Systematic Dimensional Analysis of the Scaling Relationship for Gradient and Shim Coil Design Parameters.
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| Title: | Systematic Dimensional Analysis of the Scaling Relationship for Gradient and Shim Coil Design Parameters. |
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| Authors: | Lee, Seung‐Kyun1 (AUTHOR), Bernstein, Matt A.2 (AUTHOR) mbernstein@mayo.edu |
| Source: | Magnetic Resonance in Medicine. Oct2022, Vol. 88 Issue 4, p1901-1911. 11p. |
| Subjects: | Dimensional analysis, Physical constants, Linear statistical models, Electric inductance, Permeability |
| 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. [ABSTRACT FROM AUTHOR] |
| Copyright of Magnetic Resonance in Medicine is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
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| Header | DbId: egs DbLabel: Engineering Source An: 158253996 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Systematic Dimensional Analysis of the Scaling Relationship for Gradient and Shim Coil Design Parameters. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Lee%2C+Seung‐Kyun%22">Lee, Seung‐Kyun</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Bernstein%2C+Matt+A%2E%22">Bernstein, Matt A.</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> mbernstein@mayo.edu</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Magnetic+Resonance+in+Medicine%22">Magnetic Resonance in Medicine</searchLink>. Oct2022, Vol. 88 Issue 4, p1901-1911. 11p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Dimensional+analysis%22">Dimensional analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Physical+constants%22">Physical constants</searchLink><br /><searchLink fieldCode="DE" term="%22Linear+statistical+models%22">Linear statistical models</searchLink><br /><searchLink fieldCode="DE" term="%22Electric+inductance%22">Electric inductance</searchLink><br /><searchLink fieldCode="DE" term="%22Permeability%22">Permeability</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: 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. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Magnetic Resonance in Medicine is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1002/mrm.29316 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 11 StartPage: 1901 Subjects: – SubjectFull: Dimensional analysis Type: general – SubjectFull: Physical constants Type: general – SubjectFull: Linear statistical models Type: general – SubjectFull: Electric inductance Type: general – SubjectFull: Permeability Type: general Titles: – TitleFull: Systematic Dimensional Analysis of the Scaling Relationship for Gradient and Shim Coil Design Parameters. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Lee, Seung‐Kyun – PersonEntity: Name: NameFull: Bernstein, Matt A. IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 10 Text: Oct2022 Type: published Y: 2022 Identifiers: – Type: issn-print Value: 07403194 Numbering: – Type: volume Value: 88 – Type: issue Value: 4 Titles: – TitleFull: Magnetic Resonance in Medicine Type: main |
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