S. Johanna Vannesjo^{1}

Magnetic resonance imaging relies on the ability to produce spatially linear magnetic fields (i.e. gradient fields) with a defined temporal evolution. This is achieved with room-temperature gradient coils, through which time-varying currents are passed. The resulting change in magnetic field will however induce eddy currents in nearby conducting structures according to Faraday’s Law of induction. This distorts the time-course of the gradient fields, leading to artefacts in imaging and spectroscopy. This presentation will give an overview of how eddy currents are generated, how to characterize them and how to compensate for their effects on the field.

**Introduction**

**Eddy currents**

Faraday’s law of induction states that a
change in magnetic flux, $$$\Phi$$$, through a conducting loop will induce a voltage, $$$V_{EMF}$$$,
in the loop:

$$V_{EMF}=-\frac{d\Phi}{dt}.$$

The voltage drives a current through the
loop, $$$I_{EC}$$$, which in turn produces a magnetic field. Modelling the loop as a
resistance, R, and an inductance, L, in series, yields the following differential equation:

$$RI_{EC}+L\frac{dI_{EC}}{dt}=V_{EMF}.$$

If $$$\Phi$$$ is the flux generated by a
current in the gradient coil, $$$I_G$$$, with coupling constant M:

$$\Phi=MI_G$$

a step change, $$$\Theta(t)$$$, in $$$I_G$$$ yields the induced eddy
current:

$$I_{EC}(t)=-\frac{M}{L}\Theta(t)exp^{-\frac{R}{L}t}.$$

Thus, a step change in the current through
the gradient coil induces exponentially decaying eddy currents in surrounding
conducting structures, where the amplitude depends on the coupling to the
gradient coil (varying with geometry and position of the conducting loops).

The main magnet of an MR system contains cylindrical conducting surfaces, such as heat shields in the cryostat. Due to the proximity to the liquid helium they are at a very low temperature, and therefore have a low resistance. As can be seen from Eq. 3, this leads to a long time constant for eddy currents to decay. In practice, time constants for eddy currents running on heat shields will be on the order of hundreds of milliseconds, up to around one second. Eddy currents with shorter time constants can also be induced in warmer conducting surfaces of the system. The magnetic fields produced by the induced eddy currents generally tend to be of the same spatial structure as the gradient field being switched, and they tend to oppose the original change in magnetic field.

To minimize eddy currents induced in the cryostat, modern MR systems employ active shielding on the gradient coils (1,2). In this approach, the gradient coils are surrounded by a second coil layer with opposing currents, such as to cancel the magnetic field towards the outside of the bore, thereby minimizing the magnetic flux through the cryostat. The approach can greatly reduce the coupling with the cryostat, however at the cost of reduced static coil efficiency (magnetic field inside the bore/unit current) and increased space requirements in the bore.

Active shielding greatly reduces long-living eddy currents, but does not eliminate all sources of dynamic gradient imperfections. Remaining eddy current effects are typically addressed by pre-emphasis, i.e. by filtering the input signal such as to compensate for the induced eddy currents (3–5). The pre-emphasis filters are typically made up of a sum of a few exponentially decaying terms. To determine appropriate exponential parameters, the amplitudes and time constants of the eddy currents need to be characterized. Commonly this is performed by measuring the field after a step in the gradient current, and fitting a multi-exponential model to the measured field evolution. There are several methods to measure the gradient field including phantom-based measurements and using specialized field sensors (6–8).

The multi-exponential model of the gradient
system does not capture all system responses. A more generalized description can
be given by a non-parametric linear time invariant (LTI) system model:

$$O(\omega)=I(\omega)H(\omega)$$

where $$$O(\omega)$$$ is the system output, $$$I(\omega)$$$ the input signal and $$$H(\omega)$$$ the transfer function, in the
frequency domain (9,10). The transfer function of a particular gradient system can be
determined by passing a known input to the system and measuring the resulting
output. A full LTI model can capture not only exponentially decaying eddy
currents, but also the overall low-pass behaviour of the system as well as
oscillatory field responses originating from mechanical vibrations of the
gradient coils. Building upon this, a non-parametric pre-emphasis filter can be
designed based on the inverse of the measured transfer function (11,12).

The induced eddy currents primarily generate magnetic fields of the same spatial structure as the gradient being switched. However, to a smaller degree there are also eddy current fields of different spatial structure, as well as interactions between the gradient coils. This gives rise to cross-term responses. Cross-terms between the gradients can be addressed by driving the corresponding gradient coils to counteract the induced fields, i.e. cross-term pre-emphasis. Conventionally this is set by first calculating and implementing pre-emphasis on the self-terms, and thereafter measuring and compensating for the cross-term responses. The non-parametric system model can be expanded to encompass multiple inputs and outputs (MIMO system), and a pre-emphasis filter including cross-terms can be based on an inversion of such a model (12).

Higher-order shim coils are typically not shielded as they are rarely used dynamically during MR sequences. Therefore, accurate pre-emphasis is essential for applications of higher-order dynamic shimming. Frequently, this also needs to include compensation for induced cross-term responses.

1. Mansfield P, Chapman B.
Active magnetic screening of gradient coils in NMR imaging. J Magn Reson
1986;66:573–576.

2.
Turner R. Gradient coil design: A review of methods. Magn Reson Imaging
1993;11:903–920.

3.
Morich MA, Lampman DA, Dannels WR, Goldie FT. Exact temporal eddy current
compensation in magnetic resonance imaging systems. IEEE T Med Imaging
1988;7:247–254.

4.
Jehenson P, Westphal M, Schuff N. Analytical method for the compensation of
eddy-current effects induced by pulsed magnetic field gradients in NMR systems.
J Magn Reson 1990;90:264–278.

5.
van Vaals JJ, Bergman AH. Optimization of eddy-current compensation. J Magn
Reson 1990;90:52–70.

6.
Duyn JH, Yang YH, Frank JA, van der Veen JW. Simple correction method for
k-space trajectory deviations in MRI. J. Magn. Reson. 1998;132:150–153.

7.
Balcom BJ, Bogdan M, Armstrong RL. Single-point imaging of gradient rise,
stabilization, and decay. J Magn Reson Ser A 1996;118:122–125.

8.
Barmet C, De Zanche N, Pruessmann KP. Spatiotemporal magnetic field monitoring
for MR. Magn Reson Med 2008;60:187–197.

9.
Addy NO, Wu HH, Nishimura DG. Simple method for MR gradient system
characterization and k-space trajectory estimation. Magn Reson Med 2012;68:120–129.

10.
Vannesjo SJ, Haeberlin M, Kasper L, Pavan M, Wilm BJ, Barmet C, Pruessmann KP.
Gradient system characterization by impulse response measurements with a
dynamic field camera. Magn Reson Med 2013;69:583–593.

11.
Goora FG, Colpitts BG, Balcom BJ. Arbitrary magnetic field gradient waveform
correction using an impulse response based pre-equalization technique. J Magn
Reson 2014;238:70–76.

12.
Vannesjo SJ, Duerst Y, Vionnet L, Dietrich BE, Pavan M, Gross S, Barmet C,
Pruessmann KP. Gradient and shim pre-emphasis by inversion of a linear
time-invariant system model. Magn Reson Med 2017;78:1607–1622.