*Article* **Broadband Frequency Selective Rasorber Based on Spoof Surface Plasmon Polaritons**

**Jin Bai <sup>1</sup> , Qingzhen Yang <sup>1</sup> , Yichao Liang 2,\* and Xiang Gao <sup>1</sup>**


**\*** Correspondence: echo.chao@outlook.com

**Abstract:** A broadband frequency selective rasorber (FSR) based on spoofsurface plasmon polaritons (SSPP) is proposed. The FSR is composed of a multi-layer structure comprising frequency selective surface (FSS)-polyresin (PR)-indium tin oxide (ITO)-PR-FSS and placed vertically on a metal base plate. A periodic square cavity structure is formed. The transmission characteristics of the FSR are studied by full-wave simulation and equivalent circuit method. The simulation results demonstrate that under normal incidence, the absorption rate of the structure remains 95% in the 5–30 GHz band, and the absorption rate is also 80% in the 3.5–5 GHz band. As the incident angle of the electromagnetic wave increases to 40°, the absorption rate in the 15–20 GHz band decreases to 70% in the transverse electric (TE) mode, and the absorption rate in the transverse magnetic (TM) mode is almost the same as that of vertical incidence. The transmission response of the structure is measured in an anechoic chamber. The measurement results agree well with the simulation results, proving the reliability of the design and fabrication. The structure is less sensitive to the incident angle of magnetic waves and has a better broadband absorbing ability.

**Keywords:** frequency selective rasorber (FSR); spoof surface plasmon polaritons (SSPP); frequency selective surface (FSS); broadband

## **1. Introduction**

With the rapid development of communication systems, stealth and anti-stealth technology have increasingly become decisive factors in military electronic information warfare. It is more and more essential to improve aircraft stealth. For single-station radar detection, the detection wave can be reflected to other angular domains through the shape structure design to achieve radar stealth. However, the expanding receiving angular territory for multi-station radar detection is far from enough to reflect the radar detection wave. Frequency Selective Rasorber (FSR) [1–3] directly absorbs the electromagnetic waves in the working frequency band, and the reflectivity is minimal. Therefore, FSR with a broadband wave absorption ability demonstrates an excellent application prospect.

FSR is a periodic array composed of metal patches or aperture units, which exhibit band-pass or band-stop spectral filtering characteristics for electromagnetic waves of different frequencies. It is widely used in radome [4], electromagnetic compatibility and electromagnetic shielding [5], satellite communication [6,7], and other fields. The traditional metal-dielectric Frequency Selective Surface (FSS) [8,9] is based on the scattering properties of metal resonator elements. When the electromagnetic wave is incidental on the surface of the FSS, an induced current is generated, thus generating a scattered field. The scattered and incident fields are superimposed to form an entire field with spatial filtering characteristics. This kind of FSS is generally a "sandwich" structure, which consists of a metal pattern on the top layer, a dielectric substrate in the middle, and a metal plate on the bottom layer, which is easy to process and low-cost. However, the disadvantage is that the absorption range is narrow, and the absorption rate is sensitive to the polarization mode and incidence

**Citation:** Bai, J.; Yang, Q.; Liang, Y.; Gao, X. Broadband Frequency Selective Rasorber Based on Spoof Surface Plasmon Polaritons. *Micromachines* **2022**, *13*, 1969. https://doi.org/10.3390/ mi13111969

Academic Editor: Wensheng Zhao

Received: 25 September 2022 Accepted: 8 November 2022 Published: 13 November 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

angle of electromagnetic waves. Many scholars have researched and proposed various schemes to widen the absorbing bandwidth of FSS and reduce polarization sensitivity, such as designing a multi-mode resonant unit structure or fractal structure [10–12], using a multi-layer structure to realize the superposition of the absorbing frequency band [13] and loading lumped elements to realize bandwidth expansion [14]. They are adding active controllable devices such as PIN diode or varactor diode in the FSS [15,16] and using Spoof surface plasmon polaritons (SSPP) [17–19] to confine the incident wave to the interface and then dissipate it. In [20], the dispersive properties and subwavelength field confinement properties of artificial surfaces in low-frequency bands such as microwave and terahertz waves, which are similar to electrical surface plasmons in the optical frequency band, were studied for the first time. Since then, scholars have widely used SSPP in waveguide design [21], filters [22], notch filters [23], leaky-wave antennas [24,25], and other directions. At the same time, the designed FSR based on SSPP exhibits high rejection performance in the design stopband and achieves steep cutoff and high permeability in the working passband [26–28].

SSPP is based on Surface Plasmonic Polaritons (SPPs) by etching periodic structures on the metal surface or arranging periodic metal patches on the dielectric layer. When the electromagnetic wave is incident on the interface between the metal and the medium, the free electrons in the metal conductor oscillate collectively, generating SPPs in the microwave frequency band. The electromagnetic field strength peaks at the interface between the metal and the medium. The energy propagates along the structure's surface and is completely bound near the structure's surface, thereby achieving the wave-absorbing effect. Compared with SPPs, SSPP has more vital electromagnetic wave confinement ability, and the field strength decays exponentially in the vertical direction, reducing the interference to adjacent structures and being conducive to the miniaturized design of the FSR. In contrast, the unit structure of loading lumped components makes the FSR challenging to process and has poor practicability; if active devices are loaded, an external circuit needs to be introduced, which is slightly inconvenient. This paper uses a multi-layer stacking method to couple the SSPP to design the FSR.

#### **2. Presentation of the Proposed Structure**

The FSR designed in this paper is shown in Figure 1. It consists of a periodic square cavity array composed of a metal base plate and a multi-layer FSS cross-arranged. The depth of the square cavity unit is represented as *a*, and the length, width, and depth dimensions are consistent. The multi-layer FSS is shown in Figure 2a. The structure is symmetrically distributed concerning the indium tin oxide (ITO) layer. The dielectric is placed on the sub-top layer, and the sub-bottom layer is composed of polyresin (PR). The thickness of the dielectric layer is represented as *h*, and the dielectric constant is *ε* = 4.3, tan *δ* = 0.025. The ITO layer is placed in the middle, where the square resistance of ITO is represented as *R*, its unit is Ω/, which characterizes the resistance value of a conductive material per square centimeter. Moreover, the thickness of ITO in the calculation model is 0. In reality, ITO is often formed by adhering to the surface of polyethylene glycol terephthalate (PET). Therefore, the ITO layer of the FSR is flanked by a PET layer with a thickness of *d*, and the dielectric constant of PET is *ε* = 3.0, tan *δ* = 0.18. The bottom and top layers are right-angle metal patch arrays. As shown in Figure 2b, the metal line width of the patch array is represented as *w*, the thickness is represented as *c*, and the distance between adjacent metal lines is also represented as *w*. These parameters are as follows: *R* = 200 Ω/, *a* = 10 mm, *b* = 8 mm, *c* = 35 µm, *d* = 10 µm, *w* = 125 µm, and *h* = 1 mm.

**Figure 1.** The structure of the FSR.

**Figure 2.** (**a**) Multi-layer FSS Sectional View. (**b**) The FSS patch array.

#### **3. Design Principles**

Compared with linear dipoles, the advantages of using L-shaped metal patches in this design are mainly reflected in two aspects. First, the same resonance characteristics can be obtained in both transverse electric (TE) and transverse magnetic (TM) modes. Second, one side of the L-shaped metal patch is equivalent to an inductor, and the other is equivalent to a capacitor between the adjacent patch. At the same time, the L-shaped metal patches array can achieve the most significant number with the most negligible length gradient so that the center frequency distribution of the FSS is tighter, which is more conducive to the realization of the effect of broadband wave absorption. The structure of the FSR designed in this paper is complex and cannot be directly analyzed as a whole utilizing an equivalent circuit. However, the FSR can be regarded as a multi-layer FSS with a cross-distribution in the *xoz* and *yoz* planes. According to the different incident angles of electromagnetic waves, there are two main mechanisms for the absorption principle of the FSR.

When the electromagnetic wave is vertically incident on the bottom surface of the absorber along the *z* direction, the wave vector direction is parallel to the wall surface of the square cavity. At this time, the wall surface of the FSR is equivalent to the SSPP transmission line. According to the SSPP theory, the electromagnetic wave in TE mode cannot induce polarization charges on the interface due to the absence of an electric field component perpendicular to the interface. Therefore, SSPP transmission lines only work in TM mode for specific surfaces. The FSR designed in this paper is a cross-distributed multi-layer FSS structure. For any set of mutually perpendicular multi-layer FSS, when the multi-layer FSS in the *xoz* plane is in TM mode, the multi-layer FSS in the *yoz* plane is in the TE mode, and vice versa. Therefore, relative to the whole FSR, whether it is TE mode or TM mode, the FSR can support SSPP. For a one-dimensional periodic groove array, when the groove width is much smaller than the incident wavelength (*w λ*), the dispersion relation can be expressed as Equation (1) [19]. Where *k<sup>z</sup>* is the wave number along the *z* direction of the conductor surface, the period of the groove array is represented as *D*, the width of the groove interval is represented as *A*, the depth of the groove is represented as *H*, and *k*<sup>0</sup> is the wave vector in the free space. Equation (2) is the electromagnetic wave's attenuation constant along the wave vector's propagation direction, which characterizes the attenuation ability of the transmission line to the electromagnetic field. From Equations (1) and (2), it can be concluded that in order to increase the attenuation capability of SSPP to electromagnetic waves, *k<sup>z</sup>* should be as large as possible so that the

cut-off angular frequency (*wp*) of SSPP can be obtained as shown in Equation (3). The FSR designed in this paper is no longer a simple one-dimensional groove array. Due to the reflection effect of the metal base plate and the non-negligible coupling effect between adjacent multi-layer FSS, its internal working mechanism will be more complicated.

$$k\_z^{\;\;\;\;\;\;\;\!} = k\_0^{\;\;\!\;\!\;\!\;\!\;(1 + (\frac{A}{D})^2 \tan^2(k\_0 H))}\tag{1}$$

$$
\omega\_T = \sqrt{k\_x^{\prime 2} - k\_0^{\prime 2}} = k\_0 \frac{A}{D} \tan(k\_0 H) \tag{2}
$$

$$
\omega\_p = \frac{\pi c\_0}{2H} \tag{3}
$$

When the electromagnetic wave is vertically incident on the FSS wall of the FSR along the direction *x* or *y*, the wave vector direction is perpendicular to the wall of the square cavity. At this time, the working mechanism of the filter is similar to that of the traditional multi-layer FSS. The equivalent circuit of the formed FSS is shown in Figure 3. The upper and lower surfaces of the multi-layer FSS are air, and its impedance is *Z*<sup>0</sup> = 377 Ω. The PR layer with a thickness of *h* can be equivalent to a transmission line with a length of *h*, and its impedance is represented by *Z*1, *Z*<sup>1</sup> = Z0/ √ *εr* . The PET layer of thickness *d* can be equivalent to a transmission line of length *d*. Due to its small thickness, the impedance on the corresponding transmission line can be ignored. The ITO of the center layer is equivalent to a parallel resistance, which is represented by *R*. The FSS of the upper and lower layers is equivalent to two *LC* series branches, and the inductances of the branches interact with each other. Suppose the mutual inductance value and the impedance of the dielectric layer are ignored. In that case, the equivalent impedance of the multi-layer FSS can be obtained as in Equation (4). Where *L* is the equivalent inductance, *C* is the equivalent capacitance. The square resistance of the ITO layer is *R*, *R* = 200 Ω/. The inductance corresponding to each metal patch in the FSS array is shown in Equation (5). Where *b* is half of the total length of the metal wire, *w* is the width of the metal wire, and *µ*<sup>0</sup> is the vacuum permeability. The capacitance of the metal patch is shown in Equation (6), *ε*<sup>0</sup> is the vacuum conductivity, the effective dielectric constant of the dielectric layer is shown in Equation (7) with *εe f f* , and *ε<sup>r</sup>* is the relative dielectric constant of the medium. When the interaction between adjacent metal patches is not considered, it is not difficult to find the transmission pole corresponding to a single metal patch through *Z* = ∞. By analogy, all the poles of the FSS can be found. The distance between adjacent metal patches is minimal, and their interaction cannot be ignored. At the same time, the FSR is a square cavity structure, and the interaction between adjacent walls and the reflection effect of the metal base plate cannot be ignored.

$$Z = \frac{\mathcal{R} \cdot (j\omega L\_2 + \frac{1}{j\omega \mathbb{C}\_2}) \cdot (j\omega L\_2 + \frac{1}{j\omega \mathbb{C}\_2})}{\mathcal{R} \cdot (j\omega L\_1 + \frac{1}{j\omega \mathbb{C}\_1}) + \mathcal{R} \cdot (j\omega L\_2 + \frac{1}{j\omega \mathbb{C}\_2}) + (j\omega L\_1 + \frac{1}{j\omega \mathbb{C}\_1}) \cdot (j\omega L\_2 + \frac{1}{j\omega \mathbb{C}\_2})} \tag{4}$$

$$L = -\mu\_0 \frac{b}{\pi} \ln\left[\sin(\frac{\pi w}{4b})\right] \tag{5}$$

$$\mathbf{C} = -\varepsilon\_0 \varepsilon\_{eff} \frac{4b}{\pi} \ln\left[\sin(\frac{\pi w}{4b})\right] \tag{6}$$

$$
\varepsilon\_{eff} = \frac{1}{\sqrt{(\varepsilon\_r + 1)}\_{\prime 2}} \tag{7}
$$

In fact, due to the complex structure of the FSR, the SSPP action mechanism and the traditional FSS action mechanism exist at any incident angle of electromagnetic waves, and the transmission characteristics of the absorber appear more complicated. In order to study the transmission characteristics of the absorber more accurately, we use the CST studio

software to analyze the transmission characteristics of the whole waveband and finally make the FSR and test its transmission characteristics in the microwave anechoic chamber.

**Figure 3.** Equivalent circuit diagram of multi-layer FSS.

#### **4. Analysis and Discussion**

The working frequency band of the designed FSR should cover the S, C, X, Ka, and K bands as much as possible. Therefore, the cavity size of the FSR is optimized first, and then the ITO layer directly affects the circuit structure of the multilayer FSS. Therefore, the influence of the ITO square resistance on the transmission characteristics of the absorber is studied. The PR is a lossy medium, and the thickness of the dielectric layer also has a specific influence on the performance of the FSR. Finally, the optimization analysis of the width of the FSS metal patch is carried out. When the electromagnetic wave is vertically incident on the absorber along the *z* direction, the FSR is symmetrical about the *xoz* plane and the *yoz* plane. The transmission characteristics in the TE mode and TM mode are consistent, so only the transmission characteristics of the absorber in TE mode are studied. Figure 4 shows the absorptivity distribution of FSR with different sizes in the 0–30 GHz band when electromagnetic waves are incident vertically. It can be observed that when the incident frequency is less than 9 GHz, at the same frequency, with the decrease of *a*, the absorption rate of the FSR gradually decreases. When the incident frequency is greater than 9 GHz, the fluctuation characteristics of the absorption rate are enhanced. Except for the *a* = 10 mm model, the absorption rates of other models have unstable fluctuations in different bands. When *a* is greater than 10 mm, the absorption rate gradually increases with the decrease of *a*. When the incident frequency is more than 18 GHz, the absorption rate remains unchanged with the decrease of *a*, and remains above 95%. In a comprehensive comparison, although the absorption rate of the *a* = 10 mm model is lower than that of the *a* = 15 mm and *a* = 20 mm models when the incident frequency is less than 5GHz, the reduction is not too noticeable. However, the model with *a* = 10 mm maintains the absorption rate above 95% in the 5–30 GHz band, which other models unmatch.

Figure 5 shows the absorption rate distribution of the model with the ITO layer and the model without the ITO layer in the 0–30 GHz band when the electromagnetic wave is vertically incident. It can be observed that when the incident frequency is less than 12 GHz, the FSR absorption rate is greatly improved due to the addition of the ITO layer. When the incident frequency is greater than 12 GHz, the ITO layer also dramatically reduces the fluctuation of the absorption rate of the FSR. In order to more accurately study the impact of the ITO layer on FSR's inhalation transmission characteristics, this paper studies the distribution of FSR electric fields under the resonance frequency. Figure 6 shows the FSR electric field distribution when the incident frequency is 9 GHz. Due to the addition of the ITO layer, the strength of the electric field motivated on the FSR surface is significantly enhanced; this is mainly because the conductive characteristics of the ITO layer promote the coupling of the FSS on both sides of the square wall surface. To a certain extent, the electric field can penetrate the PR medium, thereby increasing FSS resonance intensity. This paper studies the effect of different square resistances of the ITO layer on the FSR performance. It can be observed from Figure 7 that, except for the *R* = 100 Ω/ model, when the incident

frequency is less than 5 GHz, the absorptivity gradually decreases with the increasing *R*. When the incident frequency is more than 5 GHz, except when the absorption rate of the *R* = 100 Ω/ and *R* = 800 Ω/ models decreases obviously, the changes in the absorption rates of other models are small.

**Figure 4.** Effect of square cavity size on FSR absorption rate under normal incidence.

**Figure 5.** Effect of ITO layer on FSR absorption rate under normal incidence.

**Figure 6.** (**a**) *R* = 200 Ω/ model's E-field distribution at 9 GHz. (**b**) Without-ITO model's E-field distribution at 9 GHz.

Figure 8 shows the distribution of the influence of the thickness of the PR medium on the absorptivity. It can be observed that when the incident frequency is in the 3–7 GHz band, as *h* increases gradually, the absorptivity increases. When the incident frequency is in the 7–22 GHz band, the absorptivity is basically independent of *h*. When the incident frequency is in the 22–29 GHz band, the absorptivity of the *h* = 0.8 mm model and *h* = 0.6 mm model fluctuates wildly, and the absorption rate of the *h* = 0.6 mm model drops to a minimum

of 75%. Overall, the absorption rate increased gradually with the increase of *h* to more than 95%. However, when the incident frequency is more than 29 GHz, the absorptivity of the *h* = 1.2 mm model drops to 80%, while the absorptivity of the *h* = 1 mm model remains above 95%. After careful consideration, the PR medium thickness of the designed FSR is set to 1mm. This paper also studies the influence of the width of the FSS metal patch on the absorption rate. It can be observed from Figure 9 that the influence of the width of the metal patch within a specific range on the absorption rate is minimal. Except for the *w* = 500 µm mode and *w* = 250 µm model, the absorption rate is slightly lower than other models in some frequency bands, the absorption rate change is small, and both are above 90%. Considering the miniaturization design and processing difficulty of FSR, the *w* = 125 µm model is finally selected.

**Figure 7.** Effect of square resistance of the ITO layer on FSR absorption rate under normal incidence.

Finally, the absorptivity of the FSR under different incident angles is compared and analyzed. In TE mode, as the incidence angle increases, the magnetic field direction always has an angle with the metal line on the FSR wall, and the component of the incident wave parallel to the metal line decreases with the increase in the incidence angle. The absorbing principle of the FSR designed in this paper is to use the SSPP to bind the spatial electromagnetic wave to the wall of the FSR, convert it into a plane wave, and finally dissipate it. As the incidence angle increases, the electromagnetic wave component bound to the wall of the FSR will become smaller, so the absorption rate of the FSR will decrease. It can be observed from Figure 10 that in the TE mode, with the increase in the incident angle of the electromagnetic wave, the most sensitive band to the incident angle is mainly in the 15–20 GHz band. When the incident angle increases to 20°, this band's lowest point of absorptivity drop to 85%. When the incident angle increases to 40°, this band's lowest point of absorptivity drops to 70%. The absorption rate in other bands is highly insensitive to the incident angle, and the absorption rate is above 90%. In TM mode, the direction of the magnetic field is parallel to part of the metal wire. As the incidence angle increases, the metal wire in the absorption cavity will be directly irradiated by the incident wave, thus increasing the utilization rate of the metal wire, so the absorption rate increases. Nevertheless, the incidence angle within a specific range has little influence on the absorption rate. It can be observed from Figure 11 that in the TM mode, when the incident angle increases to 40°, the lowest point of the absorption rate is up to around 95%.

**Figure 8.** Effect of PR thickness on FSR absorption rate under normal incidence.

**Figure 9.** Effect of the width of the metal patch on FSR absorption rate under normal incidence.

**Figure 10.** Effect of incident angle on FSR absorption rate in TE mode.

**Figure 11.** Effect of incident angle on FSR absorption rate in TM mode.

#### **5. Fabrication and Measurement**

In order to verify the simulation results, the FSR based on SSPP is fabricated and processed using printed circuit board technology. The FSR size is 244 mm × 244 mm × 10 mm, consisting 20 × 20 square cavity cell arrays, and the processed model is shown in Figure 12. The FSS on the inner wall of the square cavity is made by a printed circuit board. The ITO layer outside the square cavity is a dense ITO film formed on the PET substrate by magnetron sputtering under high vacuum conditions. The ITO-PET layer and the PR layer are tightly bonded together by hot pressing technology.

The experimental test adopts the stepped-frequency test system, and the schematic diagram of the system is shown in Figure 13. Agilent E8363A VNA (Agilent Technologies, Palo Alto, CA, USA) generates the stepped-frequency signal. Compared to point-frequency continuous test systems, the stepped-frequency test system does not require complex hardware cancellers. The signal sent by the test system is sent by the standard gain antenna aiming at the target, and the reflected echo signal is received by another standard gain antenna and sent to the vector network analyzer. Then, the vector network analyzer is used for time domain cancellation. Limited by the laboratory antenna specifications, the FSR is tested in the 1–18 GHz band, the test angle range is −40∼40° , and the test angle interval is 0.1°.

**Figure 12.** The photo of the FSR composed of 20 × 20 square cavity units, and the enlarged view of the FSS metal patch.

**Figure 13.** Stepped-frequency test system.

Figure 14 is the ISAR image of the metal plate and FSR. It can be observed that the FSR has a strong absorption of incident electromagnetic waves. Figure 15 compares simulated and measured results of FSR absorption rates in the TE mode under normal incidence. In the 1–5 GHz band, the absorption rate gradually increases and decreases with the increase of the incident frequency. At the same incident frequency, the absorption rate measured is slightly larger than the simulated results by about 10%. In the 5–8 GHz band, the absorption rate of FSR is basically about 90%, which is slightly smaller than the simulated results. In the 8–18 GHz band, FSR's absorption rate is above 95%, consistent with the simulated results. Figure 16 compares the measured results of different incident angles in TE mode and the simulated results. It can be observed that with the increase in the incident angle of electromagnetic waves, the most sensitive bands to the incident angle are mainly in the 4.5–7 GHz band and the 13–18 GHz band. In these two bands, the absorptivity decreases roughly gradually with the increase of the incident angle. In the 4.5–7 GHz band, the fluctuation of the absorption rate is relatively small. When the incident angle increases to 40°, the lowest point of the absorption rate drops to around 77%. In the 13–18 GHz band, the absorptivity fluctuates wildly. When the incident angle increases to 20°, this band's lowest point of absorptivity drop to around 83%. When the incident angle increases to 40°, this band's lowest point of absorptivity drop to around 66%. Furthermore, absorptivity in other bands is less sensitive to the incident angle. Figure 17 compares the measured results of different incident angles in the TM mode and the simulated results. It can be observed that with the increase in the incident angle of electromagnetic waves, the absorption rate of FSR changed little, and the measured results are consistent with the simulation results.

By comparison, it is found that the experimental results are consistent with the numerical simulation results, proving that the FSR designed in this paper has excellent reliability.

**Figure 14.** (**a**) ISAR imaging of sheet metal. (**b**) ISAR imaging of FSR.

**Figure 15.** Comparison of simulated results and measured results of FSR absorption rate in TE mode under normal incidence.

**Figure 16.** Comparison of the measured results of different incident angles in TE mode and the simulated results.

**Figure 17.** Comparison of the measured results of different incident angles in TM mode and the simulated results.

Table 1 compares the proposed FSR with the reported FSR in terms of the working frequency band, band continuity, maximum incident angle, and size of the absorbing unit. The comparison shows that the proposed FSR has a wider absorption bandwidth, better band continuity, and weaker sensitivity to the incident angle. Furthermore, the smaller size of the absorbing unit, which is very conducive to the miniaturization design of FSR, makes the application prospect of FSR wider.


**Table 1.** Feature comparison between the proposed FSR and the reported FSR.

## **6. Conclusions**

In this paper, a broadband FSR based on SSPP is designed, which is composed of a square cavity array composed of multiple layers of FSS. The numerical simulation results demonstrate that the designed FSR has an excellent absorption effect. When the electromagnetic wave is incident vertically, the absorption rate of the FSR in the 5–30 GHz band is above 95%, and in the 3.5–5 GHz band, the absorption rate is also above 80%. At the same time, FSR is highly insensitive to the incident angle. When the incident angle increases by 40°, the absorption rate fluctuates wildly only in the 15–20 GHz band, and the absorption rate drops to about 70% at the lowest. In TM mode, the incidence angle has a little effect on the absorption rate of FSR. The FSR designed in this paper is fabricated, and a microwave anechoic chamber measures its transmission characteristics. The experimental results demonstrate that the FSR absorption rate is consistent with the numerical simulation results, which proves that the FSR designed in this paper is less sensitive to the incident angle of magnetic waves. It has excellent broadband absorbing ability and outstanding reliability and robustness. The proposed FSR has a high absorption rate and little reflectivity of electromagnetic waves in its operating frequency band, so that it can be applied to the stealth design for dual-station radar detection. At the same time, the structural strength of the FSR is also excellent, which can realize the integrated design of structure and stealth and has a good application prospect in the stealth design of aircraft and other targets.

**Author Contributions:** Conceptualization, J.B. and Y.L.; methodology, J.B.; software, J.B.; validation, J.B., Y.L. and Q.Y.; formal analysis, Q.Y.; investigation, X.G.; resources, Q.Y.; data curation, J.B.; writing—original draft preparation, J.B.; writing—review and editing, J.B., Y.L. and Q.Y.; visualization, J.B. and Y.L.; funding acquisition, X.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Natural Science Foundation of Shanxi Province under Grant 2022JQ052.

**Data Availability Statement:** The data that support the findings of this study are available from the corresponding author (Y.L.), upon reasonable request.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Chengfu Tian <sup>1</sup> , Shusheng Wei 2,\*, Jiayu Xie <sup>1</sup> and Tainming Bai <sup>1</sup>**

<sup>2</sup> School of Automotive Engineering, Wuhan University of Technology, Wuhan 430070, China

**\*** Correspondence: weishusheng13@163.com

**Abstract:** This paper investigates the deadbeat current controllers for isolated bidirectional dualactive-bridge dc-dc converter (IBDC), including the peak current mode (PCM) and middle current mode (MCM). The controller uses an enhanced single phase shift (ESPS) modulation method by exploiting pulse width as an extra control variable in addition to phase shift ratio. The control variables for PCM controllers are derived in detail and the two different current controllers are compared. A double-closed-loop control method is then employed, which could directly control the high-frequency inductor current and eliminate the transient DC current bias of the transformer. Furthermore, load feedforward was introduced to further enhance the dynamic of the converter. With the proposed control method, the settling time could be reduced within several PWM cycles during load disturbance without transient DC current bias. A 5 kW IBDC converter prototype was built and the settling time of 6 PWM cycles during load change with voltage regulation mode was achieved, which verifies the superior dynamic performance of the control method.

**Keywords:** dual active bridge; deadbeat controller; load feedforward

**1. Introduction**

The isolated bi-directional dual-active-bridge dc-dc converter (IBDC) has been a hot topic in recent years due to its simple structure, high efficiency and ultrafast response [1]. The transient DC current offset of the transformer and the inductor, which might saturate the transformer and increase the system's current stress during the abrupt load change, has attracted people's attention. Different dynamic modulation methods have been proposed to solve the problem [2–5]. Additionally, to increase the dynamics of IBDCs, the current mode controller could be a competitive alternative. It also has other inherent benefits including over-current protection, elimination of transient DC current offset and easy implementation of current sharing between multiple IBDCs [6].

Digital predictive current controllers based on conventional single phase shift (CSPS) modulation was proposed in [7,8], where the phase shift ratio was used to control the transformer current. In [7], the average current calculated by an analog integrator of the DC bus current was used as the feedback signal, which can achieve fast dynamic performance. However, transient DC current offset occurs during the sudden change in phase shift ratio for CSPS modulation.

The predictive duty cycle mode (PDCM) controller, shown in Figure 1b, was proposed in [6] to eliminate the transient DC current offset, which was applied in [8]. The drive signals of the primary side are fixed. The transformer current needs to be oversampled, and duty cycles d1 for S2,3 and d2 for S1,4 are calculated in turn in every half cycle. Another limitation of this method is that the controller only works in the ZVS range (IP1 > 0) shown in Figure 1b and may lose effectiveness when IP1 < 0.

**Citation:** Tian, C.; Wei, S.; Xie, J.; Bai, T. Dynamic Enhancement for Dual Active Bridge Converter with a Deadbeat Current Controller. *Micromachines* **2022**, *13*, 2048. https://doi.org/10.3390/mi13122048

Academic Editor: Wensheng Zhao

Received: 21 October 2022 Accepted: 18 November 2022 Published: 23 November 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

*Micromachines* **2022**, *13*, x FOR PEER REVIEW 2 of 13

*iL*

*P*3

*P*1

**Figure 1.** Principle of the PDCM controller. (**a**) IBDC, (**b**) waveforms. **Figure 1.** Principle of the PDCM controller. (**a**) IBDC, (**b**) waveforms. A deadbeat current controller based on the middle current and enhanced PWM mod-

A deadbeat current controller based on the middle current and enhanced PWM modulation was proposed in [9]. However, regulation of the output voltage and current was not introduced, which is more important in real application. To overcome the drawbacks, this paper investigated the deadbeat current controllers, including the peak current mode (PCM) and middle current mode (MCM). Based on the controllers, a double-closed-loop control method with load feedforward was introduced. Furthermore, a 5 kW IBDC converter prototype was built, and the settling time of 6 PWM cycles during load change could be achieved, which validates its superior dynamic performance. A deadbeat current controller based on the middle current and enhanced PWM modulation was proposed in [9]. However, regulation of the output voltage and current was not introduced, which is more important in real application. To overcome the drawbacks, this paper investigated the deadbeat current controllers, including the peak current mode (PCM) and middle current mode (MCM). Based on the controllers, a double-closed-loop control method with load feedforward was introduced. Furthermore, a 5 kW IBDC converter prototype was built, and the settling time of 6 PWM cycles during load change could be achieved, which validates its superior dynamic performance. ulation was proposed in [9]. However, regulation of the output voltage and current was not introduced, which is more important in real application. To overcome the drawbacks, this paper investigated the deadbeat current controllers, including the peak current mode (PCM) and middle current mode (MCM). Based on the controllers, a double-closed-loop control method with load feedforward was introduced. Furthermore, a 5 kW IBDC converter prototype was built, and the settling time of 6 PWM cycles during load change could be achieved, which validates its superior dynamic performance. **2. Deadbeat Peak Current Mode Controller** 

*S*3

*S*1

#### **2. Deadbeat Peak Current Mode Controller** *2.1. Basic Model of IBDC for SPS Modulation*

**2. Deadbeat Peak Current Mode Controller**  *2.1. Basic Model of IBDC for SPS Modulation* The basic model of SPS modulation-based IBDC is presented prior to introducing the

*2.1. Basic Model of IBDC for SPS Modulation*  The basic model of SPS modulation-based IBDC is presented prior to introducing the proposed current controller. Figure 2 illustrates the theoretical waveforms of the IBDC using the SPS modulation method when converter voltage gain *k* ≥ 1 , where *k* = *V*1/(n*V*2) and n is the turn ratio of the transformer. The waveforms are symmetrical for the same The basic model of SPS modulation-based IBDC is presented prior to introducing the proposed current controller. Figure 2 illustrates the theoretical waveforms of the IBDC using the SPS modulation method when converter voltage gain *k* ≥ 1, where *k* = *V*1/(n*V*2) and n is the turn ratio of the transformer. The waveforms are symmetrical for the same transmission power of two opposite directions. proposed current controller. Figure 2 illustrates the theoretical waveforms of the IBDC using the SPS modulation method when converter voltage gain *k* ≥ 1 , where *k* = *V*1/(n*V*2) and n is the turn ratio of the transformer. The waveforms are symmetrical for the same transmission power of two opposite directions.

(a) (b) **Figure 2.** Waveforms of IBDC at steady state for CSPS modulation: (**a**) forward power transmission (*P* > 0), (**b**) reverse power transmission (*P* < 0). **Figure 2.** Waveforms of IBDC at steady state for CSPS modulation: (**a**) forward power transmission (*P* > 0), (**b**) reverse power transmission (*P* < 0).

**Figure 2.** Waveforms of IBDC at steady state for CSPS modulation: (**a**) forward power transmission (*P* > 0), (**b**) reverse power transmission (*P* < 0). The symbols in Figure 2 are defined as follows: TS is the switching cycle, f is the switching frequency, *D* is the phase shift ratio and tph is the shifted time. *D* ≥ 0 (*t*ph ≥ 0) The symbols in Figure 2 are defined as follows: TS is the switching cycle, f is the switching frequency, *D* is the phase shift ratio and tph is the shifted time. *D* ≥ 0 (*t*ph ≥ 0) stands for *P* ≥ 0 and *D* < 0 (*t*ph < 0) for *P* < 0. *I*P1 and *I*P2 are the two "switching currents" and *I*P2 is the peak current when *k* > 1. The middle current IM, defined as the instantaneous The symbols in Figure 2 are defined as follows: TS is the switching cycle, f is the switching frequency, *D* is the phase shift ratio and tph is the shifted time. *D* ≥ 0 (*t*ph ≥ 0) stands for *P* ≥ 0 and *D* < 0 (*t*ph < 0) for *P* < 0. *I*P1 and *I*P2 are the two "switching currents" and *I*P2 is the peak current when *k* > 1. The middle current IM, defined as the instantaneous

stands for *P* ≥ 0 and *D* < 0 (*t*ph < 0) for *P* < 0. *I*P1 and *I*P2 are the two "switching currents" and *I*P2 is the peak current when *k* > 1. The middle current IM, defined as the instantaneous current at *T*S/2, is taken into consideration instead of the average current, which equals zero in one cycle. The basic equations for the IBDC are derived as follows: current at *T*S/2, is taken into consideration instead of the average current, which equals zero in one cycle. The basic equations for the IBDC are derived as follows:

> 2 1

$$\begin{cases} \begin{array}{c} P = \frac{V\_1^2}{2fLk}D(1 - |D|) \ \text{'} \ t\_{\text{ph}} = \frac{DT\_\text{S}}{2} \end{array} \\\ I\_{\text{P1}} = \frac{V\_1(2k|D| - k + 1)}{4fLk} \ \text{'} \ I\_{\text{P2}} = \frac{V\_1(2|D| + k - 1)}{4fLk} \ \text{'} \ I\_{\text{M}} = \frac{V\_1D}{2fLk} \end{cases} \tag{1}$$

The relationships among the variables *P*, *I*P1, *I*P2, *I*M, *D* and *t*ph at steady state can then be derived. Therefore, for a given value of one variable, other variables can be calculated. The relationships among the variables *P*, *I*P1,*I*P2, *I*M, *D* and *t*ph at steady state can then be derived. Therefore, for a given value of one variable, other variables can be calculated.

#### *2.2. Peak Current Mode Controller 2.2. Peak Current Mode Controller*

Peak current mode (PCM) controllers are introduced in this section. Figure 3a,b show the transient waveforms of a PCM current controller in one cycle for forward and reverse power transmission, respectively. *P*1, *P*2, *P*<sup>3</sup> and *P*<sup>4</sup> are the drive signals for the primary side and *S*1, *S*2, *S*<sup>3</sup> and *S*<sup>4</sup> are the drive signals for the secondary side. The variable *t*ph,ref is shifted-time at the steady state for the given *I*P2,ref which can be derived from (1). A sawtooth carrier with the same frequency of the converter was utilized to generate the reference signals. The "switching on" moment *t*<sup>1</sup> and "switching off" moment *t*<sup>2</sup> should meet the constrains as: 0 < *t*<sup>1</sup> < *T*S/2 and 3*T*S/4 < *t*<sup>2</sup> < *T*S. The variables *t*<sup>D</sup> and *t*<sup>W</sup> are defined as "delay time" and "width time", respectively. *P*ref is the power reference, and *I*P2,ref and *I*P1,ref are the references for the corresponding "switching currents", respectively. Peak current mode (PCM) controllers are introduced in this section. Figure 3a,b show the transient waveforms of a PCM current controller in one cycle for forward and reverse power transmission, respectively. *P*1, *P*2, *P*3 and *P*4 are the drive signals for the primary side and *S*1, *S*2, *S*3 and *S*4 are the drive signals for the secondary side. The variable *t*ph,ref is shifted-time at the steady state for the given *I*P2,ref which can be derived from (1). A sawtooth carrier with the same frequency of the converter was utilized to generate the reference signals. The "switching on" moment *t*1 and "switching off" moment *t*2 should meet the constrains as: 0 < *t*1 < *T*S/2 and 3*T*S/4 < *t*2 < *T*S. The variables *t*D and *t*W are defined as "delay time" and "width time", respectively. *P*ref is the power reference, and *I*P2,ref and *I*P1,ref are the references for the corresponding "switching currents", respectively.

**Figure 3.** Transient waveforms of PCM controller: (**a**) forward power transmission (*P*ref > 0), (**b**) reverse power transmission (*P*ref < 0). **Figure 3.** Transient waveforms of PCM controller: (**a**) forward power transmission (*P*ref > 0), (**b**) reverse power transmission (*P*ref < 0).

As shown in Figure 3a, there are two cases according to the initial current *I*0 and reference current *I*p2,ref: *u*2 leads *u*1 (*t*ph < 0) for the solid line waveforms and *u*2 lags *u*1 (*t*ph > 0) for dotted line waveforms. For the sake of brevity, the superposition principle was used As shown in Figure 3a, there are two cases according to the initial current *I*<sup>0</sup> and reference current *I*p2,ref: *u*<sup>2</sup> leads *u*<sup>1</sup> (*t*ph < 0) for the solid line waveforms and *u*<sup>2</sup> lags *u*<sup>1</sup> (*t*ph > 0) for dotted line waveforms. For the sake of brevity, the superposition principle was used to derivate the inductor current when calculating *t*<sup>D</sup> and *t*W.

to derivate the inductor current when calculating *t*D and *t*W. For forward power transmission, the requirement was imposed that *I*P2 = *I*P2,ref and *t*ph,ref > 0. According to the superposition principle, the inductor current ripple *<sup>L</sup>* <sup>Δ</sup>*<sup>I</sup>* during 0 and 3*T*S/4 can be calculated by adding up the two ripple currents as follows: For forward power transmission, the requirement was imposed that *I*P2 = *I*P2,ref and *t*ph,ref > 0. According to the superposition principle, the inductor current ripple ∆*I<sup>L</sup>* during 0 and 3*T*S/4 can be calculated by adding up the two ripple currents as follows:

$$\begin{cases} \Delta I\_{L} = \Delta I\_{L,\mu\_{1}} + \Delta I\_{L,\mu\_{2}} \\ \Delta I\_{L,\mu\_{1}} = \frac{-V\_{1}}{L} \cdot \frac{T\_{\text{S}}}{4} + \frac{-V\_{1}}{L} \cdot \frac{T\_{\text{S}}}{2}, \ \Delta I\_{L,\mu\_{2}} = \frac{V\_{2}}{L} \cdot t\_{D} + \frac{-V\_{2}}{L} \cdot (\frac{3T\_{\text{S}}}{4} - t\_{D}) \\ \quad I\_{\text{P2}} = I\_{0} + \Delta I\_{L} = I\_{\text{P2,ref}} \end{cases} \tag{2}$$

where ∆*IL*,*u*<sup>1</sup> and ∆*IL*,*u*<sup>2</sup> are current ripples generated by the two dependent voltage source *u*<sup>1</sup> and *u*2, respectively. *t*<sup>D</sup> is then derived as:

$$t\_D = \frac{(I\_{\text{P2,ref}} - I\_0)Lk}{2V\_1} + \frac{3-k}{8f} \tag{3}$$

Furthermore, *t*<sup>W</sup> is derived as:

$$t\_{\rm W} = \frac{\Im T\_{\rm S}}{4} - t\_{\rm D} + t\_{\rm ph,ref} \tag{4}$$

With regard to reverse power transmission, the switching current at *t*<sup>2</sup> is set to be— *I*P1,ref and *t*ph,ref < 0 as shown in Figure 4. Similar to forward power transmission, *t*<sup>D</sup> and *t*<sup>W</sup> can be obtained. Thus, the equations for the PCM controller are written as:

$$\begin{cases} \text{tD} = \frac{kL(l\text{p}\_{2\text{ref}} - \text{l}\_0)}{2V\_1} + \frac{3-\text{k}}{8f}; \, t\_{\text{W}} = \frac{kL(l\text{p}\_{2\text{ref}} + \text{l}\_0)}{2V\_1} - \frac{k-5}{8f}, \, P\_{\text{ref}} \ge 0\\\ t\_{\text{D}} = -\frac{kL(l\text{p}\_{2\text{ref}} + \text{l}\_0)}{2V\_1} + \frac{k+1}{8f}; \, t\_{\text{W}} = -\frac{kL(l\text{p}\_{2\text{ref}} - \text{l}\_0)}{2V\_1} + \frac{k+3}{8f}, \, P\_{\text{ref}} < 0 \end{cases} \tag{5}$$

**Figure 4.** Voltage control scheme based on the MCM-ESPS controller. **Figure 4.** Voltage control scheme based on the MCM-ESPS controller.

**3. Double-Closed-Loop Control with Load Feedforward**  In practice, instead of the high-frequency inductor current, the DC voltage, current or power should always be regulated. In this section, the voltage mode control strategy based on the MCM-ESPS controller is introduced. Figure 4 shows the output voltage control scheme based on the deadbeat current controller, where two control loops are involved. The load feedforward control could substantially increase the system dynamic [11,12]. In order to improve the stability of output voltage under load disturbance, load feedforward under double-closed-loop control is As in the aforementioned Equations (6) and (7), initial current *I*<sup>0</sup> is sampled to calculate the *t*<sup>D</sup> and *t*W. However, a one-cycle delay exists between the sampling instant and control update due to the algorithm implementation of the digital processor. *I*P2 for the PCM controller is sampled at 3*T*S/4, and DSP interrupt occurs to calculate the new parameters shown in Figure 3. *t*<sup>D</sup> and *t*<sup>W</sup> update at the beginning of the next cycle. Assuming DC bus voltage *V*<sup>1</sup> and *V*<sup>2</sup> are constant in two adjacent periods, the relationships between the *I*P2(*n* − 1), *I*M(*n* − 1) in the (*n* − 1)th cycle and the initial current in the *n*th switching cycle *I*0(*n*) could be derived as:

$$I\_0(n) = \begin{cases} \begin{array}{c} I\_{\rm P2}(n-1) - \frac{2V\_1(k-1)}{Lk} \left( t\_\rm D(n-1) + t\_\rm W(n-1) \right) + \frac{V\_1(7-k)}{4fLk}, \; P\_{\rm ref}(n-1) \ge 0 \\\\ I\_{\rm P2}(n-1) - \frac{V\_1(k-1)}{4fLk}, \; P\_{\rm ref}(n-1) < 0 \end{array} \end{cases} \tag{6}$$

The relationships of the middle current were derived as: 1 2 1 M 1 2 1 1 12 ( ) , 0 2 24 1 12 ( ) , 0 2 24 *<sup>V</sup> fLkP <sup>P</sup> fLk V <sup>I</sup> <sup>V</sup> fLkP <sup>P</sup> fLk V* −− ≥ = − −+ < (7) According to the power transmission directions in two adjacent cycles, four situations are considered for the PCM controller: case 1 when *P*ref(*n* − 1) ≥ 0 and *P*ref(*n*) ≥ 0; case 2 when *P*ref(*n* − 1) < 0 and *P*ref(*n*) ≥ 0; case 3 when *P*ref(*n* − 1) ≥ 0 and *P*ref(*n*) < 0; and case 4 when *P*ref(*n* − 1) < 0 and Pref(*n*) < 0. Combining (6)–(9), the control variables *t*<sup>D</sup> and *t*<sup>W</sup> with delay compensation can be derived as shown in Table 1. With the control variables in Table 1, the inductor peak current could be tracked to the reference in two cycles, which is consistent with the idea of the deadbeat control in ref [10].

Thus, the relationship between the middle current *I*M and the load current *i*o could be

1 o

−− ≥

*fLk V <sup>I</sup>*

o M

1 o

<sup>2</sup> (1 ) *fLkI i NI*

The small signal model of the system, as shown in Figure 5, can be obtained from the control block diagram in Figure 4, where *i*S is the average output current of an H bridge in a single period and *G*o(*s*) is the transfer function of capacitance voltage and capacitance

− −+ <

*<sup>V</sup> fLi <sup>i</sup> fLk V*

1 1 <sup>2</sup> ( ) , 0 2 24

*P*=*Vi*2 o (8)

(9)

o

o

*<sup>V</sup>* = − (10)

1

1

M

1

Without considering the power loss of the converter, we could obtain:

=

M

expressed as (9) and (10):

current, denoted as:


**Table 1.** Control variable calculation with one-cycle delay compensation.

#### **3. Double-Closed-Loop Control with Load Feedforward**

In practice, instead of the high-frequency inductor current, the DC voltage, current or power should always be regulated. In this section, the voltage mode control strategy based on the MCM-ESPS controller is introduced.

Figure 4 shows the output voltage control scheme based on the deadbeat current controller, where two control loops are involved. The load feedforward control could substantially increase the system dynamic [11,12]. In order to improve the stability of output voltage under load disturbance, load feedforward under double-closed-loop control is presented. As shown in Figure 4, the feedforward current *i*M,F corresponding to the load was superimposed on the current reference *i*M,VR, which is the output of the outer voltage loop, to form the final current reference value *i*M,ref.

The relationships of the middle current were derived as:

$$I\_{\mathcal{M}} = \begin{cases} \frac{V\_1}{2fLk} \left(\frac{1}{2} - \sqrt{\frac{1}{4} - \frac{2fLkP}{V\_1^2}}\right), P \ge 0\\ -\frac{V\_1}{2fLk} \left(\frac{1}{2} - \sqrt{\frac{1}{4} + \frac{2fLkP}{V\_1^2}}\right), P < 0 \end{cases} \tag{7}$$

Without considering the power loss of the converter, we could obtain:

$$P = V\_2 i\_0 \tag{8}$$

Thus, the relationship between the middle current *I*<sup>M</sup> and the load current *i*<sup>o</sup> could be expressed as (9) and (10):

$$I\_{\mathcal{M}} = \begin{cases} \frac{V\_1}{2fLk} (\frac{1}{2} - \sqrt{\frac{1}{4} - \frac{2fL i\_o}{V\_1}}) \text{, } i\_0 \ge 0\\ -\frac{V\_1}{2fLk} (\frac{1}{2} - \sqrt{\frac{1}{4} + \frac{2fL i\_o}{V\_1}}) \text{, } i\_0 < 0 \end{cases} \tag{9}$$

$$i\_{\rm o} = NI\_{\rm M}(1 - \frac{2fLkI\_{\rm M}}{V\_1})\tag{10}$$

The small signal model of the system, as shown in Figure 5, can be obtained from the control block diagram in Figure 4, where *i*<sup>S</sup> is the average output current of an H bridge in a single period and *G*o(*s*) is the transfer function of capacitance voltage and capacitance current, denoted as:

$$\mathcal{G}\_0(\mathbf{s}) = \mathbf{1}/(\mathcal{C}\_2\mathbf{s}) \tag{11}$$

**Figure 5.** Small signal model of an IBDC with the double-closed-loop control strategy. **Figure 5.** Small signal model of an IBDC with the double-closed-loop control strategy.

*G*VR(*s*) is the volatge regulation transfer function, where the conventional PI controller is always used. *K*P and *K*I are the proportional and integral coefficients of the PI regulator, respectively. Thus, we could obtain: *G*VR(*s*) is the volatge regulation transfer function, where the conventional PI controller is always used. *K*<sup>P</sup> and *K*<sup>I</sup> are the proportional and integral coefficients of the PI regulator, respectively. Thus, we could obtain:

$$\mathbf{G\_{VR}(s)} = \mathbf{K\_P} + \mathbf{K\_I s} \tag{12}$$

0 2 *G s Cs* ( ) 1/( ) = (11)

*G*F(*s*) represents the transfer function of the load feedforward and *G*MS(*s*) is the relationship between *i*M and *i*s. *G*C(*s*) is the transfer function of the deadbeat current controller. Considering a one-cylce delay, it could be written as: *G*F(*s*) represents the transfer function of the load feedforward and *G*MS(*s*) is the relationship between *i*<sup>M</sup> and *i*s. *G*C(*s*) is the transfer function of the deadbeat current controller. Considering a one-cylce delay, it could be written as:

$$G\_{\mathbb{C}}(s) = \frac{1 - e^{-sT\_s}}{s} \tag{13}$$

The feedforward transfer function *G*F(*s*) can be calculated using small-signal analysis based on Equation (10). To substitute o o <sup>o</sup> *iii*<sup>∧</sup> = + and *III* <sup>M</sup> M M ∧ = + into (10), ignoring The feedforward transfer function *G*F(*s*) can be calculated using small-signal analysis based on Equation (10). To substitute *i*<sup>o</sup> = *i*<sup>o</sup> + ∧ *i* <sup>o</sup> and *I*<sup>M</sup> = *I*<sup>M</sup> + ∧ *I*<sup>M</sup> into (10), ignoring the higher-order terms, *G*F(*s*) be derived as:

$$\mathbf{G\_{F}(s)} = \stackrel{\wedge}{I}\_{\mathbf{M}, \mathbf{F}}(s) / \stackrel{\wedge}{i}\_{\mathbf{0}}(s) = 1 / \left( \mathcal{N} (1 - 4fL k I\_{\mathbf{M}} / V\_{\mathbf{1}}) \right) \tag{14}$$

The average output current of H bridge in the secondary side is derived as:

$$i\_s = \frac{V\_1 N}{2fL} D(1 - D) \tag{15}$$

Combing (1) with (16), the *G*MS(*s*) could be derived as: Combing (1) with (16), the *G*MS(*s*) could be derived as:

$$G\_{\rm MS}(s) = \stackrel{\wedge}{i}\_S(s) / \stackrel{\wedge}{I}\_M(s) = N(1 - 4fLkI\_\mathbf{M}/V\_1) \tag{16}$$

According to the small signal model in Figure 5, the output impedance *R*o1(*s*) without and with feedforward could be calculated as (17) and (18), respectively. According to the small signal model in Figure 5, the output impedance *R*o1(*s*) without and with feedforward could be calculated as (17) and (18), respectively.

$$\mathcal{R}\_{\rm o1}(s) = \frac{\stackrel{\wedge}{V}\_{2}(s)}{\stackrel{\wedge}{i}\_{o}(s)} = -\frac{\mathcal{G}\_{\rm o}(s)}{1 + \mathcal{G}\_{\rm VR}(s)\mathcal{G}\_{\rm C}(s)\mathcal{G}\_{\rm MS}(s)\mathcal{G}\_{o}(s)}\tag{17}$$

$$R\_{\rm o2}(s) = \frac{\stackrel{\frown}{\dot{V}\_2(s)}}{\stackrel{\frown}{i\_{\rm o}(s)}} = \frac{(\mathcal{G}\_{\rm F}(s)\mathcal{G}\_{\rm C}(s)\mathcal{G}\_{\rm MS}(s) - 1)\mathcal{G}\_{\rm o}(s)}{1 + \mathcal{G}\_{\rm VR}(s)\mathcal{G}\_{\rm C}(s)\mathcal{G}\_{\rm MS}(s)\mathcal{G}\_{\rm o}(s)}\tag{18}$$

By substituting the circuit parameters and control parameters into Equations (17) and (18), baud diagrams of output impedance with different loads under the double-closed-loop control strategy can be drawn as shown in Figure 6. In this case, the inductance *L =* µH, the voltage *V*<sup>1</sup> = 300V and *V*<sup>2</sup> = 280V.The coefficients of the PI regulator are *K*<sup>P</sup> = 2 and *K*<sup>I</sup> = 4000.

2 F C MS o

VR C MS o o ( ) ( ( ) ( ) ( )-1) ( ) ( ) 1 () () () () ( ) *V s G sG sG s G s R s*

By substituting the circuit parameters and control parameters into Equations (17) and (18), baud diagrams of output impedance with different loads under the double-closedloop control strategy can be drawn as shown in Figure 6. In this case, the inductance *L =*  μH, the voltage *V*<sup>1</sup> = 300V and *V*<sup>2</sup> = 280V.The coefficients of the PI regulator are *K*<sup>P</sup> = 2 and

*G sG sG sG s i s*

(18)

**Figure 6.** Output impedance of the converter with different loads. **Figure 6.** Output impedance of the converter with different loads.

o2

*K*<sup>I</sup> = 4000.

∧

<sup>∧</sup> = = <sup>+</sup>

As shown in Figure 6, the closed-loop output impedance at low frequency decreases significantly after the feedforward is added. When *I*<sup>M</sup> = 4 A, the output impedance at a frequency of 100 Hz decreases from −15 dB to −35 dB, whereas when *I*M = 15 A, the output impedance decreases from −8 dB to −30 dB at 100 Hz. If the frequency is further reduced, the amplitude attenuation of the closed-loop output impedance brought by the feedforward control become more obvious, which indicates a more robust output voltage under the load disturbance. As shown in Figure 6, the closed-loop output impedance at low frequency decreases significantly after the feedforward is added. When *I*<sup>M</sup> = 4 A, the output impedance at a frequency of 100 Hz decreases from −15 dB to −35 dB, whereas when *I*<sup>M</sup> = 15 A, the output impedance decreases from −8 dB to −30 dB at 100 Hz. If the frequency is further reduced, the amplitude attenuation of the closed-loop output impedance brought by the feedforward control become more obvious, which indicates a more robust output voltage under the load disturbance.

#### **4. Experimental Verification**

#### **4. Experimental Verification**  *4.1. Experimental Platform*

*4.1. Experimental platform*  The laboratory IBDC experimental platform shown in Figure 7 was used to verify the proposed control method. The main circuit parameters are listed in Table 2. The current sensor LA55-P had a 200 kHz bandwidth from LEM. PE-Expert4 from Myway was utilized as the digital controller including DSP and FPGA cores. FPGA XC6SLX45 was used to generate PWM signals. The control variables were calculated in each cycle in DSP, and the corresponding PWM compare values CMP1 and CMP2 were updated at the The laboratory IBDC experimental platform shown in Figure 7 was used to verify the proposed control method. The main circuit parameters are listed in Table 2. The current sensor LA55-P had a 200 kHz bandwidth from LEM. PE-Expert4 from Myway was utilized as the digital controller including DSP and FPGA cores. FPGA XC6SLX45 was used to generate PWM signals. The control variables were calculated in each cycle in DSP, and the corresponding PWM compare values CMP1 and CMP2 were updated at the beginning of the next cycle. *Micromachines* **2022**, *13*, x FOR PEER REVIEW 8 of 13

**Figure 7.** Experimental platform. **Figure 7.** Experimental platform.

**Table 2.** System Parameters.

Turns ratio *n* 1:1 Switching frequency *f* 10 kHz Primary capacitor *C*1 2460 μF Secondary capacitor *C*2 2460 μF Inductor *L* 652 μH Equivalent resistor *Rs* 80 mΩ

The performance of the CSPS modulation-based current controller in [5] and the

Figure 8 shows the experimental waveforms of current *i*L when the references have step changes. The references could be tracked for all the controllers when the current reference steps up from 3 A to 8 A and steps down from 8 A to 3 A. The settling time *t*set in Figure 8a is obvious, while the settling time in Figure 8b is negligible. Additionally, the ESPS-PCM controller eliminates the transient DC current offset that exists in CSPS

**Figure 8.** Zoomed-out waveforms during the step change of the current reference for forward power

To verify the performance of the current controllers during bidirectional power transmission, a sequence of current references was set to investigate the response. Figure

transmission. (**a**) CSPS modulation-based current controller. (**b**) ESPS-PCM controller.

*4.2. Comparisons of Different Current Controllers for Forward Power Transmission* 

modulation-based controller, as shown in the dashed circle of Figure 8a.

(a)

*iL* (2.5A/div) 10 ms/div

(b)

proposed ESPS-PCM were compared for the forward power transmission mode.


**Table 2.** System Parameters. **Table 2.** System Parameters.

**Figure 7.** Experimental platform.

#### *4.2. Comparisons of Different Current Controllers for Forward Power Transmission 4.2. Comparisons of Different Current Controllers for Forward Power Transmission*

DC Source 1 DC Source 2 Load resistor

DAB prototype Myway controller

*Micromachines* **2022**, *13*, x FOR PEER REVIEW 8 of 13

The performance of the CSPS modulation-based current controller in [5] and the proposed ESPS-PCM were compared for the forward power transmission mode. The performance of the CSPS modulation-based current controller in [5] and the proposed ESPS-PCM were compared for the forward power transmission mode.

Figure 8 shows the experimental waveforms of current *i*<sup>L</sup> when the references have step changes. The references could be tracked for all the controllers when the current reference steps up from 3 A to 8 A and steps down from 8 A to 3 A. The settling time *t*set in Figure 8a is obvious, while the settling time in Figure 8b is negligible. Additionally, the ESPS-PCM controller eliminates the transient DC current offset that exists in CSPS modulation-based controller, as shown in the dashed circle of Figure 8a. Figure 8 shows the experimental waveforms of current *i*L when the references have step changes. The references could be tracked for all the controllers when the current reference steps up from 3 A to 8 A and steps down from 8 A to 3 A. The settling time *t*set in Figure 8a is obvious, while the settling time in Figure 8b is negligible. Additionally, the ESPS-PCM controller eliminates the transient DC current offset that exists in CSPS modulation-based controller, as shown in the dashed circle of Figure 8a.

**Figure 8.** Zoomed-out waveforms during the step change of the current reference for forward power transmission. (**a**) CSPS modulation-based current controller. (**b**) ESPS-PCM controller. **Figure 8.** Zoomed-out waveforms during the step change of the current reference for forward power transmission. (**a**) CSPS modulation-based current controller. (**b**) ESPS-PCM controller. *Micromachines* **2022**, *13*, x FOR PEER REVIEW 9 of 13

> To verify the performance of the current controllers during bidirectional power transmission, a sequence of current references was set to investigate the response. Figure 9 To verify the performance of the current controllers during bidirectional power transmission, a sequence of current references was set to investigate the response. Figure 9 shows the waveforms of the ESPS-PCM controller. The transmission power *P* stepped up from 600 W to 1450 W at *t*1, changed direction to −1450 W at *t*3, reversed direction to 1450 W at *t*<sup>5</sup> and then stepped down to 600 W at *t*7. The current references were smoothly reached with a one-cycle delay during the whole transient process, including the transition between two opposite power transmissions. 9 shows the waveforms of the ESPS-PCM controller. The transmission power *P* stepped up from 600 W to 1450 W at *t*1, changed direction to −1450 W at *t*3, reversed direction to 1450 W at *t*5 and then stepped down to 600 W at *t*7. The current references were smoothly reached with a one-cycle delay during the whole transient process, including the transition between two opposite power transmissions.

**Figure 9.** Transient waveforms of bidirectional power transmission. **Figure 9.** Transient waveforms of bidirectional power transmission.

*4.3. Dynamic Performance Comparison between Different Control Methods* 

The experimetal results with traditional single-voltage loop control, double-closed-

10V

shown in Figures 10–12. The output voltage *V*2 was 280 V, and the load increased abruptly at *t*1 with load resistance decreases from 75 Ω to 25 Ω and decreased sharply at *t*2 with load resistance increases from 25 Ω to 75 Ω. When the load increases, the DC voltage will fall, and the control loop will increase the phase shift angle to transfer more power to maintain the DC load. Simillarly, when the DC load decreases, the DC voltage will rise and the control loop will decrease the phase shift angle to reduce the transmitted power.

(**a**)

3ms 1.6ms Load step up Load step down t1 t2

图2.16. 负载突变电压电流波形

time(5 ms/div)

16 A 25.1 A

(**b**)

time(200 μs/div)

*iL* **(10 A/div)**

*iL* **(10 A/div)**

9.2V

*V***<sup>2</sup>** **(12.5 V/div)**

*V***<sup>2</sup>** **(12.5 V/div)**

#### *4.3. Dynamic Performance Comparison between Different Control Methods 4.3. Dynamic Performance Comparison between Different Control Methods*

**Figure 9.** Transient waveforms of bidirectional power transmission.

*Micromachines* **2022**, *13*, x FOR PEER REVIEW 9 of 13

transition between two opposite power transmissions.

The experimetal results with traditional single-voltage loop control, double-closedloop control and double-closed-loop control with feedforward under load disturbance are shown in Figures 10–12. The output voltage *V*<sup>2</sup> was 280 V, and the load increased abruptly at *t*<sup>1</sup> with load resistance decreases from 75 Ω to 25 Ω and decreased sharply at *t*<sup>2</sup> with load resistance increases from 25 Ω to 75 Ω. When the load increases, the DC voltage will fall, and the control loop will increase the phase shift angle to transfer more power to maintain the DC load. Simillarly, when the DC load decreases, the DC voltage will rise and the control loop will decrease the phase shift angle to reduce the transmitted power. The experimetal results with traditional single-voltage loop control, double-closedloop control and double-closed-loop control with feedforward under load disturbance are shown in Figures 10–12. The output voltage *V*2 was 280 V, and the load increased abruptly at *t*1 with load resistance decreases from 75 Ω to 25 Ω and decreased sharply at *t*2 with load resistance increases from 25 Ω to 75 Ω. When the load increases, the DC voltage will fall, and the control loop will increase the phase shift angle to transfer more power to maintain the DC load. Simillarly, when the DC load decreases, the DC voltage will rise and the control loop will decrease the phase shift angle to reduce the transmitted power.

9 shows the waveforms of the ESPS-PCM controller. The transmission power *P* stepped up from 600 W to 1450 W at *t*1, changed direction to −1450 W at *t*3, reversed direction to 1450 W at *t*5 and then stepped down to 600 W at *t*7. The current references were smoothly reached with a one-cycle delay during the whole transient process, including the

**Figure 10.** Waveforms using the conventional single-loop control. (**a**) Overall waveforms; (**b**) Zoomed-in waveforms during the load increase; (**c**) Zoomed-in waveforms during the load **Figure 10.** Waveforms using the conventional single-loop control. (**a**) Overall waveforms; (**b**) Zoomedin waveforms during the load increase; (**c**) Zoomed-in waveforms during the load decrease.

Figure 10 shows the waveform using the conventional single-loop control. The inductor current shows obvious transient DC bias during abrupt load change. When the load increased, the peak current reached 25.1 A, and the current overshoot was 9.1 A. In the experiment, both the static and dynamic performances under load increase and decrease were considered when tuning the PI parameters. The waveform showed that the voltage sag was 9.2 V, and the settling time was 3 ms when the load increased. The voltage overshoot was 10 V, and the settling time was 1.6 ms when the load decreased. Figure 11 Figure 10 shows the waveform using the conventional single-loop control. The inductor current shows obvious transient DC bias during abrupt load change. When the load increased, the peak current reached 25.1 A, and the current overshoot was 9.1 A. In the experiment, both the static and dynamic performances under load increase and decrease were considered when tuning the PI parameters. The waveform showed that the voltage sag was 9.2 V, and the settling time was 3 ms when the load increased. The voltage overshoot was 10 V, and the settling time was 1.6 ms when the load decreased. Figure 11 shows the waveform using the double-closed-loop control. The inductor current

shows the waveform using the double-closed-loop control. The inductor current was symmetrical, and the transient DC bias was eliminated. When the load increased, the

9V

(**a**)

Load step up t1 t2 Load step down time(5 ms/div)

1.9ms 1.3ms

(**b**)

time(200 μs/div)

decrease.

*V***<sup>2</sup>** **(12.5 V/div)** *iL*

*V***<sup>2</sup>** **(12.5 V/div)**

**(10 A/div)**

*iL* **(10 A/div)**

8.2V

decrease.

*V***<sup>2</sup>**

**(12.5 V/div)** *iL*

**(10 A/div)**

was symmetrical, and the transient DC bias was eliminated. When the load increased, the voltage sag was 8.2 V, and the settling time was 1.9 ms. The voltage overshoot was 9 V, and the settling time was 1.3 ms when the load decreased. symmetrical, and the transient DC bias was eliminated. When the load increased, the voltage sag was 8.2 V, and the settling time was 1.9 ms. The voltage overshoot was 9 V, and the settling time was 1.3 ms when the load decreased.

(**c**) **Figure 10.** Waveforms using the conventional single-loop control. (**a**) Overall waveforms; (**b**) Zoomed-in waveforms during the load increase; (**c**) Zoomed-in waveforms during the load

time(200 μs/div)

Figure 10 shows the waveform using the conventional single-loop control. The inductor current shows obvious transient DC bias during abrupt load change. When the load increased, the peak current reached 25.1 A, and the current overshoot was 9.1 A. In the experiment, both the static and dynamic performances under load increase and decrease were considered when tuning the PI parameters. The waveform showed that the voltage sag was 9.2 V, and the settling time was 3 ms when the load increased. The voltage overshoot was 10 V, and the settling time was 1.6 ms when the load decreased. Figure 11 shows the waveform using the double-closed-loop control. The inductor current was

*Micromachines* **2022**, *13*, x FOR PEER REVIEW 10 of 13

**Figure 11.** Waveforms using the double-closed-loop control. (**a**) Overall waveforms; (**b**) Zoomedin waveforms during the load increase; (**c**) Zoomed-in waveforms during the load decrease. **Figure 11.** Waveforms using the double-closed-loop control. (**a**) Overall waveforms; (**b**) Zoomed-in waveforms during the load increase; (**c**) Zoomed-in waveforms during the load decrease.

Figure 12 shows the waveforms using the double-closed-loop control with load feedforward. The transient DC bias was eliminated. During the transient process, the DC voltage variation was significantly reduced, and the settling time was obviously shortened compared with the double-closed-loop control without load feedforward. The voltage sag was 3 V when the load increased, and the voltage overshoot was 4.4 V when the load decreased. In the process of load surge, the recovery time for DC voltage was 0.5 ms, which is five switching cycles. The recovery time was 0.6 ms, which is six switching cycles, in the process of load decreases. Figure 12 shows the waveforms using the double-closed-loop control with load feedforward. The transient DC bias was eliminated. During the transient process, the DC voltage variation was significantly reduced, and the settling time was obviously shortened compared with the double-closed-loop control without load feedforward. The voltage sag was 3 V when the load increased, and the voltage overshoot was 4.4 V when the load decreased. In the process of load surge, the recovery time for DC voltage was 0.5 ms, which is five switching cycles. The recovery time was 0.6 ms, which is six switching cycles, in the process of load decreases.

Compared with the traditional single-voltage loop control, the double-closed-loop control utilizes the deadbeat current controller as the inner loop, which directly regulates the high-frequency AC current of the transformer. Thus, the transient DC current bias could be eliminated. Meanwhile, the dynamic performance the of the IBDC with voltage mode mainly depends on the bandwidth of the feedback signal [12]. With the feedforward control samples, the load changes directly, which could significantly increase the

4.4V

(**a**)

t1 t2

Load step up Load step down time(5 ms/div)

3V

*iL* **(10 A/div)**

*V***<sup>2</sup>** **(12.5 V/div)**

(**c**) **Figure 11.** Waveforms using the double-closed-loop control. (**a**) Overall waveforms; (**b**) Zoomedin waveforms during the load increase; (**c**) Zoomed-in waveforms during the load decrease.

time(200 μs/div)

*iL* **(10 A/div)**

cycles, in the process of load decreases.

*V***<sup>2</sup>** **(12.5 V/div)**

Figure 12 shows the waveforms using the double-closed-loop control with load feedforward. The transient DC bias was eliminated. During the transient process, the DC voltage variation was significantly reduced, and the settling time was obviously shortened compared with the double-closed-loop control without load feedforward. The voltage sag was 3 V when the load increased, and the voltage overshoot was 4.4 V when the load decreased. In the process of load surge, the recovery time for DC voltage was 0.5 ms, which is five switching cycles. The recovery time was 0.6 ms, which is six switching

Compared with the traditional single-voltage loop control, the double-closed-loop control utilizes the deadbeat current controller as the inner loop, which directly regulates the high-frequency AC current of the transformer. Thus, the transient DC current bias could be eliminated. Meanwhile, the dynamic performance the of the IBDC with voltage mode mainly depends on the bandwidth of the feedback signal [12]. With the feedforward control samples, the load changes directly, which could significantly increase the robutness of the DC voltage under the load disturblances. The dynamic performance enhancement of the proposed control can be seen by comparison of Figures 11 and 12.

**Figure 12.** Waveforms using the double-closed-loop control with load feedforward. (**a**) Overall waveforms; (**b**) Zoomed-in waveforms during the load increase; (**c**) Zoomed-in waveforms during the load decrease. **Figure 12.** Waveforms using the double-closed-loop control with load feedforward. (**a**) Overall waveforms; (**b**) Zoomed-in waveforms during the load increase; (**c**) Zoomed-in waveforms during the load decrease.

**5. Conclusions**  A double-closed-loop control strategy based on the deadbeat current controller was proposed in this paper, which directly regulates the high-frequency inductor current to the reference and eliminates the transient DC current bias during the transient process. Furthermore, load feedforward was introduced to enhance the dynamic of the converter. The proposed control method shows potential in the application of IBDC under voltage mode. With the proposed method, the settling time could be reduced to within several PWM cycles during load disturbance. A 5 kW IBDC converter prototype was built, and Compared with the traditional single-voltage loop control, the double-closed-loop control utilizes the deadbeat current controller as the inner loop, which directly regulates the high-frequency AC current of the transformer. Thus, the transient DC current bias could be eliminated. Meanwhile, the dynamic performance the of the IBDC with voltage mode mainly depends on the bandwidth of the feedback signal [12]. With the feedforward control samples, the load changes directly, which could significantly increase the robutness of the DC voltage under the load disturblances. The dynamic performance enhancement of the proposed control can be seen by comparison of Figures 11 and 12.

#### the superior dynamic performance of the proposed control strategy was verified by the experimental results. **5. Conclusions**

dc-dc converter. *IEEE Trans. Power Electron.* **2020**, *35*, 3148–3172.

**References** 

**Author Contributions:** Conceptualization, C.T. and S.W.; methodology, S.W.; software, S.W.; validation, S.W.; formal analysis, J.X. and T.B.; investigation, C.T.; resources, S.W.; data curation, A double-closed-loop control strategy based on the deadbeat current controller was proposed in this paper, which directly regulates the high-frequency inductor current to the reference and eliminates the transient DC current bias during the transient process.

J.X.; writing—original draft preparation, T.B.; writing—review and editing, S.W. and J.X.; visualization, C.T.; supervision, C.T.; project administration, C.T.; funding acquisition, C.T. All

**Conflicts of Interest:** The authors declare no conflicts of interest.

1. Hou, N.; Li, Y. Overview and comparison of modulation and control strategies for non-resonant single-phase dual-active-bridge

**Data Availability Statement:** Not applicable.

authors have read and agreed to the published version of the manuscript.

Furthermore, load feedforward was introduced to enhance the dynamic of the converter. The proposed control method shows potential in the application of IBDC under voltage mode. With the proposed method, the settling time could be reduced to within several PWM cycles during load disturbance. A 5 kW IBDC converter prototype was built, and the superior dynamic performance of the proposed control strategy was verified by the experimental results.

**Author Contributions:** Conceptualization, C.T. and S.W.; methodology, S.W.; software, S.W.; validation, S.W.; formal analysis, J.X. and T.B.; investigation, C.T.; resources, S.W.; data curation, J.X.; writing—original draft preparation, T.B.; writing—review and editing, S.W. and J.X.; visualization, C.T.; supervision, C.T.; project administration, C.T.; funding acquisition, C.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**

