An Equivalent Modeling Method for Electromagnetic Radiation of PWM Fans with Multiple Radiation Sources

Abstract

Axial flow fans, used for heat dissipation in electronic equipment, may generate significant electromagnetic interference during PWM speed regulation. Due to its multiple radiation sources and relatively smaller size compared to the equipment, the radiation prediction model for equipment-level EMC analysis often involves a huge number of grids, which leads to computational difficulties and inefficiencies, and thus an equivalent modeling method for the electromagnetic radiation of PWM fan is presented. First, a detailed field-circuit coupling model of the radiation from winding and driving circuits is established using the time-domain finite-integral method with non-uniform grids. Then, a near-field hexahedron is defined to surround the fan, and the electromagnetic field of all its surfaces is derived based on the Huygens principle and calculated. Finally, the hexahedron encapsulating all radiation sources within the fan can be used in a higher level simulation as replicable and reusable equivalent sources. The proposed method is validated by a numerical example and actual measurements and applied to predict the radiation emissions within an electronic enclosure. The results show that the equivalent model can reduce 81.4% computation time and maintain good consistency in comparison to the detailed field-circuit coupling model.

1. Introduction

As a highly reliable and environmentally adaptable heat dissipation device, axial flow fans are widely used in land-based, sea-based, aerospace, and other fields to dissipate heat from electronic equipment. As shown in Figure 1, an axial flow fan is usually composed of a fan frame, an impeller, and a motor, in which the brushless DC motor is directly connected to the impeller, driving it to rotate and forcing the air to flow in the axial direction to realize convection heat transfer. To meet the heat dissipation requirements of different working conditions, the fan will continuously adjust the speed according to the change in the heat load. Due to high rates of variations in current and voltage, the use of PWM technology in the speed regulation process can generate conducted interference and form a radiated interference through the inner components, affecting the proper operation of electronic equipment. Therefore, it is necessary to predict and control the electromagnetic radiation emissions from the fans [1,2,3,4].

Currently, numerical simulation methods are mostly used for the analysis of electromagnetic radiation interference. Commonly used numerical analysis methods include the equivalent circuit method [5,6], the finite difference time domain (FDTD) method [7], the finite element method (FEM) [8], the partial element equivalent circuit (PEEC) method [9], and the finite integration technique (FIT) [10]. The equivalent circuit method is computationally fast and suitable for small-volume structures, but it is difficult to show the details of the field distribution in space. The FDTD method can effectively deal with electromagnetic wave problems, but the computational accuracy is limited by the metric mesh step approximation. FEM is suitable for dealing with complex dielectric distributions, but the number of meshes is large and the calculation speed is slow. The PEEC method can link the transient models of multi-conductor systems to circuits, but it is more difficult to model structures with magnetic materials.

The finite integration method is a numerical solution method derived directly from the integral form of Maxwell’s equations. It defines the integral quantities of field variables on the edges or faces of a grid as the degrees of freedom to be solved. Discrete matrix equations can be directly obtained within the solution domain via this method. Compared to the FDTD method and the FEM, the FIT utilizes integral quantities as state variables, which confers a speed advantage in solving fast transient and complex structures. In addition, the finite integration method supports parallel computation resulting in reduced memory consumption and computation time. In related research, J.W.You [11] implemented the field-circuit coupling method, which integrates nonlinear circuit components with spatially discretized field models into the system equations for resolution. Other scholars have presented simulation analyses based on finite integral theory for electrostatic discharge generators, grounding networks considering frequency-variable parameters, power transformers, and motor drive systems [12,13,14,15]. The studies above indicate that the finite integral method demonstrates high efficiency and accuracy in solving electromagnetic field problems and represents a numerical method for solving complex systems efficiently.

In the process of electromagnetic radiation analysis of electronic equipment, as a local radiation source, the dimensions of the axial flow fan are typically significantly different from the entire dimensions of the equipment. Due to the complex internal structure and small size of the fan, if a refined meshing is used to discretize the entire region, it will result in a large number of grids, making the simulation difficult to complete. While non-uniform meshing techniques can help reduce the number of grids to some extent, the time step in the FIT is determined by the smallest grid size in the process of calculating radiation emissions from large and intricate structures. Therefore, there still exists a challenge of long computational time.

This paper presents the application of the Huygens equivalence principle for calculating the electromagnetic radiation of complex electronic equipment. Firstly, the main radiation sources are determined by analyzing the propagation path of the conducted interference of the fan. The field-circuit coupled FIT method with the non-uniform grid is utilized to develop a detailed model of the electromagnetic radiation of the fan, including the radiation sources of the winding and the drive circuit PCB. Then, the Huygens equivalence principle is applied to compute the near-field radiation generated by multiple radiation sources of axial fans. This near-field secondary radiation source is used as an equivalent radiation source to realize a fast calculation of electromagnetic radiation in a large space domain, and the equivalent model is validated by simulation and experiment. Finally, the electromagnetic radiation inside the electronic enclosure dissipated by multiple fans is examined as a case study to calculate the electric field distribution within the enclosure, which illustrates the effectiveness of the equivalent model in practical applications.

2. Analysis and Modeling of Radiation Sources of PWM Fan

2.1. Analysis of Interference Propagation Paths and Radiation Sources of Fan

The axial flow fan is driven by a PWM permanent magnet DC brushless motor, which consists of a driving circuit PCB and a three-phase winding. The switching action of the MOSFETs in the circuit results in voltage and current transients. These transients, interacting with the distributed inductance, capacitance, and other parasitic parameters in the PCB, generate high du/dt and di/dt values, leading to significant conducted interference. Conducted interference propagation can be categorized into common mode (CM) and differential mode (DM) conduction paths. As indicated by the red dashed line in Figure 2, CM interference is generated from transient voltage fluctuations occurring during switching operations, with currents flowing in a closed loop path through the winding and ground. Meanwhile, the MOSFET’s switching actions lead to sudden fluctuations in the currents within the bridge arms, resulting in DM interference. The conduction path for DM interference is illustrated by the blue dashed line in Figure 2. Assuming that Q1 and Q5 are in the on-state, the DM current flows from the U phase to the DC side, passes through the LISN, and flows through the V phase, and forms a closed loop through the winding.

According to the interference path and structure of the axial flow fan, the stator winding of the motor and the driving circuit PCB inside the fan have an antenna effect. As depicted in Figure 3, transient voltages and currents produced by power MOSFETs may propagate externally through these elements, forming multiple sources of radiation and causing electromagnetic interference through the apertures between the fan frame and the impeller.

2.2. Detailed Model of Fan Radiation Based on FIT

In order to obtain the electromagnetic radiation characteristics of the axial flow fan, a numerical calculation of the fan is carried out by the time-domain finite integration method. According to the finite integral theory, the Maxwell grid equation is obtained as

$$\left\{\begin{array}{l}Ce=-{\displaystyle \frac{d}{dt}}b\\ \tilde{C}h=i_C+i_S+{\displaystyle \frac{d}{dt}}d\end{array}\right.$$

(1)

where $C$ and $\tilde{C}$ are the discrete curl matrices of the main grid and the dual grid, $e$ is the voltage along the edge, and $b$ is the magnetic flux through the face elements in the main grid. $h$ is the magnetic field along the edge of the dual grid. $d$, $i_C$, and $i_S$ are the electric flux, loss current, and external current, respectively.

A central difference discretization and a leap-frog strategy are adopted for the time derivative terms of the two spinodal equations in the above equation. Then, the explicit time-domain recursive equation for the finite integration method is

$$\left\{\begin{array}{l}{e}^{n+1}=C_{AE}{e}^n+C_{AH}\left(\tilde{C}{h}^{n+{\displaystyle \frac{1}{2}}}-i_S^{n+{\displaystyle \frac{1}{2}}}\right)\\ h^{n+{\displaystyle \frac{1}{2}}}=h^{n-{\displaystyle \frac{1}{2}}}-\Delta t{D}_{\mu }^{-1}C{e}^n\end{array}\right.$$

(2)

where

$$\left\{\begin{array}{l}{C}_{AE}={\left({\displaystyle \frac{D_{\epsilon }}{\Delta t}}+{\displaystyle \frac{D_{\sigma }}{2}}\right)}^{-1}\left({\displaystyle \frac{D_{\epsilon }}{\Delta t}}-{\displaystyle \frac{D_{\sigma }}{2}}\right)\hfill \\ C_{AH}={\left({\displaystyle \frac{D_{\epsilon }}{\Delta t}}+{\displaystyle \frac{D_{\sigma }}{2}}\right)}^{-1}\hfill \end{array}\right.$$

(3)

where $D_{\epsilon }$, $D_{\sigma }$, and $D_{\mu }$ are the diagonal matrices of dielectric constant, electrical conductivity, and magnetic permeability, respectively.

The transient electromagnetic process of the axial flow fan can be calculated according to (2) and (3), which is accomplished by CST Studio Suite 2023. The 3D model of the fan radiation calculation based on the FIT method is shown in Figure 4. The model mainly consists of the winding, the motor core, the driving circuit PCB, and a shell consisting of the fan frame and impeller, in which the stator winding and drive circuit PCB are directly integrated. The shell has a certain shielding effect from radiation.

The PCB model, which carries circuit components, was obtained in the universal ODB++ format. In CST, the network containing the interference input path (power path) and output path (winding path) was selected, as indicated by the blue boxes in Figure 5. By extracting the parasitic parameters of the network, a field-circuit coupling model of the PCB was established.

The winding model was constructed using copper enameled wire with a diameter of 0.71 mm, with 25 turns wound around each stator slot. SolidWorks 2021 was used to create the 3D models of the slots, windings, and permanent magnets. The three-phase windings were connected in a star configuration, and the resulting 3D winding model is shown in Figure 6. The permanent magnets were made of neodymium iron boron (NdFeB) material, and the iron core was formed by laminating silicon steel sheets.

Inside the fan, there are numerous circuit components and complex structures that cannot be characterized by 3D field models. Therefore, the components in the circuit are linked to the field model through an external circuit, where the FIT method in the time domain with field-circuit coupling is used to solve the problem. As shown in Figure 7, an external driving circuit is used to achieve the PWM control of MOSFET to calculate the excitation current of the electromagnetic field. The excitation current is transferred to the field model through a coupled interface between the circuit model and the field model.

To achieve field-circuit coupling, the excitation current is represented by adding the current $i_L$ to the right side of Ampere’s law, i.e.,

$$\tilde{C}{h}^{n+{\displaystyle \frac{1}{2}}}=D_{\sigma }{\displaystyle \frac{e^{n+1}+e^n}{2}}+D_{\epsilon }{\displaystyle \frac{e^{n+1}-e^n}{\Delta t}}+{i_L}^{n+{\displaystyle \frac{1}{2}}}$$

(4)

The field-circuit coupled transient process can be calculated according to (4). Grid refinement is required to maintain the accuracy of the numerical model. However, this will increase the total number of grids. Furthermore, the reduction in explicit recursion time step results in increased computation resources and an extension of the computation time.

The non-uniform grid technique achieves a relative balance between computational accuracy and efficiency by employing a fine grid with small dimensions in areas of the solution domain where the electromagnetic field changes significantly while employing a coarse grid in the remaining areas. Grid refinement is applied to the windings and driving PCB as shown in Figure 8.

The detailed model was numerically calculated with a fan size of 120 mm × 120 mm × 38 mm within a calculation domain of 2000 mm × 2000 mm × 2000 mm. To ensure consistency with the experiments, the boundary condition in the simulation was set to “open”. The number of grids in the computational domain reached 4,572,216, the minimum grid size was 0.198 mm, and the total computational time was up to 42 h 59 min. It is evident that when employing the time domain finite integration method with a non-uniform grid for analyzing electromagnetic radiation in a large space, challenges exist due to a large number of grids and prolonged computational time consumption.

3. Modeling of Equivalent Radiation Sources Based on the Huygens Principle

The Huygens principle states that the fields at points on any closed surface surrounding a field source can serve as a secondary source to radiate electromagnetic fields outside the closed surface again. That is, the electromagnetic field at any point outside the closed surface is generated by the contribution of all the electromagnetic fields present on the closed surface. Hence, the computation of electromagnetic radiation can be achieved through twice numerical calculations. Firstly, the near-field electromagnetic field of the axial flow fan is calculated by the field-circuit coupled FIT. Subsequently, the near-field radiation source obtained is utilized as the secondary excitation source and the electromagnetic field distribution characteristics in the target domain are further calculated.

Figure 9 presents a schematic diagram of the equivalent modeling of the near-far field radiation from the fan. The near-field radiation source of the fan is computed by the field-circuit coupling FIT method from Equations (2)–(4) for the enclosed hexahedral domain surrounding the fan. The surface current source $J_s$ and the surface magnetic flux source $J_s^m$ of the near-field closed domain S are obtained:

$$\begin{array}{l}{J}_s=e_n\times \left(H_s-{\mathbf{H}}_s^{\prime }\right)\\ J_s^m=\left(E_s-{\mathbf{E}}_s^{\prime }\right)\times e_n\end{array}$$

(5)

where $e_n$ is the unit vector in the direction normal to the outside of the closed surface S; ${\mathbf{E}}_s^{\prime }$ and ${\mathbf{H}}_s^{\prime }$ are the electric and magnetic fields in the closed surface of the radiation source, respectively; and $E_s$ and $H_s$ are the electric and magnetic fields on the outside of the closed surface S. In this case, both the inner and outer electromagnetic fields are generated by the equivalent source $J_s$ and $J_s^m$. The electromagnetic field generated by the equivalent source can be calculated by the vector potential function:

$$\begin{array}{l}A(r)=\mu {\displaystyle {\int }_{\mathrm{s}}{J}_{\mathrm{s}}}(r)G\left(r,r^{\prime }\right)ds\\ A_{\mathrm{m}}(r)=\epsilon {\displaystyle {\int }_{\mathrm{s}}{J}_{\mathrm{s}}}(r)G\left(r,r^{\prime }\right)ds\end{array}$$

(6)

where $G\left(r,r^{\prime }\right)$ is the free-space Green’s function. According to the principle of equivalent sources, the electromagnetic field in a closed surface can be assumed arbitrarily. Setting ${\mathbf{E}}_s^{\prime }$ and ${\mathbf{H}}_s^{\prime }$ to zero, (5) can then be simplified as follows:

$$\begin{array}{l}{J}_s=e_n\times H_s\\ J_s^m=E_s\times e_n\end{array}$$

(7)

Equation (7) shows that an equivalent source on the closed surface surrounding the radiating source can generate an equivalent space electromagnetic field outside the closed surface, indicating that points on the closed surface can serve as a secondary excitation source to continue radiating equivalent fields outward. The equivalent surface current and magnetic flux on the closed surface satisfy (8):

$$\begin{array}{l}\nabla \times \nabla \times E(r)-k^{2}E(r)=-j\omega \mu J_S-\nabla \times J_S^{\mathrm{m}}\\ \nabla \times \nabla \times H(r)-k^{2}H(r)=-j\omega \epsilon J_S^{\mathrm{m}}+\nabla \times J_S\end{array}$$

(8)

where $E(r)$ and $H(r)$ are the electric and magnetic fields in the target space, respectively. k is the wave number, $\omega$ is the angular frequency, $\epsilon$ is the dielectric constant, and $\mu$ is the magnetic permeability. According to the second vector Green’s theorem, $e_{\mathrm{c}}$ is a unit constant vector; then, there is

$$\begin{array}{c}{\displaystyle {\int }_V\left\{e_{\mathrm{c}}G\left(r,r^{\prime }\right)\cdot [\nabla \times \nabla \times E(r)]-E(r)·\right.}\left.\nabla \times \nabla \times \left[e_{\mathrm{c}}G\left(r,r^{\prime }\right)\right]\right\}dV\\ ={\displaystyle {\oint }_{S+S_0}\left\{E(r)\times \nabla \times \left[e_{\mathrm{c}}G\left(r,r^{\prime }\right)\right]-e_{\mathrm{c}}G\left(r,r^{\prime }\right)\times \right.}\nabla \times E(r)\}dS\end{array}$$

(9)

Further transformation yields the following:

$$\begin{array}{c}E(r)={\displaystyle {\int }_V\left[-j\omega \mu J-J^{\mathrm{m}}\left(r^{\prime }\right)\times {\nabla }^{\prime }+\frac{\rho \left(r^{\prime }\right)}{\epsilon }\right]}G\left(r,r^{\prime }\right)dV+\\ {\displaystyle {\oint }_S\left[-j\omega \mu \left(e_{\mathrm{n}}\times H_S\right)+\left(e_{\mathrm{n}}\times E_S\right)\times {\nabla }^{\prime }+\right.}\left.\left(e_{\mathrm{n}}\cdot E_S\right)\times {\nabla }^{\prime }\right]G\left(r,r^{\prime }\right)dS\end{array}$$

(10)

In free space, the volume fraction in the above equation is zero, so the expression for the electric field intensity can be obtained as follows:

$$E(r)={\displaystyle {\oint }_S\left[-j\omega \mu \left(e_{\mathrm{n}}\times H_S\right)+\left(e_{\mathrm{n}}\times E_S\right)\times {\nabla }^{\prime }+\right.}\left.\left(e_{\mathrm{n}}\cdot E_S\right)\times {\nabla }^{\prime }\right]G\left(r,r^{\prime }\right)dS$$

(11)

Similarly, the expression for the intensity of the magnetic field can be obtained as follows:

$$H(r)={\displaystyle {\oint }_S\left[-j\omega \mu \left(e_{\mathrm{n}}\times E_S\right)+\left(e_{\mathrm{n}}\times H_S\right)\times {\nabla }^{\prime }+\right.}\left.\left(e_{\mathrm{n}}\cdot H_S\right)\times {\nabla }^{\prime }\right]G\left(r,r^{\prime }\right)dS$$

(12)

Therefore, when calculating the electromagnetic field distribution in a large space, the surface electromagnetic field distribution $E_s$ and $H_s$ can be obtained by calculating the near-field radiation source of the fan numerically. The calculation results are then used as an excitation source to obtain the electromagnetic field distribution in the target space through secondary calculation. During the numerical calculations, the computational domain is divided into two parts: the near-field domain and the target computational domain. The near-field domain is first refined with grid discretization and computation. Then, the obtained surface electromagnetic field data from the near-field source are imported into the target computational domain as a secondary excitation source.

To simulate the electromagnetic field within the target domain, it is necessary to compute the surface electromagnetic field data propagating in each direction within the near-field domain. The equivalent currents and magnetic flux on six surfaces are calculated utilizing the hexahedral near-field domain, where the equivalent surface current $J_{e,x}$ and the equivalent magnetic flux $J_{m,x}$ in the positive direction of the x-axis are as follows:

$$\begin{array}{l}{J}_{e,x}=e_x\times H=e_x\times \left(0,H_y,H_z\right)=-H_z{e}_y+H_y{e}_z\\ J_{m,x}=-e_x\times E=-e_x\times \left(0,E_y,E_z\right)=-E_y{e}_z+E_z{e}_y\end{array}$$

(13)

The equivalent surface current $J_{e,-x}$ and the equivalent magnetic flux $J_{m,-x}$ in the negative direction of the x-axis are as follows:

$$\begin{array}{l}{J}_{e,-x}=-e_x\times H=e_x\times \left(0,H_y,H_z\right)=H_z{e}_y-H_y{e}_z\\ J_{m,-x}=e_x\times E=-e_x\times \left(0,E_y,E_z\right)=E_y{e}_z-E_z{e}_y\end{array}$$

(14)

The equivalent surface current $J_{e,y}$ and the equivalent magnetic flux $J_{m,y}$ in the positive direction of the y-axis are as follows:

$$\begin{array}{l}{J}_{e,y}=e_y\times H=e_y\times \left(H_x,0,H_z\right)=H_z{e}_x-H_x{e}_z\\ J_{m,y}=-e_y\times E=-e_y\times \left(E_x,0,E_z\right)=E_z{e}_x-E_x{e}_z\end{array}$$

(15)

The equivalent surface current $J_{e,-y}$ and the equivalent magnetic flux $J_{m,-y}$ in the negative direction of the y-axis are as follows:

$$\begin{array}{l}{J}_{e,-y}=-e_y\times H=-e_y\times \left(H_x,0,H_z\right)=H_x{e}_z-H_z{e}_x\\ J_{m,-y}=e_y\times E=e_y\times \left(E_x,0,E_z\right)=E_z{e}_x-E_x{e}_z\end{array}$$

(16)

The equivalent surface current $J_{e,z}$ and the equivalent magnetic flux $J_{m,z}$ in the positive direction of the z-axis are as follows:

$$\begin{array}{l}{J}_{e,z}=e_z\times H=e_z\times \left(H_x,H_y,0\right)=H_x{e}_y-H_y{e}_x\\ J_{m,z}=-e_z\times E=-e_z\times \left(E_x,E_y,0\right)=E_y{e}_x-E_x{e}_y\end{array}$$

(17)

The equivalent surface current $J_{e,-z}$ and the equivalent magnetic flux $J_{m,-z}$ in the negative direction of the z-axis are as follows:

$$\begin{array}{l}{J}_{e,-z}=-e_z\times H=-e_z\times \left(H_x,H_y,0\right)=-H_x{e}_y+H_y{e}_x\\ J_{m,-z}=e_z\times E=e_z\times \left(E_x,E_y,0\right)=-E_y{e}_x+E_x{e}_y\end{array}$$

(18)

Therefore, in the numerical calculation of electromagnetic fields within the near-field domain, it is required to calculate the surface electromagnetic field data propagating along each direction. Then, these data are imported into the target computational domain as an equivalent field source to acquire the electromagnetic field distribution characteristics within the target domain. Compared with the numerical computation directly within the target domain, both computational models feature the same minimum grid size. However, due to the smaller size of the near-field domain, the number of grids is reduced, allowing for a quicker acquisition of the near-field surface electromagnetic field. Moreover, the secondary computation can significantly reduce the number of grids and increase the time step, which can effectively reduce computational memory usage and improve the efficiency of computation.

In the equivalent model calculation process, the axial flow fan is initially analyzed in the near-field domain to determine the surface electromagnetic field distribution. Subsequently, these surface electromagnetic field data are used as a near-field source and imported into the larger target computational domain (2000 mm × 2000 mm × 2000 mm) to calculate the electromagnetic field distribution. The computational domain for both calculations of the equivalent model is illustrated in Figure 10. The equivalent model has the same boundary conditions as the detailed model.

4. Validation of Equivalent Field Source Model

4.1. Simulation Validation of the Equivalent Model

To validate the accuracy of the equivalent model, the detailed model is employed to directly calculate the electromagnetic field in free space. The results are then compared with those obtained from the equivalent model. For the electric field analysis, a plane located 500 mm from the front of the axial flow fan within the computational domain is selected. As illustrated in Figure 11, Figure 11a,b depict the electric field distributions at a frequency of 100 MHz, calculated by the detailed model and the equivalent model, respectively.

It is observed that when the axial flow fan is in operation in free space, it serves as a radiation source, emitting electromagnetic radiation interference externally. Additionally, the results from both computational models depicting the variation in electric field intensity with frequency at a distance of 1 m from the fan are shown in Figure 12.

The Feature Selective Validation (FSV) technique is a method utilized for validating computational electromagnetics and predictive electromagnetic compatibility data evaluations. In this paper, the FSV technique is applied to evaluate the variance between the computational results of the detailed model and the equivalent model. The FSV technique can evaluate the “quality” of the association between two sets of data according to specific criteria, mainly focusing on the total amplitude difference measure (ADM_tot), the total feature difference measure (FDM_tot), and the total global difference measure (GDM_tot). These variances are presented as numerical values and are described using natural language as six levels depending on the range of the data: excellent (0–0.1), very good (0.1–0.2), good (0.2–0.4), fair (0.4–0.8), poor (0.8–1.6), and very poor (>1.6).

The evaluation results of FSV in Figure 10 are shown in

Table 1. FSV evaluation results of the two computational models.

ADMtot	FDMtot	GDMtot
0.1801/very good	0.3002/good	0.384/good

. The proportion of “good” or higher in ADM reaches 89.33%, the proportion of “good” or higher in FDM reaches 75.86%, and the proportion of “good” or higher in GDM reaches 67.26%. The FSV evaluation results indicate that the results derived from the equivalent model show greater consistency in comparison to the electric field results obtained from the detailed model.

Table 2. Comparison of grid parameters and computation time for two models.

Model Type	Dimension of the Calculation Domain	No. of Grid	Computation Time
Detailed model	2000 × 2000 × 2000	4,572,216	42 h 59 min
Equivalent model	220 × 220 × 138	1,813,965	11 h 3 min
	2000 × 2000 × 2000	126,720	7 min

lists the grid and computation time comparison of the two models. The detailed model requires a total of 4,572,216 grids for calculations within the domain. In contrast, when utilizing the equivalent field source model, the number of grids is reduced by 60.3% due to the smaller calculation domain. In the secondary calculations, due to the replacement of the detailed model with the equivalent model, the size of the radiation source changes. This change increases the minimum grid size within the computational domain from 0.198 mm to 9.769 mm and significantly reduces the total number of grids. Consequently, there is a noteworthy 74.0% decrease in computational time, highlighting the substantial enhancement in computational efficiency achieved through the use of an equivalent model compared to a detailed model across the entire domain.

4.2. Experimental Validation of the Equivalent Model

Measurements of radiation emissions from the axial flow fan are executed in a shielded darkroom. The radiated interference measurement process is based on the GJB-151B standard. The experiment layout is shown in Figure 13 and Figure 14. The receiving antenna is placed 1 m from the device under test. Within the rated operating range of the fan, the antenna measures the intensity of the electric field in the frequency band from 30 MHz to 200 MHz. The device involved in the measurement includes a LISN, a receiving antenna, and a measurement receiver.

The measurement procedures are as follows:

a. Check the measurement system integrity.

b. Verify that the ambient requirements specified meet the standard. Ensure that the ambient conditions do not compromise the test results.

c. Turn on the measurement equipment, and allow sufficient time for stabilization.

d. Turn on the equipment under test, and allow sufficient time for stabilization.

e. Using the measurement path of Figure 13, determine the radiated emissions from the equipment under test and its associated cabling.

According to the layout of the experiment, a detailed model of the radiation emission of the fan is established, including the axial flow fan model, DC lines, LISN, and the grounded plane, as shown in Figure 15. The DC line current of the fan is used as the excitation source of cable radiation, which is measured by a current probe and an oscilloscope, as shown in Figure 16.

A comparison is presented in Figure 17 between the measured electric field intensity results and the calculated results from both models. The evaluation results from the FSV analysis are tabulated in

Table 3. FSV evaluation of computational models and measurements.

	ADMtot	FDMtot	GDMtot
Evaluation of the two models	0.0814/very good	0.3695/good	0.3971/good
Evaluation of equivalent model and measurements	0.1165/very good	0.4920/fair	0.5354/fair

. It is evident that there is a high level of agreement between the results obtained from the equivalent model and the detailed model. Furthermore, the calculated results align well with the measured data.

5. Enclosure Radiation Interference Prediction Using Equivalent Model

In practical applications, multiple axial flow fans are employed to dissipate heat from components inside the enclosure of electronic equipment. As shown in Figure 18, the enclosure features apertures on its rear and sides, and two axial fans are installed inside the enclosure via a metal plate. These fans create high-speed airflow at the rear of the enclosure, which passes through the apertures to provide air-cooling and facilitate the dissipation of heat generated by the electronic equipment within the enclosure.

The dimension of the electronic enclosure is 600 mm × 300 mm × 900 mm. The equivalent source containing the fan is calculated independently, with a distance of 50 mm from the fan as the equivalent surface. A comparison of the electric field distribution inside the enclosure is depicted in Figure 19.

According to the calculation results, the axial fan inside the enclosure generates radiated interference externally. The electric field distribution inside the enclosure, as determined by both models, shows significant consistency. Discrepancies exist in the electric field distribution near the fan installation position because the equivalent model only contains electromagnetic field data on the surface of the closed domain from internal radiation sources. The internal electric field distribution results in the equivalent model are derived by extrapolating the surface electromagnetic field inward, which may be influenced by other interference sources and cannot accurately represent the actual internal field distribution of the field source. In the external domain surrounding the equivalent surface, the outward electromagnetic radiation from both models is considered equivalent.

An electric field probe is set inside the aperture at the rear side of the enclosure. The curve of the electric field strength with frequency is shown in Figure 20, and it is obvious that the electromagnetic energy leaks outward through the aperture of the enclosure. The evaluation results are shown in

Table 4. FSV evaluation results of the two computational models.

ADMtot	FDMtot	GDMtot
0.1199/very good	0.2343/good	0.2887/good

. The FSV results also illustrate the consistency between the equivalent model and the detailed model.

Table 5. Comparison of grid parameters and computation time for two models.

Model Type	Computational Object	No. of Grid	Computation Time
Detailed model	Enclosure with fans	9,719,864	59 h 21 min
Equivalent model	Equivalent source	1,813,965	11 h 3 min
	Enclosure	136,864	9 min

presents a comparison of the grid parameters and computation time for the two computational models. The equivalent model leads to an 81.4% reduction in computation time. Notably, the equivalent model only requires a single calculation of the interference source to determine the electromagnetic field and can be used repeatedly, while the detailed model necessitates multiple iterations of the field circuit for each interference source. Thus, the equivalent model offers improved computational efficiency for analyzing radiation emissions from multiple interference sources.

6. Conclusions

This paper introduces a detailed electromagnetic radiation model of an axial flow fan, incorporating components such as the winding, driving circuit PCB, and shell. Numerical computations are conducted using the time-domain finite integral method with field-circuit coupling to analyze the fan’s electromagnetic radiation characteristics. To enhance computational efficiency and reduce resource consumption, the Huygens principle is applied to establish an equivalent radiation source model for the fan. The electromagnetic radiation calculation process involves two numerical computations. The comparison of results obtained from both the detailed electromagnetic radiation model and the equivalent radiation source model is evaluated by the FSV technique, which confirms the accuracy and high efficiency of the equivalent calculation model.

The accuracy of the equivalent model is validated by experimental measurements of the radiated emissions of the axial flow fan, which demonstrates the feasibility and validity of the equivalent source model established based on the Huygens principle in predicting electromagnetic radiation interference from the fan.

In practical applications, axial flow fans are commonly installed inside electronic enclosures. Through a comparison of the electric field distributions generated by the detailed model and the equivalent model of the axial flow fan within the enclosure, as well as considering the calculation time, the practicality and efficiency of the equivalent method are showcased. In scenarios where multiple interference sources coexist within the enclosure, the equivalent calculation model demonstrates more computational efficiency advantages.

In conclusion, the electromagnetic radiation model of fans established by the finite integration method holds significant importance in the analysis and prediction of electromagnetic radiation from electronic equipment. The equivalent field source model based on the Huygens principle not only accurately predicts the electromagnetic radiation emission characteristics of the fan but also enhances computational efficiency. This provides an effective method for electromagnetic compatibility analysis and design.