in the present invention , armature force is controlled by implementing a closed flux loop around each half station . these flux loops ensure that the magnetic flux density in the gaps is linearly proportional to a commanded input . to linearize the flux - squared term a bias technique is used . this consists of maintaining a constant bias flux in each half - station gap . the net force on the armature due to this bias flux is zero , since both sides are equal in magnitude and diametrically opposing . force on the armature is controlled by adding a flux command of a predetermined magnitude to one half - station and subtracting the same flux command magnitude from the diametrically opposite half station . the result is that the net armature force is proportional to the product of the bias flux and commanded flux as shown below : for a linear medium , force is proportional to the square of the magnetic flux densities in the gaps between the half - stations and armature , as given in equation ( 3 ): ## equ1 ## where f net = net force exerted on armature ( newtons ) b 1 , b 2 = magnetic flux density at each half - station pole face ( tesla ) since the bias flux is held constant , the net force is directly proportional to the commanded flux . therefore , the force produced by the differential electromagnet can be measured and controlled in a closed flux feedback loop by sensing the magnetic flux density produced by each half - station . referring first to fig1 a conceptual block diagram is shown of one feedback loop for driving a half - station . it will be appreciated that two such loops will be required for energizing two half - stations comprising a magnetic actuator . the magnetic actuator is comprising four major components : a differential electromagnet 10 , a flux sensor 12 , the compensation electronics 14 , and an amplifier 16 , all of which are described in further detail below with respect to fig4 . referring next to fig2 the differential magnetic actuator is comprising an electromagnet having two identical stator pieces 20 , 22 and a soft iron armature 24 . the stator pieces are each made up of two identical coils 26 , 27 and 28 , 29 connected in series and wound around a magnetic core 30 , 31 . the armature 24 is an essentially planar piece of soft iron or other magnetic material placed between the pole faces 32 - 35 . the flux density is preferably sensed by a hall effect device 36 , 37 placed on one pole face of each stator . a block diagram of the electronics required to drive the hall effect device and obtain an output is shown in fig3 . a current source 40 establishes a constant current in two leads 42 , 43 of the hall effect device 44 . the voltage across the output terminals 45 , 46 of the device is proportional to the magnetic flux density . this voltage is amplified by a high gain differential amplifier 48 to produce a suitable signal 49 proportional to the flux density for processing by the control circuitry . fig4 shows the system of fig1 with details of scaling , bias , and frequency compensating networks for driving the electromagnets of an opposing pair of half - stations . the function of this circuitry is to accept a voltage proportional to the desired force output f c , convert this voltage into two voltages b 1c , b 2c proportional to the flux density levels required to produce this force , compare the required flux density levels with the measured flux density levels at sensors 77 , 79 to provide resultant outputs 62 , 64 , and through the compensation amplifiers 66 , 68 command the drive amplifiers 70 , 72 to apply current to the coils 74 , 76 to produce the required flux density . the force command f c is converted to flux density commands by gain scaling block 50 and adding bias flux density commands 52 . the function of the bias command is to force the actuator to respond linearly to the force command . in operation , the unidirectional force exerted by each half station on the armature is given by : ## equ3 ## where f 1 , f 2 = the forces exerted by each half station ( newtons ) b 1 , b 2 = the flux densities at the half - station pole faces ( tesla ) since each half - station is diametrically opposing with respect to the armature , the net armature force is the difference of the forces exerted by each half - station and can be described by the closed flux loops 78 , 80 around each half - station of the magnetic actuator ensure a flux density level in the magnetic gaps equal to a referenced input b 1c , b 2c . this input is obtained by applying the sum and difference of a flux density bias 52 and scaled command input b cmd to summing junctions 54 and 58 , respectively : substituting these relationships into force equation ( 8 ) and simplifying yields ## equ5 ## by holding b bias at a predetermined value the scaling factor 50 between the force command ( f c ) and the flux density command ( b cmd ) is : ## equ6 ## thus , the bias flux density added to the scaled force command produces flux density commands which energize the compensation and power amplifiers to produce a force equal to the commanded force . the bias flux density may be selected to minimize the peak power required by the actuator . peak power occurs when the maximum force is commanded , therefore : ## equ7 ## which , assuming a linear actuator , is also given by ## equ8 ## where , g = the magnetic gap ( meters ) thus , the peak power is ## equ9 ## taking the derivative of p max gives ## equ10 ## setting the derivative equal to zero yields ## equ11 ## therefore , the above value of b bias will minimize the peak actuator power . referring now to fig5 the compensation amplifier u 1 comprises an integrator and a lead - lag network c 1 r 2 c 2 . the integrator is used to reduce the dc force error and the lead - lag network is used to obtain increased bandwidth . the lead frequency is selected to cancel the low - frequency pole produced by the resistance and inductance of the electromagnet and the lag frequency is selected to assure control loop stability . the compensation network parameters chosen for the flux loops around each half station are dependent on the desired frequency response . typically the closed loop response chosen is of the form : ## equ12 ## where ω 0 = the open loop crossover frequency to design compensation circuitry for the desired closed loop response the frequency characteristics of the magnetic actuator must be known . these characteristics are dependent upon the geometry of the actuator and are in the form shown below for each half - station : ## equ13 ## where b out = the flux density at the pole face for a given input coil voltage k p = is a gain constant with units of ( tesla / volt ) p 1 , p 2 = are poles associated with the transfer function for most actuator designs p 2 occurs at sufficiently high frequencies ( typically & gt ; 2 khz ) that it can be neglected ; whereas p 1 occurs at low frequencies ( typically & lt ; 10 hz ) and must be compensated for . thus each half station is conveniently viewed as having a frequency characteristic in form ## equ14 ## to obtain the closed loop response identified in equation ( 18 ), the compensation design would be in the form : ## equ15 ## where the zero ( z 1 ) is chosen to cancel the low frequency pole ( p 1 ) of the actuator and compensation gain ( k c ) and compensation pole ( p c ) are chosen for the desired open loop crossover bandwidth ( ω 0 ) and damping coefficient ( δ ). a typical compensation amplifier frequency characteristic curve is shown in fig7 . from ( 18 ) for unity gain of the power amplifier and flux sensing blocks , the closed loop characteristics would be : ## equ16 ## where one possible operational amplifier realization of this compensation design is shown in fig5 . the output of the compensation amplifier 66 , 68 is preferably connected to an optically coupled pulse - width modulated driver ( fig6 ) to produce the desired fluxes and provide isolation between the low level command signals and the high current power amplifier . as current in either direction in the coil produces force in the same direction , since the force is a function of the square of the current , the amplifiers may be unipolar . the pulse width modulation results in high efficiency . the pulse width modulator is simply a comparator referenced to a triangle wave as shown in fig6 . the output pulse widths of the comparator are a function of the input signal level . a suitable component for the comparator would be an integrated circuit lm111 as manufactured by national semiconductor corp ., santa clara , ca 95051 . a suitable component for the opto - isolator would be the type 4n24 , as made by texas instruments , inc ., dallas tx 75265 . the operation of the system is shown most clearly in fig4 . a desired force f c is applied to block 50 , where it is gain scaled and applied to a summing junction 54 , 58 . a bias voltage from source 52 is algebraically combined in junction 54 and junction 58 with scaled signal b cmd to provide force command signals b 1c and b 2c , respectively . signals b 1c and b 2c energize coils 74 and 76 and the flux densities produced thereby are sensed by flux sensors 77 , 79 . the flux density signals are combined in junctions 56 and 60 with the corresponding force command signals b 1c and b 2c to drive power amplifiers 70 , 72 through compensation amplifiers 66 , 68 . the sensed flux values are thereby compared with the commanded flux density to increase or decrease the driving signal to the power amplifier , thereby producing the desired force level in linear relationship to the output force command signal . in summary , the present invention offers the following advantages over previous linearization techniques : ( 2 ) no requirement for transmitting feedback information across the magnetic gap by the use of harness assemblies or complicated non - contacting telemetry systems . ( 3 ) no requirement for position sensors or multiplication circuitry in force linearization . ( 4 ) the ability to use core materials exhibiting hysteresis and high saturation levels to realize smaller , lighter , and less costly actuators . ( 5 ) high level power and low - level signal power are easily isolated using pulse width modulation techniques and optical couplers . while the invention has been described in its preferred embodiments , it is to be understood that words which have been used are words of description rather than limitation and that changes may be made within the purview of the appended claims without departing from the true scope and spirit of the invention in its broader aspects .