Systematic Dimensional Analysis of the Scaling Relationship for Gradient and Shim Coil Design Parameters

Purpose: To demonstrate systematic, linear algebra–based, dimensional analysis to derive a scaling relationship among the design parameters of MRI gradient and harmonic shim coils. Theory and Methods: The dimensions of five physical quantities relevant for gradient coil design (inductance, gradient amplitude, inner diameter [(Formula presented.)], current, and the permeability of free space) were decomposed into fundamental units, and their exponents were arranged into a dimensional matrix. The resulting set of homogenous equations was solved using standard linear algebraic methods. Inclusion of the number of turns as an additional unit yielded a 5 × 5 dimensional matrix with a unique, nontrivial solution. The analysis was extended to harmonic shim coils. The gradient coil scaling relationship was compared with data from 24 published gradient coil sets. Results: Only when the unit of turns was included did the linear algebra–based analysis uniquely produce the known scaling relationship that gradient inductance is proportional to gradient efficiency squared times (Formula presented.). By applying the same methodology to an lth order shim coil, a novel result is obtained: Shim inductance is proportional to its efficiency squared times (Formula presented.). The predicted power-law relationship between inductance-normalized gradient efficiency and the diameter accounted for > 92% of the efficiency variation of the surveyed gradient coils. A dimensionless parameter is proposed as an intrinsic figure-of-merit of gradient coil efficiency. Conclusion: Systematic application of linear algebra–based dimensional analysis can provide new insight in gradient and shim coil design by revealing fundamental scaling relations and helping to guide the design and comparison of coils with different diameters.