with a core on the mover, only a small current needs to flow to obtain the same torque as compared with the coreless type, and a low-loss linear motor can be obtained. Cooling jacket holes are provided at the four corners of the frame that supports the armature, so even if leakage of refrigerant occurs, it can be easily bypassed without removing the armature composed of the armature, frame, etc. Maintenance of pipes and cooling jacket holes can be performed, and maintenance costs can be reduced. Such an effect can be obtained.

[Brief description of the drawings]

FIG. 1 is a front view showing a cylindrical magnetic field type linear motor according to an embodiment of the present invention, in which a bracket and a sub tooth of FIG. 2 are removed.

FIG. 2 is a side sectional view of the cylindrical field type linear motor of FIG. 1;

FIG. 3 is a perspective view of the cylindrical field linear motor of FIG. 1;

FIG. 4 is a perspective view showing two specific examples of the ring-shaped magnet in FIG.

FIG. 5 is a plan view of a conventional linear slider viewed from above.

FIG. 6 is a side view as viewed from the direction of arrow C in FIG. 5;

FIG. 7 is a cross-sectional view as viewed from the direction of arrows AA in FIG. 6;

FIG. 8 is a diagram illustrating a known cylindrical linear motor.

[Explanation of symbols]

 11: Ring-shaped coil 12: Armature core 13: Ring-shaped magnet 14: Magnetic rod 15: Linear guide 16: Sub-teeth 17: Frame 18: Resin 19: Fan-shaped magnet 20: Radial anisotropic ring magnet 21: Bypass Pipe 22: Cooling jacket hole 23: Non-magnetic spacer 24: Slot 25: Bracket

 ──────────────────────────────────────────────────続 き Continuing on the front page (72) Inventor Masahiko Tanabe 2-1 Kurosaki Castle Stone, Yawatanishi-ku, Kitakyushu City, Fukuoka Prefecture Inside Yaskawa Electric Co., Ltd. (72) Inventor Yukio Tsutsui 2-1 Kurosaki Castle Stone, Yawatanishi-ku, Kitakyushu City, Fukuoka Prefecture F term in Yaskawa Electric Co., Ltd. (reference) 5H641 BB06 BB14 BB18 GG03 GG04 GG08 GG11 GG12 GG15 GG17 HH02 HH05 HH08 HH10 HH12 HH13 HH14 JB05

Claims (12)

[Claims] 1. A cylindrical field linear motor comprising a mover and a stator, wherein the mover comprises a plurality of ring-shaped coils each having a winding wound in a ring shape and having a constant pitch in an axial direction (thrust direction). A ring-shaped coil group arranged by
A plurality of armature cores each having a plurality of slots for accommodating the ring-shaped coil group in the thrust direction are provided over the entire circumference of the ring-shaped coil, and the surface of the armature core on the magnetic gap side is a regular polygon. An armature core group arranged so as to have a groove having a groove for positioning the armature core group in the circumferential direction, wherein the stator is provided on a central axis of the ring-shaped coil group. And a plurality of hollow magnetic rods arranged concentrically on the outer peripheral side of the hollow magnetic rod at equal pitches in the thrust direction and provided with a magnetic gap with respect to the inner surface of the regular polygon of the armature core. And a ring-shaped magnet. 2. The length of the ring-shaped magnet laid in the thrust direction is a sum of “the length (L) of the cylindrical armature in the thrust direction” and “drive stroke”. 2. The cylindrical field type linear motor according to 1. 3. The height of teeth located at both ends of each armature core is lower than the height of teeth of a main armature core located between the both ends. Cylindrical field type linear motor. 4. An armature portion is formed by fitting an armature core into the ring-shaped coil, and the entire armature portion is fitted into the groove of the frame, and the armature portion excluding the inner diameter portion is removed. The cylindrical field type linear motor according to any one of claims 1 to 3, wherein the resin is resin-molded. 5. A cylindrical field type linear motor according to claim 1, wherein a slot for inserting a nut is provided on an outer periphery of said frame. 6. The process according to claim 1, wherein the crossover and connection of the ring-shaped coil are performed in a fan-shaped space between the armature cores arranged in the regular polygon. The cylindrical field type linear motor as described. 7. The cylindrical field type linear motor according to claim 1, wherein the ring-shaped magnet is constituted by a radially anisotropic ring magnet integrally formed. 8. The cylindrical field according to claim 1, wherein the ring-shaped magnet is a fan-shaped magnet formed by joining a plurality of fan-shaped magnet pieces into a ring shape. Magnetic linear motor. 9. The cooling jacket according to claim 1, wherein cooling jacket holes are formed at four corners of the frame, and the cooling jacket holes at the four corners are communicated with each other. Item 3. A cylindrical field type linear motor according to item 1. 10. The cylindrical field linear motor according to claim 1, wherein the shape of the regular polygonal armature is at least a regular decagon. 11. A cylindrical field type linear motor according to claim 10, wherein said ring-shaped coil is a regular polygon having the same angle as the regular polygonal armature core. 12. The cylindrical magnetic field type linear motor according to claim 1, wherein an outer peripheral surface of said ring-shaped magnet is processed into an arc shape when viewed in a radial cross section.