Med. & Biol. Eng. & Comput., 1977, 15, 446-449
Linearity of radio-frequency transducers K. M . J a c k s o n Department of Physics, Guy's Hospital Medical School, London Bridge, London SE1 9RT, England.
A b s t r a c t - - T h e paper outlines a method of monitoring changes in ioint angle for a 2-dimensional movement of the limbs. The transducer described is acceptably linear over a range of 100 ~ It provides a useful alternative to goniometry in situations where the attachment of a goniometer across the ioint would cause undue interference to the movement being studied. The transducer detects spatial movement by measuring the voltage induced in a sensing coil from a 1 MHz magnetic field produced by a small solenoid, 3-dimensional monitoring of body movements is discussed and shown to be feasible using a more complex system. Keywords--Goniometer, Radio-frequency transducers
Nomenclature A B D F 10 Q r
= = = = -= =
magnetic vector potential magnetic flux distance from oscillator to axis of rotation distance from solenoid to axis of rotation zero-to-peak amplitude of solenoid current element of sensor-coil area radial distance to sensor-coil element of area r' = vector distance from dl to Q t = time x, y, z = co-ordinate axes d! = element of length of solenoid winding 0 = angle to be measured ~u = angular distance to sensor-coil element of area /to = relative permeability of free space 09 = frequency
2 Theory Consider a system with one limb rotating about a joint, and two coils placed as in Fig. I: coil-A
oscillator
1 Introduction A PERENNIALproblem in biomechanical research is the measurement of limb movements without the complications of interference or restrictions to the movement, or laborious interpretation of cinematic or television data. Both of these limitations can be overcome by the judicious use of low-strength magnetic fields. The main problems involved in using such systems are the absorption or reflection of the electromagnetic radiation by surrounding structures, and nonlinearity in the response of the transducing system. These difficulties can be overcome by use of long-wavelength radiation, so that the dimensions of the human body are far smaller than the wavelength involved and the tissues are not 'seen' by the radiation (relative permeability -~ 1), coupled First receive d 26th Aug ust and in final form 19th November 1976
446
with careful design, construction and placing of the radio-frequency (r.f.) coils to provide the maximum possible linear region. Simple magnetic-dipole theory is useless in any practical transducer, as the approximation that the distance between two coils is far greater than the coil dimensions cannot be used. Thus a complete analysis o f any magnetic fields must be used. The following analysis shows how computing techniques can be used to predict the response of a goniometer which utilises a small solenoid producing a 1 MHz magnetic field and a small detector coil.
I upper limb
I I 1 0 I
sens
\
Fig. 1 Mounting positions for oscillatJr and sensor. In this case for measurement of flexion of the elbow
Medical and Biological Engineering & C o m p u t i n g
July 1977
having a sinusoidal current passing through it. Coil-B has a current induced in it, which is dependent upon the amplitude and frequency of the current in coil-A and the relative position of the two coils. If the magnetic flux lines in the vicinity of coil-B were parallel, the response when the lower limb was rotated would be sinusoidal, giving a linear range in the measurement of 0 of + 10~ ~ depending upon the accuracy required. Extended linearity regions can be obtained if coil-A is a solenoid, and the relative sizes of D, F and the coil dimensions are such that the flux lines curve back in towards coil-A, as 0 is increased, at the position where sinusoidal departures from linearity are occurring. If the current in the solenoid shown in Fig. 2 is Io #,or, the vector potential at any point in space is given by A - 12o Io e jot
dl
where d! is an element of a solenoid loop, and r' is a vector from d! to the relevant point in space. This can be used to calculate the vector potential at any element of the sensor coil (denoted by r and ~ in Fig. 2). Using the co-ordinate system shown in Fig. 2, A= = 0 and as the sensor coil is perpendicular to the xz-plane, the y-component of the magnetic flux induces no current in the sensor.
f
Therefore the magnetic flux =B = curl A
-Y;-z] +t' \ ~
~y )
The current induced in the sensor coil from a sensor coil element with area Q is/~. Q. But B is a sinusoidal function of time, therefore = jogBe J*'
Thus, for a fixed frequency, /~ is proportional to B and 90 ~ out of phase. Using appropriate electronics, only the amplitude and direction of the current induced in the sensor coil are measured; thus constant phase changes are of no consequence. As there are no singularities in the magnetic field across the whole of the spatial region used, the rate of change of A with respect to any co-ordinate axis can be found by linear interpolation if the interpolative distance is made small (HERSHEY et al., 1967). Ax was calculated at two positions + Ayawayfrom the sensor coil element parallel to the y-axis {Ax(+ Ay), Ax(- Ay)}. Therefore ~A~ ~y
a x ( + A y ) - Ax(- Ay) 2Ay
dt
Z
/
/01
/ /
/
t I X
~sen$oP a
b
Fig. 2 Oscillator and sensor in the co-ordinate system (a) Section taken in the xz-plane, the y-axis being perpendicular to the page. The sensor is always perpendicular to the xz-plane and rotates about point P in this plane. The distances from the solenoid and the sensor to the axis of rotation are D and F, respectively (b) Sensor perpendicular to both the xz-plane and the face of the sensor. Area element Q is defined by the radial distance r and the angle r" is a line vector from solenoid line element d l to sensor area element Q
Medical and Biological Engineering & Computing J
July 1977
447
Ay was reduced until its size had no meaningful effect on the value calculated for ~A~/~y. Similar calculations were made to find ~Ay/~x and ~Ay/?z at + Ax and + Az parallel to the appropriate axis. /~ was then calculated by summing the above calculation over all elements of the sensor coil for all the loops on the solenoid. Ax, Ay, Az and the size of sensor coil element were all reduced in size such that any further reduction had no meaningful effect on the calculation of B. The whole calculation was then repeated at varying values of 0.
1[
/!
~0
i
o ~ ~
~"q'~
.0"4 -1
-90
0
90 0
Fig. 3 Comparison of the transducer's response to a computer prediction of the response. The transducer output is normalised ( + 1) over the range - - 9 0 ~ to + 90 ~
Fig. 3 shows the agreement between the calculated and observed responses when the solenoid and sensor coil are mounted upon non-metal hinged rods, and the rods are rotated, this good agreement being despite the fact that the oscillator solenoid was hand-wound and therefore not consistent along its length.
3 Discussion
A suitable oscillator and sensing system in current use is very similar to that used by CHANDLER et al. (1972) for detecting movements of the fingers, which uses an oscillator frequency of approximately 1 MHz. The coil in the oscillator circuit is wound as a solenoid, with the associated components in the oscillator circuit mounted in a very small package close to the solenoid. This prevents any stray r.f. radiation, which could radiate from cables connecting the solenoid to the oscillator electronics, interfering with the measurement field pattern around the coil. A phase-sensitive detector eliminated the necessity of using tuned sensing coils to remove extraneous r.f. interference. The output is in the form of a direct voltage level proportional to the magnitude, and having the sign of the current induced in the sensor coil. This system is in use at the moment in the measurement of forearm movement relative to the upper-arm 448
during walking, the output being calibrated against a pendulum goniometer. The main circuitry is all housed in a 40 • 70 • 100 mm metal box attached to the subject's belt, a 50 Hz notch filter being used to remove extraneous mains interterence. The output signal is of a sufficiently high level to be fed to a recording system via a cable trailing behind the subject while he walks. Fig. 3 shows that the computer program used can predict accurately the response of the transducer system mentioned above and thus is capable of producing an accurate 'map' of the magnetic field around the solenoid. Therefore 3-dimensional monitoring of the orientation of one limb with respect to another becomes feasible, using multiple oscillator solenoids placed at known fixed distances from each other and three mutually perpendicular sensor coils forming each sensor set. The output from these sensor coils can be squared and added to give a measure of the modulus of the magnetic field strength at that point in space, no matter what the orientation of the sensor-coil set happens to be. At least three oscillator solenoids are necessary to discriminate in three dimensions, more being useful to remove duplicity and improve accuracy. Information from each oscillator solenoid can be discriminated by either powering the oscillators at different frequencies or sequentially pulsing them. Computer processing of the data would be essential, but presents no great difficulties. Among the numerous advantages of such a system, as opposed to the present almost universal reliance on cinematography, or potentiometric recordings about a joint, would be its compactness, the ability of a subject to perform normal tasks unhindered, and the ability to perform working tasks anywhere and not just in the field of view of a camera.
Acknowledgment--The author gratefully acknowledges the financial support of Guy's Hospital Endowment Fund.
References
CHANDLER,S. A. G., NIGHTINGALE,J. M. and SEDGWICK, E. M. (1972) A multi-channel radio frequency goniometer, J. Physiol. 226, llP-12P. COBBOLD, R. S. C. (1974) Transducers for biomedical measurements: Principles and applications. Wiley. HERSHEY, H. C., ZAKIN, J. L. and SIMHA, R. (1967) Numerical differentiation of equally spaced and not equally spaced experimental data. Ind. Eng. Chem. Fund. 6, 413-421.
Medical and Biological Engineering & Computing
July 1977
Lin~arit6 des transducteurs & radiofr6quences Sommaire---Cet article pose les grandes lignes d'une m~thode de d6termination des changements d'angle des articulations pour un mouvement des membres dans un plan donn6. Le transducteur d6crit poss~de une lin6arit6 acceptable sur 100 ~ I1 constitue une solution utile en remplacement de la goniom6trie dans les cas oO la fixation d'un goniom6tre sur l'articulation interfere trop avec le mouvement 6tudi6. Le transducteur d6tecte les mouvements spatiaux en mesurant le voltage induit dans une bobine d6tectrice par un champ magn6tique de 1 MHz produit par un petit sol6noide. L'article discute 6galement de la d6tection des mouvements du corps en trois dimensions et montre qu'elle est possible avec un syst6me plus complexe.
Die Linearit~it von Hochfrequenzumformern Zusammenfassung--Dieser Aufsatz umreil3t eine Methode zur [3berwachung yon Anderungen im Gelenkwinkel bei einer zweidimensionalen Bewegung der Gliedmagen. Der beschriebene Umformer ist annehmbar linear fiber einen Bereich yon 100 ~ Es handelt sich um eine nfitzliche Alternative zur Winkelmessung in Situationen, in denen ein fiber dem Gelenk angebrachter Goniometer die zu studierende Bewegung unangemessen st6ren wfirde. Der Umformer stellt rgumliche Bewegung fest, indem er die Spannung migt, die von einem 1 MHz-Magnetfeld, alas von einer kleinen Magnetspule erzeugt wird, in eine Ffihlspule induziert wird. Die dreidimensionale ~berwachung von KiSrperbewegungen wird diskutiert und deren Durchffihrbarkeit unter Verwendung eines komplizierteren Systems nachgewiesen.
Medical and Biological Engineering & Computing
July 1977
449
Linearity of radio-frequency transducers K. M . J a c k s o n Department of Physics, Guy's Hospital Medical School, London Bridge, London SE1 9RT, England.
A b s t r a c t - - T h e paper outlines a method of monitoring changes in ioint angle for a 2-dimensional movement of the limbs. The transducer described is acceptably linear over a range of 100 ~ It provides a useful alternative to goniometry in situations where the attachment of a goniometer across the ioint would cause undue interference to the movement being studied. The transducer detects spatial movement by measuring the voltage induced in a sensing coil from a 1 MHz magnetic field produced by a small solenoid, 3-dimensional monitoring of body movements is discussed and shown to be feasible using a more complex system. Keywords--Goniometer, Radio-frequency transducers
Nomenclature A B D F 10 Q r
= = = = -= =
magnetic vector potential magnetic flux distance from oscillator to axis of rotation distance from solenoid to axis of rotation zero-to-peak amplitude of solenoid current element of sensor-coil area radial distance to sensor-coil element of area r' = vector distance from dl to Q t = time x, y, z = co-ordinate axes d! = element of length of solenoid winding 0 = angle to be measured ~u = angular distance to sensor-coil element of area /to = relative permeability of free space 09 = frequency
2 Theory Consider a system with one limb rotating about a joint, and two coils placed as in Fig. I: coil-A
oscillator
1 Introduction A PERENNIALproblem in biomechanical research is the measurement of limb movements without the complications of interference or restrictions to the movement, or laborious interpretation of cinematic or television data. Both of these limitations can be overcome by the judicious use of low-strength magnetic fields. The main problems involved in using such systems are the absorption or reflection of the electromagnetic radiation by surrounding structures, and nonlinearity in the response of the transducing system. These difficulties can be overcome by use of long-wavelength radiation, so that the dimensions of the human body are far smaller than the wavelength involved and the tissues are not 'seen' by the radiation (relative permeability -~ 1), coupled First receive d 26th Aug ust and in final form 19th November 1976
446
with careful design, construction and placing of the radio-frequency (r.f.) coils to provide the maximum possible linear region. Simple magnetic-dipole theory is useless in any practical transducer, as the approximation that the distance between two coils is far greater than the coil dimensions cannot be used. Thus a complete analysis o f any magnetic fields must be used. The following analysis shows how computing techniques can be used to predict the response of a goniometer which utilises a small solenoid producing a 1 MHz magnetic field and a small detector coil.
I upper limb
I I 1 0 I
sens
\
Fig. 1 Mounting positions for oscillatJr and sensor. In this case for measurement of flexion of the elbow
Medical and Biological Engineering & C o m p u t i n g
July 1977
having a sinusoidal current passing through it. Coil-B has a current induced in it, which is dependent upon the amplitude and frequency of the current in coil-A and the relative position of the two coils. If the magnetic flux lines in the vicinity of coil-B were parallel, the response when the lower limb was rotated would be sinusoidal, giving a linear range in the measurement of 0 of + 10~ ~ depending upon the accuracy required. Extended linearity regions can be obtained if coil-A is a solenoid, and the relative sizes of D, F and the coil dimensions are such that the flux lines curve back in towards coil-A, as 0 is increased, at the position where sinusoidal departures from linearity are occurring. If the current in the solenoid shown in Fig. 2 is Io #,or, the vector potential at any point in space is given by A - 12o Io e jot
dl
where d! is an element of a solenoid loop, and r' is a vector from d! to the relevant point in space. This can be used to calculate the vector potential at any element of the sensor coil (denoted by r and ~ in Fig. 2). Using the co-ordinate system shown in Fig. 2, A= = 0 and as the sensor coil is perpendicular to the xz-plane, the y-component of the magnetic flux induces no current in the sensor.
f
Therefore the magnetic flux =B = curl A
-Y;-z] +t' \ ~
~y )
The current induced in the sensor coil from a sensor coil element with area Q is/~. Q. But B is a sinusoidal function of time, therefore = jogBe J*'
Thus, for a fixed frequency, /~ is proportional to B and 90 ~ out of phase. Using appropriate electronics, only the amplitude and direction of the current induced in the sensor coil are measured; thus constant phase changes are of no consequence. As there are no singularities in the magnetic field across the whole of the spatial region used, the rate of change of A with respect to any co-ordinate axis can be found by linear interpolation if the interpolative distance is made small (HERSHEY et al., 1967). Ax was calculated at two positions + Ayawayfrom the sensor coil element parallel to the y-axis {Ax(+ Ay), Ax(- Ay)}. Therefore ~A~ ~y
a x ( + A y ) - Ax(- Ay) 2Ay
dt
Z
/
/01
/ /
/
t I X
~sen$oP a
b
Fig. 2 Oscillator and sensor in the co-ordinate system (a) Section taken in the xz-plane, the y-axis being perpendicular to the page. The sensor is always perpendicular to the xz-plane and rotates about point P in this plane. The distances from the solenoid and the sensor to the axis of rotation are D and F, respectively (b) Sensor perpendicular to both the xz-plane and the face of the sensor. Area element Q is defined by the radial distance r and the angle r" is a line vector from solenoid line element d l to sensor area element Q
Medical and Biological Engineering & Computing J
July 1977
447
Ay was reduced until its size had no meaningful effect on the value calculated for ~A~/~y. Similar calculations were made to find ~Ay/~x and ~Ay/?z at + Ax and + Az parallel to the appropriate axis. /~ was then calculated by summing the above calculation over all elements of the sensor coil for all the loops on the solenoid. Ax, Ay, Az and the size of sensor coil element were all reduced in size such that any further reduction had no meaningful effect on the calculation of B. The whole calculation was then repeated at varying values of 0.
1[
/!
~0
i
o ~ ~
~"q'~
.0"4 -1
-90
0
90 0
Fig. 3 Comparison of the transducer's response to a computer prediction of the response. The transducer output is normalised ( + 1) over the range - - 9 0 ~ to + 90 ~
Fig. 3 shows the agreement between the calculated and observed responses when the solenoid and sensor coil are mounted upon non-metal hinged rods, and the rods are rotated, this good agreement being despite the fact that the oscillator solenoid was hand-wound and therefore not consistent along its length.
3 Discussion
A suitable oscillator and sensing system in current use is very similar to that used by CHANDLER et al. (1972) for detecting movements of the fingers, which uses an oscillator frequency of approximately 1 MHz. The coil in the oscillator circuit is wound as a solenoid, with the associated components in the oscillator circuit mounted in a very small package close to the solenoid. This prevents any stray r.f. radiation, which could radiate from cables connecting the solenoid to the oscillator electronics, interfering with the measurement field pattern around the coil. A phase-sensitive detector eliminated the necessity of using tuned sensing coils to remove extraneous r.f. interference. The output is in the form of a direct voltage level proportional to the magnitude, and having the sign of the current induced in the sensor coil. This system is in use at the moment in the measurement of forearm movement relative to the upper-arm 448
during walking, the output being calibrated against a pendulum goniometer. The main circuitry is all housed in a 40 • 70 • 100 mm metal box attached to the subject's belt, a 50 Hz notch filter being used to remove extraneous mains interterence. The output signal is of a sufficiently high level to be fed to a recording system via a cable trailing behind the subject while he walks. Fig. 3 shows that the computer program used can predict accurately the response of the transducer system mentioned above and thus is capable of producing an accurate 'map' of the magnetic field around the solenoid. Therefore 3-dimensional monitoring of the orientation of one limb with respect to another becomes feasible, using multiple oscillator solenoids placed at known fixed distances from each other and three mutually perpendicular sensor coils forming each sensor set. The output from these sensor coils can be squared and added to give a measure of the modulus of the magnetic field strength at that point in space, no matter what the orientation of the sensor-coil set happens to be. At least three oscillator solenoids are necessary to discriminate in three dimensions, more being useful to remove duplicity and improve accuracy. Information from each oscillator solenoid can be discriminated by either powering the oscillators at different frequencies or sequentially pulsing them. Computer processing of the data would be essential, but presents no great difficulties. Among the numerous advantages of such a system, as opposed to the present almost universal reliance on cinematography, or potentiometric recordings about a joint, would be its compactness, the ability of a subject to perform normal tasks unhindered, and the ability to perform working tasks anywhere and not just in the field of view of a camera.
Acknowledgment--The author gratefully acknowledges the financial support of Guy's Hospital Endowment Fund.
References
CHANDLER,S. A. G., NIGHTINGALE,J. M. and SEDGWICK, E. M. (1972) A multi-channel radio frequency goniometer, J. Physiol. 226, llP-12P. COBBOLD, R. S. C. (1974) Transducers for biomedical measurements: Principles and applications. Wiley. HERSHEY, H. C., ZAKIN, J. L. and SIMHA, R. (1967) Numerical differentiation of equally spaced and not equally spaced experimental data. Ind. Eng. Chem. Fund. 6, 413-421.
Medical and Biological Engineering & Computing
July 1977
Lin~arit6 des transducteurs & radiofr6quences Sommaire---Cet article pose les grandes lignes d'une m~thode de d6termination des changements d'angle des articulations pour un mouvement des membres dans un plan donn6. Le transducteur d6crit poss~de une lin6arit6 acceptable sur 100 ~ I1 constitue une solution utile en remplacement de la goniom6trie dans les cas oO la fixation d'un goniom6tre sur l'articulation interfere trop avec le mouvement 6tudi6. Le transducteur d6tecte les mouvements spatiaux en mesurant le voltage induit dans une bobine d6tectrice par un champ magn6tique de 1 MHz produit par un petit sol6noide. L'article discute 6galement de la d6tection des mouvements du corps en trois dimensions et montre qu'elle est possible avec un syst6me plus complexe.
Die Linearit~it von Hochfrequenzumformern Zusammenfassung--Dieser Aufsatz umreil3t eine Methode zur [3berwachung yon Anderungen im Gelenkwinkel bei einer zweidimensionalen Bewegung der Gliedmagen. Der beschriebene Umformer ist annehmbar linear fiber einen Bereich yon 100 ~ Es handelt sich um eine nfitzliche Alternative zur Winkelmessung in Situationen, in denen ein fiber dem Gelenk angebrachter Goniometer die zu studierende Bewegung unangemessen st6ren wfirde. Der Umformer stellt rgumliche Bewegung fest, indem er die Spannung migt, die von einem 1 MHz-Magnetfeld, alas von einer kleinen Magnetspule erzeugt wird, in eine Ffihlspule induziert wird. Die dreidimensionale ~berwachung von KiSrperbewegungen wird diskutiert und deren Durchffihrbarkeit unter Verwendung eines komplizierteren Systems nachgewiesen.
Medical and Biological Engineering & Computing
July 1977
449
