Magnetic Forces and Torques on Local Gradient Coils: An Example Using a Head Gradient Coil (2001)

Introduction The magnetic forces and torques acting on local gradient coils have been calculated and also measured using a specially developed experimental apparatus. Good agreement has been found. Most of the work was done using a local head gradient coil built by Medical Advances, Inc (Milwaukee, WI) which was tested in a 1.5 T cylindrical supeconducting imager (General Electric, Milwaukee, WI)., but the results are relevant to the general issues involved in the mounting of local gradient coils. This work is important as local gradient coils offer several potential advantages in terms of inductance, slew rate, maximum gradient strength and nerve stimulation. However, their utilization has been somewhat limited because of safety concerns regarding motion caused by magnetic forces and torques. Body gradient coils are rigidly attached to the frame of the magnet and do not lead to the same level of concern. We have analyzed and measured the forces and torques on several versions of an insertable head gradient coil. This work has involved mathematical modeling of the forces and torques and direct physical measurements. Near the central axis of the scanner the z-gradient coils respond only to the radial components of Bo and these are small unless the coil is near the opening of the of the magnet bore. Therefore, this abstract concentrates on the transverse gradient coils (TGCs) as their behavior is more complex than that of the z-gradient coils. Mathematical Analysis The force dF on an area element dA of a given coil surface current density, J, (A/m) in a magnetic field, Bo is given by dF=JxBodA. Although the force on any given current element may be large, it can be shown that any current loop in a uniform magnetic field has zero net translational force. However, there is a torque on such loops, which tends to align the dipole moment of the coil (i.e., the normal vector to the current loop) with the applied field. These torques can be substantial if the coil area, total current and Bo are large. Furthermore, away from the isocenter of the magnet, where the Bo field is nonuniform, substantial translational forces and torques can occur. Ordinarily, TGCs are composed four current patches with two of each polarity. Therefore, the total dipole moment of a complete (symmetric) TGC is zero and the total torque on a TGC considered as a rigid body positioned at isocenter is zero. Away from isocenter the total forces and torques are not zero and potentially dangerous coil motion may occur if adequate restraints are not present. As long as Bo is uniform the torque is independent of the line chosen as the axis of rotation. (1) To guard against these hazards an interlock mechanism (light source, mirror and sensor) may be used to prevent the energizing of the coil unless it is located at isocenter. If the the current patches comprising a transverse gradient coil are not symmetic around the plane z=0, or if an electrical fault, e.g., an internal open or short circuit, occurs that causes current to flow in only one portion of the coil, a substantial net torque will be present even if the Bo field is uniform. For specificity, consider a TGC (radius a, length 2L, N turns per patch, current I) of the streamline or fingerprint conductor pattern (2). In a worst case scenario only the two patches on the same side of the plane z=0 are activated. If the winding density is uniform, a closed-form expression for the torque on one half of an x-gradient coil is τx=(π/2)aBoNIL. At 1.5 T with a=0.15 m, L=0.25 m, N=12 turns and I=200 A, τx=212 N-m=156 ft-lbf (1 ftlbf = 1.3558 N-m). In this case the full coil will produce an x-gradient of approximately 80 mT/m (8 G/cm). The motion induced by such a torque will depend on the properties of the pulse sequence. For a train of current pulses with fundamental amplitude τx and period 2π/ω the equation of motion for the angular motion around the x-axis is (Ix is the moment of inertia).