Introduction

If space debris is not actively removed from Earth orbits, humankind risks losing access to space itself. Collisions between objects in orbit can lead to Kessler Syndrome1 and the proliferation of space debris, preventing satellite deployment and space travel. Omnidirectional magnetic field generators, or Omnimagnets, have recently gained attention as a potential solution for non-contact space debris removal2,3,4. Omnimagnets are novel electromagnetic devices that remotely manipulate non-magnetic, electrically-conductive objects using eddy currents5. Omnimagnets have primarily found use in biomedical6,7,8 and aerospace2,3,4 applications where non-contact manipulation is desirable. An Omnimagnet’s electromagnetic strength is fundamentally coupled with its thermal performance due to the onset of overheating5. Large voltages and currents are required to power these devices, generating considerable Joule heating. The concentric geometry of Omnimagnets thermally insulates the innermost components and leads to elevated local temperatures. All of these effects limit the current that can be safely supplied to an Omnimagnet, decreasing its magnetic strength. Improving the design of Omnimagnets to increase heat dissipation and reduce heat generation will delay the onset of overheating failure and thus increase electromagnetic performance.

The links between Omnimagnet performance and thermal effects, electromagnetic strength, and design parameters are not well understood9,10,11. Further, the definition of Omnimagnet performance is highly dependent on application: Omnimagnets used in biomedical applications are designed to generate very powerful magnetic fields over small amounts of time7 whereas aerospace applications focus on long-term magnetic field strength per mass10 due to mass limitations of rocket payloads. The different constraints of each application have been investigated separately, but it remains unclear how to design the best Omnimagnets for new applications and new sets of constraints. Convective cooling has been the preferred method of thermal management for biomedical applications. Transient lumped-capacitance thermal analysis was employed to predict Omnimagnet overheating for convectively-cooled Omnimagnets11,12. Other cooling methods, such as phase-change materials and internal convective cooling of the frames, have also been investigated11. It was shown that increasing the size of the Omnimagnet is thermally beneficial and increases the time before overheating occurs. These modeling efforts report quantitative thermal improvements but fail to directly relate the thermal and magnetic effects. The fundamental relationship between the thermal and magnetic behavior of convectively-cooled Omnimagnets remains unknown. Omnimagnets for aerospace applications are designed to operate at a quasi-steady-state condition in Earth orbits. Radiative cooling is the primary mode of heat rejection for these applications. Thermal resistance networks10,13 and finite-element analysis9 have been used to study the thermal behavior of radiatively-cooled Omnimagnets. Effects of discrete design changes have been investigated, such as additional radiators13, scaling of Omnimagnet size9, and optimal ferromagnetic core size10. The reported results are discrete observations of a coupled thermal-electromagnetic relationship, but prior work falls short of quantifying the nature of this relationship. Even more, it is unclear how to merge findings from convectively- and radiatively-cooled Omnimagnets. Separate work has demonstrated thermal advantages for increasing Omnimagnet size, but it is unclear if this benefit is tied to the same physics, or if all Omnimagnets should be as large as possible in all applications. It is very difficult to develop a general Omnimagnet design philosophy using focused, specific results.

The task of optimizing an electromagnet is difficult because system performance and constraints are highly contextual. Magnetic field strength, mass, electrical power requirements, volume, and thermal performance can all be key parameters for different applications. Previous electromagnet optimization work tends to focus on only one or two of these parameters, thus limiting the value of the findings to one specific application. Cylindrical electromagnets and solenoids are the simplest and most studied geometry. The maximum magnetic field per electromagnet volume has been identified for cylindrical electromagnets14. The cylindrical electromagnet geometry was related to the electromagnetic strength, but mass and thermal limitations were not considered. The optimal cylindrical geometry has been demonstrated to maximize electromagnetic strength for a constant power input by relating field strength H to Joule heating via the supplied current15. Thermal contributions from Joule heating were considered, but the impact of different cooling mechanisms was not investigated. The magnetic field strength has been maximized specifically for cylindrical electromagnets cooled via integrated thermally conductive cold discs16. The presence of the cold discs inhibits magnetic field generation but simultaneously improves cooling, allowing an increased current to be supplied. However, the findings cannot be generalized to deduce how conductive cold discs could be used for another geometry, or how other cooling mechanisms may compare for the same geometry. For mass-constrained applications, cylindrical electromagnet mass was reduced by 70% for only a 10% increase in supplied electrical power17. Recent work has used coupled thermal and electromagnetic models to better understand the design of electrical machines, such as electric motors and generators18,19. Scaling analysis has been used to make recommendations for the size20 and torque-speed curves21 of electric vehicle motors. Cooling of induction motors was improved via thermal analysis22 and lithium-ion battery monitoring was accomplished using a coupled electrochemical-magnetic-thermal model23. Prior research has clearly improved electromagnet design principles, but it is difficult to combine individual contributions and develop a holistic understanding of electromagnet design for different applications.

There exists a need to improve the Omnimagnet design process by studying the relationships between thermal effects, electromagnetic effects, and common design parameters. Current approaches use advanced modeling to investigate very specific applications, which cannot be synthesized into an understanding of general Omnimagnet operation or design. A highly generalized and adaptable approach is necessary to capture application-driven differences such as alternative cooling mechanisms, mass constraints, and power requirements. To address this knowledge gap, this work develops a universal framework to estimate the performance of an Omnimagnet, regardless of application, by defining a figure of merit to capture key Omnimagnet design and performance variables. This approach calculates Omnimagnet performance from base principles and can be adjusted to account for different applications, cooling mechanisms, and more, which prior work has never accomplished. One coupling equation is developed to link Omnimagnet thermal effects, electromagnetic strength, and design parameters via supplied current density J. This framework is then used to develop guidelines for Omnimagnet design for a variety of applications. Despite differences in cooling mechanism and electromagnetic quantities of interest across Omnimagnet applications, the fundamental thermal-electromagnetic coupling remains constant and can be utilized for many general applications. Using this general coupling, the impact of Omnimagnet length \(L_{max}\), maximum temperature before overheating \(T_{max}\), convective heat transfer coefficient h, and cooling fluid temperature \(T_{\infty }\) are investigated on m produced by convectively-cooled Omnimagnets using a previously published coupled thermal-electromagnetic (T-EM) model10. The effect of each variable is combined into a convective figure of merit, \(\eta _{conv}\), which estimates m produced by a given convectively-cooled Omnimagnet. Increasing an Omnimagent design \(\eta _{conv}\) will generally increase m as well. Using \(\eta _{conv}\) allows researchers and even non-experts to estimate the performance of a convectively-cooled Omnimagnet using simple, fundamental equations rather than advanced modeling techniques. This process is repeated for radiatively-cooled Omnimagnets: the effects of \(L_{max}\), \(T_{max}\), frame emissivity \(\varepsilon _{F}\) and solenoid emissivity \(\varepsilon _{S}\) are measured on m/mass of the Omnimagnet, which is a key performance metric for space-bound Omnimagnets. A figure of merit, \(\eta _{rad}\), is formulated using the scaling of each individual design parameter. Finally, the framework is then applied to a new Omnimagnet application that has never been explored before, maximizing m per power input for a radiatively-cooled Omnimagnet. A figure of merit, \(\eta _{power}\), is easily formulated that can inform researchers how best to design an Omnimagnet for this new application. The coupled thermal-electromagnetic relationship shown here is widely applicable to any thermally-limited Omnimagnet. Comparisons can be rapidly made between different design directions, and even directly compare different cooling mechanisms, without the need for advanced simulations. This work represents a step forward not only in the theoretical understanding of Omnimagnet operation but also in the practical implementation of Omnimagnets for real-world manipulation tasks. This process may be applicable to permanent magnet electrical machines, such as motors, which are similar devices that also experience coupled thermal-electromagnetic behavior and are limited by overheating.

Methodology

Omnimagnet design and operation

An Omnimagnet is formed by surrounding a ferromagnetic core with three orthogonal, concentric solenoids (Fig. 1). Supplying the solenoids with current generates a magnetic field. Each solenoid can be independently supplied with current to precisely control the magnitude and direction of the generated magnetic field. Aluminum frames are used to rigidly support the solenoids. Solenoids and frames are numbered starting from the center of the Omnimagnet and moving outward. The axial lengths of solenoids 1, 2, and 3 are defined as \(L_1\), \(L_2\), and \(L_3\), respectively. The side length of the ferromagnetic core is denoted \(L_c\).

Fig. 1
figure 1

Omnimagnet geometry schematic. (a–d) Concentric layers of the Omnimagnet, starting from the innermost layer and moving radially outward. (e) Side-view photograph of Omnimagnet. (f) Top-view photograph of Omnimagnet.

Full size image

The coupled relationship of interest is between Omnimagnet thermal and electromagnetic effects. Intuitively, the supplied current lies at the center of the link between thermal and electromagnetic effects. Both the amount of heat generated via Joule heating, \(q_{Joule}\), and magnetic dipole moment, m, depend on the supplied current. Magnetic dipole moment m is conventionally used to measure the magnetic strength of an Omnimagnet5,10. The thermal limitations of the device place constraints on the current, which then limits the electromagnetic strength of the device. The quantitative scaling of this coupled relationship is obtained by evaluating current using both thermal and magnetic equations. Then, the two expressions can be set equal to determine a quantitative coupling equation for thermal and magnetic effects.

Omnimagnet thermal scaling analysis

Heat is generated via Joule heating whenever a current is supplied to a solenoid. Joule heating is calculated using Joule’s Law24: \(q_{Joule} = I^2R\), where q is the heat generation rate, I is current, and R is electrical resistance. Omnimagnets are conventionally designed by specifying a current density J rather than current I, which are related by the wire cross-sectional area, \(I=JA_{wire}\). Current density J does not change with wire gauge, simplifying the geometric design variables of an Omnimagnet. The total heat generation of an Omnimagnet thus expressed as

$$\begin{aligned} q_{Joule} \propto J^2R \end{aligned}$$
(1)

Electrical resistance R of a solenoid is dependent on the material resistivity and the size and shape of the solenoid. Changing the resistivity is not of interest for this work, as copper solenoids already have very low resistivity relative to other materials. Previous analysis5 demonstrated that \(R \propto L_{max}^3\), where \(L_{max}\) is the maximum dimension of an Omnimagnet (i.e., size of a minimum bounding cube). This approximation holds for all Omnimagnets with square cross-section solenoids, which is the most common Omnimagnet geometry. Substituting this expression into Eq. (1) and rearranging to solve for J gives an expression for current density in terms of thermal quantities:

$$\begin{aligned} J \propto q_{Joule}^{0.5}L_{max}^{-1.5} \end{aligned}$$
(2)

Omnimagnet electromagnetic scaling analysis

Omnimagnet strength is conventionally evaluated using the magnetic dipole moment m due to the dipole-like fields that are generated5,10. The magnetic dipole moment produced by one square solenoid surrounding a cubic ferromagnetic core is10

$$\begin{aligned} m_i = J_i \Bigg [ \frac{6}{\pi }L_iL_{c}^3 \int _{\beta _{i,1}}^{\beta _{i,2}} \arctan \Bigg (\frac{1}{\sqrt{1+2\zeta ^2}}\Bigg )d\zeta + \frac{L_i^4}{6}\Big (\beta _{i,2}^3-\beta _{i,1}^3\Big ) \Bigg ] \end{aligned}$$
(3)

where m is the dipole moment, J is the supplied current density, L is the axial length of a solenoid, \(L_{c}\) is the length of the cubic ferromagnetic core, \(\beta _1\) and \(\beta _2\) are aspect ratios, and \(\zeta\) is an arbitrary integration variable. Figure 1 demonstrates how these variables relate to Omnimagnet geometry. The subscript i indicates a specific solenoid, i.e. \(m_{1}\), \(J_{1}\), and \(L_{1}\) correspond to solenoid 1. Rearranging Eq. (3) to solve for J is trivial, but this would represent J in terms of very abstract quantities. To help clarify the magnetic effect in terms of tangible design inputs, scaling arguments are used to simplify Eq. (3) into a more tractable form. Equation (3) shows that magnetic dipole moment is calculated by multiplying J by two terms: the first integral term represents the field produced by a magnetized cubic core. The second \(L_{i}^4\) term represents the field produced by the powered solenoid. These terms will be referred to as the core component and solenoid magnetic component, respectively.

First, the order of magnitude of the core component and solenoid component are compared to determine if any terms are negligible. The core and solenoid components are multiplied by constants 6/\(\pi\) and 1/6, respectively. The aspect ratios \(\beta _1\) and \(\beta _2\) also generate constants. Omnimagnet geometry is defined such that \(\beta _1\) and \(\beta _2\) are limited to small ranges: 0 \(\le \beta _1 \le\) 1 and 1.291 \(\ge \beta _2 \ge\) 15. Across this range of values, the integral term of the core component in Eq. (3) is a constant between 0 and 0.8166. Likewise, the difference of cubed aspect ratios in the solenoid component takes a value between 0 and 2.15. Multiplying these geometry-defined constants by the appropriate constants 6/\(\pi\) and 1/6 yields ranges of 0–1.56 and 0–0.36, respectively. These constants have the same order of magnitude and thus show that both the core and solenoid components are important to consider for a scaling analysis. After eliminating the constant terms, the dipole moment can be approximated using

$$\begin{aligned} m_i \propto J_i(L_iL_{c}^3 + L_i^4) \end{aligned}$$
(4)

The various lengths in Eq. (4) can be simplified further. By definition, the length of solenoid 3 is equal to the length of one side of the Omnimagnet, \(L_{3} = L_{max}\). The length of the core must also be equal to the length of solenoid 1, \(L_{c} = L_{1}\). For Omnimagnet geometries with reasonably large ferromagnetic cores (i.e. \(L_{c} / L_{max} \ge\) 0.1), the lengths of the core, the Omnimagnet, and all three solenoids will be on the same order of magnitude, and all can be represented by a single length L. Previous work demonstrated that well-designed Omnimagnets utilize ferromagnetic cores in the range 0.4 \(\le L_{c}/L_{max} \le\) 0.810. For geometries in this range, it is guaranteed that the lengths of the core, solenoid, and overall dimensions will be on the same order of magnitude. Thus, the dipole moment can be simplified to \(m_i \propto J_iL_{max}^4\). Current densities supplied to the solenoids are either identical5 or very similar9,10, and all current densities can be represented by a general current density J. The scaling arguments suggest that the magnetic dipole moment of all three solenoids can be expressed as:

$$\begin{aligned} m \propto JL_{max}^4 \end{aligned}$$
(5)

This expression agrees with previous Omnimagnet research on the length scaling of Omnimagnets9. Rearranging to solve for J defines the current density as a function of electromagnetic quantities:

$$\begin{aligned} J \propto mL_{max}^{-4} \end{aligned}$$
(6)

Omnimagnet coupled thermal-electromagnetic relationship

Substituting Eq. (2) into Eq. (6) yields the relationship between Omnimagnet thermal and electromagnetic effects:

$$\begin{aligned} mL_{max}^{-4} \propto q_{Joule}^{0.5}L_{max}^{-1.5} \end{aligned}$$
(7)

For clarity, this expression is rearranged to directly relate m to Omnimagnet design variables and thermal effects:

$$\begin{aligned} m \propto q_{Joule}^{0.5}L^{2.5} \end{aligned}$$
(8)

This expression captures the fundamental relationship between Omnimagnet strength via dipole moment m and the heat generation \(q_{Joule}\). The form of Eq. (8) is confirmed via the Buckingham-\(\pi\) theorem in the Supplementary Information. Critically, steady-state conditions require that Joule heating is equal to the heat rejected from the device, \(q_{Joule} = q_{out}\). This equality means that m produced by an Omnimagnet is directly related to the mechanism of heat rejection, such as convection or radiation. Modeling convectively-cooled Omnimagnets can be accomplished using \(q_{out} = hA(T-T_{\infty })\), where h is the convective heat transfer coefficient, A is the surface area of convection, T is the Omnimagnet surface temperature, and \(T_{\infty }\) is the working fluid temperature25. Radiatively-cooled Omnimagnets can be investigated with Eq. (8) using \(q_{out} = \varepsilon \sigma A (T^4-T_{\infty }^4)\), where \(\varepsilon\) is the average emissivity and \(\sigma\) is the Stefan–Boltzmann constant26. This approach can be extended to any Omnimagnet application where \(q_{out}\) can be attributed to a specific thermal management strategy. Importantly, very few assumptions were made to reach Eq. (8); Relevant Omnimagnets must use square cross-section solenoids (which help to maximize the solenoid packing density5), must be roughly cubic (\(L_{max}\) can define geometry in x-, y-, and z-directions), must use relatively large ferromagnetic cores (\(L_{c}/L_{max} \le\) 0.1), and must operate at steady-state. To the authors’ knowledge, every Omnimagnet that has been manufactured meets the first three criteria.

Results and discussion

Convectively-cooled omnimagnet scaling

Equation (8) captures the thermal and electromagnetic tradeoff that exists during Omnimagnet operation with very few assumptions. While prior experiments and coupled models are inherently limited in scope to specific operating modes and cooling mechanisms, Eq. (8) can be used not only to easily model a wide variety of Omnimagnet systems but also draw important conclusions from the results. The relation between thermal and electromagnetic effects in Eq. (8) is verified using an existing coupled thermal-electromagnetic (T-EM) model10. This model calculates the temperature distribution and resulting dipole moment m for any Omnimagnet geometry, any operating condition, and any supplied current. The generated m calculated by the T-EM model is compared to the estimate from Eq. (8) for verification. Notably, the T-EM model does not assume any of the scaling relations discussed in this work. The T-EM model rigorously considers the electromagnetic and thermal effects during Omnimagnet operation. The estimates via Eq. (8) can be calculated two to three orders of magnitude faster than the T-EM model.

Omnimagnets cooled via convection are typically designed to produce the maximum possible dipole moment m. It follows that the relationship between m and thermal and geometric parameters should be investigated to inform Omnimagnet design and maximize m. For an Omnimagnet that rejects heat using only convection to an ambient temperature \(T_{\infty }\), heat rejection can be represented as \(q_{conv} = hA_s(T_{max}-T_{\infty })\), where h is the prescribed heat transfer coefficient on the exposed surfaces of solenoid 2, solenoid 3, and frame 4, \(A_s\) is the convective surface area of those same components, \(T_{max}\) is the maximum safe temperature before overheating failure occurs, and \(T_{\infty }\) is the cooling fluid temperature25. By recognizing the convective area \(A_s\) is \(A_s \propto L_{max}^2\) for the roughly cube-shaped Omnimagnet, Eq. (8) becomes

$$\begin{aligned} m_{conv} \propto h^{0.5}(T-T_{\infty })^{0.5}L_{max}^{3.5} \end{aligned}$$
(9)

The T-EM model is used to evaluate m while \(L_{max}\), \(T_{max}\), \(T_{\infty }\), and h are parametrically varied. The minimum, reference, and maximum values are detailed in Table 1. The reference values of h, \(T_{max}\), and \(T_{\infty }\) are based on prior Omnimagnet convective cooling analysis11. The reference value of \(L_{max}\) is based on an existing prototype9,11. Changes to the reference values would likely change the linear constants in the following analyses, but would not affect the scaling of the parameters.

Table 1 Parametric variable ranges for convectively-cooled omnimagnet.
Full size table

Using Eq. (9), when h, \(T_{max}\), and \(T_{\infty }\) are held constant, the predicted relationship simplifies to \(m \propto L_{max}^{3.5}\). This prediction is shown along with the T-EM model results for 11 Omnimagnets of various sizes of \(L_{max}\) in Fig. 2. The magnetic dipole moment m is a strong function of Omnimagnet length \(L_{max}\) for convectively-cooled Omnimagnets. Permanent magnets typically scale according to \(L_{max}^{3}\). The additional \(L_{max}^{0.5}\) comes from the \(q^{0.5}\) in Eq. (8). T-EM model m data fit exceptionally well to \(L_{max}^{3.5}\). Regression analysis is used to determine a constant \(k_{L} =\) 1.18\(\times\)10\(^5\) A/m\(^{1.5}\) such that \(m = k_{L}L_{max}^{3.5}\) closely matches the T-EM data (\(r^{2}=\) 0.999). This equation reveals that small increases in \(L_{max}\) can dramatically increase m produced by an Omnimagnet and that \(L_{max}\) is a critical parameter to consider for Omnimagnet design.

Fig. 2
figure 2

Verification of predicted expression \(m \propto L_{max}^{3.5}\) for a convectively-cooled Omnimagnet. The expression closely matches the T-EM model results for 11 Omnimagnets (\(r^2 = 0.999\)). Small increases of \(L_{max}\) dramatically increase m produced by an Omnimagnet.

Full size image

The same procedure is followed to verify the predicted expressions between m and temperatures \(T_{max}\) and \(T_{\infty }\). \(T_{max}\) represents the maximum safe temperature that the Omnimagnet can achieve before overheating, and \(T_{\infty }\) is the temperature of the convection cooling fluid. Simplifying Eq. (9) for constant \(L_{max}\) and h, the expression becomes \(m \propto (T_{max}-T_{\infty })^{0.5}\). This expression is further simplified to \(m \propto \Delta T^{0.5}\), where \(\Delta T\) is the temperature difference \(T_{max} - T_{\infty }\). For eight values of \(T_{max}\) and twelve values of \(T_{\infty }\), m produced by an Omnimagnet is calculated via the T-EM model and compared to the predicted expression (Fig. 3). The T-EM data matches \(\Delta T^{0.5}\) nearly perfectly, regardless of whether \(T_{max}\) or \(T_{\infty }\) is varied. A regression analysis yields \(k_{T} =\) 22.1 A\(\cdot\)m\(^2\)/K\(^{0.5}\) (\(r^2=\) 1.000). Designing an Omnimagnet to increase \(\Delta T\) can noticeably increase the produced m. This enhancement can be accomplished either by changing materials to increase \(T_{max}\) or using a cooled working fluid. Notably, the T-EM model shows that all of the convectively-cooled solenoids or frames operate below \(T_{max}\) due to conduction thermal resistances within the device. However, this discrepancy is accounted for using \(k_{T}\) in the scaling model, and the actual temperature distribution is not necessary to accurately predict m using the expression.

Fig. 3
figure 3

Verification of predicted expression \(m \propto \Delta T^{0.5}\) for a convectively-cooled Omnimagnet. The expression closely matches the T-EM model data for 20 Omnimagnets that vary both \(T_{max}\) and \(T_{\infty }\) (\(r^2 = 1.000\)). Increasing \(\Delta T\) leads to substantial increases of m produced by an Omnimagnet.

Full size image

Omnimagnet designs are compared for a wide range of heat transfer coefficients h. The T-EM model data and predicted expression \(m \propto h^{0.5}\) do not agree as well as the previous two variables (Fig. 4a). For \(h \le\) 500 W/m\(^2\cdot\)K, the scaling relation and T-EM model data agree. For \(h>\) 500 W/m\(^2\cdot\)K the T-EM model calculates almost no change in m, while the expression \(m \propto h^{0.5}\) continues to increase. This difference is due to an assumption of the scaling analysis: Omnimagnet cooling is completely controlled by convection, and increasing h will always decrease the maximum temperatures. This assumption does not account for any conduction thermal resistance between solenoids. At low h, the ratio of the thermal resistance of convection to the thermal resistance of conduction is much greater than one, and the assumption holds. As h increases, the ratio of convection to conduction thermal resistances decreases. At large h values the ratio becomes less than unity, and the thermal resistance of conduction is non-negligible. This explanation can be readily demonstrated using the temperature profiles inside the Omnimagnet (Fig. 4b). The temperatures of all three solenoids calculated by the T-EM model are shown as a function of h. For \(h>\) 500 W/m\(^2\cdot\)K, increasing h does not appreciably change the solenoid temperatures, because the temperature difference between solenoids is dominated by conduction thermal resistances. The expression \(m\propto h^{0.5}\) predicts the heat transfer in this regime quite poorly, but quite well in the convection-limited regime. Thus, the predicted expression is valid only for convection-limited cooling, which in this case is \(h \le\) 500 W/m\(^2\cdot\)K. This range of h corresponds to free or forced convection using gases or free convection using liquids. A best-fit line is generated via regression for all points \(h\le\) 500 W/m\(^2\cdot\)K, yielding constant \(k_{h}=\) 17.8 A\(\cdot\)m\(^3\cdot\)K\(^{0.5}\)/W\(^{0.5}\) (\(r^2=\) 0.720). This finding has interesting implications for Omnimagnet design principles. Increasing h only increases m in convective-limited operation. The relative contribution of convection and conduction thermal resistances should be well understood to maximize m without excessive h.

Fig. 4
figure 4

Verification of predicted expression \(m \propto h^{0.5}\) for a convectively-cooled Omnimagnet. (a) The expression matches T-EM model data for the convection-limited cooling regime, \(h \le 500\) W/m\(^2\cdot\)K, but does not match for conduction-limited cooling. (b) Temperature distribution of all three solenoids that demonstrates the shift from convection-limited to conduction-limited cooling.

Full size image

Predictive thermal-electromagnetic coupling equation for convectively-cooled omnimagnets

Understanding the effects of parametrically varying \(L_{max}\), \(T_{max}\), \(T_{\infty }\), and h on m is useful but mostly intuitive. Newton’s Law of Cooling can be used to understand that cooling is increased when, for example, \(T_{max}\) increases. Newton’s Law of Cooling cannot ascertain the correct scaling exponents, but the enhancement/reduction of cooling is captured. The scaling approach shown in this work goes further: individual expressions can be combined into one expression that directly links the thermal performance to m, the desired electromagnet variable for this application using Eq. (9) and a new variable \(\eta _{conv}\):

$$\begin{aligned} m \propto \eta _{conv} = h^{0.5}(T_{max}-T_{\infty })^{0.5}L_{max}^{3.5}~~~[\text {W}^{0.5}\text {m}^{2.5}] \end{aligned}$$
(10)

The value of \(\eta _{conv}\) incorporates the independent scaling of \(L_{max}\), \(T_{max}\), \(T_{\infty }\), and h concurrently. Because \(m\propto \eta _{conv}\) the value of \(\eta _{conv}\) can serve as a figure of merit to compare the electromagnetic and thermal performance of different Omnimagnet designs. For two completely different Omnimagnet designs with different \(\eta _{conv}\) values, the higher \(\eta _{conv}\) design will generally produce a larger m as well. In other words, design changes that increase \(\eta _{conv}\) will also increase m, and \(\eta _{conv}\) serves as a figure of merit for estimating the success of an Omnimagnet design. The calculation of \(\eta _{conv}\) also accounts for the relative importance of each parameter. Using \(\eta _{conv}\) allows designers to compare, for example, a 10% increase in \(L_{max}\) versus a 50% increase in h, which cannot be done using intuition alone. Newton’s Law of Cooling does not give this level of quantitative insight. Designing effective Omnimagnets becomes a simple task of generating multiple designs and comparing the figure of merit \(\eta _{conv}\) for each case.

The ability to compare the absolute m produced by different Omnimagnets is highly desirable. When designing Omnimagnets for new applications it is critical to estimate m produced because the value of m is necessary to calculate the generated forces, torques, and time to complete manipulation tasks. The feasibility of using an Omnimagnet for a given application is unknown until m can be quantitatively estimated. Establishing an equality between \(\eta _{conv}\) and m removes the need for advanced modeling in the feasibility and early design stages.

To establish an equation for calculating m from \(\eta _{conv}\), a total of 250 Omnimagnet designs are randomly generated across all variables (\(L_{max}\), \(T_{max}\), \(T_{\infty }\), h) within their full ranges as shown in Table 1. The exception is h, which is limited to \(h\le\) 500 W/m\(^2\cdot\)K according to its valid range. \(\eta _{conv}\) is evaluated for each of the 250 Omnimagnets, and the T-EM model is used to calculate m generated by each case. The coupling variable \(\eta _{conv}\) is then compared to the calculated m (Fig. 5). From visual examination the data are highly linear, indicating a good fit. A line of best fit is obtained via linear regression, which yields \(k_{conv} =\) 530 A/W\(^{0.5}\cdot\)m\(^{0.5}\) (\(r^2 =\) 0.942). The shaded region indicates ±20% error from this coupling equation, which contains nearly all 250 Omnimagnet designs. While precision within ±20% may not be sufficient for finalized designs in biomedical and aerospace applications, it can certainly speed up preliminary system design and suggest operational limits for Omnimagnet systems. All data points that lie outside of the ±20% region are from large Omnimagnets (0.40 m \(< L_{max} < 0.60\) m), and most are near the largest possible size (0.55 m \(< L_{max} < 0.60\) m). These data all use large temperature differences (200 K \(<\Delta T < 310\) K) with moderate to low convective heat transfer coefficients (50 W/m\(^2\cdot\)K \(< h < 200\) W/m\(^2\cdot\)K). Under these conditions, the convective cooling is underpredicted by the coupling equation, and the predicted m is lower than m from the T-EM model. Despite these niche cases of disagreement, it is clear that for a wide range of h, \(L_{max}\), \(T_{max}\), and \(T_{\infty }\), the dipole moment an Omnimagnet generates can be predicted. The value \(k_{conv} =\) 530 A/W\(^{0.5}\cdot\)m\(^{0.5}\) is multiplied by \(\eta _{conv}\) in Equation 10 to yield the following equation:

$$\begin{aligned} m_{conv} = 530\eta _{conv} = 530h^{0.5}(T_{max}-T_{\infty })^{0.5}L_{max}^{3.5}~~~[\text {A}\cdot \text {m}^2] \end{aligned}$$
(11)
Fig. 5
figure 5

Thermal-electromagnetic coupling equation to estimate m from scaling variable \(\eta _{conv}\). The best-fit relationship is created by simulating 250 Omnimagnets of random \(L_{max}\), \(T_{max}\), \(T_{\infty }\), and h. \(\eta _{conv}\) is able to accurately estimate m within ±20% for most cases, and can be used to quantitatively compare completely different Omnimagnet designs.

Full size image

This prediction capability represents a significant advancement in the understanding of Omnimagnet operation. Vastly different Omnimagnet designs can be compared using the figure of merit \(\eta _{conv}\). m generated by each design can be estimated using Eq. (11) to quantitatively predict the performance of convectively-cooled Omnimagnets without the use of advanced simulation techniques. The improvements to m are tied directly to the fundamental operation of Omnimagnets, giving further insight into their behavior.

Radiatively-cooled omnimagnet scaling

The above coupling equation only applies for Omnimagnets cooled via convection that are designed to maximize m. Other Omnimagnet applications, such as space debris detumbling2,3, possess radically different design goals and constraints. The strength and flexibility of the figure of merit approach is key, because the same approach can be followed to generate a figure of merit \(\eta _{rad}\) to link thermal and electromagnetic effects for new applications. An Omnimagnet’s dipole moment per unit mass, m/mass, is critical to maximize for space applications due to the costs associated with launching heavy objects into orbit. These Omnimagnets are conventionally cooled via radiation, which can be modeled using the Stefan-Boltzmann Law26, \(q_{rad} = \varepsilon A_s \sigma (T^4-T_{\infty }^4)\), where \(\varepsilon\) is the average emissivity, \(A_s\) is the radiating surface area of solenoid 2, solenoid 3, and frame 4, \(\sigma\) is the Stefan–Boltzmann constant, T is the radiating temperature, and \(T_{\infty }\) is the temperature of the surroundings. The surface area can be approximated as \(A_s \propto L_{max}^2\), and the radiating temperature as \(T \approx T_{max}\). Combining this expression for radiative cooling with Eq. (8) yields the expression for radiatively-cooled Omnimagnets:

$$\begin{aligned} m \propto \varepsilon ^{0.5}\sigma ^{0.5}(T_{max}^4-T_{\infty }^4)^{0.5}L_{max}^{3.5} \end{aligned}$$
(12)

Because \(\sigma\) is a constant, it can be combined with the proportionality constant. For space applications \(T_{\infty } \approx\) 4 K and is therefore negligible. Finally, the expression should be framed in terms of m/mass instead of m. Mass can be expressed as density multiplied by volume, which for a cube is \(\approx \rho L_{max}^3\). The density of an Omnimagnet is dominated by the ferromagnetic core and is assumed to be constant for most solenoid or frame materials. These simplifications yield the following expression for radiatively-cooled Omnimagnets in space:

$$\begin{aligned} m / \text {mass} \propto \varepsilon ^{0.5} T_{max}^2L_{max}^{0.5} \end{aligned}$$
(13)

To test this proportionality expression the scaling of each individual component is examined parametrically. The minimum, maximum, and reference values of these variables are shown in Table 2. Reference values are based on a previously constructed prototype9,11 and previous analysis of radiatively-cooled Omnimagnets10. The expressions are compared to m values calculated by a coupled thermal-electromagnetic (T-EM) model.

Table 2 Parametric variable ranges for radiatively-cooled omnimagnet.
Full size table

When \(\varepsilon\) and \(T_{max}\) are held constant, Eq. (13) predicts an expression m/mass \(\propto L_{max}^{0.5}\). Eleven different \(L_{max}\) values are used to generate a wide range of Omnimagnet sizes. The m value calculated by the T-EM model is then compared to the m estimate using \(L_{max}^{0.5}\) (Fig. 6). The data from the two methods show tremendous agreement, confirming the predicted expression. Regression analysis is applied to the data to determine a fitting constant \(k_{L} =\) 12.5 A\(\cdot\)m\(^{1.5}\)/kg (\(r^2 =\) 1.000). This relationship indicates that even for mass-sensitive applications, it is advantageous to use larger Omnimagnets. The additional m produced by large Omnimagnets offsets the extra mass that is required. Additionally, the extra surface area enables additional radiative cooling that contributes to the increased m.

Fig. 6
figure 6

Verification of predicted expression \(m/\text {mass} \propto L_{max}^{0.5}\) for a radiatively-cooled Omnimagnet. The expression closely matches the T-EM model data for 11 Omnimagnets (\(r^2 = 1.000\)). Increasing \(L_{max}\) results in additional mass and additional m generation, but the electromagnetic benefits outweigh the mass penalties.

Full size image

For constant \(\varepsilon\) and \(L_{max}\), the predictive expression for maximum allowable Omnimagnet temperature \(T_{max}\) is m/mass \(\propto T_{max}^2\). Notably, this is a departure from the convectively-cooled Omnimagnet, which predicts \(m \propto T_{max}^{0.5}\). Maximum temperature \(T_{max}\) is varied parametrically to study the effect on m and verify the predicted m/mass \(\propto T_{max}^2\) relationship (Fig. 7). Once again, the expression predicts the electromagnetic performance incredibly well. Regression analysis yields the constant \(k_{T} =\) 2.28\(\times 10^{-5}\) A\(\cdot\)m\(^2\)/kg\(\cdot\)K\(^2\) (r\(^2 =\) 1.000). Intuitively, choosing Omnimagnet materials that increase \(T_{max}\) will lead to an increase in m.

Fig. 7
figure 7

Verification of predicted expression \(m/\text {mass} \propto T_{max}^{2}\) for a radiatively-cooled Omnimagnet. The expression closely matches the T-EM model data for nine Omnimagnets (\(r^2 = 1.000\)). Choosing materials that allow \(T_{max}\) to increase will increase m accordingly.

Full size image

An Omnimagnet possesses two distinct emissivities: an emissivity of the solenoids, \(\varepsilon _S\), and an emissivity of the frames, \(\varepsilon _F\). However, for constant \(L_{max}\) and \(T_{max}\) the proposed expression takes the form \(m \propto \varepsilon ^{0.5}\), utilizing only a single emissivity. The two separate emissivities are combined into an effective Omnimagnet emissivity based on the fraction of radiating surface area for each component: \(\varepsilon _{eff} = A_{S}\varepsilon _{S}/A_{total} + A_{F}\varepsilon _F/A_{total}\). The radiating surface area fractions of the solenoids and frames amount to 83% and 17%, respectively, and \(\varepsilon _{eff} = 0.83\varepsilon _S + 0.17\varepsilon _F\). Each of these two emissivities is varied across 0 \(< \varepsilon \le\) 1 independently while holding the other at the reference value. Both parameters closely match the proposed expression (Fig. 8). Varying \(\varepsilon _S\) corresponds to a much wider range of \(\varepsilon _{eff}\) because it is the dominant term. Regression analysis shows that a constant \(k_{\varepsilon } =\) 7 A\(\cdot\)m\(^2\)/kg yields an excellent fit (\(r^2 =\) 0.994).

Fig. 8
figure 8

Verification of predicted expression \(m/\text {mass} \propto \varepsilon _{eff}^{0.5}\) for a radiatively-cooled Omnimagnet. The expression closely matches the T-EM model data for 21 Omnimagnets that vary both \(\varepsilon _F\) and \(\varepsilon _S\) (\(r^2 = 0.994\)). Variations in both the frame emissivity and solenoid emissivity are well captured by the scaling analysis.

Full size image

Predictive thermal-electromagnetic coupling equation for radiatively-cooled omnimagnets

The previous section parametrically validates the expressions of m/mass with \(L_{max}\), \(T_{max}\), \(\varepsilon _S\), and \(\varepsilon _F\). As with the convectively-cooled Omnimagnet case, it is advantageous to combine all separate scaling relationships into a single expression \(\eta _{rad}\):

$$\begin{aligned} m/\text {mass} \propto \eta _{rad} = \varepsilon _{eff}^{0.5}T_{max}^2L_{max}^{0.5} \end{aligned}$$
(14)

The value of \(\eta _{rad}\) can be used as a figure of merit to compare the relative m/mass for different radiatively-cooled Omnimagnet designs. Design changes that increase \(\eta _{rad}\) will tend to increase m/mass and vice versa. It is desirable to move from a proportionality expression to an equality between \(\eta _{rad}\) and m/mass. This allows designers to predict Omnimagnet perfomance rather than compare across designs. 250 Omnimagnet designs are generated using random values of \(L_{max}\), \(T_{max}\), \(\varepsilon _S\), and \(\varepsilon _F\) between their minimum and maximum values as shown in Table 2. The T-EM model is used to calculate m/mass of each Omnimagnet design. This calculated m/mass is compared to \(\eta _{rad}\) for each case (Fig. 9). Each black data point corresponds to a single randomly generated Omnimagnet design. The data are highly linear, and \(\eta _{rad}\) is able to predict m accurately. A linear regression analysis results in a line of best fit using \(k_{rad} =\) 2.40\(\times 10^{-5}\) A\(\cdot\)m\(^{1.5}\)/K\(^{2}\)/kg (\(r^2 =\) 0.942). The shaded region indicates a deviation of ±10% from the best-fit line and encapsulates nearly all 250 Omnimagnet designs. To predict m/mass for radiatively-cooled Omnimagnets, the coupling equation is:

Fig. 9
figure 9

Coupled thermal-electromagnetic relation to estimate m/mass from scaling variable \(\eta _{rad}\). The best-fit relationship is created by simulating 250 Omnimagnets of random \(L_{max}\), \(T_{max}\), \(\varepsilon _S\), and \(\varepsilon _F\). \(\eta _{rad}\) is able to accurately estimate m within ±10% for all cases, and can be used to quantitatively compare completely different Omnimagnet designs.

Full size image
$$\begin{aligned} m/\text {mass} = (2.40\times 10^{-5})\varepsilon _{eff}^{0.5}T_{max}^2L_{max}^{0.5}~~~[\text {A}\cdot \text {m}^2] \end{aligned}$$
(15)

Equation (15) enables engineers and researchers to determine Omnimagnet m/mass for any radiatively-cooled design without the need for advanced simulation techniques. Using fundamental Omnimagnet design parameters and Eq. (15) it becomes trivial to determine the performance of any radiatively-cooled Omnimagnet based on m/mass metrics. The implementation and design of such Omnimagnets is streamlined using the simple but powerful relations in Eq. (15).

Formulating figure of merit expressions for new applications

This section presents the validation of two Omnimagnet expressions that couple thermal and electromagnetic behavior: \(m \propto \eta _{conv}\) for convectively-cooled Omnimagnets and m/mass \(\propto \eta _{rad}\) for radiatively-cooled Omnimagnets. Many other Omnimagnet applications exist, and these applications may use alternative cooling mechanisms or prioritize different electromagnetic quantities. In these cases, Eqs. (11) and (15) are not useful. However, the scaling analysis methodology presented in this work does not assume any application, and Eq. (8) will remain true. The same methodology can be used to generate a new figure of merit expression for any Omnimagnet application. The first step is to determine an expression for the cooling heat transfer rate, \(q_{out}\). The cooling mechanisms could be convection, conduction, radiation, or a combination of two or more. Second, substitute the expression for \(q_{out}\) into Eq. (8). Finally, manipulate both sides of the equation to solve for the electromagnetic quantity of interest. This quantity of interest could be m or m/mass as previously investigated, or focus on an entirely new aspect of Omnimagnet design. As an example, for applications with low power availability or strict power supply requirements, it may be desirable to maximize dipole moment m per electrical power input \(P_{in}\). With the understanding that \(P_{in} = q_{in} = q_{out}\) at steady-state, the Omnimagnets can be described using:

$$\begin{aligned} m / P_{in} \propto q_{out}^{-0.5}L_{max}^{2.5} \end{aligned}$$
(16)

At this point, both convection and radiative cooling can be easily compared via changes to \(q_{out}\). This method gives Omnimagnet designers a straightforward way of comparing designs for applications that have not been previously investigated. If the choice is made to cool these Omnimagnets with radiative cooling, the expression can be rearranged to define \(\eta _{power}\), a new figure of merit:

$$\begin{aligned} m / P_{in} \propto \eta _{power} \propto \varepsilon _{eff}^{-0.5}T_{max}^{-2}L_{max}^{1.5} \end{aligned}$$
(17)

This expression can be used to quickly identify key Omnimagnet design parameters. To maximize m/\(P_{in}\) it is clear that \(T_{max}\) and \(\varepsilon _{eff}\) should be minimized, which is counter-intuitive for most radiation applications. Rather than suggesting that radiative heat transfer should be minimized, this expression demonstrates that the Omnimagnet should be run at a lower current and maintained at a lower steady-state temperature, which in turn decreases \(P_{in}\). Increases to \(L_{max}\) also benefit m/\(P_{in}\). This case study demonstrates that the methodology in this work can be used to easily generate meaningful relationships between the electromagnetic and thermal effects for Omnimagnets of any application. Engineers considering the use of an Omnimagnet can readily generate these relationships without prior Omnimagnet design experience. Developing such relationships for a wide variety of Omnimagnet applications and cooling mechanisms will expedite the use of Omnimagnets for novel manipulation tasks. Additionally, these expressions provide insight into the fundamental operation of Omnimagnets, bringing attention to the specific design decisions that affect thermal and electromagnetic performance. This methodology ultimately provides a universal framework for Omnimagnet designing across many varied applications.

Conclusion

Current Omnimagnet design processes require advanced modeling capabilities to navigate the non-intuitive thermal-electromagnetic coupling of the device. This work presents a novel method to link the thermal and electromagnetic effects of an Omnimagnet via scaling analysis. This link is critical for understanding how Omnimagnet performance scales with different design parameters. The equations in this work represent a significant step forward in both the understanding of the thermal-electromagnetic coupling and the simplification of the Omnimagnet design process. This approach is adaptable for different cooling mechanisms, such as radiation or convection, as well as different electromagnetic optimizations, such as dipole moment m or m/mass. Developers are now, for the first time, able to design effective Omnimagnets for any application.

Scaling analysis is used to solve for current density J in terms of Joule heating, \(q_{Joule}\), and magnetic dipole moment, m. The two expressions are combined to fundamentally link thermal and electromagnetic effects via Eq. (8): \(m \propto q_{Joule}^{0.5}L_{max}^{2.5}\). The cooling of an Omnimagnet is equal to the Joule heating at steady-state, meaning \(q_{out} = q_{Joule}\). For any cooling mechanism where \(q_{out}\) can be defined, Omnimagnet m can be tied directly to the thermal management of the system. This is a universal approach that quantitatively links thermal and electromagnetic performance of an Omnimagnet for any application.

The proposed scaling relationships were verified for a convectively-cooled Omnimagnet by comparing the predicted scaling expressions to the results of a previously published coupled thermal-electromagnetic (T-EM) model. All anticipated relationships showed good agreement with the T-EM model. The individual scaling relationships were combined to define a figure of merit \(\eta _{conv}\) that is proportional to m: \(\eta _{conv} = h^{0.5}(T_{max}-T_{\infty })^{0.5}L_{max}^{3.5}\). Using 250 randomly generated Omnimagnet designs, the proportionality was explicitly defined as an equality \(m = k_{conv}\eta _{conv}\). The dipole moment produced by most convectively-cooled Omnimagnets can be estimated within ±20% using \(m = 530\eta _{conv}\). This equation allows a convectively-cooled Omnimagnet dipole moment m to be predicted for many different designs without the use of advanced simulations.

The proposed expressions were also verified for a radiatively-cooled Omnimagnet. Once again, the T-EM model was used for validation. This analysis aimed to understand the link between thermal effects and the dipole moment per mass, m/mass. Equation (15), \(m/\text {mass}=2.40\times 10^{-5}\eta _{rad}\) was shown to estimate m/mass within \(\pm 10\)% for all 250 random Omnimagnet designs. This leads to a figure of merit \(\eta _{rad}=\varepsilon _{eff}^{0.5}T_{max}^{2}L_{max}^{0.5}\) that quantifies the Omnimagnet performance across many design changes. Although the operation and thermal constraints are fundamentally different than the convectively-cooled case, the same methodology was successfully applied to yield insightful expressions linking thermal and electromagnetic performance.

Equation (8) relates Omnimagnet electromagnetic and thermal performance for any Omnimagnet application and cooling mechanism. For any Omnimagnet where the device cooling rate \(q_{out}\) is defined, Equation (8) can be used to generate a figure of merit. As a demonstration, Omnimagnet design was investigated for an entirely new application to maximize m per input electrical power, \(P_{in}\), and a new figure of merit \(\eta _{power}\) was established. Regardless of application, Omnimagnet design can now be facilitated by a fundamental understanding of the coupled physics instead of the previous slow, iterative methods. This new method cannot be applied to all Omnimagnet systems: due to the steady-state heat transfer assumption, it is not possible to consider transient changes. Future work to establish a thermal-electromagnetic coupling under transient conditions would be instrumental to the next wave of Omnimagnet design improvements. Additionally, it may be possible to extend this analysis to the design of similar devices, such as the design of permanent magnet electrical machines or helmholtz coils.