Open AccessArticle

A Complementary Split-Ring Resonator (CSRR)-Based 2D Displacement Sensor

Kun Ren

Pengwen Zhu

Taotao Sun

Junchao Wang

Dawei Wang

Jun Liu

and

Wensheng Zhao

Zhejiang Provincial Key Lab of Large-Scale Integrated Circuit Design, School of Electronics and Information, Hangzhou Dianzi University, Hangzhou 310018, China

Author to whom correspondence should be addressed.

Symmetry 2022, 14(6), 1116; https://doi.org/10.3390/sym14061116

Submission received: 28 April 2022 / Revised: 20 May 2022 / Accepted: 26 May 2022 / Published: 29 May 2022

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Figure 1
(a) Schematic of CSRR structure and its (b) equivalent circuit model. "> Figure 2
Schematic of the 1D displacement sensor based on CSRR structure. "> Figure 3
Simulated results of the 1D displacement sensor with different values of g. "> Figure 4
Simulated results of the 1D displacement sensor with different triangle base lengths. "> Figure 5
Schematic of 2D displacement sensor: (a) top view and (b) bottom view. "> Figure 6
The influence of the mover in the two-dimensional direction on the CSRR slit. "> Figure 7
Sensor measurement error caused by direction-finding displacement. "> Figure 8
Schematic diagram of the improved CSRR. "> Figure 9
Flow chart of particle swarm optimization device parameters. "> Figure 10
Simulated sensor response with (a) vertical displacement, and (b) horizontal displacement. "> Figure 11
(a) The linear relationship between y displacement and 1st resonant frequency. (b) The linear relationship between x displacement and 2nd resonant frequency. "> Figure 12
CSRR 2D sensor prototype. "> Figure 13
Sensor Test System. "> Figure 14
Measured sensor response with (a) vertical displacement and (b) horizontal displacement. ">

Versions Notes

Abstract

In this paper, a two-dimensional (2D) displacement sensor based on the diagonal symmetry complementary split-ring resonator (CSRR) structure is proposed. The one-dimensional (1D) displacement sensor is initially developed by etching a ring on the ground plane, and a triangle metal patch is used as the mover. When the mover is attached, a CSRR is formed, thereby inducing a resonant frequency. The displacement of the mover can be retrieved by the variation in the resonant frequency. Then, the structure is extended for designing a 2D displacement sensor. To further reduce the error caused by the lateral displacement, the shape of the sensor is further optimized using the particle swarm algorithm. Finally, the structure of the CSRR displacement sensor is simulated and analyzed, and a physical prototype is made. The measured results are consistent with the theoretical analysis, and its sensitivity is 160 MHz/mm.

Keywords:

split-ring resonator (SRR); complementary split-ring resonator (CSRR); displacement sensor; particle swarm optimization (PSO) algorithm

1. Introduction

With the development of radio-frequency technology, the application of microwave sensors to determine different physical quantities is becoming more and more widespread. Planar structures, such as microstrip lines and coplanar waveguides, have been widely used in integrated microwave resonant sensor design. Microwave resonant sensors can be divided into the frequency variation sensor, the coupled modulation sensor, the frequency division sensor, the differential sensor and so on. The principle of a frequency variation sensor is to establish the mapping relationship between the measurement characteristics and the resonant frequency of the sensor [1]. Moreover, due to the characteristics of stable performance in different environments and low cost, microwave displacement sensors have gained the attention of researchers, and there are a variety of sensors with planar microstrip structures [2,3,4,5]. Compared to displacement sensors based on other technologies, the main advantages of the displacement sensors based on microwave technology include small size, easy design, low fabrication costs, and high sensitivity. In addition, the microwave displacement sensor will not have any additional impact on the object during measuring process, and non-ionizing radiation is safe for human beings.

A reusable, environmentally friendly planar resonant liquid sensor was proposed in [6] with the fluid channel fabricated using 3D printing technology. By introducing embedded pores in substrate-integrated waveguide (SIW) to distribute moist air in the sensing area [7], the mapping relationship between resonant frequency and ambient humidity was achieved. Further, a microwave temperature sensor was designed and implemented in [8]. Microwave resonant sensors could also be used to detect bacterial growth, human health monitoring in real time, and pipe-coating defects [9,10,11]. Split-ring resonators (SRR) were first proposed in 1999 [12], and the structure consists of two metal rings printed on the dielectric layer. Due to its simple structure and low cost, it has a wide range of applications. Then, it derives complementary split-ring resonators (CSRR) [13] with a similar structure, that is, grooves are carved on the metal plane, which is complementary to the SRR structure. Saadat-Safa M. et al. designed a sensor with CSRR structure and calculated the electromagnetic properties, such as the dielectric constant of the material, to be measured according to the position of the resonance point [14]. CSRR structural sensors have also been widely used in the microfluidic field, and can be used to measure the concentration of glucose in the solution to be detected [15,16]. In recent years, sensors for detecting displacement using CSRR structures have attracted a lot of interest. Different from most sensors that use notch depth and frequency offset as measurement methods, the angular displacement sensor based on CSRR structure uses phase as a measurement method, which enables the sensor to determine whether the target to be measured is rotating clockwise or counterclockwise [14,16]. So far, various microwave sensor structures based on the reflection frequency response or the transmission frequency response have been proposed. For example, refs. [17,18] discussed the 1D and 2D sensor structures of the SRR structure; ref. [19] presented a two-dimensional displacement sensor based on the two-layer frequency selective surface (FSS); ref. [20] studied the 1D displacement and rotation sensors based on the electromagnetic bandgap (EBG) phenomenon; and ref. [21] reported broadside-coupled split-ring resonators (BC-SRRs), which principle of operation is based on a shift in resonance frequency caused by a break of symmetry.

The research on 2D displacement sensors using the diagonal symmetry CSRR structure has not been seen previously published in the literature. In this paper, a new type of microwave sensor structure is proposed, the quantitative relationship between the sensor resonance frequency and the 2D micro-displacement value was investigated, and further, the structure of the sensor with a particle swarm algorithm was optimized, and finally the 2D displacement detection accuracy of the sensor was improved. Experimental results showed that the CSRR structure 2D displacement sensor proposed in this paper was able to achieve the measurement of 2D small displacement in the plane.

2. CSRR-Based Displacement Sensor

The SRR unit is essentially a magnetic resonance unit with high-quality value in the microwave frequency band. By exchanging the dielectric material of SRR with metal structure, as shown in Figure 1, the complementary structure of SRR, i.e., CSRR unit, can be obtained. If the incident electric field is parallel to the normal line of the plane where the CSRR unit is located, an induced current will be generated on the outer metal and inner metal, resulting in the inductive effect, and the capacitive coupling effect will appear between the outer metal and inner metal. Therefore, the CSRR unit can be modeled as a resonant circuit, with the resonant frequency given as [22]

f_{CSRR} = \frac{1}{2 π \sqrt{L_{0} C_{c}}}

(1)

where

L_{0}

and

C_{c}

represent the equivalent inductance and capacitance of the CSRR.

2.1. 1D Displacement Sensor

Firstly, the CSRR structure is employed for the design of a 1D displacement sensor. Figure 2 shows the 1D displacement sensor based on CSRR structure, which is composed of an etched ring on the ground plane of microstrip line. A triangular metal patch is used as mover, and when it is attached on the ground plane, the CSRR unit would be formed, thereby resulting in a resonant frequency. Here, the substrate is set as Rogers 4350 with a dielectric constant of 3.66 and loss tangent of 0.004. The substrate thickness is 0.762 mm, the metal thickness is 0.035 mm, and the microstrip line width is 1.69 mm to achieve 50 Ω impedance matching. The size of the etched ring is 6 mm × 6 mm, and the width is 0.4 mm. The mover is also designed with Rogers 4350 substrate, and the height and base length of the triangle shown in Figure 2 are 3 mm and 2 mm, respectively.

Since the resonant frequency of the CSRR is highly dependent on the structural parameters, it is expected that the mover’s movement could be retrieved by the variation in the resonant frequency. As shown in Figure 3, when the mover moves upward, the value of

g

increases, thereby inducing an upward shift of resonant frequency. As displacement of the mover increases from 0 to 2 mm, the resonant frequency increases from 4.48 GHz to 5.25 GHz, corresponding to a sensitivity of 382 MHz/mm. Further, the influence of mover’s triangle base length

L

on the sensor response is examined. As shown in Figure 4, as

L

increases from 2 mm to 4 mm, the sensor sensitivity can be improved significantly.

2.2. 2D Displacement Sensor

It is expected that the 1D displacement sensor can be extended to sensing the 2D displacement. As shown in Figure 5, two rings are etched on the ground plane, and two movers are attached to form CSRR structures. In order to distinguish the displacements in two directions, the geometrical parameters of two CSRRs are different, thereby producing two resonant frequencies. Here, the sizes of two CSRRs are denoted by

c s r r_x_l \times c s r r_x_w

and

c s r r_y_l \times c s r r_y_w

, respectively. In the applications, two movers are simultaneously bonded to the object under test. It is anticipated that the position of the object under test could be obtained by the combinations of the variations in two resonant frequencies. To excite both CSRRs in different directions, the microstrip line is designed with a 90° bend, and the corner is cut to reduce parasitic capacitance.

In addition, we find that the object to be measured moves on a 2D plane, and its movements could be decomposed into horizontal x and vertical y directions. The displacement in the x-direction will drive the horizontal movement of the mover in the CSRR in the x-direction, causing the change in the slit value of CSRR in the x-direction. Similarly, the movement in the y-direction will also cause a change in the slit value of the CSRR in the y-direction. However, some errors may be caused here. Taking the y-direction as an example, the movement in the x-direction will cause the mover of the CSRR in the y-direction to move in the horizontal direction, and the resonance point of the CSRR structure will not only be affected by the slit width but is also slightly affected by the position of the slit, as shown in Figure 6.

The total movement displacement of the mover is denoted as d, and the displacement components in the horizontal and vertical directions are denoted as d_x and d_y, respectively. This displacement widens the slit of CSRR but also makes the slit move in the horizontal direction. The change in frequency point caused by horizontal movement is undesirable because it introduces measurement error. For the measurement in the vertical direction, the movement in the horizontal direction is called a lateral movement, and the error caused by this movement is called lateral error.

We performed a simulation analysis of the error here. We set the CSRR size to 5 mm × 3 mm, the slit width to 0.8 mm, and the bottom side of the triangle mover to be 4 mm long and 4 mm high. As shown in Figure 7, the measurement error caused by a lateral movement of 1 mm is 0.15 GHz. This lateral error has no linear relationship with the lateral displacement. When the lateral displacement is small, there is almost no lateral error, while when the lateral displacement is large, the lateral error will increase a lot.

To reduce the measurement error small enough, we need to make improvements to the structure of CSRR so that the results of two-dimensional displacement measurement can be guaranteed. Here, an improved CSRR structure is designed for such a goal, as shown in Figure 8. The design objective is to keep the resonant frequency unchanged when the mover moves in the horizontal direction. By virtue of the proposed structure, both the equivalent inductance and equivalent capacitance change with the displacement in horizontal direction, and their variations could cancel each other out. The distance between the metal layer and the printed circuit board (PCB) boundary is w. When the width of the metal layer is reduced by delta, an arc structure with a height of h should be added to compensate. The principle is that the displacement of the slit caused by the lateral displacement causes an error in the measurement on the two-dimensional plane. Therefore, the error caused by lateral displacement can be compensated by adjusting the groove width on the upper side of CSRR to make it change unevenly with the lateral displacement direction of the mover.

3. Numerical Optimization

For the structure of the CSRR 2D displacement sensor proposed above, it is difficult to obtain the best parameter combination through the analytical expression of the CSRR structure to achieve a small error in the 2D measurement. To find the parameter combination with the smallest error in the design space, this paper uses the particle swarm optimization algorithm to find the solution with the smallest error in the design space. The workflow is shown in Figure 9.

Ten parameters can be used to uniquely mark a 2D displacement sensor composed of an improved CSRR, that is,

w i d t h_{x}

l e n g t h_{x}

w_{x}

h_{x}

, and

d e l t a_{x}

represent the CSRR structural parameters used to measure the horizontal direction, and

w i d t h_{y}

l e n g t h_{y}

w_{y}

h_{y}

, and

d e l t a_{y}

represent the CSRR structure parameters used to measure the vertical direction as shown in Figure 8. The subscripts x and y represent the respective corresponding qualities of the CSRR structures used to detect displacements in the x and y directions. Considering that the effect of the particle swarm optimization algorithm may be poor when the dimension is too high, the difference in CSRR structure size in the other two directions is mainly to distinguish the two resonance points. Therefore, the width and length of the x-direction and y-direction are determined in advance. The length and width of the CSRR in the x-direction are determined in advance to be 18 mm and 10 mm; the length and width of the CSRR in the y-direction are 14 mm and 6 mm, respectively. This 6D vector is optimized by particle swarm optimization, and the goal is to make the error of the 2D measurement as small as possible. Therefore, the fitness function can be defined as described below.

In the process of each round of particle swarm iteration, five finite-element simulations are performed, and the displacement conditions are shown in Table 1. Taking the fitness of displacement in the horizontal direction (x-direction) as an example, the error brought by the displacement in the y-direction to the x-direction can be determined according to the three simulation results of (a), (d), and (e). The specific strategy is to calculate according to the sum of the difference between the frequencies

f_{x d}

and

f_{x e}

of the resonance point in the x-direction in the simulation results of (d) and (e) compared with the resonance point

f_{x a}

in the x-direction in (a). The loss function in the x-direction is:

X_{c o s t} = (f_{x d} - f_{x a}) + (f_{x e} - f_{x a})

(2)

Similarly,

f_{y a}

f_{y b}

, and

f_{y c}

are the frequency of the resonance point of (a), (b), and (c) in the y direction. The loss function in the Y direction can be defined as:

Y_{c o s t} = (f_{y b} - f_{y a}) + (f_{y c} - f_{y a})

(3)

In addition, it is found in the optimization process that although the CSRR sizes in the two directions have been determined in advance, sometimes the resonance point in the x-direction is too close to the resonance point in the y-direction. We want to prevent this from happening so as not to cause confusion when the resonant frequency shifts. Here, an additional quantity

d e l t a T Z

is introduced, and it denotes the difference between two resonant frequencies.

d e l t a T Z

should be large enough to distinguish two resonant frequencies. The fitness function, which takes a candidate solution to the problem as input and produces its cost value as output, is defined as follows:

F i t n e s s = d e l t a T Z + \frac{4}{X_{c o s t} + Y_{c o s t} + 1}

(4)

The goal is to reduce the error in the x- and y-directions as much as possible, while the difference between two resonant frequencies should exceed a certain value. So, the denominator in the equation is set as “

X_{c o s t} + Y_{c o s t} + 1

”.

The basic idea of particle swarm optimization is to randomly generate a series of initial populations, in this case, a series of random sensor parameter combinations. In the subsequent process of randomly changing the population, the current position of each particle is evaluated every time, that is, the loss function value of each sensor parameter combination in the population, and each particle moves to the neighborhood of the current position with random quantity under the guidance of individual optimal position and population optimal position. Specifically, the size and direction that each particle moves in this cycle are determined by the velocity matrix v, and v is determined by the current velocity v, the best position of the individual, and the best position of the population. The update to the velocity of each particle in the population is as follows:

v_{i}^{d} = ω v_{i}^{d} + c_{1} r_{1} (p_{i}^{d} - x_{i}^{d}) + c_{2} r_{2} (p_{g}^{d} - x_{i}^{d})

(5)

The update to the position of each particle in the population is as follows:

x_{i}^{d} = x_{i}^{d} + α v_{i}^{d}

(6)

In Equations (5) and (6), w is the inertia factor, which is used to control the range of particle motion and is related to the convergence speed;

p_{i}, p_{g}

are the best position of the individual and the best position of the population, respectively;

c_{1}

c_{2}

are called self-learning factors and group learning factors. Controlling particles are affected by the individual optimal position and group optimal position, respectively;

r_{1}

r_{2}

are random numbers on [0, 1],

α

controls the weight of the speed. The sensor parameter results of particle swarm optimization are shown in Table 2:

The structure of this optimized 2D sensor is simulated and analyzed. The influence of the horizontal and vertical displacement on the CSRR resonance point are shown in Figure 10a,b. The error in the horizontal or vertical direction is very small, the resonance point used to measure the horizontal displacement is unchanged when the mover is displaced vertically, and the resonance point used to measure the vertical displacement is at the horizontal position of the mover. It remains unchanged during displacement. The different loss functions of the final optimization results of the algorithm may be quite different. The goal here is to optimize the error value of the sensor in the 2D measurement, and this error value is defined as the sum of the resonant frequency offsets of 1 mm and 2 mm of lateral movement in the x and y directions, respectively. The simulated results are shown in Figure 11. The relationship of the resonant frequency and the displacement is linear, while another resonant frequency is nearly unchanged. Here, the quality factor is defined as the ratio of resonant frequency and 3 dB bandwidth. The quality factors of the first and second resonances are 80.8 and 83.6, respectively.

4. Experimental Validation

To verify the feasibility of the proposed 2D sensor, we fabricated a prototype of the 2D sensor for experimental verification. The substrate uses Rogers 4350 with a thickness of 0.762 mm, and the specific object is shown in Figure 12. The two ports of the sensor are respectively connected to 50 Ω subminiature version A (SMA) connectors and connected to a vector network analyzer (VNA) to test the performance of the sensor. The actual measurement diagram is shown in Figure 13. Displace 1~2 mm in the x and y directions, respectively, and observe the frequency offset and error of the two resonance points of the sensor. Among them, the experimental results of fixing the x-direction and making the mover move in the y-direction are shown in Figure 14a, and the experimental results of fixing the y-direction and making the mover move in the x-direction are shown in Figure 14b. The experimental results are consistent with the simulation results, and the errors caused by the lateral displacement of 1 mm in the x and y directions are all less than 0.05 GHz.

The CSRR proposed sensor is compared with six previously published microwave displacement sensors in Table 3. Here, the sensitivity of the frequency-variation sensor is defined as the ratio of relative frequency shift and displacement. Some amplitude-variation sensors are also listed in the table, and the corresponding sensitivity is defined as the ratio of amplitude variation and displacement. The CSRR sensor in this work exhibits lower losses and better matching than sensors in [17,18]. Compared with FSS sensor in [19], our proposed sensor has higher relative sensitivity. The CSRR has a simple structure and more sensitive performance than the proposed sensors in [20,21].

5. Conclusions

This paper firstly proposes the design of a 1D displacement sensor based on CSRR and conducts simulation analysis on it. Subsequently, this design was extended to 2D planar displacement measurement, and the structure of CSRR was improved to reduce the error of the sensor in 2D displacement measurement. Since the parameters of CSRR structure struggle to obtain the optimal solution directly from the analytical formula, this paper applies the particle swarm algorithm to find a parameter combination with the smallest error in the 2D plane measurement in the constrained design space. Finally, the design is verified experimentally. The measurement data prove that the sensitivity of the sensor in both x and y directions is about 160 MHz/mm, which is consistent with the theoretical prediction.

Author Contributions

Conceptualization, W.Z.; methodology, K.R. and P.Z.; validation, T.S., J.W. and J.L.; writing—original draft preparation, K.R. and P.Z.; writing—review and editing, W.Z. and D.W. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded in part by the National Key R&D Program of China under Grant 2019YFB2205002, the National Natural Science Foundation of China, grant number 61874020, 61934006, 62101170, and 61874038.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. (a) Schematic of CSRR structure and its (b) equivalent circuit model.

Figure 2. Schematic of the 1D displacement sensor based on CSRR structure.

Figure 3. Simulated results of the 1D displacement sensor with different values of g.

Figure 4. Simulated results of the 1D displacement sensor with different triangle base lengths.

Figure 5. Schematic of 2D displacement sensor: (a) top view and (b) bottom view.

Figure 6. The influence of the mover in the two-dimensional direction on the CSRR slit.

Figure 7. Sensor measurement error caused by direction-finding displacement.

Figure 8. Schematic diagram of the improved CSRR.

Figure 9. Flow chart of particle swarm optimization device parameters.

Figure 10. Simulated sensor response with (a) vertical displacement, and (b) horizontal displacement.

Figure 11. (a) The linear relationship between y displacement and 1st resonant frequency. (b) The linear relationship between x displacement and 2nd resonant frequency.

Figure 12. CSRR 2D sensor prototype.

Figure 13. Sensor Test System.

Figure 14. Measured sensor response with (a) vertical displacement and (b) horizontal displacement.

Table 1. Finite Element Simulation Condition Table.

The Serial Number	X-Direction Displacement (mm)	Y-Direction Displacement (mm)
a	0	0
b	1	0
c	2	0
d	0	1
d	0	2

Table 2. The Geometric Parameters of The Optimized CSRR 2D Displacement Sensor.

Parameters	w_x (mm)	h_x (mm)	delta_x (mm)	w_y (mm)	h_y (mm)	delta_y (mm)
	0.4	1.1445	3.4863	0.27	0.5487	1.8077

Table 3. A Comparison Between Published Microwave Displacement Sensors.

Ref	Dimension	Sensor Type	Sensitivity of X	Sensitivity of Y	Operating Frequency
This work	2D	CSRR	0.09/mm	0.073/mm	1.4~2.2 GHz
[19]	2D	FSS	0.014/mm	0.027/mm	11~14 GHz
[21]	2D	BC-SRR	0.04/mm	0.032/mm	2.5 GHz
[20]	1D	EBG	40 dB/mm	N/A	13 GHz
[18]	2D	SRR	18.7 dB/mm	18.7 dB/mm	4.25 GHz
[17]	1D	SRR	23 dB/mm	N/A	1.2 GHz

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Ren, K.; Zhu, P.; Sun, T.; Wang, J.; Wang, D.; Liu, J.; Zhao, W. A Complementary Split-Ring Resonator (CSRR)-Based 2D Displacement Sensor. Symmetry 2022, 14, 1116. https://doi.org/10.3390/sym14061116

AMA Style

Ren K, Zhu P, Sun T, Wang J, Wang D, Liu J, Zhao W. A Complementary Split-Ring Resonator (CSRR)-Based 2D Displacement Sensor. Symmetry. 2022; 14(6):1116. https://doi.org/10.3390/sym14061116

Chicago/Turabian Style

Ren, Kun, Pengwen Zhu, Taotao Sun, Junchao Wang, Dawei Wang, Jun Liu, and Wensheng Zhao. 2022. "A Complementary Split-Ring Resonator (CSRR)-Based 2D Displacement Sensor" Symmetry 14, no. 6: 1116. https://doi.org/10.3390/sym14061116

APA Style

Ren, K., Zhu, P., Sun, T., Wang, J., Wang, D., Liu, J., & Zhao, W. (2022). A Complementary Split-Ring Resonator (CSRR)-Based 2D Displacement Sensor. Symmetry, 14(6), 1116. https://doi.org/10.3390/sym14061116

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A Complementary Split-Ring Resonator (CSRR)-Based 2D Displacement Sensor

Abstract

1. Introduction

2. CSRR-Based Displacement Sensor

2.1. 1D Displacement Sensor

2.2. 2D Displacement Sensor

3. Numerical Optimization

4. Experimental Validation

5. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

References

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