Uniform temperature distribution in microwave heating achieved via rotating electric field

Abstract

Microwave heating (MWH) is widely used in various industrial and research applications due to its advantages such as rapid heating rates, energy efficiency, and low overall processing temperatures. However, a critical challenge in conventional MWH systems is temperature non-uniformity, which arises from the formation of nodes and antinodes due to standing wave patterns inside the cavity. In this study, we propose an improved MWH system that enhances temperature uniformity through cavity system optimization. To achieve a uniform electric field distribution, we implemented a rotating electric field by employing a multi-waveguide system combined with a phase-shifting technique. Furthermore, we increased the wavelength within the cavity by adjusting the waveguide height, resulting in further uniformity enhancement. Simulation and experimental results confirmed that the proposed system achieves uniform heating over a 150 mm area, with the electric field distribution showing less than 5% variation. The temperature coefficient of variation (COV) was successfully reduced to below 5%, clearly demonstrating the system’s capability to significantly improve uniformity in microwave heating. These results validate the effectiveness of the proposed system in achieving stable and uniform thermal processing, even for large-area samples.

Introduction

Microwave energy has emerged as an innovative energy source in recent years. Microwave heating technology and related equipment have been widely applied in both industrial and small-scale applications, such as the microwave sintering of metals and alloys, as well as the processing of semiconductors, ceramics, biomaterials, chemicals, and agricultural products^1–6. Microwave heating’s versatility and efficiency have made it a preferred solution for many industries, significantly reducing energy consumption and processing times compared to conventional heating methods^7–9.

The primary advantage of microwave heating lies in its ability to heat materials internally, which conserves energy by eliminating the need to heat the surrounding environment¹⁰. This method is particularly effective for materials that are difficult to heat uniformly through conventional means, as it allows for direct interaction between the microwave energy and the material’s molecular structure^11,12. As a result, heat is generated from within the material, facilitating more efficient heating at lower temperatures and shorter processing times, which enhances both energy conservation and material quality^13,14.

Furthermore, microwave energy’s rapid heating capability has opened new avenues in advanced manufacturing processes such as additive manufacturing, rapid prototyping, and the sintering of advanced ceramics and composites^15–18. These processes benefit from the precise control that microwave heating offers, allowing manufacturers to tailor the heating process to achieve specific material properties and characteristics, often unattainable with traditional methods. Despite its many benefits, microwave heating faces significant challenges that limit its application in large-scale processing. Specifically, the formation of standing waves can create nodes and antinodes, leading to non-uniform temperature distributions within the materials being processed^11,12. One of the most critical factors influencing the internal field distribution is the waveguide, which governs how microwave energy is introduced into the cavity. The waveguide’s geometric parameters, including its dimensions, placement, and supported electromagnetic modes, directly affect the field symmetry and propagation characteristics. For instance, rectangular waveguides are typically designed to operate in the dominant TE₁₀ mode, where the waveguide width determines the cut-off frequency and transmission efficiency^10,22–24. Improper design or misalignment of the waveguides can result in mode mismatch or localized field concentrations, which can further aggravate non-uniform heating.

In this study, we carefully designed a multi-waveguide system with symmetric placement and integrated phase-shifting control to intentionally generate a time-rotating electric field, thereby minimizing standing wave effects and achieving spatial uniformity across the target area. A detailed discussion of the waveguide design strategy and its role in field control is included and further elaborated in the following section.

Various other strategies have previously been explored to mitigate the effects of standing waves, including modifying the cavity’s internal structure, employing multiple waveguides, implementing mechanical rotation, and using antenna-based solutions^19–24. However, many of these approaches require physical movement within the heating system, increasing the risk of particle contamination and complicating the achievement of uniform temperature distribution across larger areas. In contrast, the proposed system achieves high uniformity without mechanical complexity, offering a more robust and contamination-free solution. By utilizing a multi-waveguide system with phase-shifting techniques and adjusting the waveguide height, we have developed a rotating electric field that reduces system complexity and potential contamination risks. Our experimental and simulation results have confirmed that this new system maintains a highly uniform electric field across an area of 150 mm, achieving an exemplary coefficient of variation (COV) of 0.05. These findings illustrate that our improved MWH system effectively addresses the limitations of traditional microwave heating methods and significantly enhances temperature consistency, making it suitable for large-scale applications that require precise thermal control and uniformity.

Result

Microwave heating uniformity with single waveguide

The principle of microwave heating can be explained by the coupling of electromagnetics and solid heat transfer. In electromagnetics, the following Maxwell’s equations serve as the fundamental equations describing the electromagnetic field distribution inside the microwave cavity^24,25. :

1

where and are the electric field and magnetic field vector, is the electric displacement vector, is the magnetic flux density vector, and is the conductive current density vector. During the heating of materials by microwave, heat is generated by the dielectric loss P_e and hysteresis loss P_m of the material^24,25:

2

where P is the microwave dissipation power and ω is the angular frequency. Energy dissipation due to microwave absorption leads to a temperature rise governed by heat conduction. The thermal response of the material is described by the transient heat conduction Eqs^24,25:

3

In this equation, ρ is the material density (kg/m³), C_p is the specific heat capacity (J/kg·K), T is the temperature (K), and κ is the thermal conductivity (W/m·K). The left-hand side represents the net thermal energy change within the material: the first term corresponds to the internal energy change due to temperature rise, while the second term accounts for heat diffusion via conduction. The right-hand side represents the power density supplied by the microwave source.

For non-magnetic materials, magnetic losses P_m are typically negligible, and microwave heating is dominated by dielectric losses²⁴. Therefore, by combining Eqs. (2) and (3), the governing equation for temperature distribution becomes:

4

This expression clearly shows that the spatial and temporal evolution of temperature within the material is directly driven by the local electric field intensity and the dielectric properties of the material. The source term on the right-hand side quantitatively represents the volumetric heat generation from microwave absorption. From Eq. (4), it can be concluded that the distribution of the electric field is an important parameter for the temperature change of the material. Therefore, we performed electric field distribution simulations with commercial software Ansys HFSS simulation software, which solves numerical simulations based on finite element method. To ensure the accuracy and numerical stability of the simulation results, a mesh convergence study was conducted using the adaptive refinement feature in Ansys HFSS. The initial mesh was generated with a λ/3 refinement target based on the operating frequency of 2.45 GHz and refined through five adaptive passes. As shown in Table 1, the change in S-parameters (ΔS) decreased consistently with each pass and reached 0.00896 at Pass 5, which satisfies the commonly used convergence threshold of 0.02. The number of solved mesh elements increased steadily from 3,606 to 9,639, and no significant variation in the electric field distribution was observed between the last two passes. This confirms that the simulation results are mesh-independent and numerically robust.

Table 1.

Change in S-parameters (ΔS) between successive mesh refinement passes during the convergence study. The ΔS value decreased steadily and dropped below the threshold of 0.02 at Pass 5, confirming that the simulation results are numerically converged.

Pass number	ΔS (the change in S-parameters)
1	N/A
2	0.07489
3	0.02877
4	0.02306
5	0.00896

We optimized physical parameters such as the position, length, and height of the waveguide, as shown in Fig. 1. To evaluate the characteristics of a conventional microwave heating system, we first constructed a reference structure using a standard WR-340 waveguide, which is designed for operation at 2.45 GHz. A magnetron source operating at the same frequency was used to generate the microwave. Figure 2(a) illustrates the configuration of this single-waveguide system, where the cavity has dimensions of 300 mm in width and length, and 86.4 mm in height. The waveguide used in this setup has an internal cross-section of 43.2 mm × 86.4 mm, which corresponds to the standard size of WR-340¹⁰.

Fig. 1.

Illustration of the optimized waveguide length, and waveguide position in the proposed microwave heating system.

Fig. 2.

(a) Schematic of the single waveguide system. (b) Simulation results of the time-dependent electric field distribution in the single waveguide system.

At 2.45 GHz, the wavelength in this waveguide is approximately 163 mm. Consequently, due to standing wave formation, antinodes and nodes appear every half-wavelength (81.5 mm), resulting in a highly non-uniform electric field distribution. This phenomenon is clearly observed in Fig. 2(b), where the field shows strong spatial variations with multiple high and low intensity regions. Furthermore, the electric field distribution also fluctuates significantly over time.

According to Eq. (4), the rate of temperature change (ΔT) is proportional to the square of the electric field magnitude. Therefore, the resulting temperature distribution closely follows the spatial non-uniformity of the electric field. As shown in Fig. 2(b), the presence of numerous field peaks and nulls leads to large variations in localized heating. This can result in a highly non-uniform temperature profile across the material.

Based on this analysis, we conclude that the single-waveguide structure is not suitable for uniformly heating large-area materials. This finding provided the motivation for optimizing physical parameters such as the waveguide position, length, and height in the design of the proposed system presented in the subsequent sections.

Impact of multiple waveguides and phase shift method

To address the non-uniformity characteristics of a single waveguide system, we proposed a multiple waveguide system, placing four waveguides at the corners of the cavity, as shown in Fig. 3(a). Compared to a single waveguide system, the introduction of multiple microwaves results in a more uniform electric field distribution. In particular, the use of four sources, as shown in Fig. 3(b), increases the electric field intensity within the cavity, which was verified through simulations. However, due to the influence of nodes and antinodes, the electric field distribution still shows significant non-uniformity. Additionally, regions of weak electric field intensity appear at certain times. Such characteristics can lead to extended heating times and non-uniform temperature distribution across different spots. This temporal non-uniformity in the electric field distribution arises from the inherent characteristics of the sine wave. Specifically, the sine wave’s nature leads to points with zero electric field strength at every half period, which in turn results in variations in the field intensity over time. Therefore, we utilized a phase shift method for each port to compensate for areas with weak electric field strength, thereby resolving temporal variations in the electric field distribution.

Fig. 3.

(a) Schematic of the multi-waveguide system. (b) Simulation results of the time-dependent electric field distribution in the multi-waveguide system.

Figure 4(a) shows the microwave heating system in which the phase of the incident waves is adjusted by modifying the length of each port’s waveguide. The lengths of Ports 2 and 4 were increased by a quarter wavelength (λ/4) to introduce a phase shift to the incident microwaves. As a result, the phase-shifted system achieves a more uniform electric field distribution over time compared to when microwaves are input simultaneously from all four ports, as shown in Fig. 4(b). Specifically, to analyze the electric field distribution over time, each port was activated and deactivated to observe the electric field distribution inside the cavity. Figure 4(c) shows the electric field distribution in the cavity when Ports 1, 3 and Ports 2, 4 were respectively activated. At 0 degrees, it was observed that the electric field strength from Ports 2 and 4 was low, whereas the electric field strength from Ports 1 and 3 was the highest. Thus, the phase shift approach was confirmed through simulation to effectively compensate for areas of weak field strength at each port, thereby eliminating regions of low electric field strength over time, as shown in Fig. 4(c). Moreover, when this phase shift method was applied, constructive interference between ports resulted in a temporal electric field distribution that appeared to rotate clockwise, as shown in Fig. 4(b). This ensures a highly uniform electric field distribution at the center of the cavity, allowing for uniform heating of the target material without requiring physical rotation.

Fig. 4.

(a) Schematic of the multi-waveguide system with phase shift applied. (b) Time-dependent electric field distribution in the multi-waveguide system with phase shift. (c) Electric field distribution when activating Port 1, 3 and Port 2, 4 respectively, showing uniformity over time with all ports activated. (d) Time-dependent electric field distribution in the multi-waveguide system, showing non-uniformity due to standing waves at half-wavelength intervals.

Proposed model: analysis of patterns in rotating electric field

Specifically, we present a mathematical model to quantitatively analyze the rotating electric field distribution over time. Figure 5(a) shows a schematic model based on the top view of the proposed microwave heating system. Microwaves from Ports 1 and 3 propagate in directions parallel to the y-axis, while those from Ports 2 and 4 propagate in directions parallel to the x-axis. The equations representing the electric field from each port can be expressed as follows.

5

Fig. 5.

Mathematical modeling of the rotating electric field distribution. (a) Schematic illustrating the top view of the cavity in the proposed microwave system. (b) Graph showing the time-dependent variation of the electric field vector angle at the center of the cavity.

where E_pn​(t) represents the electric field from Port n, k is the wave number, r is the distance from the source, and ωt is the phase change over time. Ports 2 and 4 have a phase difference of π/2 compared to Ports 1 and 3 due to the phase shift. Furthermore, since the distances from the measurement point to all the sources are identical, r₁​ r₂, r₃, and r₄​ have the same value. To provide a more intuitive understanding of the rotating electric field, we extended the expression in Eq. (5) into a vector-based formulation. The total electric field vector at the center of the cavity can be written as:

6

where is the unit direction vector from the n-th port to the center of the cavity, and ϕ_n ​ is the phase delay applied to each port. This representation emphasizes that the total field is the vector sum of directional contributions, which collectively form a rotating electric field over time.

The time-varying nature of the field components causes the direction of the electric field vector to rotate continuously over time. Therefore, by analyzing how the angle (ϕ) of the electric field vector at the measurement point changes over time, we can predict the direction of its rotation. For example, if the angle decreases in the negative direction over time, the electric field distribution will rotate clockwise. Conversely, if the angle increases in the positive direction, the electric field distribution will rotate counterclockwise. The angle of the electric field vector is calculated using a cylindrical coordinate system approach, defined by the ratio of the y-coordinate to the x-coordinate in the Cartesian plane, and is calculated using the arctangent function as follows.

7

Substituting the expressions for and into Eq. (7), the vector angle simplifies to:

8

Figure 5(b) shows the change in the angle of the electric field vector over time. The vector angle decreases in the negative direction over time, indicating that the electric field distribution rotates in a clockwise direction. The results of this mathematical modeling align with our simulation data, as illustrated in Fig. 4(b), which shows the clockwise rotation of the electric field over time. This consistency between the analytical model and the simulation results validates our design approach and confirms its effectiveness in generating a spatially uniform, time-rotating electric field. Consequently, this method ensures that the heating process remains consistent and uniform, even in large-area materials, without the need for physical movement or mechanical stirring.

Impact of guide wavelength modulation

However, since we used the standard WR-340 waveguide, we observed that nodes and antinodes were formed at intervals of approximately 80 mm, which is half of the guided wavelength resulting in a non-uniform electric field distribution, as shown in Fig. 4(d). This non-uniformity is an obstacle for large area heating materials and leads to uneven heating in different regions. To overcome this, we increased the guided wavelength by adjusting the height of the waveguide. The equation for the guided wavelength is as follows²⁶.

9

where λ_g is the guided wavelength, λ₀ is the free-space wavelength, c is the speed of light in free space, and h is height of the waveguide. In Eq. (9), the guided wavelength can be increased by adjusting h and f₀, as both are inversely proportional to the guided wavelength.

Using a magnetron microwave source with a fixed frequency of 2.45 GHz, we optimized the adjustable parameter h, as shown in Fig. 6(a). Notably, as h approaches the cut-off range, the guided wavelength increases dramatically, as shown in Fig. 6(b). Therefore, to ensure a sufficient guided wavelength while minimizing losses, we optimized the height of the waveguide to an appropriate value of 62 mm. As a result, the guided wavelength increased to approximately 776.61 mm, which was sufficient to achieve a uniform electric field distribution across the 300 × 300 mm cavity. As shown in Fig. 6(c), a uniform electric field distribution throughout the cavity was confirmed through simulation. Additionally, the electric field distribution remained highly uniform over time, particularly in the central region of the cavity, as shown in Fig. 6(d). By adjusting the waveguide position, phase shift, and guided wavelength within the cavity, we designed a microwave heating system with a uniform electric field even for large-area materials.

Fig. 6.

(a) Schematic of the proposed microwave heating system, including the method of increasing guide wavelength by adjusting waveguide height. (b) Graph of guided wavelength as a function of height (h). (c) Time-dependent electric field distribution in the proposed microwave heating system. (d) Time-dependent electric field distribution along the cavity scan line of the proposed microwave heating system, showing a uniform electric field distribution at the center regardless of time.

Figure 7(a) shows the difference in the electric field distribution between the proposed microwave heating system and a conventional system. In the case of the conventional system, a highly non-uniform electric field distribution is observed due to the formation of nodes and antinodes. In contrast, the proposed microwave heating system demonstrated a significantly more uniform electric field distribution. This characteristic ensures a uniform temperature distribution in large-area materials. Specifically, the distance over which the electric field distribution remains within a 5% error margin of the maximum electric field is approximately 105 mm, while the distance for a 10% error margin is approximately 150 mm, as shown in Fig. 7(b). This represents an approximately 2.5-fold increase in the distance with a uniform electric field distribution compared to the conventional system. Consequently, the proposed microwave heating system maintains a uniform electric field distribution up to 105 mm, as shown in Fig. 7(c) unlike the conventional system, where the electric field distribution changes over time. This characteristic is advantageous because it ensures a uniform electric field distribution, regardless of the material’s position or the heating duration.

Fig. 7.

(a) Electric field distribution before (black) and after (red) engineering of the microwave heating system. (b) Uniform electric field distribution within a 5% error margin over 105 mm and within a 10% error margin over 150 mm. (c) 3D plot of the time-dependent electric field distribution, showing a highly uniform electric field in the central region.

Experimental demonstration of proposed microwave heating system

We fabricated and conducted measurements to experimentally verify the proposed microwave heating system. Figure 8(a) presents the design schematic of the proposed microwave heating system. We fabricated a cavity with dimensions of 300 × 300 mm, and the lengths of the waveguides were made to differ by a quarter wavelength, specifically 150 mm and 180.6 mm. The temperature was measured by observing the heating element externally using an infrared (IR) thermal imaging camera through small holes located above the cavity, as shown in Fig. 8(b). To ensure efficient energy coupling and maintain the desired field distribution near the cavity entrance, the waveguide used in the experiment was fabricated with a tapered profile. As shown in Fig. 8(c), the waveguide height decreases gradually along the propagation direction and becomes fixed at 62 mm where it interfaces with the cavity. This design closely follows the optimized parameters used in the simulation and ensures consistency between the simulated and experimental setups. Figure 8(d) and (e) show photographs of the connection between the waveguide and the magnetron source and the placement of a 4-inch Si wafer in the proposed microwave heating system. We used four 1 kW magnetron microwave sources to heat the system for 20 s.

Fig. 8.

(a) Illustration of the proposed microwave heating system. (b) Top view of the microwave heating system with mesh plate and waveguides. (c) Cross-sectional photograph of the fabricated waveguide used in the experimental setup. The waveguide is tapered along the propagation direction, and the height gradually decreases, reaching a fixed height of 62 mm at the junction with the cavity. (d) Connection between waveguide and magnetron source. (e) 4-inch Si wafer, used as the heated element, in the proposed microwave heating system.

Figure 9(a) shows the measurement result of the Si wafer before microwave heating, indicating that the average temperature of the Si wafer was approximately 25 °C, similar to room temperature. After microwave heating for 20 s, the temperature of the Si wafer increased by about 343 °C to 368 °C, as shown in Fig. 9(b). For quantitative analysis of temperature uniformity, temperature coefficient of variation (COV) is calculated using the following Eqs^27,28.

10

Fig. 9.

Temperature measurement images of the heated element within the microwave system using an IR camera (a) before microwave heating, (b) after 20 s of microwave heating, and (c) spot for calculating the coefficient of variation (COV).

where T_a, T₀ are the average and initial temperature of the selected area. A smaller COV value indicates a more uniform temperature distribution of the heated material. In the proposed microwave heating system, the COV for the 4-inch Si wafer was approximately 0.05, demonstrating a highly uniform temperature distribution. In the proposed microwave heating system, the COV for the 4-inch Si wafer was approximately 0.05, demonstrating a highly uniform temperature distribution. This COV value was obtained from temperature measurements at 18 points across the wafer, as shown in Fig. 9(c). Table 2 compares the characteristics of the proposed microwave heating system with those of previous studies. The proposed system exhibited excellent COV characteristics, which significantly improved uniformity. This indicates that the proposed microwave heating system, by maximizing constructive interference through the phase shift method, staggered waveguide arrangement, and guided wavelength adjustment, enables uniform heating even for large areas.

Table 2.

Comparison of key parameters between the proposed microwave heating system and previously reported studies.

Ref	MW Power (W)	Freq. (GHz)	Source	Uniform area (mm2)	COV
20	800	2.45	Magnetron	50 × 50	0.23
29	300	2.45	Solid-state	40 × 40	0.26
30	700	2.45	Magnetron	D: 165 mm(146 × 146)	0.2
31	100	2.45	Magnetron	40 × 40	0.1
Prop.	4000	2.45	Magnetron	D: 150 mm(133 × 133)	0.05

Conclusion

In this study, we developed and validated an optimized microwave heating system that significantly improves temperature uniformity by addressing the fundamental limitations of conventional single-waveguide configurations. The proposed system incorporates a multi-waveguide structure with phase-shifting control to generate a time-rotating electric field distribution. In addition, the waveguide height was adjusted to increase the guided wavelength, thereby reducing the formation of standing wave nodes and antinodes.

Simulation and experimental results demonstrated that the proposed design achieves a uniform electric field distribution and a highly consistent temperature profile. Specifically, temperature measurements on a 4-inch silicon wafer showed a coefficient of variation (COV) of 0.05, confirming the system’s ability to minimize thermal gradients. In contrast, the reference system using a standard WR-340 waveguide exhibited significant field and temperature non-uniformities due to standing wave effects.

These findings confirm that the combined use of cavity design optimization, phase-shifted multi-waveguide input, and guided wavelength adjustment enables uniform and efficient microwave heating. The proposed system offers a practical solution for industrial and scientific applications requiring precise thermal control over large-area materials.

Author contributions

H.S.H. conceived and the directed the research. J.S. conducted the experiment and simulation. J.S., and G.H. analysed theresults. J.S. wrote the manuscript. All authors reviewed the manuscript.

Data availability

The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.