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 am...
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Published in | Magnetic resonance in medicine Vol. 88; no. 4; pp. 1901 - 1911 |
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Main Authors | , |
Format | Journal Article |
Language | English |
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01.10.2022
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Abstract | 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 [d$$ d $$], 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 d5$$ {d}^5 $$. 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 d2l+3$$ {d}^{2l+3} $$. 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. |
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AbstractList | To demonstrate systematic, linear algebra-based, dimensional analysis to derive a scaling relationship among the design parameters of MRI gradient and harmonic shim coils.
The dimensions of five physical quantities relevant for gradient coil design (inductance, gradient amplitude, inner diameter [
], 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.
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
. 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
. 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.
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. PURPOSETo 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 METHODSThe dimensions of five physical quantities relevant for gradient coil design (inductance, gradient amplitude, inner diameter [ d$$ d $$ ], 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. RESULTSOnly 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 d5$$ {d}^5 $$ . 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 d2l+3$$ {d}^{2l+3} $$ . 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. CONCLUSIONSystematic 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. 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 [d$$ d $$], 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 d5$$ {d}^5 $$. 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 d2l+3$$ {d}^{2l+3} $$. 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. 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 [], 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 . By applying the same methodology to an l th order shim coil, a novel result is obtained: Shim inductance is proportional to its efficiency squared times . 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. Click here for author‐reader discussions |
Author | Lee, Seung‐Kyun Bernstein, Matt A. |
AuthorAffiliation | 1. GE Global Research, Niskayuna, NY 12309 2. Department of Radiology, Mayo Clinic, Rochester, MN 55905 |
AuthorAffiliation_xml | – name: 1. GE Global Research, Niskayuna, NY 12309 – name: 2. Department of Radiology, Mayo Clinic, Rochester, MN 55905 |
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BackLink | https://www.ncbi.nlm.nih.gov/pubmed/35666832$$D View this record in MEDLINE/PubMed |
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Cites_doi | 10.1016/j.neuroimage.2013.05.078 10.1002/mrm.26954 10.1002/mrm.1910260202 10.1002/mrm.10263 10.1097/00004728-199911000-00002 10.1002/mrm.27382 10.1002/mrm.26044 10.1002/mrm.27175 10.1002/cmr.b.20082 10.1088/1361-6560/ab99e2 10.1002/cmr.a.20163 10.1002/mrm.1910390214 10.1002/mrm.27110 10.1088/2057-1976/ab8d97 10.1002/mrm.28087 10.1016/0730-725X(93)90209-V 10.1016/j.jmr.2011.07.005 10.1002/mrm.22850 10.1002/mrm.24766 10.1002/mrm.1910190210 10.1002/9780470495032 10.1016/j.neuroimage.2017.06.013 |
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Notes | Funding information National Institutes of Health, Grant/Award Number: U01‐EB024450 Seung‐Kyun Lee and Matt A. Bernstein contributed equally to this work. ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 The authors contributed equally to this work. |
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To demonstrate systematic, linear algebra–based, dimensional analysis to derive a scaling relationship among the design parameters of MRI gradient and... To demonstrate systematic, linear algebra-based, dimensional analysis to derive a scaling relationship among the design parameters of MRI gradient and harmonic... PurposeTo demonstrate systematic, linear algebra–based, dimensional analysis to derive a scaling relationship among the design parameters of MRI gradient and... PURPOSETo demonstrate systematic, linear algebra-based, dimensional analysis to derive a scaling relationship among the design parameters of MRI gradient and... |
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SubjectTerms | Design Design parameters Diameters Dimensional analysis Efficiency Equipment Design gradient coil Inductance Linear algebra Magnetic Resonance Imaging - methods Mathematical analysis Matrices (mathematics) Matrix algebra Permeability Scaling shim coil Shim coils turns |
Title | Systematic Dimensional Analysis of the Scaling Relationship for Gradient and Shim Coil Design Parameters |
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