Abstract:
A cancellation circuit removes interfering signals from desired signals in electrical systems having antennas or other electromagnetic pickup systems. The cancellation circuit provides amplitude adjustment and phase adjustment to electrical signals induced in an electrical system by received electromagnetic signals. The cancellation circuit may be adapted to process received electromagnetic waves having multiple frequencies. Phase adjustment compensates for electrical path-length differences between signals resulting from propagation delays, circuit delays, and multipath. The amplitude-adjusted and phase-adjusted signals are combined to cancel the effects of electromagnetic interference. In an electromagnetic receiver, a plurality of receiver elements provide the cancellation circuit with different complex proportions of desired and interfering signals to enable removal of the interfering signals. An electromagnetic-wave transmitter having multiple transmitter elements is provided with a cancellation circuit for canceling electromagnetic signals in at least one predetermined region of space. A compensation circuit enables the cancellation circuit to compensate for frequency-dependent phase and amplitude differences in received signals and/or transmitted electromagnetic waves having multiple frequencies and/or broadband characteristics.

Description:
This application is a division of Ser. No. 08/279,050, filed Jul. 22, 1994, now U.S. Pat. No. 6,208,135, which is related to application Ser. No. 08/097,272, filed Jul. 23, 1993, now U.S. Pat. No. 5,523,526. 
    
    
     BACKGROUND OF THE INVENTION 
     The present invention relates to electromagnetic shielding for shielding electromagnetic pickups, other types of electronic equipment, and specific regions of space from electromagnetic radiation, and more particularly to active electromagnetic shielding for providing an electrical cancellation signal for canceling electromagnetic radiation or canceling the response of an electronic device to electromagnetic radiation. 
     It has long been known that voltages are induced in all conductors exposed to changing magnetic fields regardless of the configuration of such conductors. Electromagnetic radiation will induce electrical signals in electronic devices according to the laws of magnetic induction. Thus it has been desirable in some applications of electronic instrumentation to reduce the inductive noise caused by electromagnetic radiation. 
     A common method for providing electromagnetic shielding involves the use of passive electromagnetic shielding. A passive shield consisting of layers of high and low permeability material may be used to attenuate electromagnetic radiation passing through it. However, this passive electromagnetic shielding adds substantial bulk and weight to the system that it shields. 
     Another method for providing electromagnetic shielding is to utilize cancellation coils for generating a canceling electromagnetic radiation in opposition to incident radiation produced by external sources in order to cancel the effects of the incident radiation. In U.S. Pat. No. 5,066,891, Harrold presents a magnetic field sensing and canceling circuit for use with a cathode ray tube (CRT). Magnetic flux gate sensors provide output signals that are functions of detected fields. These signals are then used to control the current in cancellation coils that produce a cancellation magnetic field. Harold explains that it is of great importance that the CRT in a color monitor be protected from the effects of external magnetic fields, and, in particular, time-varying magnetic fields. However, this method provides no compensation to the frequency-dependent amplitude and phase responses of the sensor that picks up incident electromagnetic radiation and the device that generates the cancellation radiation. 
     Likewise, in U.S. Pat. No. 5,132,618, Sugimoto shows a magnetic resonance imaging system that includes active shield gradient coils for magnetically canceling leakage fields that would otherwise produce eddy currents in the heat-shield tube. 
     A common method for providing shielding to an electromagnetic pickup is to utilize identical pickup coils connected in series or in parallel so as to cancel the effects of uniform electromagnetic radiation. Pizzarello shows such a system in U.S. Pat. No. 5,045,784 for reducing inductive noise in a tachometer coil. An electric tachometer is a coil of wire that may be attached to a moving part of a motor that passes through a stationary magnetic field. The motion of the wire through the magnetic field induces a voltage that is indicative of the motor&#39;s speed. However, if the motor is powered by electricity, changes in the current powering the motor will cause a magnetic flux that also produces a voltage in the coil. Pizzarello shows a stationary pickup coil that is responsive to magnetic flux, and a means for subtracting the pickup voltage from the tachometer voltage. 
     Likewise, in U.S. Pat. No. 4,901,015, Pospischil shows a cancellation circuit for canceling the response of a magnetic pickup to ambient electromagnetic fields. Pospischil describes first and second pickups that are positioned in parallel with the wavefronts of an interfering electromagnetic field. With such placement, the electromagnetic field impinges simultaneously upon the first and second pickups. The pickups are connected in opposition. Thus, simultaneous impingement of the electromagnetic field upon the pickups is expected to produce a 180-degree phase displacement of the received signals. 
     If the electrical path lengths of the received signals in Pospischil&#39;s cancellation system are different where they are combined (summed), the relative phase difference between the received signals will not have 180-degree phase displacement. Thus, the signals will not cancel. Pospischil shows that differences in the electrical path length occur when the propagation path lengths of the signals received by the pickups are different (e.g., the signals do not impinge upon the pickups simultaneously). These differences in propagation path lengths can result from reflections, multipath delay, superpositions of multiple received signal components, or received electromagnetic signals having non-perpendicular angles of arrival. 
     Pospischil does not identify nor compensate for electrical path-length differences (e.g., differences in impedance) that occur between different electromagnetic receivers (pickups). Such pickup assemblies are also used with electric guitars and are known as “hum-bucking” pickups. This technique is not effective for providing a high degree of cancellation because slight differences between the pickups, even pickups that are substantially identical, cause the frequency-dependent amplitude and phase response of the pickups to differ significantly from each other. Thus the pickup signals will not be exactly out of phase and equal in amplitude when they are combined. 
     A prior-art method for providing shielding to an electromagnetic pickup from an electromagnetic source that produces a non-uniform field is to “unbalance” either the pickup device or the electromagnetic source. Such a method is described by Hoover in U.S. Pat. No. 4,941,388. Hoover uses amplitude-adjustment techniques to compensate for amplitude variations between the responses of separate pickups to electromagnetic radiation generated by an electromagnetic sustaining device that drives the vibrations of a string on an electric guitar. However, Hoover does not compensate for differences in the pickup coils which cause the amplitude-variation of the responses of the pickups to be frequency-dependent. Thus, Hoover&#39;s proposed solution results in poor cancellation over a broad range of frequency. Furthermore, Hoover does not compensate for phase-variations that occur between different pickup coils. The resulting cancellation from the unbalancing method is poor. 
     Hoover describes the operation of negative feedback in a system where a magnetic pickup provides an electrical signal to a magnetic driver that generates an electromagnetic field to which the pickup responds. Hoover mentions that the system tends to drift from the negative feedback condition at higher frequencies, and identifies the cause of this drift as distortions in the phase-response of the system resulting from the pickup, driver, and amplifier in the system. Hoover does not present an effective method for controlling the phase-response of the system, nor does Hoover present the mathematical relationships between phase and frequency resulting from the driver and pickup coils. Rather, Hoover proposes the use of a low-pass filter to reduce the gain of the system at which the negative feedback condition breaks down. 
     Methods of active phase-compensation are described by Rose in U.S. Pat. No. 4,907,483, U.S. Pat. No. 5,123,324, and U.S. Pat. No. 5,233,123. Rose uses active circuits for determining the frequency or frequency range of an electrical signal from an electromagnetic pickup. Active phase-adjustment is applied to the pickup signal, which is used to power an electromagnetic driver that generates an electromagnetic driving force on a vibratory ferromagnetic element of a musical instrument. The purpose of the phase-adjustment of the pickup signal is to provide a driving force to the vibratory element that is substantially in-phase with its natural motion. Because the purpose of Rose&#39;s invention is to improve the efficiency of the electromagnetic drive force on the element, it is apparent that a passive phase-compensation circuit would be preferable to Rose&#39;s active phase-compensation circuit. However, Rose does not realize the mathematical relationships between phase and frequency that provide the basis for constructing a passive phase-compensation network. Furthermore, Rose&#39;s invention does not provide simultaneous phase-compensation to more than one harmonic. 
     Another method for providing electromagnetic shielding is to orient the angle of a pickup coil to incident electromagnetic radiation such that the electrical current induced in the coil by the electromagnetic radiation will substantially cancel. One application of this method is shown by Burke in the  Handbook of Magnetic Phenomena , published in 1986. Burke uses a transmitting coil that produces electromagnetic radiation and a receive coil that senses radiation. The two coils can be configured in such a way that no energy is transferred between the transmitting and receiving coils. Burke shows the receiving coil oriented with the axis of its turns at right angles to the direction of the magnetic field produced by the transmitting coil. Burke explains that the instantaneous generated voltage of the receive coil is determined by the instantaneous rate of change of the magnetic flux passing through the coil. If the flux is directed at right angles to the coil&#39;s axis, none of it is intercepted by the coil, and the instantaneous rate of change through the coil is zero. This method of cancellation was used in an electromagnetic sustain device for electric guitars marketed by T Tauri Research of Wilmette Ill. in November, 1988, and patented by Tumura, European Patent Application No. 92307423.1 filed on Aug. 13, 1992, and U.S. Pat. No. 5,292,999. The actual effectiveness of this technique is limited by several factors, such as the uniformity of the pickup coil&#39;s windings, the uniformity of the electromagnetic radiation near the pickup, interference due to other nearby conducting materials, and the difficulty of precisely positioning a pickup coil in a field whose intensity varies as the inverse square of the distance from its source. 
     Another method for providing active electromagnetic shielding is the differential transformer also shown by Burke. The differential transformer comprises a drive coil for generating a magnetic flux, and two pickup coils wrapped around a ferromagnetic core that includes a moveable armature that, when moved, varies the reluctance of the magnetic path associated with each pickup coil. If the two pickup coils are identical, and if the two magnetic paths about which they are wound are identical, the voltages induced in each pickup coil will be the same. However, Burke explains that the two pickup coils nor the two magnetic paths can be made exactly the same, therefore a differential transformer will always have some output voltage under zero stimulus. 
     Coils of wire whose currents support magnetic fields in space function as antennas radiating electromagnetic energy. There are several cancellation methods used with antennas that act as electromagnetic shielding. One of these methods is the basis of operation for a sidelobe canceller that uses an auxiliary antenna in addition to a main antenna. Combining the outputs from the two antennas results in cancellation of the antenna beam pattern in the direction of a noise source so that the effective gain of the antenna in that direction is very small. Likewise, the multiple sidelobe canceller addresses the problem of multiple noise sources. 
     Delay-line cancellers are used in systems where multiple radar pulses are transmitted. These cancellers are used to detect moving objects. In a single-element delay-line canceller, a received pulse is delayed and added to another pulse received later so that the pulses reflected by stationary objects are out of phase and thus cancel, whereas the pulses reflected by moving objects do not cancel. 
     Several methods are used to allow an antenna to simultaneously transmit and receive electromagnetic radiation. For example, in a continuous wavelength radar system, a single antenna may be employed since the necessary isolation between transmitted and received signals is achieved via separation in frequency as a result of the Doppler effect. The received signal enters the radar via the antenna and is heterodyned in a mixer with a portion of the transmitted signal to produce a Doppler beat frequency. 
     An intermediate-frequency receiver may use separate antennas for transmission and reception. A portion of the transmitted signal is mixed with an intermediate frequency, and then a narrow-band filter selects one of the side bands as the reference signal, which is mixed with the signal from the receiver antenna. 
     It is one object of the present invention to provide active electromagnetic shielding for canceling the effects of electromagnetic induction in electrical circuits. It is a related object of the present invention to reduce interference between transmitters and receivers of electromagnetic radiation that operate simultaneously. It is another object of the present invention to provide a cancellation circuit that allows a single antenna element to simultaneously transmit and receive electromagnetic radiation. It is still another object of the present invention to compensate for frequency-dependent amplitude and phase responses of electromagnetic receivers and transmitters. 
     SUMMARY OF THE INVENTION 
     In accordance with the present invention, a cancellation circuit is provided for canceling the inductive effects of electromagnetic radiation. The cancellation circuit comprises a means for acquiring or generating an electrical reference signal that is similar in shape to the inductive electrical signal produced by the electromagnetic radiation, an amplitude-adjustment circuit that adjusts the amplitude of either or both the reference signal and an electrical pickup signal containing an inductive noise component, a phase-adjustment circuit that adjusts the relative phase between the reference signal and the pickup signal such that when these signals are combined, the inductive noise component will be canceled, and a combining circuit that combines the reference and pickup signals to produce a pickup signal that is substantially free from inductive noise. 
     In one aspect of the present invention, the reference signal is obtained from an electromagnetic pickup that is responsive to external magnetic flux. In another aspect of the present invention, the reference signal is obtained from part of the electrical signal that is used to generate the external magnetic flux. In still another aspect of the present invention, a signal generator generates the reference signal. 
     The present invention provides substantial electromagnetic shielding capabilities compared to prior-art shielding devices. Because the present invention actively shields from electromagnetic flux, it is non-intrusive compared to passive shielding technologies, which use materials that are heavy and bulky and require complete enclosure in order to provide optimum shielding. Thus the present invention may be used in order to reduce or eliminate the need for passive electromagnetic shielding in certain applications. Furthermore, in addition to being superior for shielding electromagnetic radiation compared to prior-art active electromagnetic shielding technologies, the present invention may be adapted to prior-art shielding devices to improve their performance. 
     The cancellation effect of the present invention allows electromagnetic pickups to operate in environments containing high levels of electromagnetic noise. For example, the present invention may be integrated into a sustaining device for a stringed musical instrument (as described by Rose and Hoover) to provide a very small sustain device that both picks up and drives the vibrations of a string on the musical instrument. This sustain device would be much smaller than the devices shown by either Rose or Hoover because the improved shielding capability of the present invention allows for the electromagnetic pickups (which pick up string vibrations) and the driver (which generates an electromagnetic flux to drive those vibrations) to be placed very close together (or even share the same structure) without the effects of electromagnetic interference. Other applications of the present invention include electric tachometers that operate near devices that generate large amounts of magnetic flux, and other electromagnetic receivers such as radars that operate near sources of electromagnetic radiation. This aspect of the present invention allows an electromagnetic antenna to simultaneously operate as a transmitter and receiver by decoupling the receiver-response to the transmitted signal. The present invention may also be used to cancel the response of a radar to ground clutter. 
     Another aspect of the present invention further includes a compensation circuit for adjusting the pickup signal&#39;s amplitude and/or phase in order to compensate for frequency-dependent amplitude and phase responses of the pickup. The compensation circuit may also compensate for frequency-dependent amplitude and/or phase variations of electromagnetic flux generated by an electromagnetic-flux generator, such as a drive coil. The present invention may be integrated into a prior-art active magnetic shielding circuit that generates a canceling magnetic flux for canceling external magnetic flux. The present invention provides a more accurate response to external magnetic flux, and thereby improves the cancellation effect of the circuit. Such a circuit may be used to provide active electromagnetic shielding to instruments that are sensitive to magnetic or electromagnetic fields. The invention has applications as a shielding device for atomic clocks, magnetic resonance imaging apparatus, tactical instrumentation, cathode ray tubes, satellites, and spacecraft. 
     In another embodiment of the present invention, the electromagnetic flux generated by the drive coil provides a magnetic force upon a moving ferromagnetic element. The phase of the electromagnetic flux generated by the system may be adjusted to provide electromagnetic damping to the ferromagnetic element, and thus act as a stabilizer for that element. The electromagnetic flux may be adjusted in phase to drive the oscillations of the ferromagnetic element (as discussed by Rose) The invention allows a broad range of driving frequencies to be compensated, thus allowing for the driving of the harmonics as well as the fundamental frequency of the element. 
     These and other aspects of the present invention will become apparent to those skilled in the art upon consideration of the following detailed descriptions of the preferred embodiments. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a schematic view of a prior-art cancellation circuit. 
     FIG. 2 is a schematic view of a cancellation circuit of the present invention. 
     FIG. 3A is a schematic view of a phase-adjustment circuit that may be used in the cancellation circuit of the present invention. 
     FIG. 3B is a schematic view of a phase-adjustment circuit that may be used in the cancellation circuit of the present invention. 
     FIG. 3C is a schematic view of a circuit of an embodiment of the present invention. 
     FIG. 4A is a schematic view of a cancellation circuit of the present invention that illustrates another method of phase-adjustment. 
     FIG. 4B is a schematic view of a phase-adjustment circuit that may be used in the cancellation circuit of the present invention. 
     FIG. 4C is a schematic view of a phase-adjustment circuit that may be used in the cancellation circuit of the present invention. 
     FIG. 5 is a plot of cancellation relative to signal frequency for three different cancellation circuits. 
     FIG. 6 is a schematic view of a cancellation circuit of the present invention that generates an electromagnetic flux in response to a pickup signal, and includes a compensation circuit for compensating for frequency-dependent phase and/or amplitude variations in electrical signals used to generate the electromagnetic flux. 
     FIG. 7 is a schematic view of a cancellation circuit of the present invention that includes a compensation circuit and provides an electromagnetic drive force to a ferromagnetic element. 
     FIG. 8A is a schematic view of a compensation circuit of the present invention. 
     FIG. 8B is a schematic view of a compensation circuit of the present invention. 
     FIG. 8C is a schematic view of a compensation circuit of the present invention. 
     FIG. 9 is a schematic view of a cancellation circuit of the present invention wherein cancellation of incident electromagnetic flux is achieved by generating an out-of-phase electromagnetic flux. 
     FIG. 10 is a schematic view of a cancellation circuit of the present invention wherein a pickup coil and a drive coil are wrapped around the same core. 
     FIG. 11 is a schematic view of a cancellation circuit of the present invention wherein pickup and drive coils are wrapped around the same core. 
     FIG. 12 is a schematic view of a cancellation circuit of the present invention wherein a reference signal is obtained from a signal generator used to provide a drive signal that generates the electromagnetic flux. 
     FIG. 13 is a schematic view of a cancellation circuit of the present invention wherein a reference signal is obtained from splitting the drive signal used to generate an electromagnetic flux. 
     FIG. 14A is a schematic view of a cancellation circuit of the present invention for a single-element transmit/receive system that includes a harmonic-compensation circuit for canceling the non-linear response of nearby magnetically permeable materials. 
     FIG. 14B is a schematic for a harmonic-compensation circuit of the present invention. 
     FIG. 14C is a schematic for a harmonic-compensation circuit of the present invention. 
     FIG. 14D is a schematic for a harmonic-compensation circuit of the present invention. 
     FIG. 15 is a schematic view of a cancellation circuit of the present invention for a single-element transmit/receive system. 
     FIG. 16 is a schematic view of a cancellation circuit of the present invention used in a system that compensates for magnetic fields. 
     FIG. 17 is a schematic view of a cancellation circuit of the present invention used in a system that compensates for magnetic fields. 
     FIG. 18 is a schematic view of a cancellation circuit of the present invention used in a single-element transmit/receive system. 
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     A prior-art balancing device for a pair of electromagnetic pickups shown in FIG. 1 includes a two coil assemblies,  10  and  12 , two amplifiers,  14  and  16 , and a combining means  18 . The first pickup coil  10  has values of resistance and inductance of R 1  and L 1 , respectively. The second pickup coil  12  has values of resistance and inductance of R 2  and L 2 , respectively. The pickup coil  10  is connected to the input of the amplifier  14  and the pickup coil  12  is connected to the input of the other amplifier  16 . The inputs to the amplifiers  14  and  16  each have a capacitor C 1  and C 2 , respectively, connected to electrical ground, as is commonly done to filter out high-frequency noise and interference from the pickup signals. The outputs of the amplifiers  14  and  16  are combined by a combining circuit  18 , which may be a voltage divider, a summing amplifier, or a differential amplifier. 
     The pickup coils  10  and  12  are responsive to external magnetic flux that induces a first electrical pickup signal in coil  10  and a second electrical pickup signal in coil  12 . Due to coil-positioning with respect to the external magnetic flux, coil properties, and properties of materials (not shown) which the coils  10  and  12  may surround, the amplitude of the first electrical pickup Fan signal will most likely differ from the amplitude of the second electrical pickup signal. Thus amplifiers  14  and  16  may be used to change the amplitude of either or both of the first and second electrical pickup signals. If the pickup signals are out of phase, the combining circuit  18  is a voltage divider or a summing amplifier. If the signals are in phase, then the combining circuit  18  is a differential amplifier. However, the relative phase between the first and second electrical signals will tend to be substantially different than 0 or 180 degrees, thus providing poor cancellation of the signals at the output of the combining circuit  18 . 
     The impedance Z 1  of the first pickup coil  10  is related to the coil&#39;s  10  resistance R 1  and inductance L 1 : Z 1 =R 1 +iωL 1 . Likewise, the impedance Z 2  of the second pickup coil  12  has the value: Z 2 =R 2 +iωL 2 , where ω represents the frequency of the pickup signals multiplied by 2 Pi. The voltage V 1in  at the input of the first amplifier  14  is 
     
       
         V 1in =V 1 [(1−ω 2 C 1 L 1 )−iωC 1 R 1 ]/[(1−ω 2 C 1 L 1 ) 2 +ω 2 C 1   2 R 1   2 ], 
       
     
     where V 1  is the voltage induced in the coil  10  by external magnetic flux. The voltage V 2in  of the second pickup signal is 
     
       
         V 2in =V 2 [(1−ω 2 C 2 L 2 )−iωC 2 R 2 ]/[(1−ω 2 C 2 L 2 ) 2 +ω 2 C 2   2 R 2   2 ], 
       
     
     where V 2  is the voltage induced in the coil  12  by external magnetic flux. Incidentally, V 1  and V 2  are proportional to the magnitude of external magnetic flux at the locus of each pickup coil  10  and  12 . The gain imparted to one or both pickup signal voltages V 1  and V 2  by the amplifiers  14  and  16  can correct for differences in the amplitude between V 1  and V 2  but can not correct for phase differences between those signals. The phase of the first pickup signal voltage V 1in  has the value 
     
       
         Ø 1 =−Arc Tan(ωC 1 R 1 /(1−ω 2 C 1 L 1 )), 
       
     
     and the phase of the second pickup signal voltage V 2in  has the value 
     
       
         Ø 2 =−Arc Tan(ωC 1 R 2 /(2−ω 2 C 2 L 2 )), 
       
     
     It is typical for the values of L 1  and R 1  to differ substantially from the values of L 2  and R 2 , respectively, even for pickup coils having identical numbers of windings. For example, two coils of 34 gauge copper wire, each wound 330 times around identical cores yielded values of resistance of 16.5 and 16.7 ohms, and values of inductance of 205 uH and 194 uH, respectively. Thus when only the amplitudes of the two signals are adjusted so that they are equivalent, the relative phase between the signals prevents optimal cancellation of the signals. 
     The expressions shown for the pickup voltages V 1in  and V 2in  are very accurate, but not exact representations for illustrating the differences in the phase variations between the pickup coils  10  and  12 . The exact impedance relations for the pickup coils  10  and  12  should also include capacitive effects. Other factors that may contribute to phase variations between the signals produced by the pickup coils  10  and  12  include ground oscillations, complexities resulting from the fact that each of the pickup coils  10  and  12  acts as a source for the electrical pickup signals, possible electrical loading between the two pickup coils  10  and  12 , and variations in how the voltage leads the current in the coils  10  and  12  resulting from inductance and capacitance in each of the coils  10  and  12 . Although the expressions shown for the pickup voltages V 1in  and V 2in  do not provide exact representations for the differences in the phase between the pickup coils  10  and  12 , these expressions are accurate to a great degree, and represent the basis from which extremely effective cancellation circuits can be designed. It will be appreciated that even more precise representations of the electrical signals induced in electromagnetic pickups can enable the design of cancellation circuits that are even more effective. 
     An embodiment for a cancellation circuit of the present invention is shown in FIG. 2 as including a pair of pickup coils  20  and  22 , a pair of amplitude-adjustment circuits  24  and  26 , a phase-adjustment circuit  25 , and a combining circuit  28 . The pickup coil  20  is connected to a phase-adjustment circuit  25 . The output of the phase-adjustment circuit  25  is connected to the input of an amplitude-adjustment circuit  24 . The pickup coil  22  is connected to the input of an amplitude-adjustment circuit  26 . The outputs of both amplitude-adjustment circuits,  24  and  26 , are connected to a combining circuit  28 . The output of the combining circuit  28  provides an electrical signal that is substantially free from the effects of electrical noise caused by the response of the pickup coils  20  and  22  to external magnetic flux. 
     The pickup coils  20  and  22  are responsive to external magnetic flux that induces a first signal voltage V 1  at the output of the first pickup coil  20 , and a second signal voltage V 2  at the output of the second pickup coil  22 . The phase of signal voltage V 1  is Ø 1  and the phase of signal voltage V 2  is Ø 2 . Because both Ø 1  and Ø 2  are functions of signal frequency ω, we will write Ø 1 (ω) and Ø 2 (ω). The pickup coil  20  is connected to the input of a phase-adjustment circuit  25 , which provides a phase-shift F(ω))=(Ø 2 (ω))−Ø 1 (ω)) to V 1  that compensates for the phase-difference between signals V 1  and V 2 . The nature of the phase-adjustment circuit  25  is determined by the frequency range of signal cancellation required. The output of the phase-adjustment circuit  25  is connected to the input of an amplitude-adjustment circuit  24 . The pickup coil  22  is connected to an amplitude-adjustment circuit  26 . Both amplitude-adjustment circuits,  24  and  26 , may provide amplitude-adjustment to the pickup signals V 1  and V 2 , respectively. In an alternative embodiment, only one of the amplitude-adjustment circuits  24  or  26  may provide amplitude adjustment while the other circuit  26  or  24  acts only as a buffer. Because phase-adjustment circuits (such as phase-adjustment circuit  25 ) typically change signal-amplitude as well as phase, it is preferable that the amplitude-adjustment circuits,  24  and  26 , have little effect on signal phase. Thus the amplitude-adjustment circuits,  24  and  26 , may comprise non-inverting amplifiers. The outputs of the amplitude-adjustment circuits,  24  and  26 , are combined in the combining circuit  28  in order to cancel the effects of external magnetic flux picked up by coils  20  and  22 . Depending on whether the output signals of the amplitude-adjustment circuits,  24  and  26 , are in phase or out of phase, the combining circuit  28  may comprise a voltage divider, a summing amplifier, or a differential amplifier. It will be appreciated that the coils  20  and  22  may be wrapped around a core (not shown), such as a core comprising a ferromagnetic material. It will also be appreciated that one or more additional phase-adjustment circuits may be included in series with coil  20  and/or coil  22 . Furthermore, it will be appreciated that amplitude-adjustment circuits such as amplitude-adjustment circuit  24  may precede phase-adjustment circuits, such as phase-adjustment circuit  25 . 
     Several phase-adjustment circuits shown in FIG. 3 may be used in the circuit shown in FIG.  2 . The circuit in FIG. 3A is commonly referred to as an “all-pass filter.” The all-pass filter provides a phase-shift of Ø=180°−2 ArcTan (ωR 6 C 3 ) while producing little amplitude-variation with respect to signal frequency. The circuit shown in FIG. 3B is also an all-pass filter. It produces a phase-shift of Ø=2 ArcTan (ωR 8 C 4 ). The all-pass filters in FIGS. 3A and 3B may be preceded by a buffer amplifier (not shown). 
     It is sometimes desirable to have substantial noise-cancellation over only a narrow frequency-range. This is called “notch-cancellation” and may be used in a single-frequency or band-limited system. One application for notch-cancellation is when an external magnetic flux contains a weak signal having a significantly different frequency than the noise that accompanies it, then a cancellation circuit that cancels a narrow frequency range including the noise, but not the desired signal, is preferable. Ferromagnetic materials used for pickup cores have the property of non-linear responsiveness to magnetic flux. This non-linear responsiveness is observed in a pickup signal as a higher harmonic or intermodulation product of the frequency of the magnetic flux. In order to observe the extent of the core material&#39;s non-linearity, it is preferable to cancel only the primary pickup signal, which has the same frequency ω as the applied magnetic flux. Typically, the higher harmonic signatures caused by a core&#39;s non-linearity is at least several orders of magnitude less than the intensity of the primary signal induced in the coil. Thus the method of notch-cancellation provides an advantage over conventional electrical filtering techniques in both simplicity and performance. 
     FIG. 3C shows another embodiment for a cancellation circuit of the present invention. Two pickup coils  30  and  32  are each connected to a phase-adjustment circuit  35  and  37 , respectively. The phase-adjustment circuits  35  and  37  are each connected to the input of an amplitude-adjustment circuit  34  and  36 , respectively. The outputs of the amplitude-adjustment circuits  34  and  36  are combined by a combining circuit  38 . The output of the combining circuit  38  is substantially free from inductive noise. The phase-adjustment circuits shown in FIG.  3 A and FIG. 3B may be used as the phase-adjustment circuits  35  and  37  shown in FIG.  3 C. Furthermore, the phase-adjustment circuits shown in FIG.  3 A and FIG. 3B include a means for adjusting the amplitude of electrical signals via adjustment of the resistors R 5  and R 7  in FIG. 3A, and resistors R 9  and R 10  in FIG.  3 B. 
     For the case of notch-cancellation in which the relative phase between the pickup signals from the first and second pickup coils  30  and  32  is very small, such as when the coils  30  and  32  are very close to being identical or when a phase-adjustment circuit (not shown) has already created this condition, it is preferable to select types of phase-adjustment circuits  35  and  37  that cause a very narrow frequency-range in which the cancellation is substantial. If the pickup signals from the two pickup coils  30  and  32  are in phase, this may be accomplished by selecting the all-pass filter shown in FIG. 3A as one phase-adjustment circuit  35  and selecting the all-pass filter shown in FIG. 3B as the other phase-adjustment circuit  37 . This selection is suggested because as the signal-frequency ω changes, the phase of the pickup signal of one of the phase-adjustment circuits  35  increases while the phase of the pickup signal of the other phase-adjustment circuit  37  decreases, thus causing a rapid change in the relative phase with respect to frequency ω. To maximize the change in the relative phase near the “notch frequency” where the cancellation is most substantial, one could select values of R 6  and C 3  in FIG.  3 A and values of R 8  and C 4  in FIG. 3B such that ω n R 6 C 3  and ω n R 8 C 4  are nearly equal to 1 for the notch frequency ω n . To further narrow the cancellation notch about the notch frequency ω n , the phase-adjustment circuits  35  and  37  may each include multiple all-pass filters as shown in FIG.  3 A and FIG.  3 B. 
     For the case in which cancellation is desired over a broad frequency range (such as when the pickups  30  and  32  are part of a feedback circuit that is prone to oscillate, making it necessary to cancel the higher harmonic terms that will accompany the primary signal), phase-adjustment circuits  35  and  37  may be selected to broaden the cancellation notch about the notch frequency ω n . For example, the choice of phase-adjustment circuits  35  and  37  for two pickup signals that are in phase may both be of the type of all-pass filters shown in FIG. 3A or FIG. 3B where the values of resistance R 6  or R 8  and capacitance C 3  or C 4  are chosen to minimize the relative phase and amplitude variations with respect to frequency between the two pickup signals. 
     FIG. 4A shows how two phase circuits may be integrated into a circuit comprising two pickup coils  40  and  42 . Coil  40  is connected to series element  45 , which may include resistors and/or inductors (not shown) connected in series with the coil  40 . Likewise, coil  42  is connected to series element  47 , which may include resistors and/or inductors (not shown) connected in series with the coil  42 . Series element  45  is connected to the input A of an amplitude-adjustment circuit  44  and series element  47  is connected to the input B of an amplitude-adjustment circuit  46 . Input A includes a resistor R 3  connected to electrical ground, and input B includes a resistor R 4  connected to electrical ground. Together, the series element  45  and resistor R 3  form one phase-adjustment circuit, and the series element  47  and the resistor R 4  form another phase-adjustment circuit. The outputs of the amplitude-adjustment circuits  44  and  46  are connected to the input of a combining circuit  48  that combines the output signals of the amplitude-adjustment circuits such that the noise-signal caused by external magnetic flux substantially cancels. 
     The effective impedance of the coil  40  at the input A of the amplitude-adjustment circuit  44  includes the actual impedance of the coil  40  added to the impedance of the series element  45 , and is represented by Z 1 =R 1 +iωL 1 . The effective impedance of the coil  42  at the input B of the amplitude-adjustment circuit  46  includes the actual impedance of the coil  42  added to the impedance of the series element  47 , and is represented by Z 2 =R 2 +iωL 2 . The voltage of the pickup signal induced in the coil  40  by external magnetic flux having frequency ω, measured at the input A is 
     
       
         V 1A =R 3 V 1 /(R 1 +R 3 +iωL 1 ), 
       
     
     where V 1  is the voltage-magnitude of the signal induced in the pickup coil  40  by the external magnetic flux. The signal voltage induced in the coil  40  by external magnetic flux having frequency ω, measured at the input B is 
     
       
         V 2B =R 4 V 2 /(R 2 +R 4 +iωL 2 ), 
       
     
     where V 2  is the voltage-magnitude of the signal induced in the pickup coil  42  by the external magnetic flux. The phase of the voltage of the pickup signal at the input A is 
     
       
         Ø 1 =Arc Tan(−ωL 1 /(R 3 +R 1 )), 
       
     
     and the phase of the voltage of the pickup signal at the input A is 
     
       
         Ø 2 =Arc Tan(−ωL 2 /(R 4 +R 2 )). 
       
     
     In order that Ø 1 =Ø 2  for a broad range of signal-frequencies ω, it is necessary that the series element  45  and/or series element  47  be adjusted such that L 1 /(R 3 +R 1 )=L 2 /(R 4 +R 2 ). This may also be accomplished by adjusting resistors R 3  and/or R 4 . However, if we look at the equations for signal voltage at the inputs A and B of the amplitude-adjustment circuits  44  and  46 , respectively, we note that equivalence of the ratios just discussed does not, by itself, provide the condition whereby the magnitude of the voltage difference V 1A −V 2 B remains substantially constant as o changes. Thus, in order to assure optimal cancellation over a broad range of signal frequencies, it is necessary that the series elements  45  and  47  are adjusted such that the effective resistances R 1  and R 2  are equivalent and the effective inductances L 1  and L 2  are equivalent. It is also necessary that resistance R 3  equal resistance R 4 . It is possible to replace resistors R 3  and R 4  with capacitors (not shown) for filtering out high-frequency noise, However, for optimal cancellation over a broad range of signal frequencies, it is necessary that both capacitors (not shown) have substantially equal values. 
     It will be appreciated from the equations representing the voltages V 1A  and V 2B  at the amplifier inputs A and B, respectively, that the series elements  45  and  47  may each include a large value of series resistance so as to increase the effective resistances R 1  and R 2  of the pickup coils  40  and  42 , respectively. This reduces the frequency-dependent amplitude and phase variations of the pickup signals V 1A  and V 2B . However, it is preferable that the increase in the effective resistances R 1  and R 2  of the pickup coils  40  and  42 , respectively, not be the only means of phase-adjustment used in the circuit as other phase effects that are unrelated to the signal-voltage equations for V 1A  and V 2B  tend to occur. 
     Consider the circuit shown in FIG. 4A for a case in which it is not optimized for canceling the effects of external magnetic flux. An applied phase shift between voltages V 1A  and V 2B  that matches their phases is 
     
       
         F(ω)=Arc Tan(−ωL 1 /(R 3 +R 1 ))−Arc Tan(−ωL 2 /(R 4 +R 2 )). 
       
     
     A phase-adjustment circuit that provides the required phase-shift may include a buffered input and precede either or both amplitude-adjustment circuits  44  and  46 , or may follow either or both amplitude-adjustment circuits  44  and  46 . The components of this phase-adjustment circuit are shown in FIG.  4 B and FIG.  4 C. 
     The phase-adjustment circuit shown in FIG. 4B is a passive lead network. An input voltage V in  is applied across terminals C and G. The output voltage V out  of this circuit is measured across terminals D and G. The output voltage is 
      V out =V in R 12 (iωC 10 R 11 +1)/(R 12 (iωC 10 R 11 +1)+R 11 ). 
     If R 11 &gt;&gt;R 12 , then the phase-shift is Ø=Arc Tan(ωR 11 C 10 ). 
     The phase-adjustment circuit shown in FIG. 4C is a passive lag network. An input voltage V in  is applied across terminals E and G. The output voltage V out  of the circuit is measured across terminals F and G. The output voltage is 
     
       
         V out =V in R 14 /((R 14 +R 13 )+iωC 15 R 14 R 13 ). 
       
     
     If R 14 &gt;&gt;R 13 , then the phase-shift is Ø=−Arc Tan(ωR 14 C 15 ). 
     The circuits in FIG.  4 B and FIG. 4C provide the basis for constructing F(ω). The passive lead network shown in FIG. 4B can be combined in series with the passive lead network shown in FIG.  4 C and use appropriate buffering between the lead and lag networks (such as a buffered amplifier (not shown)). Values of R 14  and C 15  are preferably selected such that R 14 C 15 =L 1 /(R 3 +R 1 ) and values of R 11  and C 10  are preferably selected such that R 11 C 10 =L 2 /(R 4 +R 2 ). 
     It should be appreciated that the phase-adjustment circuits shown in FIG.  3  and FIG. 4 are only a few of the many phase-adjustment circuits that can be used in a cancellation circuit to substantially eliminate electrical noise in a pickup signal caused by external magnetic flux. The phase-adjustment circuits shown, as well as other phase-adjustment circuits, may be used in combination for either broadening or narrowing the frequency range where a high degree of signal cancellation occurs. Phase-adjustment circuits may also be used for adjusting amplitudes of electrical signals, and thus may be employed as combined phase and amplitude-adjustment circuits. 
     A graphical analysis of different types of cancellation is shown in FIG.  5 . This graph shows three plots of cancellation (in decibels) of a combined output of two pickups at an output of a combining circuit For example, one or more plots in FIG. 5 may show the voltage magnitude V out  of combining circuit  48  shown in FIG. 4A, divided by the voltage magnitude of the pickup signal at one of the inputs to the combining circuit  48 . The graph in FIG. 5 shows cancellation plotted relative to signal frequency (Hz). 
     Plot  1  of FIG. 5 represents cancellation obtained by a circuit that does not compensate for phase-differences between two pickup signals, such as the prior-art circuit shown in FIG.  1 . Plot  2  of FIG. 5 illustrates “notch-cancellation” as explained above with reference to the cancellation circuit shown in FIG.  3 C. The frequency at which the notch occurs can be changed by adjusting the phase-adjustment circuits  35  and  37 . Plot  3  of FIG. 5 represents cancellation obtained by a cancellation circuit (such as the cancellation circuit shown in FIG. 4A) that provides substantial cancellation of external magnetic flux over a relatively broad range of frequency. This curve illustrates a very broad notch centered at a notch frequency ω n . At frequencies below and above the notch frequency ω n , the degree of cancellation begins to diminish. In the case where one uses a cancellation circuit, such as the one shown in FIG. 4A, it is possible to improve the cancellation at frequencies below the notch frequency ω n  by adjusting the resistance in either or both series elements  45  and  47  so that R 1  better approximates the value of R 2 . Thus, Z 1  better approximates Z 2  at low frequencies. It is also possible to improve the cancellation at frequencies above the notch frequency ω n  by adjusting the inductance in either or both series elements  45  and  47  so that L 1  better approximates the value of L 2 . Thus, Z 1  better approximates Z 2  at high frequencies. Furthermore, the overall level of cancellation over the entire frequency range may be improved by adjusting the values of resistors R 3  and R 4  such that the values of these resistances better approximate each other. It will be appreciated that additional phase circuits (not shown) may be used to provide compensating phase-shifts at low and/or high frequencies in order to broaden the notch centered at the notch frequency ω n . 
     FIG. 6 shows a cancellation circuit of the present invention that includes two pickup coils  60  and  62 , phase-adjustment circuits  65  and  67 , amplitude-adjustment circuits  64  and  66 , a combining circuit  68 , a preamplifier  74 , a power amplifier  76 , and a drive coil  70 . The pickup coils  60  and  62  and/or the drive coil  70  may also include one or more ferromagnetic cores (not shown). 
     Pickup signals are induced in the pickup coils  60  and  62  by external magnetic flux. In this case, the external magnetic flux is generated by the drive coil  70 . The phase of the pickup signal from the first pickup  60  is adjusted by the phase-adjustment circuit  65 . The phase of the pickup signal from the second pickup  62  is adjusted by the phase-adjustment circuit  67  such that the phases of the two pickup signals are substantially in phase (0 degrees) or out of phase (180 degrees) for a broad range of signal frequencies. The amplitude of the first pickup signal is adjusted by the amplitude-adjustment circuit  64 . The amplitude of the second pickup signal may be adjusted by the amplitude-adjustment circuit  66 . However, the amplitude-adjustment circuit  66  may act only as a buffer and provide no amplitude adjustment to the second pickup signal. It will be appreciated that the amplitude-adjustment circuits  64  and  66  may provide either gain or attenuation to the pickup signals. It will also be appreciated that the amplitude-adjustment circuits  64  and  66  may be replaced by a means for adjusting the position and/or orientation of the pickup coils in order to provide adjustment to the amplitude of the pickup signals induced in either or both pickup coils  60  and  62 . The outputs of both amplitude-adjustment circuits  64  and  66  are received by a combining circuit  68  that combines the pickup signals such that the pickup signals induced by the external magnetic flux generated by the driver  70  substantially cancel. The output of the combining circuit  68  is amplified by a preamplifier  74 . The output of the preamplifier  74  is amplified into a drive signal by a power amplifier  76 . The drive signal is an amplified pickup response of pickup coils  60  and  62  to an external magnetic flux other than the external magnetic flux generated by the drive coil  70 . This other external magnetic flux may be from an electromagnetic source, such as another drive coil (not shown), or may be caused by the motion of a ferromagnetic element (not shown) in a magnetic field. The drive signal flows through the drive coil  70  and generates an external magnetic flux that cancels the magnetic flux generated by the other magnetic source (not shown). The drive coil  70  may generate a magnetic flux in response to a magnetic flux caused by motion of a ferromagnetic element (not shown). The drive coil may either drive or damp the motion of the ferromagnetic element (not shown). 
     For the circuit shown in FIG. 6, it is necessary that a high degree of cancellation is obtained for a broad range of frequencies, else the circuit will undergo oscillation due to direct magnetic feedback. In general, a circuit will oscillate at a frequency at which the feedback gain is positive (i.e., when the circuit gain exceeds the circuit losses). If the circuit in FIG. 6 achieves a cancellation profile similar to plot  3  in FIG. 5, the circuit may oscillate at a relatively high frequency where the cancellation is not as effective. However, a low-pass filter may be included in the feedback circuit to reduce the feedback gain of the circuit. For example, preamplifier  74  may comprise an active low-pass filter. Likewise, one or more high-pass or bandpass filters may be used to eliminate circuit oscillation. It is also possible that the phase-adjustment circuits  65  and  67  and/or amplitude-adjustment circuits  64  and  66  of the cancellation circuit may be designed specifically to filter certain frequencies. 
     The phase-adjustment circuits  65  and  67  are designed specifically for compensating for frequency-dependent phase-variations between the pickup signals from the pickups  60  and  62 . However, the phase-adjustment circuits  65  and  67  may provide an overall phase-shift to the combined pickup signal at the output of the combining circuit  68 . This overall phase-shift may compensate for the phase shift introduced to the drive signal as a result of the frequency responses of the pickups  60  and  62  and the driver  70 . The phase-adjustment circuits  65  and  67  may also be used to compensate for phase shifts in the circuit caused by other circuit elements (not shown) that may precede the drive coil  70 . The phase-adjustment circuits  65  and  67 , are preferably preceded by a buffer (not shown), and may precede or follow the amplitude-adjustment circuits  64  and  66 . 
     An embodiment for a cancellation circuit of the present invention is shown in FIG.  7 . The circuit in FIG. 7 includes a magnetic element  81 , a ferromagnetic core  83 , two pickup coils  80  and  82  wrapped around the core  83 , and a phase-adjustment circuit  85 . The input to the phase-adjustment circuit is shown connected to electrical ground by a resistor R 3 . The output of the phase-adjustment circuit is shown connected to the input of an amplitude-adjustment circuit  84 . The pickup coil  82  is shown connected to the input of an amplitude-adjustment circuit  86 . This input also includes a resistor R 4  connected to electrical ground. The outputs of each amplitude-adjustment circuit  84  and  86  are connected to a combining circuit  88 . The output of the combining circuit  88  is connected to a compensation circuit  89 . The compensation circuit  89  is connected to an amplifier  96  that amplifies an input signal into a drive signal that flows through a drive coil  90  wrapped around a ferromagnetic core  93 . 
     The pickup coils  80  and  82  are shown approximately equidistant to the drive coil  90 , which generates an external magnetic flux F. The pickup coils  80  and  82  receive approximately equal intensities of the external magnetic flux F generated by the drive coil  90 . Amplitude adjustment of the pickup signals induced in the pickup coils  80  and  82  by the drive coil&#39;s  90  generated external magnetic flux F compensates for differences that may occur between the pickup signals, such as differences in the intensities of the drive coil&#39;s  90  generated magnetic flux F at the location of each pickup coil  80  and  82 , distortions in the drive coil&#39;s  90  generated magnetic flux F resulting from nearby conducting or magnetically permeable materials (such as ferromagnetic element  95 ) and differences in the amplitude-responses (including frequency-dependent amplitude responses) of the pickup coils  80  and  82  to magnetic flux. Likewise, the pickup signals induced in the pickup coils  80  and  82  by uniform external magnetic flux and magnetic flux generated by other magnetic sources (not shown) disposed in the plane that is the perpendicular bisector of the height dimension of the drive coil  90  are also substantially equal in amplitude. Thus, after phase-shifting by the phase-adjustment circuit  85  and combining by the combining circuit  88 , the signals induced in the pickup coils  80  and  82  by uniform external magnetic flux, the magnetic flux F generated by the drive coil  90 , and magnetic flux generated by any other magnetic sources (not shown) disposed in the plane that is the perpendicular bisector of the height dimension of the drive coil  90  cancel. 
     The cancellation of the effects of the drive coil&#39;s  90  generated magnetic flux F on the combined pickup signal is altered if either a permeable or conducting object enters the space shared by the field patterns F of the drive coil  90  and the pickup coils  80  and  82 . If the intruding object is permeable, the field pattern F surrounding the pickup coils  80  and  82  is distorted, and energy passes directly from the drive coil  90  to the pickup coils  80  and  82  through the distorted field. Thus if the ferromagnetic element  95  vibrates, the frequency of its motion is reproduced in the combined pickup signal. Because of the cancellation of external magnetic flux, the output of the combining circuit  88  comprises only the pickup signals induced by the motion of the ferromagnetic element  95 . 
     The compensation circuit  89  provides a phase-shift to the output signal of the combining circuit  88  in order to compensate for the frequency-dependent phase variations between the magnetic flux F applied to the ferromagnetic element  95  and the response of the pickups  80  and  82  to the motion of the ferromagnetic element  95 . The output signal of the phase-adjustment circuit  88  is amplified by the amplifier  96  into a drive signal which flows through the drive coil  90  and generates the magnetic flux F. This magnetic flux F may be used to either drive or damp the motion of the ferromagnetic element  95  depending on the phase of the drive signal. It is preferable to provide drive forces to the ferromagnetic element  95  such that the phase-relationship of the drive force to the motion of the ferromagnetic element  95  is not changed by the frequency of the element&#39;s  95  motion. This is particularly important if the motion of the ferromagnetic element  95  comprises a plurality of different frequencies. 
     For the circuit shown in FIG. 7, the signals from the pickup coils  80  and  82 , after being combined, may have a total phase-shift of 
     
       
         Ø p =Arc Tan(−ωL 2 /(R 4 +R 2 )) 
       
     
     The phase-shift of the drive signal at the drive coil  90  is 
     
       
         Ø d =Arc Tan(ωL d /R d ), 
       
     
     where R d  and L d  are the resistance and inductance, respectively, of the drive coil  90 . Ignoring any phase-shifts caused by other elements in the circuit, the total phase-shift between the magnetic flux F generated by the drive coil  90  and the response of the pickups  80  and  82  to the magnetic flux F generated by the drive coil  90  is ω t =ω p +ω d . By adjusting the values of resistances R 2 , R 4 , and R d , and/or the values of inductances L 2  and L p , or combinations thereof, it is possible to cause the ratios L 2 /(R 4 +R 2 ) and L d /R d  be substantially equal. Thus, the value of ω t  can be made substantially zero for a broad range of signal frequencies, ω. 
     The circuits shown in FIG.  8 A and FIG. 8B may be used in a cancellation circuit as phase-adjustment circuits (such as phase-adjustment circuit  85 ) for providing phase-shifts to pickup signals before they are combined. The circuits shown in FIG.  8 A and/or FIG. 8B may be used in the feedback loop of FIG. 7 as a phase-compensation circuit, such as phase-compensation circuit  89 . The circuit shown in FIG. 8A is a inverting amplifier. A buffer amplifier  94  may precede the first impedance element Z 11 . The second impedance element Z 12  provides feedback between the output and inverting input of amplifier  98 . The gain resulting from this inverting amplifier is G 1 =Z 12 /Z 11 . Thus the ratio Z 12 /Z 11  maybe adjusted to compensate for phase-shifts. The circuit shown in FIG. 8B is a non-inverting amplifier comprising an amplifier  99  and gain-control impedance values Z 13  and Z 14 . The gain of the non-inverting amplifier is G 1 =1+Z 14 /Z 13 . Likewise, the ratio Z 14 /Z 13  may be adjusted to compensate for phase-shifts. 
     The circuits shown in FIG.  8 A and FIG. 8B also provide a means for compensating for frequency-dependent amplitude variations in the feedback signal caused by the frequency response of the pickup coils  80  and  82 , the drive coil  90 , and any other circuit elements in the feedback loop. For example, the circuit shown in FIG. 7 will produce a pickup signal voltage V P =V B R/(R 2 +R 4 +iωL 2 ) at the input of amplifier  86 , where V B  is the voltage induced in the pickup coil  82  by external magnetic flux B. The voltage V B  is proportional to the magnitude of the magnetic flux B. The magnitude of magnetic flux B is proportional to the amplitude of drive current I d  in the drive coil  90 , where I d =V d /(R d +iωL d ) and V D  is the drive voltage. Thus, the pickup signal voltage V P  is proportional to V B R/(R 2 +R 4 +iωL 2 )(R d +iωL d ). One can observe from the equation for V P  that as ω increases, the pickup signal voltage V P  decreases. 
     An example of a circuit design that can compensate for the frequency-dependent amplitude variation of the pickup signal voltage V P  includes two inverting amplifiers (as shown in FIG. 8A) connected in series. The impedance element Z 12  of the first amplifier  98 A comprises a resistor R 12  (not shown) and inductor L 12  (not shown) connected in series. The value of the resistor R 12  is (R 2 +R 4 ) and the value of the inductor L 12  is L 2 . The value of the impedance element Z 11  of the first amplifier  98 A is αR 4 , where α is a scalar constant. The impedance element Z 12  of the second amplifier  98 B comprises a resistor R 22  (not shown) and inductor L 22  (not shown) connected in series. The value of the resistor R 22  is R d  and the value of the inductor L 22  is L d . The value of the impedance element Z 21  of the second amplifier is αR 4 . The gain G 1  of the first amplifier is (R 2 +R 4 +iωL 2 )/αR 4 . The gain G 2  of the second amplifier is (R d +iωL d )/αR 4 . The total gain of this circuit is G t =G 1 G 2 . The gain G t  multiplies the pickup voltage V P  so that the frequency-dependent nature of the pickup voltage amplitude is compensated. It should be noted that the values of R 2 , R 4 , and R d  may be increased to reduce the frequency-dependent effects on the pickup signal voltage V P . However increasing the value of R d  substantially reduces the magnitude of magnetic flux generated by the drive coil  90 . 
     The frequency-dependent phase-shifts and amplitude variations that typically occur between a pickup coil and drive coil may be substantially compensated over a broad range of signal frequencies ω via the selection of the values for electrical components in the compensation circuits that adjust both amplitude and phase-response. Typically, for a feedback system (such as shown in FIG. 7) in which the motion of a ferromagnetic element is driven by an external magnetic flux generated by the drive coil  90 , the frequency-dependence of the phase of the drive signal is of more interest than the frequency-dependence of the amplitude of the drive signal. However, for a feedback system in which a drive coil generates a specific magnetic flux in response to an external magnetic flux, such as in order to cancel an external magnetic flux in a specific region of space, it is important to control both the phase and amplitude of the drive signal. 
     Another embodiment for a cancellation circuit of the present invention is shown in FIG.  9 . The circuit in FIG. 9 includes a pickup coil  104  wrapped around a pickup core  105 . The pickup core  105  may be made of a ferromagnetic material, and the core  105  may be magnetized. The pickup coil  104  is connected to a compensation circuit  106  that is connected to a splitting circuit  108 . The splitting circuit  108  has a first output connected to a first phase-adjustment circuit  109  and a second output connected to a second phase-adjustment circuit  110 . The first phase-adjustment circuit  109  is connected to the input of a first amplifier  111  and the second phase-adjustment circuit  110  is connected to the input of a second amplifier  112 . The output of the first amplifier  111  is connected to a first drive coil  100 , and the output of the second amplifier  112  is connected to a second drive coil  102 . Both the first and second drive coils  100  and  102 , respectively, are wrapped around a drive core  101 , and they generate a magnetic flux F. The drive core  101  may be made of a ferromagnetic material, and it may be magnetized. In this case, the drive core  101  is shaped so that both of its endpoles are in close proximity to a ferromagnetic element  115 . The ferromagnetic element  115  induces a current in the pickup coil  104  when its motion disturbs the distribution of magnetic flux F that passes through the pickup coil  104 . The shape of the drive core  101  concentrates the magnetic flux lines F generated by electrical current in the drive coils  100  and  102  so as to provide a more efficient magnetic drive force to the ferromagnetic element  115 . 
     A first electrical pickup signal V D1  is induced in the pickup coil  104  by magnetic flux generated by the first drive coil  100 , and a second pickup signal V D2  is induced in the pickup coil  104  by magnetic flux generated by the second drive coil  102 . An electrical pickup signal V Pickup  is induced in the pickup coil  104  by magnetic flux produced by other sources, such as the ferromagnetic element  115  moving through a static magnetic field. The electrical signals induced in the pickup coil  104  pass through the compensation circuit  106  to the splitting circuit  108 , which splits the pickup signal into two drive signals. One of the drive signals passes through the first phase-adjustment circuit  109  and is amplified by the first amplifier  111 . The other drive signal passes through the second phase-adjustment circuit  110  and is amplified by the second amplifier  112 . The first drive coil  100  has an effective resistance of R D , and an effective inductance L D1 , which results in a total impedance of Z 1 =R D1 +iωL D1 , where ω is the frequency of the drive signal. The second drive coil  102  has an effective resistance of R D2  and an effective inductance L D2  resulting in a total impedance of Z 2 =R D2 +iωL D2 . Because the impedance values Z 1  and Z 2  of drive coils  100  and  102 , respectively, tend to differ from each other, a drive signal flowing through the first drive coil  100  having the same frequency ω as a drive signal flowing through the second drive coil  102  tends to differ in phase and amplitude from the second drive signal. 
     The phase-adjustment circuits  109  and  110  compensate for frequency-dependent phase differences between the first and second drive signals. The amplifiers  111  and  112  may provide amplitude adjustment to either or both of the drive signals to compensate for amplitude differences between the signals. The phase adjustment and amplitude adjustment is performed such that the signals V D1  and V D2  induced in the pickup coil  104  by the first and second drive coils  100  and  102 , respectively, are substantially equal in magnitude and 180 degrees out of phase so they cancel. It will be appreciated that the splitting circuit  108  may be used to adjust the relative magnitudes of the drive signals in the drive coils  100  and  102 . It will also be appreciated that the phase-adjustment circuits  109  and  110  may be positioned so that they each follow the amplifiers  111  and  112 , respectively. Because phase adjustment and amplitude adjustment need only be applied to one of the two drive signals, this allows for removal of one of the phase-adjustment circuits  109  or  110 . 
     Another method of amplitude and phase adjustment involves changing the effective resistance R D1  and R D2  and/or the effective inductance L D1  and L D2  of either or both drive coils  100  and  102 . Therefore, the phase-adjustment circuits  109  and  110  follow the amplifiers  111  and  112 , respectively. The phase-adjustment circuits  109  and  110  may comprise resistors (not shown) and/or inductors (not shown) connected in series with the drive coils  100  and  102  so as to adjust their effective resistance R D1  and R D2  and/or effective inductance L D1  and L D2 . In order for the relative phase between the magnetic flux generated by each drive coil  100  and  102  to be substantially 180 degrees, the relationship Arc Tan(ωL D1 /R D1 )=Arc Tan(ωL D2 /R D2 ) must hold for a wide range of signal frequencies ω. Thus, L D1 /R D1 =L D2 /R D2 . However, in order for the relative amplitudes between the magnetic flux generated by each drive coil  100  and  102  to be substantially equal for a broad range of signal frequencies ω, it is preferable that the effective resistance R D1 =R D2  and the effective inductance L D1 =L D2 . It will be appreciated that because the effective resistance R D1  and R D2  and the effective inductance L D1  and L D2  can be adjusted to control both the phase and amplitude of the drive signals in the drive coils  100  and  102 , only a single amplifier (not shown) for amplifying a pickup signal into a drive signal is necessary. An alternative cancellation circuit may include a single amplifier circuit (not shown) placed between the compensation circuit  106  and the splitting circuit  108 , and the amplifiers  111  and  112  may be removed. The phase-adjustment circuits  109  and  110  can provide both amplitude and phase-adjustment to the drive signals going to each drive coil  100  and  102 , as described above. 
     An embodiment for a cancellation circuit of the present invention is shown in FIG.  10 . The circuit in FIG. 10 includes a first pickup coil  120  wrapped around a pickup core  121 , a second pickup coil  122  wrapped around a second core  131 , and a drive coil  130  wrapped around the second core  131 . The first pickup coil  120  is connected to the input of an amplitude-adjustment circuit  124 . The output of the amplitude-adjustment circuit  124  is connected to the input of a phase-adjustment circuit  125 . The second pickup coil  122  is connected to the input of an amplitude-adjustment circuit  126 . The output of the amplitude-adjustment circuit  126  is connected to a phase-adjustment circuit  127 . The outputs of the phase-adjustment circuits  125  and  127  are connected to a combining circuit  128 . The output of the combining circuit  128  is connected to a compensation circuit  129 . The compensation circuit  129  is connected to an amplifier  136  that amplifies the input signal into a drive signal that flows through the drive coil  130  and generates a magnetic flux. 
     The pickup coils  120  and  122  are responsive to the magnetic flux generated by the drive coil  130 . However, due to the proximity of the second pickup coil  122  to the drive coil  130 , the second pickup coil  122  receives a greater magnitude of magnetic flux generated by the drive coil  130  than does the first pickup coil  120 . It will be appreciated that the second pickup coil  122  may be located inside of the drive coil  130 , or the pickup coils  120  and  122  may be positioned, shielded or otherwise designed such that the second pickup coil  122  receives greater magnetic flux generated by the drive coil  130  than does the first pickup coil  120 . The amplitudes of the pickup responses of the first and second pickup coils  120  and  122  induced by the magnetic flux generated by the drive coil  130  are made equivalent by either or both of the amplitude-adjustment circuits  124  and  126 . The phases of the pickup responses of the first and second pickup coils  120  and  122  induced by the magnetic flux generated by the drive coil  130  are compensated by either or both of the phase-adjustment circuits  125  and  127  so that when the pickup signals are combined in the combining circuit  128 , they substantially cancel. However, the response of the pickup coils  120  and  122  to uniform external magnetic flux results in a non-zero contribution to the combined signal at the output of the combining circuit  128 . The compensation circuit  129  may comprise either or both phase-adjustment and amplitude-adjustment circuits (not shown) for adjusting the phase response and/or amplitude response of the drive signal. The drive signal flows through the drive coil  130  and generates a uniform magnetic flux inside the drive coil  130  that substantially cancels the uniform magnetic flux inside the drive coil  130  generated by other sources (not shown). 
     The core  121  may be a ferromagnetic core. However, ferromagnetic materials tend to have a non-linear response to magnetic flux, resulting in pickup signals comprising higher-harmonic signals. The core  131  is preferably comprised of a non-ferromagnetic material having a hollow center. If the core  121  is made of a ferromagnetic material, then it is preferable that the core  131  be made of a similar ferromagnetic material so that the non-linear responses of the cores  121  and  131  of the pickups  120  and  122  substantially cancel. 
     Because the drive coil  130  generates a very uniform magnetic flux along its axis it is preferable that the region of space in which cancellation of magnetic flux is desired be surrounded by the drive coil  130 . However, it will be appreciated that the region of space in which cancellation is desired may be external to the drive coil  130 . It will also be appreciated that the second pickup coil  122  could be wrapped around the core  131  without being interwoven with the drive coil  130 , as shown. It will also be appreciated that these methods for canceling magnetic flux may be used along with a device that generates a static magnetic field to cancel external static magnetic fields. 
     Another embodiment for a cancellation circuit of the present invention is shown in FIG.  11 . The circuit in FIG. 11 includes a first pickup coil  140  wrapped around a core  141 , a second pickup coil  142  wrapped around the core  141 , and a drive coil  150  wrapped around the core  141 . The first pickup coil  140  is connected to the input of a phase-adjustment circuit  145 . The output of the phase-adjustment circuit  145  is connected to the input of an amplitude-adjustment circuit  144 . The second pickup coil  142  is connected to the input of an amplitude-adjustment circuit  146 . It will be appreciated that either amplitude-adjustment circuit  144  or  146  may act only as a buffer, as amplitude-adjustment of only one of the pickup coil  140  and  142  outputs may be necessary. The output of the amplitude-adjustment circuits  144  and  146  are connected to a combining circuit  148 . The output of the combining circuit  148  is connected to a compensation circuit  149 . The compensation circuit  149  is connected to an amplifier  156 . The amplifier  156  amplifies its input signal into a drive signal that flows through the drive coil  150 , and generates a magnetic flux. 
     The pickup coils  140  and  142  are responsive to the external magnetic flux generated by the drive coil  150 . The pickup coils  140  and  142  maybe positioned relative to the drive coil  150  as shown in FIG. 11 such that one of the pickup coils, such as pickup coil  140 , receives a greater amount of the magnetic flux generated by the drive coil  150  than does the second pickup coil  142 . Thus when the amplitudes and phases of the pickup signals from each of the pickup coils  140  and  142  are adjusted so that the contributions of magnetic flux generated by the drive coil  150  cancel at the combining circuit  148 , the combined response of the pickup coils  140  and  142  to uniform external magnetic flux are substantially non-zero. It will be appreciated that other methods may be used to adjust the responses of the pickup coils  140  and  142  to external magnetic flux, such as utilizing different numbers of coil windings in the pickups  140  and  142 , and/or changing the size, shape or material of the core  141  which the pickup coils  140  and  142  are wrapped around. 
     The compensation circuit  149  may comprise either or both phase-adjustment circuits (not shown) and amplitude-adjustment circuits (not shown) for adjusting the phase and/or amplitude response of the drive signal so that the drive signal has a specific amplitude and phase relationship to the external magnetic flux impinging on the pickup coils  140  and  142 . The drive signal flows through the drive coil  150  and generates a uniform magnetic flux inside the drive coil  150  that cancels the external magnetic flux inside the drive coil  150 . 
     It will be appreciated that the circuit shown in FIG. 11 may be used to drive or damp the motion of a ferromagnetic element (not shown) that generates a magnetic flux as it moves through a magnetic field. In this case, the core  141  may be made of a ferromagnetic material, and it may be shaped so that both endpoles of the core  141  are in close proximity to the ferromagnetic element (not shown) for providing a more powerful and concentrated driving (or damping) force to the ferromagnetic element. To produce a driving force, the drive coil  150  preferably generates a magnetic flux that is in phase with the motion of the ferromagnetic element, increasing in strength as the speed of the ferromagnetic element toward the core increases. To produce a damping force, the drive coil  150  preferably generates a magnetic flux that is out of phase, hence opposing the motion of the ferromagnetic element. 
     An embodiment for a cancellation circuit of the present invention is shown in FIG.  12 . The circuit in FIG. 12 includes a pickup coil  160  wrapped around a pickup core  161 , a signal generator  162 , and a drive coil  170  wrapped around a second core  171 . The pickup coil  160  is connected to the input of an amplitude-adjustment circuit  164 . The signal generator  162  provides a signal to an amplifier  176  that amplifies the signal to produce a drive signal. The drive signal flows through the drive coil  170  and generates a magnetic flux. The signal generator  162  is connected to the input of a phase-adjustment circuit  167 . The output of the phase-adjustment circuit  167  is connected to an amplitude-adjustment circuit  166 . The outputs of the amplitude-adjustment circuits  164  and  166  are connected to a combining circuit  168 . The output of the combining circuit  168  provides a pickup signal that is substantially free from the response of the pickup coil  160  to the magnetic flux generated by the drive coil  170 . 
     The cancellation circuit shown in FIG. 12 demonstrates that in order to provide a pickup with a cancellation signal, it is not necessary to have a second pickup device. In fact, any electrical representation of a drive signal that has the proper phase and amplitude characteristics may be used to cancel the response of the pickup to external magnetic flux generated by that drive signal. In this case, the waveform of the drive signal is generated by a signal generator  162 . The output of the signal generator  162  is adjusted by a phase-adjustment circuit  167  and an amplitude-adjustment circuit  166  before it is combined with the output of the pickup coil  160 . An amplitude-adjustment circuit  164  is shown connected to the output of the pickup coil  160  with a resistor R connected to electrical ground. It will be appreciated that either amplitude-adjustment circuit  164  or  166  may act only as a buffer, as amplitude-adjustment of only one of the signals from either the pickup coil  160  or the signal generator  162  is necessary. It will also be appreciated that the output signal V out  of the combining circuit  168  may be supplied to an input of the signal generator  162  to generate signals or control the frequency and amplitude of the generated signals output to the amplifier  176 . 
     The magnetic flux generated by the drive coil  170  induces a voltage V B  in the pickup coil  160  that is proportional to the magnitude of the magnetic flux. The magnitude of the magnetic flux generated by the drive coil  170  is proportional to the drive current I D  in the drive coil  170 . The drive current is I D =V D /(R D +iωL D ), where R D  and L D  are the effective resistance and inductance, respectively, of the drive coil  170 , and V D  is the drive voltage. The drive voltage is V D =G V O , where G is the gain of the amplifier  176  and V O  is the signal voltage produced by the signal generator  162 . The voltage V P  of the pickup coil  160  at the input of the amplitude-adjustment circuit  164  is V P =V B R/(R+R P +iωL P )=B V O R/(R+R P +iωL P )(R D +iωL D ), where R P  and L P  are the effective resistance and inductance, respectively, of the pickup coil  160 , and B is a proportionality constant that represents the contribution of gains and losses in the circuit. The signal voltage at the input of the combining circuit  168  connected to the output of the amplitude-adjustment circuit  164  is 
     
       
         V CP =A V P =AB V O R/(R+R P +iωL P )(R D +iωL D ), 
       
     
     where A is the gain of the amplitude-adjustment circuit  164 . In the case where the combining circuit  168  is a differential amplifier circuit, it is desirable that the input signal, V CO , from the output of the amplitude-adjustment circuit  166  be substantially identical to V CP  in order for cancellation to occur. 
     One possible design for a circuit that may be used as the phase-adjustment and amplitude-adjustment circuits  167  and  166 , respectively, includes two inverting amplifier circuits connected in series, as shown in FIG.  8 C. The impedance element Z 11  of the first inverting amplifier circuit may comprise a resistor (not shown) having the value (R+R P ) and an inductor (not shown) having the value L P , the resistor and inductor being connected in series. The impedance element Z 11  of the second inverting amplifier circuit may comprise a resistor (not shown) having the value R D  and an inductor (not shown) having the value L D , the resistor and inductor being connected in series. The impedance element Z 12  of the first inverting amplifier  98 A, and the impedance element Z 11  of the second inverting amplifier  98 B may have values such that their product equals the value: A B R. Any buffer amplifiers, such as amplifier  94 , may provide unity gain. Thus the input signal V CO  into the combining circuit  168  is substantially identical to the other input signal V CP , and thus cancel. It will be appreciated that the example shown illustrates only one of many designs for combined phase and amplitude-adjustment circuits that may be used as amplitude and phase-adjustment circuits  166  and  167 , respectively. Furthermore, it will also be appreciated that either or both phase and amplitude adjustment may be performed on the signal from the pickup coil  160 , such as pickup signal V P , in addition to, or instead of the phase adjustment and amplitude adjustment performed on the signal V O  generated by the signal generator  162 . For example, the amplitude-adjustment  164  circuit may be preceded or followed by, or comprise a phase-adjustment circuit (not shown) that adjusts the phase of the signal from the pickup coil  160 . 
     Another embodiment for a cancellation circuit of the present invention is shown in FIG.  13 . The receiver and transmitter elements in FIG. 13 include a pickup coil  180  wrapped around a pickup core  181  and a drive coil  190  wrapped around a second core  191 . The pickup coil  180  is connected to the input of an amplitude-adjustment circuit  184  at Terminal A. An amplifier  196  generates a drive signal at Terminal B that flows through the drive coil  190  and generates a magnetic flux. The output of the amplifier  196  is connected to the input of a phase-adjustment circuit  187 . Preferably, there is some sort of buffer (not shown) as part of the phase-adjustment circuit  187 . For example, the buffer (not shown) may include a large value of resistance that forms a voltage divider with the drive coil  190  and attenuates the input signal to the phase-adjustment circuit  187 . The output of the phase-adjustment circuit  187  is connected to an amplitude-adjustment circuit  186 . The outputs of the amplitude-adjustment circuits  184  and  186  are connected to a combining circuit  188 . The output of the combining circuit  188  provides a pickup signal at the input of the amplifier  196  (Terminal C) that is substantially free from the response of the pickup coil  180  to the magnetic flux generated by the drive coil  190 . 
     The cancellation circuit shown in FIG. 13 demonstrates that a cancellation signal can be generated without requiring a second pickup device. Part of a drive signal used to generate an external magnetic flux may be combined with a pickup signal to cancel the response of a pickup device to the external magnetic flux. The signal V A  at Terminal A represents a pickup signal βV D  induced by the magnetic flux generated by the drive coil  190 , where V D  is the voltage of the drive signal flowing through the drive coil  190  and β is a scaling factor that represents losses between the magnitude of the drive signal V D  in the drive coil  190  and the magnitude of the pickup coil&#39;s  180  response to that drive signal V D  after being amplified by the amplitude-adjustment circuit  184 . The signal V A  at Terminal A also includes an additional pickup signal V Pickup  induced by magnetic flux generated by other sources other than the drive coil  190 . Thus, V A =βV D +V Pickup . The signal at Terminal B is V B =V D +V DPickup , where V DPickup  is a signal that the drive coil  190  picks up due to external sources of magnetic flux. If V D  is much greater than V DPickup , V DPickup  can be ignored. Thus, V B ≈V D . The signal V B  is attenuated by a factor of β and its phase is adjusted in a feedback loop comprising amplitude-adjustment and phase-adjustment circuits  186  and  187 , respectively. Signal V B  is combined at the combining circuit  188  with the pickup signal V A =βV D +V Pickup  from Terminal A. The resulting output of the combining circuit  188  (Terminal C) is V C =V Pickup . Thus, feedback resulting from the pickup coil&#39;s  180  response to the magnetic flux generated by the drive coil  190  is substantially eliminated. 
     A cancellation circuit of the present invention is shown in FIG. 14A. A signal generator  202  generates an electrical signal that is amplified by an amplifier  203  to produce a drive signal at Terminal A. The drive signal flows through a drive coil  200  and generates a magnetic flux. The drive coil  200  may be wrapped around a core, such as core  201 . Terminal A is connected to an amplitude-adjustment circuit  204 , which preferably draws only a small portion of the drive signal. Thus, the input of the amplitude-adjustment circuit  204  may include a buffer (not shown), such as a high-value resistor. The output of the amplitude-adjustment circuit  204  is connected to the input of a phase-adjustment circuit  205 . The signal generator  202  has a second output (Terminal B) that produces a signal similar to the signal input to the amplifier  203 . However, the second output signal may differ in phase and/or amplitude from the signal input to the amplifier  203 . The second output of the signal generator  202  may be an output that is split from the input of the amplifier  203  by a splitting circuit (not shown). Terminal B is connected to the input of a harmonic-compensation circuit  210 . The output of the harmonic-compensation circuit  210  is connected to Terminal C, which is connected to an amplitude-adjustment circuit  206 . The output of the amplitude-adjustment circuit  206  is connected to the input of a phase-adjustment circuit  207 . The outputs of both phase-adjustment circuits  205  and  207  are connected to separate inputs of a combining circuit  208 . The combining circuit  208  produces an output signal V out  that results from cancellation of the drive signals generated by the signal generator  202 . 
     The drive coil  200  is responsive to external magnetic flux, even while a drive signal is flowing through the coil  200 . If the drive coil  200  is not in close proximity to materials that have a non-linear response to external magnetic flux, then higher harmonic effects in the drive signal and/or pickup signal of the drive coil  200  are substantially negligible. Consequently, the harmonic compensation circuit  210  may be replaced by a short circuit connecting Terminal B to Terminal C. The signal voltage at Terminal A is V A =V D +V Pickup , where V D  is the voltage of the drive signal and V Pickup  is the voltage of the induced pickup signal in the drive coil  200  resulting from external magnetic flux. The signal voltage at Terminal B is: V B =Q V D , where Q is a proportionality constant. Because the drive coil  200  has a complex impedance Z D =R D +iωL D , and other circuit elements (not shown) associated with the drive coil  200  may also contribute frequency-dependent terms to the effective impedance of the coil  200 , it is necessary that phase adjustment be performed to compensate for phase variations between the drive components V D  of the signal voltages V A  and V B  at Terminals A and B, respectively. After phase adjustment and amplitude adjustment have been performed on either or both of the signals V A  and V B , these signals are combined in the combining circuit  208  such that the signal components related to the drive voltage V D  cancel, leaving only a signal that is related to the pickup signal V Pickup . This method of cancellation allows a single element, such as the drive coil  200  shown in FIG. 14A, to simultaneously transmit and receive electromagnetic signals. It will be appreciated that the output signal V 0 ut may include an interface (not shown) to the signal generator  202  to control the amplitude, frequency, and phase of the signal generated by the signal generator  202 . 
     It will be further appreciated that if the core  201  is made of a ferromagnetic material, a voltage V H  resulting from the non-linear response of that material to the magnetic flux generated by the drive coil  200  is included in the signal voltage V A  at Terminal A: V A =V D +V Pickup +V H . Such nonlinearity creates even harmonics V H  in the fundamental driving frequency V D  that are not reducible to zero by prior-art cancellation techniques. Thus, it may be necessary to eliminate the signal voltage V H  resulting from the non-linear response of the core  201  material by providing the circuit with a filter (not shown) to filter out harmonic effects. The circuit may include a harmonic-compensation circuit, such as harmonic compensation circuit  210 . 
     FIG. 14B shows a circuit that provides harmonic compensation. The circuit in FIG. 14B includes a coil  212  wrapped around a core  211  that has a substantially identical non-linear response to magnetic flux as does the core  201  shown in FIG.  14 A. The coil  212  maybe oriented with respect to magnetic flux generated by the coil  200  such that the electrical signals induced in the coil  212  by the magnetic flux substantially cancel. The coil  212  may be provided with active magnetic shielding such as a cancellation circuit (not shown), or magnetic shielding materials (not shown) for substantially reducing the response of the coil  212  to magnetic flux generated by external magnetic sources (not shown). The material of the core  211  responds to the magnetic flux generated by the signal output V B  of the signal generator  202  at terminal B and substantially reproduces the non-linear effect in the signal V B  that is used to cancel the signal V A . 
     FIG. 14C shows a circuit that may be used as the harmonic compensation circuit  210 . The circuit in FIG. 14C includes a splitting circuit  213  for splitting the input signal from Terminal B into two signals at output Terminals B 1  and B 2 . The output terminal (Terminal B 2 ) is connected to an input terminal (Terminal C 2 ) of a combining circuit  223 . Terminal B 1  is connected to a harmonic-generator circuit  214  that substantially reproduces the shape of the harmonic signal V H  in the signal voltage V A  at Terminal A. The harmonic-generator circuit  214  may include one or more harmonic-generator circuits (not shown) known to persons skilled in the art as “frequency-doublers” or “frequency-triplers.” The output of the harmonic-generator circuit  214  is connected to a phase-adjustment circuit  218 . The phase-adjustment circuit  218  is connected to an input of an amplitude-adjustment circuit  219 . The output of the amplitude-adjustment circuit  219  is connected to an input terminal (Terminal C 1 ) of the combining circuit  223 . A signal that is proportional to the drive signal V D  is input to the combining circuit  223  at Terminal C 2 . A signal that is proportional to the harmonic signal V H  is adjusted in phase and/or amplitude by the phase-adjustment circuit  218  and the amplitude-adjustment circuit  219  before being input to the combining circuit  223  at Terminal C 1 . The signal voltage V C  at the output (Terminal C) of the combining circuit  223  has amplitude and phase relationships between signals V D  and V H  such that when that signal V C  is combined with the amplitude-adjusted, phase-adjusted signal V A  at the combining circuit  218 , the contributions of the harmonic terms V H  and the drive signal terms V D  substantially cancel. 
     FIG. 14D shows a circuit that may be used as the harmonic compensation circuit  210 . The circuit in FIG. 14D includes a splitting circuit  213  for splitting the input signal from Terminal B into three signals at output terminals B 1 , B 2 , and B 3 . Terminal B 3  is connected to an input terminal (Terminal C 3 ) of the combining circuit  223 . Terminal B 2  is connected to the input of an amplitude-adjustment circuit  216 . Terminal B 1  is connected to a coil  212  that is wrapped around a core  211 . The core  211  is made of a material that has a non-linear response to magnetic flux that is substantially identical to the non-linear response of the core  201  material. Thus, the output of the coil  212  comprises a signal voltage having a component that is proportional to the drive signal V D  and a component that is proportional to the harmonic signal V H . The coil  212  is connected to the input of a phase-adjustment circuit  214 . The output of the phase-adjustment circuit  214  is connected to the input of a an amplitude-adjustment circuit  215 . The outputs of the amplitude-adjustment circuits  215  and  216  are connected to a combining circuit  217  that combines the outputs such that the contributions that are proportional to the drive signal voltage V D  substantially cancel, providing a signal that is proportional to the harmonic signal V H . The output of the combining circuit  217  is connected to a phase-adjustment circuit  218 . The output of the phase-adjustment circuit  218  is connected to the input of an amplitude-adjustment circuit  219 . The output of the amplitude-adjustment circuit  219  is connected to the input Terminal C 1  of the combining circuit  223 . The combining circuit  223  combines the harmonic signal V H  and the drive signal V D  such that their relative proportion and phase are substantially identical to the relative proportion and phase of the harmonic signal V H  and drive signal V D  in the signal V A . 
     It will be appreciated that the splitting circuit  213  and the combining circuits  217  and  223  shown in FIG.  14 C and FIG. 14D may control relative amplitudes between the split and combined electrical signals, thereby eliminating the need for amplitude-adjustment circuits  215 ,  216 , and  219 . In FIG.  14 C and FIG. 14D, the output of the amplitude-adjustment circuit  219  is shown connected to an input terminal of the combining circuit  223 . However, it will be appreciated that the output of the amplitude-adjustment circuit  219  may be connected to the output V out  of the combining circuit  208 . 
     The circuit shown in FIG. 15 is an embodiment of a cancellation circuit of the present invention for a simultaneous transmit/receive system. An electrical signal V B  at a terminal (Terminal B) is amplified into a drive signal V D  at an output (Terminal A) of an amplifier  233 . Terminal A is connected to a drive coil  230  that may be wrapped around a core  231 . Terminal A is connected to an amplitude-adjustment circuit  234  that may attenuate the output of the amplitude-adjustment circuit  234  so that it is substantially lower in amplitude than the drive signal V D . The output of the amplitude-adjustment circuit  234  is connected to the input of a phase-adjustment circuit  235 . Terminal B is connected to the input of an amplitude-adjustment circuit  236 . The output of the amplitude-adjustment circuit  236  and the output of the phase-adjustment circuit  235  are connected to a combining circuit  238 . The output of the combining circuit  238  is connected to a preamplifier  232 . The output of the preamplifier  232  is connected to Terminal B. 
     The signal voltage V A  at Terminal A comprises a drive voltage V D , that flows through the drive coil  230  to generate a magnetic flux, and a pickup voltage V Pickup  induced in the drive coil  230  by other sources (not shown) of magnetic flux. In this case, it is preferable that the material comprising the core  231  has a substantially linear response to the magnetic flux so as to minimize additive harmonic signatures V H  caused by non-linear responses of the core  231  material to magnetic flux. The signal-voltage V B  at Terminal B represents the drive signal V D  before it is amplified by the amplifier  233 . The phase-adjustment circuit  235  and the amplitude-adjustment circuits  234  and  236  adjust the relative phase and amplitude of the signals V A  and V B  so that the components in the signals V A  and V B  related to the drive signal V D  substantially cancel when they are combined at the combining circuit  238 . 
     The total gain of a feedback loop is calculated by summing the gains and losses of each component in the feedback loop. For example, the gain of the first feedback loop in FIG. 15 is measured starting at Terminal B and moving through the amplitude-adjustment circuit  236  to the combining circuit  238 , then through the preamplifier  232  back to Terminal B. The amplitude-adjustment circuit  236  and the preamplifier  232  may provide gain or attenuation to the electrical signals. The combining circuit  238  provides an effective attenuation to the electrical signals by canceling the electrical signals representing the drive signal V D . Thus, if the total gain of the first feedback loop is less than one, this part of the circuit will not cause oscillation. 
     The gain of the second feedback loop in the circuit shown in FIG. 15 is measured starting at Terminal A and including the amplitude-adjustment circuit  234 , the phase-adjustment circuit  235 , the combining circuit  238 , the preamplifier  232 , and the amplifier  233 . The amplitude-adjustment circuit  234  includes a means for attenuating the signal V A  at Terminal A. The preamplifier  232  and the amplifier  233  can provide substantial gain to the electrical signal. The phase-adjustment circuit  235  generally has little effect on the amplitude of the electrical signal. The combining circuit  238  will provide a substantial effective attenuation to the electrical signal flowing through it by canceling the signal voltage that is related to the drive signal V D . If this cancellation is large enough, it will cause the total gain of the second feedback loop to be less than one. Thus, the circuit will not oscillate. 
     The pickup signal V Pickup  induced in the drive coil  230  is amplified and returned to the drive coil  230  to generate a magnetic flux. However, the feedback effects of the drive signal V D  in the circuit are canceled to prevent oscillation. This allows the drive coil  230  to simultaneously transmit and receive electromagnetic signal. It will be appreciated that if the core  231  is made of a material that has a non-linear response to magnetic flux, the cancellation circuit may be designed to cancel the electrical signals resulting from the non-linear response of the core  231  material. Different configurations of the cancellation circuit are shown in FIG.  14 B through FIG.  14 D. It will also be appreciated that either or both of the signals V A  and V B  from Terminals A and B, respectively, may have amplitude adjustment and/or phase adjustment applied to them such that the signal components related to the drive signal V D  will cancel at the combining circuit  238 . Furthermore, it will be appreciated that a compensation circuit (not shown) may be included in the feedback loop or may precede the drive coil  230  to provide a specific phase and/or amplitude relationship between the pickup signal V Pickup  and the drive signal V D . 
     The circuit shown in FIG. 16 is an embodiment of a cancellation circuit of the present invention that cancels both static magnetic fields and magnetic flux in a specific region of space. The circuit in FIG. 16 includes two magnetic-field sensors  240  and  242  that generate electrical signals that are proportional to the scalar magnitude of the magnetic field strength in a specific direction at the location of each sensor  240  and  242 . The sensors  240  and  242  may be flux gate sensors or the like. The phase of the signal from the first sensor  240  is adjusted by a phase-adjustment circuit  241 . The amplitude of the signal from the first sensor  240  is adjusted by an amplitude-adjustment circuit  244 . The phase of the signal from the second sensor  242  is adjusted by a phase-adjustment circuit  243 . The amplitude of the signal from sensor  242  is adjusted by an amplitude-adjustment circuit  246 . The outputs of the amplitude-adjustment circuits  244  and  246  are connected to a combining circuit  247 . The output of the combining circuit  247  is connected to a compensation circuit  248 , that is connected to the input of an amplifier  249 . The amplifier  249  amplifies an input signal to produce a drive signal V D  that flows through the drive coil  250  and produces a magnetic field that is substantially parallel to, but opposite to the magnetic field sensed by the sensors  240  and  242 . 
     The first sensor  240  is positioned in close proximity to the coil  250  or inside the coil  250  to sense both the magnetic field generated by the coil  250  and the magnetic field generated by external sources (not shown). The second sensor  242  is positioned in such a manner so that it is more sensitive to magnetic fields generated by external sources (not shown) than to the magnetic field generated by the coil  250 . Each of the sensors  240  and  242  has a specific ratio of response between the magnetic field generated by the coil  250  and the magnetic field generated by the external magnetic sources (not shown). It will be appreciated that there are many ways to change the ratio of response of one of the sensors  240  or  242 , For example, the position of one of the sensors may be adjusted. The important point is to provide one of the sensors  240  or  242  with a different ratio of response than the other sensor  242  or  240 . The amplitude and/or phase of the signals produced by the sensors  240  and  242  are adjusted by amplitude-adjustment circuits  244  and  246 , respectively, and phase-adjustment circuits  241  and  243 , respectively, such that the components of the signals related to the drive signal V D  substantially cancel at the combining circuit  247 . The output of the combining circuit will comprise a voltage V Ext  that is proportional to the magnetic field intensity generated by the external sources (not shown). 
     The amplitude of the signals produced by the sensors  240  and  242  may be adjusted in order to cancel a dc signal resulting from the magnetic field generated by the drive coil  250 . However, as the magnitude of the drive signal V D  changes in response to a changing external magnetic field, there may be some response anomalies between the two sensors  240  and  242  related to the rate-of-change (flux) of the drive signal V D . Thus, it may be necessary to compensate for flux-dependent amplitude differences and rate-of-response (phase) differences between the two sensors  240  and  242 . 
     The drive coil  250  generates a highly uniform magnetic field within the region of space that it encloses. Thus, it is preferable to utilize the interior of the coil  250  as the space in which uniform magnetic fields will be canceled. However, due to the inductive properties of the coil  250  and the possible flux-dependent amplitude and phase characteristics of the sensors  240  and  242 , it is necessary to provide phase and/or amplitude compensation using a compensation circuit (such as compensation circuit  248 ) in order to provide substantial cancellation of magnetic flux. 
     The circuit shown in FIG. 17 is another embodiment of a cancellation circuit of the present invention that cancels both static magnetic fields and magnetic flux in a specific region of space. The circuit in FIG. 17 includes a magnetic field sensor  262  that generates an electrical signal that is proportional to the scalar magnitude of magnetic-field strength in a specific direction at the location of the sensor  262 . It will be appreciated that the sensor  262  may be a flux gate sensor or the like. The signal produced by the sensor  262  is sent to an automatic control unit  264 . The automatic-control unit  264  controls the gain of an amplifier  266  connected to a dc-level generator  265 . The output of the amplifier  266  is an amplified or attenuated dc-level drive signal V D  that is passed through a compensation circuit  267  to a drive coil  260 , which generates a magnetic field. 
     The magnetic-field sensor  262  is preferably positioned inside the region of space where cancellation of magnetic fields is desired. The magnetic-field sensor  262  produces a signal that is proportional to the amplitude of the magnetic field it senses. The magnetic field comprises the magnetic field generated by the drive coil  260  and magnetic fields generated by other sources (not shown). The automatic-control unit  264  determines if a magnetic field is present at the sensor  262  and controls the amplifier  266  so that the drive signal V D  in the drive coil  260  produces a magnetic field that cancels the magnetic field at the sensor  262 . Because the drive coil  260  has inductive properties, there will be an inductive lag in the drive signal V D  flowing through the drive coil  260  when the amplitude of that signal changes. Likewise, the effective impedance Z D =R D +iωL D  of the coil  260  changes with signal frequency ω. Thus, a signal flux results in an amplitude variation of the magnetic field generated by the drive coil  260 . The compensation circuit  267  provides phase adjustment and amplitude adjustment to the drive signal V D  so that the drive coil  260  cancels both static and dynamic magnetic fields. 
     It will be appreciated that the compensation circuit  267  may also provide compensation for any flux-dependent amplitude and phase variations in the response of the sensor  262 . It will also be appreciated that the automatic control unit  264  provide amplitude and phase compensation to the drive signal V D . The circuits shown in FIG.  16  and FIG. 17 show systems that cancel magnetic fields along a single axis. However, a superposition of three such circuits, each along an orthogonal axis, can provide complete cancellation of magnetic fields in three dimensions. 
     Because coils of wire whose currents support magnetic fields in space function as antennas radiating electromagnetic energy, it is obvious that the cancellation and/or compensation circuits shown above may be used in radar systems for providing interference cancellation and simultaneous transmit/receive capability. 
     The circuit shown in FIG. 18 is a preferred embodiment of a cancellation circuit of the present invention. A signal generator  276  generates an electrical generator signal V G  that is amplified by a power amplifier  274  and passed through a junction  272  to an antenna element  270 . Antenna element  270  both emits and receives electromagnetic radiation. The antenna element  270  is responsive to other sources (not shown) of electromagnetic radiation, thus producing an electrical pickup signal V P . The junction  272  is connected to an input of a combining circuit  275  that receives the pickup signal V P  along with a leakage signal V L  from the power amplifier  274 , The leakage signal V L  is a portion of the generator signal V G . The signal generator  276  also produces a reference signal V R  that is similar in shape to the generator signal V G . The reference signal V R  passes through an amplitude-adjustment circuit  271  and a phase-adjustment circuit  273  to an input of the combining circuit  275 . An amplifier  278  amplifies the output of the combining circuit  275 . 
     The junction  272 , which the generator signal V G  passes through on its way to the antenna element  270 , may be a circulator (not shown) that directs most of the power from the power amplifier  274  to the antenna element  270 . However, because the efficiency of a circulator is frequency-dependent on the electrical signals passing through it, the performance of the circulator (not shown) is degraded by the use of large signal-bandwidths or multiple frequencies. Therefore, some of the energy from the power amplifier  274  leaks into the combining circuit  275 . The amplitude-adjustment circuit  271  provides frequency-dependent amplitude adjustment to the reference signal V R  such that its amplitude is substantially identical to the amplitude of the leakage signal V L  leaked from the power amplifier  274  into the combining circuit  275 . It will be appreciated that the amplitude-adjustment circuit  271  may be a circulator that is similar to the circulator used as the junction  272 . Thus, the output of the amplitude-adjustment circuit  271  is substantially proportional to the amplitude of the leakage signal V L . The phase-adjustment circuit  273  adjusts the phase of the reference signal V R  such that it cancels the leakage signal V L  at the combining circuit  275 . Preferably, the phase-adjustment circuit  273  produces a substantially constant phase between the leakage signal V L  and the reference signal V R  over the desired frequency range of generated signals V G . Thus, the output of the combining circuit  275  will comprise a pickup signal V P  that is substantially free from the effects of leakage signal V L  originating from the power amplifier  274 . 
     It will be appreciated that many possible designs exist for cancellation circuits that cancel the effects of interference between the transmitting and receiving elements of a radiating system. The circuit shown in FIG. 18 is only one of these designs. Amplitude and/or phase-adjustment circuits (not shown) may be interposed in the circuit between the junction  272  and the combining circuit  275 . Furthermore, in the case where amplitude-modulated and/or frequency-modulated signals are generated and received, the circuit may include filters (not shown) for filtering out the carrier frequency before the cancellation circuit removes the transmitted signal from the received signal. 
     The phase-adjustment circuit  273  may include a delay apparatus (not shown), such as from nearby objects (such as ground clutter) is canceled from the pickup signal V P . The reference signal V R  may also include electrical signals that are similar in shape to signals induced by other noise sources (not shown) in the antenna element  270 . Separate amplitude adjustment and phase adjustment may be performed to cancel the response of the antenna element  270  to the other noise sources (not shown). Furthermore, the antenna element  270  may be responsive to incident radiation for producing a drive signal that allows an antenna, such as the antenna element  270 , to transmit electromagnetic radiation that cancels the reflection of the incident radiation off of the antenna element  270  or some other object (not shown). Thus, the reflected radiation may be canceled at a distant receiver (not shown). 
     Magnetic pickups comprising pickup coils are shown in the circuits of FIG.  2  through FIG. 15, however any type of electromagnetic pickup may be used with these types of cancellation circuits. Likewise, a compensation circuit may be used to compensate for amplitude and phase variations arising from any pickup device that produces an electrical pickup signal. A cancellation circuit may be used to cancel the electrical signals arising from the response of the pickup device to noise. For example, a cancellation circuit may be used for canceling the signals generated by an optical sensor&#39;s electrical response to background electromagnetic radiation. In many of the figures, a drive coil is illustrated as the element that generates an electromagnetic field. However, compensation circuits may be used to compensate for frequency-response characteristics exhibited by any structures that generate electromagnetic fields. 
     The preferred methods of amplitude adjustment were shown to be electrical gain and attenuation controls. However, it will be appreciated that other methods of amplitude adjustment may be used, such as adjusting the relative position of the pickup coils, the drive coils, the cores for the pickup coils and drive coils, and/or nearby permeable and/or conducting materials. It will also be appreciated that the inductance of a coil may be changed by changing the reluctance of the path seen by that coil&#39;s magnetic field. 
     The magnitude of electrical current in the pickup coils was considered to be very small. Thus, the formulation of the equations representing the electrical pickup signals induced in the pickup coils by magnetic flux have not included the inductive effects that the pickup coils may have on each other. However, the scope and spirit of the present invention would not be challenged by considering the inductive effects between pickup coils when designing the cancellation circuits. Furthermore, consideration of the more subtle electromagnetic effects (such as how capacitance in the pickup coils affects the induction of electrical signals in the pickup coils) and how the cancellation and compensation circuits that may be designed accordingly is anticipated by this invention. 
     Although the invention has been described in detail with reference to the illustrated preferred embodiments, variations and modifications exist within the scope and spirit of the invention as described and as defined in the following claims.