Eddy Currents & Interactions: Characterization & Compensation
S. Johanna Vannesjo1

1Wellcome Centre for Integrative Neuroimaging, FMRIB, University of Oxford, United Kingdom

### Synopsis

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

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. The induced eddy currents in turn produce magnetic fields, which distort the time-course of the gradient fields leading to artefacts in imaging and spectroscopy. To counteract the effects of eddy currents, both hardware and software solutions have been implemented. This presentation will give an overview of how eddy currents are generated, how to characterize the gradient system response and how to compensate for imperfections.

### 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.

### Compensation

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).

### Cross-terms & higher-order shims

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.

### Summary

In summary, eddy currents are induced by Faraday induction upon switching of the currents through the gradient or shim coils. Modern gradient coils employ active shielding to minimize long-living eddy currents. Remaining eddy current effects are compensated for by pre-emphasis filters, which conventionally constitute a sum of exponentially decaying terms. A non-parametric LTI model can provide a more generalized model of the gradient system, which can also serve as a basis for pre-emphasis.

### Acknowledgements

No acknowledgement found.

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Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)