fig1 is an isometric view of a rotor and stator assembly of a limited rotation actuator . fig2 is a side sectional view of the actuator of fig1 at 2 -- 2 also showing rotor shaft bearings . fig3 is an enlarged side sectional view , broken apart and partially cut away , of the actuator of fig1 . fig5 is a symbolic representation of the pole faces and gaps of the actuator of fig1 . fig7 is a schematic side view of another actuator embodiment . referring to fig1 a limited rotation actuator 10 includes a ferromagnetic rotor 12 which rotates within a cylindrical space defined by a stator assembly 14 . assembly 14 has two stacks 16 , 18 of ferromagnetic laminations joined by a pair of permanent magnets 20 , 22 . stacks 16 , 18 define four stator pole faces a , b , c , d which all lie on an imaginary cylindrical surface 35 . magnets 20 , 22 provide bias flux . a pair of drive coils 24 , 26 are wound respectively on stacks 16 , 18 to provide drive flux that interracts with the bias flux to produce driving torques to cause rotation of rotor 12 through a limited angle of excursion . cables 28 , 30 connect coils 24 , 26 to a control amplifier 32 which provides drive currents to cables 28 , 30 based on position feedback signals received over a cable 34 from a position sensor ( not shown ). the currents generated by amplifier 32 can , for example , be controlled to cause rotation of the angular orientation of rotor 12 through a limited excursion in either direction for driving or pointing an optical scanning element ( not shown ) attached to an end of shaft 36 of rotor 12 . referring to fig2 , rotor shaft 36 is fixed at its opposite ends to the inner sleeve 40 , 42 of instrument grade ball bearings 44 , 46 . outer sleeves 48 , 50 of bearings 44 , 46 are both held fixed relative to stator assembly 14 . steel balls 56 are carried in 52 , 54 in the raceways formed between grooves in inner sleeves 40 , 42 , and outer sleeves 48 , 50 . bearings 44 , 46 have a degree of radial free play between sleeves 40 and 48 and between sleeves 42 , 50 . the two pole faces 60 , 62 of rotor 12 lie on an imaginary cylindrical surface . the central axis 64 of the cylindrical surface on which rotor pole faces 60 , 62 lie is tilted by an angle α from the central axis 66 of the stator bore represented by imaginary cylindrical surface 35 on which stator pole faces a , b , c , d lie . the central axis 66 of surface 35 is approximately coaxial with shaft 36 . the cylindrical surface on which the rotor pole faces lie is smaller in diameter than the cylindrical surface on which the stator pole faces lie . as a result of this configuration , at one end of rotor 12 ( e . g ., the top end in fig3 ), the gap between faces a or b and 62 is smaller than between faces c or d and 60 , and vice versa at the other end of rotor 12 . as explained below , the magnetic flux operates across those gaps to radially load the two bearings 44 , 46 in opposite directions , thus minimizing radial free play and hence radial movement of rotor 12 . at the top of rotor 12 in fig3 a force f 1 minimizes the free play 68 between the left side of fixed outer sleeve 48 and the left side of floating inner sleeve 40 . simultaneously , a force f 2 at the bottom of rotor 12 minimizes the free play 70 between the right sides of sleeves 50 , 42 . referring to fig4 rotor pole faces 60 , 62 respectively span stator pole faces c , d and a , b in the neutral position shown and also in each other angular position within some maximum excursion of ± e from the neutral position . the bias flux established by magnet 20 has a path that crosses the gap from face c to face 60 and then the gap from face 60 to face d . similarly , the path of the bias flux established by magnet 22 crosses the gap from face a to face 62 and then the gap from face 62 to face b . referring to fig5 the pole faces and gaps of the actuator can be represented symbolically ( for purposes of calculating the torques that produce the forces f 1 , f 2 in fig3 ), by eliminating the curvature of the pole faces . three parallel imaginary planes ( all normal to axis 66 ) are shown at different positions along the 67 axis . the y = 0 plane lies midway between the planes ( y =+ y and y =- y ) at the top and bottom of rotor 12 . all of the pole face gaps at the y = 0 plane are the same ( g 0 ). the radial loading torque exerted on the rotor as a result of the permanent magnet bias flux in general is ## equ1 ## where u is the total stored in the four gaps a - 62 , b - 62 , c - 60 , d - 60 , which we will call gap a , gap b , gap c , and gap d , and α is the inter - axis angle ( fig3 ) ( the magnetic field of the permanent magnet is several times larger than the magnetic field induced by the control current , and therefore only the effects of the permanent magnet are considered in this analysis ). the magnetic energy in gap a , u a , is the volume integral where h a is the magnetic field in gap a supported by magnet 22 , μ is the permeability of the air in the gap , and dv is an incremental volume . h a can be determined using ampere &# 39 ; s law where no currents are present . the line integral along a flux path , e . g ., a path n - a - 62 - b - s - n in a plane having a given y position assuming that the iron rotor and stator poles all have infinite permeability , the line integral simplifies to where magnet field h m and magnet dimension l m are constant , and g a and g b are the widths of gaps a and b respectively at the given y position . because the gap and pole structure is symmetrical above and below the y = 0 plane , it is convenient to use that plane as a reference plane so that where g a ( 0 ) means the width of gap a at the y = 0 plane . because gaps g a ( 0 ) and g b ( 0 ) are identical and connected in parallel through identical pole faces that are themselves magnetically connected applying ampere &# 39 ; s law to the path jklm gives the magnetic field under pole a as a function of y as ## equ2 ## using these expressions for the magnetic field , the magnetic energy under each pole ( a , b , c , d ) can be calculated using equation 2 as follows ## equ4 ## the incremental volume dv is a function of y : ## equ5 ## where w is the width along the gap in a direction normal to the directions in which y and g 0 are measured . thus equation 11 for u a simplifies to ## equ6 ## which expands to ## equ7 ## keeping in mind that u a = u b , we must now determine u c . u d will be equal to u c . for determining u c , the gap then ## equ8 ## which expands to ## equ9 ## which is equal to u a . thus u a = u b = u c = u d and the total magnetic energy ## equ10 ## this can be simplified by expressing the ratio x of the pole tilt α to the maximum possible pole tilt ( g 0 / y ). then ## equ11 ## because energy is not a polarized quantity and because all of the gaps have identical shape and volume when rotor 12 is at its neutral position , it is apparent that the energy under the pole faces a , b , c , d are all equal . similarly , the last derived expression for u remains valid when the rotor is at angular positions other than the neutral position , so long as each rotor pole continues to span both of its associated stator poles . from the expression for torque t = du / dα , where dα =[ g 0 / y ] dx , it can be seen that ## equ12 ## the taylor series expansion of the logarithmic term gives ## equ13 ## which can be rearranged as ## equ14 ## which simplifies as ## equ15 ## this expression is consistent with expectations . for example , when x = α = 0 ( i . e ., axis 66 is coaxial with axis 64 ) torque is null , and when x = 1 ( i . e ., α = g 0 , meaning that the rotor ends touch the stator pole faces ) the torque reaches infinity ( assuming no saturation takes place ). where x & lt ; 1 / 2 the series in brackets in equation 26 can be represented by its first term alone with a resulting error of onyl 1 / 2 %. thus torque can be represented in typical cases as ## equ16 ## the following table shows values of the term ## equ17 ## for various tilt ratios x . ______________________________________ x ## str1 ## ______________________________________0 . 0 0 0 . 05 0 . 100410 . 1 0 . 20360 . 2 0 . 42990 . 3 0 . 70620 . 4 1 . 07140 . 5 1 . 58840 . 6 2 . 3749______________________________________ the torque increases rapidly with x and can be chosen to produce an appropriate pair of forces f 1 , f 2 ( fig3 ), thus radially loading the rotor as desired . ______________________________________total height in y direction 1 . 45 &# 34 ; diameter of stator pole face cylinder 0 . 4500 &# 34 ; diameter of rotor pole face cylinder 0 . 492 &# 34 ; tilt angle ( α ) 0 . 0010 radiansx 0 . 35bearing radial free play 0 . 0004 &# 34 ; magnet field strength 1 teslaresulting force on top bearing of rotor 1952 grams______________________________________ for example , referring to fig6 the rotor pole faces can each straddle two stator pole faces of the same polarity . such a configuration can be analyzed in a similar manner to show that h a ( y )= h a &# 39 ;( y )= h b &# 39 ;(- y )= h c &# 39 ;(- y )= h d &# 39 ;( y ) and that u a = u a &# 39 ;= u b &# 39 ;= u c &# 39 ;= u d &# 39 ;, where prime values are those pertaining to fig6 . instead of tilting shaft 36 relative to rotor pole faces 60 , 62 , as in fig2 shaft 36 can be coaxial with the axis of rotor pole faces 60 , 62 and tilted relative to axis 66 , by offsetting the bearings relative to axis 66 . alternatively , referring to fig7 bearings 70 with relatively large free play can be used . one element 72 of each bearing is fixed to the stator assembly 14 while the other floating element 74 is fixed to the rotor 12 . if the difference in diameters of the rotor and stator is p , then the free play in each direction ( q ) can be as much as 20 % of p . for example , if p is 0 . 002 inches , then the radial free play in one direction can be 0 . 0004 , for a total radial free play in both directions of 0 . 0008 inches . in that case , the magnetic flux will cause the rotor to maintain a tilted position relative to the stator assembly ( as indicated by dashed lines ), keeping the bearings preloaded as desired .