# Shield Optimization and Formulation of Regression Equations for Split-Ring Resonator

Authors: Tahir Ejaz; Hamood Ur Rahman; T. Tauqeer; Adnan Masood; Tahir Zaidi
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

Microwave resonators are widely used for numerous applications including communication, biomedical and chemical applications, material testing, and food grading. Split-ring resonators in both planar and nonplanar forms are a simple structure which has been in use for several decades. This type of resonator is characterized with low cost, ease of fabrication, moderate quality factor, low external noise interference, high stability, and so forth. Due to these attractive features and ease in handling, nonplanar form of structure has been utilized for material characterization in 1–5 GHz range. Resonant frequency and quality factor are two important parameters for determination of material properties utilizing perturbation theory. Shield made of conducting material is utilized to enclose split-ring resonator which enhances quality factor. This work presents a novel technique to develop shield around a predesigned nonplanar split-ring resonator to yield optimized quality factor. Based on this technique and statistical analysis regression equations have also been formulated for resonant frequency and quality factor which is a major outcome of this work. These equations quantify dependence of output parameters on various factors of shield made of different materials. Such analysis is instrumental in development of devices/designs where improved/optimum result is required.


---

## Body

## 1. Introduction

Split-ring resonator (SRR) [1], loop-gap resonator (LGR) [2–4], or open-loop resonator (OLR) [5] is an important component which is utilized in electronic and nonelectronic applications/utilities. These include devices like oscillators, filters, tuned amplifiers, frequency meters, and processes like permittivity measurement, compositional analysis, quality control, and so forth. SRR is characterized with low phase noise, moderate quality (Q) factor, low cost, and ease of fabrication. Material characterization and compositional analysis especially for polar liquids in microfluidic channels [6, 7] have also been performed with the help of this structure in 1–5 GHz range.Compositional analysis for determining volume fraction of dielectric materials in a compound/composite is often performed in biomedical, chemical, pharmaceutical, and petrochemical applications, food processing/grading, forensic testing, and other applications/areas. Classical methods are time consuming and often require expensive equipment, and above all different methods have to be adopted for various materials. It has been shown that microwave method using split-ring resonator can be used for highly sensitive compositional analysis [6, 7] for various materials. In this technique material under test (MUT) is placed inside SRR in a region of maximum electric field strength. When volume fraction of materials in MUT is changed complex permittivity of composite changes. Variation in complex permittivity of MUT is sensed through changes in resonant frequency and Q factor. Due to placement of MUT in region of maximum electric field very small changes in composite composition have been detected [6]. The larger the change in these parameters for small compositional variation the higher the sensitivity of device and hence the higher the resolution in analysis. Key to enhancing sensitivity and resolution in analysis through this technique is the optimization of Q factor pertaining to SRR structure.SRR is enclosed inside a conducting cylindrical shield/cavity to prevent radiation loss and enhanceQ factor [1, 8]. Resonant frequency, Q factor, and other parameters are affected by shield dimensions and material characteristics like conductivity, dielectric constant, mass density, and dielectric loss [9]. Various models [1–4, 10–12] have been developed to quantize output parameters of SRR structure. Subsequent analyses [13, 14] of these models highlight a number of shield parameters which include material and dimensions which can be utilized to develop a structural design to optimize Q factor. This work focuses on analyzing effects of these parameters of shield on resonant frequency and Q factor of SRR structure. Analysis of these effects was used to design shield around SRR to yield optimized Q factor [15, 16]. Regression equations were also formulated through statistical analysis to determine dependence of resonant frequency and Q factor on shield parameters. Such analysis is instrumental in the development of an optimized SRR structure/device for performing highly sensitive compositional analysis of dielectric materials in a mixture/composite.


## 2. SRR Structure for Analysis

A simplest form of SRR structure with resonator enclosed in conducting cylindrical shield is shown in Figure1 [3, 4]. It is comprised of a metallic cylinder with a longitudinal gap. SRR can have a number of variations including planar forms [5, 17], geometrical shapes [17, 18], multiple rings [18], and multiple gaps [10]. It can be considered as a single-turn inductor connected with a gap capacitor [5] as shown in the figure. SRR has to be coupled with transmission line and needs mechanical support [10]. Inductive, capacitive, or aperture/hole coupling [19] of SRR can be done based on its utilization.Figure 1
Split-ring resonator structure and its cross-sectional view.Earlier research [6, 7, 11, 20] provided guidance in formulation of a base design. Gap was selected to accommodate microfluidic channel [7]. Guidelines have been provided for selecting various shield design parameters [3, 4]. SRR was fabricated using copper [6]. Design parameters of SRR structure are shown in Table 1.Table 1
Parameters of base design.
Design parameters
Dimensions (mm)
Inner radius of shield “R0”
24
Inner radius of resonator “r0”
6
Width of resonator “W”
6
Length of resonator “Z”
6
Gap of resonator “t”
2
Inner height of shield “H”
24
Thickness of shield “T” (in all directions)
4


## 3. Simulation

### 3.1. Effect of Shield Material

Simulations were performed to analyze shield material effect on resonant frequency andQ factor of base design. Shield was designed using aluminum (AL), brass (BR) [6], stainless steel (SS), cast iron (CI), and tin (SN). Some useful properties of these materials are shown in Table 2. Parametric results are presented in Table 3. Developed structure of SRR enclosed in shield is shown in Figure 2.Table 2
Properties of shield material.
Shield material
Relative permeabilityμr
Bulk conductivityσ, Siemens
Mass densityρ, kg/m3
AL
1.000021
38000000
2689
BR
1
15000000
8600
SS
1
1100000
8055
CI
60
1500000
7200
SN
1
8670000
7304Table 3
Simulated parametric results.
Shield material
Resonant frequencyf0 [GHz]
Q factor
AL
2.10280
2406.20
BR
2.10278
2338.35
SS
2.10269
1942.16
CI
2.10198
846.31
SN
2.10277
2283.73Figure 2
HFSS simulation of SRR.Figure3 shows variations in output parameters of SRR with shields of different materials. SRR structures with AL and CI shields have shown maximum and minimum resonant frequency, respectively. A mean value of 2.1026 GHz with 0.000351 GHz as standard deviation was achieved. Maximum variation in resonant frequency for SRR shield made of different material is 0.039 percent. This suggests that resonant frequency is independent of shield material. In this case, resonant frequency is a function of SRR and shield dimensions but is independent of shield material properties. However, Q factor is considerably affected with change in shield material. Maximum value for SRR structure with AL shield was achieved while minimum value with CI shield was achieved. Mean value is 1963 with 650 as standard deviation. Moreover, maximum variation is 64.83 percent which is quite large. This suggests that variation is due to material properties/characteristics mentioned in Table 2 [1, 3, 4].Figure 3
Parametric variations due to shield material.Further statistical analysis was carried out utilizing design of experiment (DOE) method [21–23] with data as given in Table 4. The table has been formed by combining material properties of Table 2 and parametric output data of Table 3. The table forms basis for analyzing effects of material properties on resonant frequency and quality factor. Regression analysis [22, 23] was performed which yielded results for resonant frequency and Q factor as shown in Tables 5 and 6, respectively. Regression equations [22, 23] were obtained for full factorial DOE. Equations show effects of shield material properties on resonant frequency and Q factor as(1)f0=2.103-0.000013×μr-0.000000×σ-0.000000×ρ+0.000000×σ×ρQ=3664.00-21.7100×μr-0.000042×σ-0.21540×ρ+0.000000×σ×ρ.Table 4
Statistical analysis: shield material.
Shield material
Relative permeabilityμr
Bulk conductivityσ, Siemens
Mass densityρ, kg/m3
Resonant frequency (GHz)
Q factor
AL
1.000021
38000000
2689
2.10280
2406.20
BR
1
15000000
8600
2.10278
2338.35
SS
1
1100000
8055
2.10269
1942.16
CI
60
1500000
7200
2.10198
846.31
SN
1
8670000
7304
2.10277
2283.73Table 5
Regression analysis: resonant frequency.
Factors
DF
SS
MS = SS/DF
F-value
P value
Regression
3
0.000000
0.000000
98.22
0.074
μ
r
1
0.000000
0.000000
150.48
0.052
σ
1
0.000000
0.000000
1.97
0.394
ρ
1
0.000000
0.000000
0.56
0.592
Error,E
1
0.000000
0.000000
Total
4
0.000000
DF: degree of freedom, SS: sum of squares, and MS: mean square.Table 6
Regression analysis:Q factor.
Factors
DF
SS
MS = SS/DF
F-value
P value
Regression
3
1657382
552461
18.28
0.170
μ
r
1
631835
631835
20.90
0.137
σ
1
64150
64150
2.12
0.383
ρ
1
20117
20117
0.67
0.564
Error,E
1
30229
30229
Total
4
1687612These equations also verify earlier investigations [2–4]. It can be noted that resonant frequency decreases negligibly with increase in relative permeability of shield material while it remained unaffected by bulk conductivity and mass density of shield material. Moreover, Q factor decreases considerably with increase in relative permeability; however it slightly decreases with increase in mass density and bulk conductivity. It remained unaffected against bulk conductivity and mass density of resonator material. Analysis show that Q factor can be optimized by selecting a shield material for which relative permeability is low enough along with appropriate bulk conductivity and mass density.


### 3.2. Materialwise Effect of Height and Thickness of Shield

For analyzing effects on resonant frequency andQ factor against variation in shield dimensions, five simulations one for each shield material were designed. Inner radius of shield was kept as value given in Table 1. Range of parametric variations introduced is shown in Table 7. Height of shield has seven levels while wall thickness has five levels. Each simulation resulted in thirty-five solutions. Results of these simulations for SRR with AL shield are presented in Figure 4.Table 7
Range of parametric variations.
Parameters
Range of variation
Minimum (mm)
Maximum (mm)
Increment (mm)
Height “H”
24
30
1
Thickness “T”
4
8
1Figure 4
Parametric variations due to AL shield.
(a)
Resonant frequency versus height of shield
(b)
Q factor versus height of shield
(c)
Resonant frequency versus thickness of shield
(d)
Q factor versus thickness of shieldFigure4(a) presents effect on resonant frequency against shield height for various thicknesses pertaining to AL shield whereas Figure 4(b) presents effect on Q factor for the same variations. Figures 4(c) and 4(d) depict parametric variations but plotted against shield thickness for various shield heights. Resonant frequency was highest for a 25 mm high and 5 mm thick wall shield, whereas the lowest value was yielded for shield dimensions of 25 mm high and 7 mm wall thickness. Highest Q factor was observed with 25 mm high and 5 mm thick shield, whereas the lowest value was achieved for 24 mm high and 8 mm thick wall shield.Effects on resonant frequency andQ factor due to variations in height and wall thickness of shield pertaining to other materials were studied in a similar manner. Extreme values of resonant frequency and Q factor along with related shield dimensions pertaining to different materials are summarized in Table 8.Table 8
Extreme values related to shield dimensions.
Shield material
Parametric output
Shield dimensions (mm)
Height
Thickness
AL
f
0 (max) (GHz)
2.2167
25
5
f
0 (min) (GHz)
2.0410
25
6
Q factor (max)
3115.55
25
5
Q factor (min)
2323.86
24
8
BR
f
0 (max) (GHz)
2.2167
25
5
f
0 (min) (GHz)
2.0170
25
7
Q factor (max)
3019.78
25
5
Q factor (min)
2260.69
24
8
SS
f
0 (max) (GHz)
2.2090
24
7
f
0 (min) (GHz)
2.0142
26
4
Q factor (max)
2573.31
30
4
Q factor (min)
1889.18
24
8
CI
f
0 (max) (GHz)
2.2083
24
7
f
0 (min) (GHz)
2.0136
25
7
Q factor (max)
1230.26
30
4
Q factor (min)
836.70
24
8
SN
f
0 (max) (GHz)
2.2022
24
7
f
0 (min) (GHz)
2.0169
27
7
Q factor (max)
2951.80
28
4
Q factor (min)
2209.26
24
8Parametric outputs achieved for statistical analysis for SRR structure with AL shield are given in Table9. Statistical analysis data reveals mean value of 2.1155 GHz for resonant frequency with 0.0531 GHz as standard deviation. Maximum variation in resonant frequency for different shield dimensions is 7.92 percent. Q factor has a mean value of 2784.8 with standard deviation of 223.3 whereas 63.99 percent was maximum variation observed. Regression analysis [22, 23] for resonant frequency and Q factor on its dependencies upon dimensions of AL shield are given in Tables 10 and 11, respectively, whereas main effects’ plot for resonant frequency and Q factor is shown in Figure 5.Table 9
Statistical analysis for SRR structure with AL shield.
Height (mm)
Thickness(mm)
Resonant frequency (GHz)
Q factor
24
4
2.102801
2406.20
24
5
2.162182
2885.54
24
6
2.145050
2814.10
24
7
2.209145
2998.04
24
8
2.155604
2323.86
25
4
2.059838
2467.30
25
5
2.216705
3115.55
25
6
2.041012
2374.01
25
7
2.017052
2377.98
25
8
2.118541
2962.74
26
4
2.174556
2904.15
26
5
2.147571
2783.36
26
6
2.173646
2880.35
26
7
2.159383
2940.05
26
8
2.073591
2853.99
27
4
2.157872
3017.62
27
5
2.146808
2711.32
27
6
2.146426
2951.71
27
7
2.016988
2547.46
27
8
2.069348
2602.48
28
4
2.192393
3089.93
28
5
2.050508
2877.79
28
6
2.050680
2719.64
28
7
2.093319
2552.03
28
8
2.148180
2901.54
29
4
2.154959
3008.97
29
5
2.114135
2850.55
29
6
2.090548
2884.91
29
7
2.084256
3006.12
29
8
2.110622
2913.54
30
4
2.129782
3040.76
30
5
2.124868
2491.20
30
6
2.079355
2829.28
30
7
2.072270
2654.94
30
8
2.051706
2727.87Table 10
Regression analysis: resonant frequency.
Factors
DF
SS
MS = SS/DF
F-value
P value
Regression
2
0.015463
0.007731
3.07
0.060
Height
1
0.006329
0.006329
2.52
0.122
Thickness
1
0.009133
0.009133
3.63
0.066
Error,E
32
0.080482
0.002515
Total
34
0.095944Table 11
Regression analysis:Q factor.
Factors
DF
SS
MS = SS/DF
F-value
P value
Regression
2
139132
69566
1.43
0.254
Height
1
85561
85561
1.76
0.194
Thickness
1
53570
53570
1.10
0.302
Error,E
32
1556319
48635
Total
34
1695450Figure 5
Main effects plot: resonant frequency andQ factor—AL shield.
(a)
Mean ofQ factor and resonant frequency versus height of shield
(b)
Mean ofQ factor and resonant frequency versus thickness of shieldTwo-level factorial regression equations were derived for full factorial DOE [23]. Equations for resonant frequency and Q factor pertaining to SRR structure with AL shield are given as(2)f0=1.780+0.0150×H+0.0862×T-0.00361×H×TQ=799+79.7×H+220×T-9.2×H×T,where resonant frequency is in GHz. Similar data as shown in Table 9 was achieved for analysis of SRRs enclosed in shields of other materials. Regression analysis was carried out similar to the one shown for AL shield in Tables 10 and 11. Two-level factorial regression equations for full factorial DOE were generated [23] for each structure and given below.Regression equations for SRR structure with BR shield are(3)f0=1.956+0.0076×H+0.0660×T-0.00274×H×TQ=1527+45.8×H+112×T-4.5×H×T.Regression equations for SRR structure with SS shield are(4)f0=1.637+0.0182×H+0.1073×T-0.00411×H×TQ=75+82.7×H+160×T-6.07×H×T.Regression equations for SRR structure with CI shield are(5)f0=1.721+0.0149×H+0.0951×T-0.00362×H×TQ=-431+56.4×H+21.4×T-0.91×H×T.Regression equations for SRR structure with SN shield are(6)f0=1.762+0.0156×H+0.0944×T-0.00390×H×TQ=434+87.7×H+243×T-9.9×H×T.From (2)–(6) resonant frequency and Q factors can be predicted for each category of structure with predesigned SRR. It can be observed that resonant frequency was slightly affected due to change in height and thickness of shield whereas Q factor was considerably affected due to dimensional changes in shield as already shown in Table 8.


### 3.3. Materialwise Effect of Height, Radius, and Thickness of Shield

Inner radius of shield was kept constant in Section3.2 to analyze effects separately. Another set of analysis was designed for SRR structures with shield of different materials to analyze effects on resonant frequency and Q factor against additional dimension, that is, shield radius. Each analysis was designed to have three hundred fifteen solutions due to additional nine levels of shield radius. Range of parametric variations introduced in this analysis is given in Table 12.Table 12
Range of parametric variations.
Parameters
Range of variation
Minimum (mm)
Maximum (mm)
Increment (mm)
Height
24
30
1
Radius
22
30
1
Thickness
4
8
1Results of SRR structure with shield of different materials were obtained in a manner described in previous section. Data pertaining to these simulations were used for statistical analysis. Extreme values of parametric output with related shield dimensions are presented in Table13.Table 13
Extreme values related to shield dimensions.
Shield material
Parametric output
Shield dimensions (mm)
Height
Radius
Thickness
AL
f
0 (max) (GHz)
2.2184
24
23
4
f
0 (min) (GHz)
1.9322
29
30
5
Q factor (max)
3226.05
28
27
4
Q factor (min)
2129.90
29
30
4
BR
f
0 (max) (GHz)
2.2184
24
23
4
f
0 (min) (GHz)
1.9322
29
30
5
Q factor (max)
3151.12
28
27
4
Q factor (min)
2100.71
29
30
4
SS
f
0 (max) (GHz)
2.2183
24
23
4
f
0 (min) (GHz)
1.9322
29
30
5
Q factor (max)
2697.54
28
27
4
Q factor (min)
1799.00
24
29
5
CI
f
0 (max) (GHz)
2.2176
24
23
4
f
0 (min) (GHz)
1.7938
30
29
4
Q factor (max)
1388.51
30
29
5
Q factor (min)
830.21
24
23
5
SN
f
0 (max) (GHz)
2.2184
24
23
4
f
0 (min) (GHz)
1.9322
29
30
5
Q factor (max)
3090.23
28
27
4
Q factor (min)
2066.21
27
22
7Maximum resonant frequency for all SRR structures was found almost equal. Interestingly, this value was obtained for the same shield dimensions. This observation also showed that resonant frequency remains unaffected by shield material. Similar observation was made for minimum value of resonant frequency and maximum value ofQ factor except for structure with CI shield. Minimum Q factor was obtained for the same shield dimensions made of AL and BR. Highest Q factor was noted for structure with AL shield. An improvement of around 26 percent in Q factor was noted from original design along with slight increase in resonant frequency. Regression analysis for resonant frequency and Q factor was performed for full factorial DOE to analyze dependence of these factors on shield dimensions as shown in Tables 14 and 15, respectively, whereas main effects’ plot for resonant frequency and Q factor is shown in Figure 6.Table 14
Regression analysis: resonant frequency.
Factors
DF
SS
MS = SS/DF
F-value
P value
Regression
3
0.17150
0.057167
19.21
0.000
Height
1
0.13821
0.138208
46.45
0.000
Radius
1
0.00113
0.001125
0.38
0.539
Thickness
1
0.03217
0.032166
10.81
0.001
Error,E
311
0.92527
0.002975
Total
314
1.09677Table 15
Regression analysis:Q factor.
Factors
DF
SS
MS = SS/DF
F-value
P value
Regression
3
8852
2950.5
0.06
0.981
Height
1
6674
6674.5
0.14
0.713
Radius
1
562
561.8
0.01
0.915
Thickness
1
1615
1615.3
0.03
0.857
Error,E
311
15358691
49384.9
Total
314
15367542Figure 6
Main effects plot: resonant frequency andQ factor—AL shield.
(a)
Mean ofQ factor and resonant frequency versus height of shield
(b)
Mean ofQ factor and resonant frequency versus thickness of shield
(c)
Mean ofQ factor and resonant frequency versus radius of shieldTwo-level factorial regression equations were derived for full factorial DOE [23]. Equations obtained for resonant frequency and Q factor of AL shield are given as(7)f0=-0.22+0.0996×H+0.0909×R0+0.469×T-0.00383×H×R0-0.0195×H×T-0.0165×R0×T+0.000683×H×R0×TQ=-7677+421×H+333×R0+1866×T-13.6×H×R0-74.8×H×T-60.8×R0×T+2.47×H×R0×T.Regression analysis for SRRs enclosed in shields made of other materials was carried out. Two-level factorial regression equations for full factorial DOE were generated for each structure and are given below.Regression equations for SRR structure with BR shield are(8)f0=0.25+0.0829×H+0.0705×R0+0.403×T-0.00309×H×R0-0.0171×H×T-0.0134×R0×T+0.000571×H×R0×TQ=-6215+363×H+265×R0+1626×T-11.1×H×R0-65.7×H×T-50.6×R0×T+2.08×H×R0×T.Regression equations for SRR structure with SS shield are(9)f0=0.85+0.0597×H+0.0460×R0+0.341×T-0.00217×H×R0-0.0146×H×T-0.0109×R0×T+0.000473×H×R0×TQ=-3448+227×H+134×R0+1086×T-5.5×H×R0-43.9×H×T-32.8×R0×T+1.36×H×R0×T.Regression equations for SRR structure with CI shield are(10)f0=0.62+0.0767×H+0.0655×R0+0.364×T-0.00325×H×R0-0.0167×H×T-0.0135×R0×T+0.000619×H×R0×TQ=-18+34.3×H-19.4×R0+283×T+0.99×H×R0-12.0×H×T-9.5×R0×T+0.410×H×R0×T.Regression equations for SRR structure with SN shield are(11)f0=0.34+0.0810×H+0.0655×R0+0.394×T-0.00325×H×R0-0.0170×H×T-0.0131×R0×T+0.000567×H×R0×TQ=-6613+377×H-276×R0+1613×T-11.5×H×R0-65.7×H×T-50.4×R0×T+2.10×H×R0×T.Equations (7)–(11) present dependence of output parameters on shield dimensions. Values of resonant frequency and Q factor can be predicted for various structures with predesigned SRR as shown in Table 1.


## 3.1. Effect of Shield Material

Simulations were performed to analyze shield material effect on resonant frequency andQ factor of base design. Shield was designed using aluminum (AL), brass (BR) [6], stainless steel (SS), cast iron (CI), and tin (SN). Some useful properties of these materials are shown in Table 2. Parametric results are presented in Table 3. Developed structure of SRR enclosed in shield is shown in Figure 2.Table 2
Properties of shield material.
Shield material
Relative permeabilityμr
Bulk conductivityσ, Siemens
Mass densityρ, kg/m3
AL
1.000021
38000000
2689
BR
1
15000000
8600
SS
1
1100000
8055
CI
60
1500000
7200
SN
1
8670000
7304Table 3
Simulated parametric results.
Shield material
Resonant frequencyf0 [GHz]
Q factor
AL
2.10280
2406.20
BR
2.10278
2338.35
SS
2.10269
1942.16
CI
2.10198
846.31
SN
2.10277
2283.73Figure 2
HFSS simulation of SRR.Figure3 shows variations in output parameters of SRR with shields of different materials. SRR structures with AL and CI shields have shown maximum and minimum resonant frequency, respectively. A mean value of 2.1026 GHz with 0.000351 GHz as standard deviation was achieved. Maximum variation in resonant frequency for SRR shield made of different material is 0.039 percent. This suggests that resonant frequency is independent of shield material. In this case, resonant frequency is a function of SRR and shield dimensions but is independent of shield material properties. However, Q factor is considerably affected with change in shield material. Maximum value for SRR structure with AL shield was achieved while minimum value with CI shield was achieved. Mean value is 1963 with 650 as standard deviation. Moreover, maximum variation is 64.83 percent which is quite large. This suggests that variation is due to material properties/characteristics mentioned in Table 2 [1, 3, 4].Figure 3
Parametric variations due to shield material.Further statistical analysis was carried out utilizing design of experiment (DOE) method [21–23] with data as given in Table 4. The table has been formed by combining material properties of Table 2 and parametric output data of Table 3. The table forms basis for analyzing effects of material properties on resonant frequency and quality factor. Regression analysis [22, 23] was performed which yielded results for resonant frequency and Q factor as shown in Tables 5 and 6, respectively. Regression equations [22, 23] were obtained for full factorial DOE. Equations show effects of shield material properties on resonant frequency and Q factor as(1)f0=2.103-0.000013×μr-0.000000×σ-0.000000×ρ+0.000000×σ×ρQ=3664.00-21.7100×μr-0.000042×σ-0.21540×ρ+0.000000×σ×ρ.Table 4
Statistical analysis: shield material.
Shield material
Relative permeabilityμr
Bulk conductivityσ, Siemens
Mass densityρ, kg/m3
Resonant frequency (GHz)
Q factor
AL
1.000021
38000000
2689
2.10280
2406.20
BR
1
15000000
8600
2.10278
2338.35
SS
1
1100000
8055
2.10269
1942.16
CI
60
1500000
7200
2.10198
846.31
SN
1
8670000
7304
2.10277
2283.73Table 5
Regression analysis: resonant frequency.
Factors
DF
SS
MS = SS/DF
F-value
P value
Regression
3
0.000000
0.000000
98.22
0.074
μ
r
1
0.000000
0.000000
150.48
0.052
σ
1
0.000000
0.000000
1.97
0.394
ρ
1
0.000000
0.000000
0.56
0.592
Error,E
1
0.000000
0.000000
Total
4
0.000000
DF: degree of freedom, SS: sum of squares, and MS: mean square.Table 6
Regression analysis:Q factor.
Factors
DF
SS
MS = SS/DF
F-value
P value
Regression
3
1657382
552461
18.28
0.170
μ
r
1
631835
631835
20.90
0.137
σ
1
64150
64150
2.12
0.383
ρ
1
20117
20117
0.67
0.564
Error,E
1
30229
30229
Total
4
1687612These equations also verify earlier investigations [2–4]. It can be noted that resonant frequency decreases negligibly with increase in relative permeability of shield material while it remained unaffected by bulk conductivity and mass density of shield material. Moreover, Q factor decreases considerably with increase in relative permeability; however it slightly decreases with increase in mass density and bulk conductivity. It remained unaffected against bulk conductivity and mass density of resonator material. Analysis show that Q factor can be optimized by selecting a shield material for which relative permeability is low enough along with appropriate bulk conductivity and mass density.


## 3.2. Materialwise Effect of Height and Thickness of Shield

For analyzing effects on resonant frequency andQ factor against variation in shield dimensions, five simulations one for each shield material were designed. Inner radius of shield was kept as value given in Table 1. Range of parametric variations introduced is shown in Table 7. Height of shield has seven levels while wall thickness has five levels. Each simulation resulted in thirty-five solutions. Results of these simulations for SRR with AL shield are presented in Figure 4.Table 7
Range of parametric variations.
Parameters
Range of variation
Minimum (mm)
Maximum (mm)
Increment (mm)
Height “H”
24
30
1
Thickness “T”
4
8
1Figure 4
Parametric variations due to AL shield.
(a)
Resonant frequency versus height of shield
(b)
Q factor versus height of shield
(c)
Resonant frequency versus thickness of shield
(d)
Q factor versus thickness of shieldFigure4(a) presents effect on resonant frequency against shield height for various thicknesses pertaining to AL shield whereas Figure 4(b) presents effect on Q factor for the same variations. Figures 4(c) and 4(d) depict parametric variations but plotted against shield thickness for various shield heights. Resonant frequency was highest for a 25 mm high and 5 mm thick wall shield, whereas the lowest value was yielded for shield dimensions of 25 mm high and 7 mm wall thickness. Highest Q factor was observed with 25 mm high and 5 mm thick shield, whereas the lowest value was achieved for 24 mm high and 8 mm thick wall shield.Effects on resonant frequency andQ factor due to variations in height and wall thickness of shield pertaining to other materials were studied in a similar manner. Extreme values of resonant frequency and Q factor along with related shield dimensions pertaining to different materials are summarized in Table 8.Table 8
Extreme values related to shield dimensions.
Shield material
Parametric output
Shield dimensions (mm)
Height
Thickness
AL
f
0 (max) (GHz)
2.2167
25
5
f
0 (min) (GHz)
2.0410
25
6
Q factor (max)
3115.55
25
5
Q factor (min)
2323.86
24
8
BR
f
0 (max) (GHz)
2.2167
25
5
f
0 (min) (GHz)
2.0170
25
7
Q factor (max)
3019.78
25
5
Q factor (min)
2260.69
24
8
SS
f
0 (max) (GHz)
2.2090
24
7
f
0 (min) (GHz)
2.0142
26
4
Q factor (max)
2573.31
30
4
Q factor (min)
1889.18
24
8
CI
f
0 (max) (GHz)
2.2083
24
7
f
0 (min) (GHz)
2.0136
25
7
Q factor (max)
1230.26
30
4
Q factor (min)
836.70
24
8
SN
f
0 (max) (GHz)
2.2022
24
7
f
0 (min) (GHz)
2.0169
27
7
Q factor (max)
2951.80
28
4
Q factor (min)
2209.26
24
8Parametric outputs achieved for statistical analysis for SRR structure with AL shield are given in Table9. Statistical analysis data reveals mean value of 2.1155 GHz for resonant frequency with 0.0531 GHz as standard deviation. Maximum variation in resonant frequency for different shield dimensions is 7.92 percent. Q factor has a mean value of 2784.8 with standard deviation of 223.3 whereas 63.99 percent was maximum variation observed. Regression analysis [22, 23] for resonant frequency and Q factor on its dependencies upon dimensions of AL shield are given in Tables 10 and 11, respectively, whereas main effects’ plot for resonant frequency and Q factor is shown in Figure 5.Table 9
Statistical analysis for SRR structure with AL shield.
Height (mm)
Thickness(mm)
Resonant frequency (GHz)
Q factor
24
4
2.102801
2406.20
24
5
2.162182
2885.54
24
6
2.145050
2814.10
24
7
2.209145
2998.04
24
8
2.155604
2323.86
25
4
2.059838
2467.30
25
5
2.216705
3115.55
25
6
2.041012
2374.01
25
7
2.017052
2377.98
25
8
2.118541
2962.74
26
4
2.174556
2904.15
26
5
2.147571
2783.36
26
6
2.173646
2880.35
26
7
2.159383
2940.05
26
8
2.073591
2853.99
27
4
2.157872
3017.62
27
5
2.146808
2711.32
27
6
2.146426
2951.71
27
7
2.016988
2547.46
27
8
2.069348
2602.48
28
4
2.192393
3089.93
28
5
2.050508
2877.79
28
6
2.050680
2719.64
28
7
2.093319
2552.03
28
8
2.148180
2901.54
29
4
2.154959
3008.97
29
5
2.114135
2850.55
29
6
2.090548
2884.91
29
7
2.084256
3006.12
29
8
2.110622
2913.54
30
4
2.129782
3040.76
30
5
2.124868
2491.20
30
6
2.079355
2829.28
30
7
2.072270
2654.94
30
8
2.051706
2727.87Table 10
Regression analysis: resonant frequency.
Factors
DF
SS
MS = SS/DF
F-value
P value
Regression
2
0.015463
0.007731
3.07
0.060
Height
1
0.006329
0.006329
2.52
0.122
Thickness
1
0.009133
0.009133
3.63
0.066
Error,E
32
0.080482
0.002515
Total
34
0.095944Table 11
Regression analysis:Q factor.
Factors
DF
SS
MS = SS/DF
F-value
P value
Regression
2
139132
69566
1.43
0.254
Height
1
85561
85561
1.76
0.194
Thickness
1
53570
53570
1.10
0.302
Error,E
32
1556319
48635
Total
34
1695450Figure 5
Main effects plot: resonant frequency andQ factor—AL shield.
(a)
Mean ofQ factor and resonant frequency versus height of shield
(b)
Mean ofQ factor and resonant frequency versus thickness of shieldTwo-level factorial regression equations were derived for full factorial DOE [23]. Equations for resonant frequency and Q factor pertaining to SRR structure with AL shield are given as(2)f0=1.780+0.0150×H+0.0862×T-0.00361×H×TQ=799+79.7×H+220×T-9.2×H×T,where resonant frequency is in GHz. Similar data as shown in Table 9 was achieved for analysis of SRRs enclosed in shields of other materials. Regression analysis was carried out similar to the one shown for AL shield in Tables 10 and 11. Two-level factorial regression equations for full factorial DOE were generated [23] for each structure and given below.Regression equations for SRR structure with BR shield are(3)f0=1.956+0.0076×H+0.0660×T-0.00274×H×TQ=1527+45.8×H+112×T-4.5×H×T.Regression equations for SRR structure with SS shield are(4)f0=1.637+0.0182×H+0.1073×T-0.00411×H×TQ=75+82.7×H+160×T-6.07×H×T.Regression equations for SRR structure with CI shield are(5)f0=1.721+0.0149×H+0.0951×T-0.00362×H×TQ=-431+56.4×H+21.4×T-0.91×H×T.Regression equations for SRR structure with SN shield are(6)f0=1.762+0.0156×H+0.0944×T-0.00390×H×TQ=434+87.7×H+243×T-9.9×H×T.From (2)–(6) resonant frequency and Q factors can be predicted for each category of structure with predesigned SRR. It can be observed that resonant frequency was slightly affected due to change in height and thickness of shield whereas Q factor was considerably affected due to dimensional changes in shield as already shown in Table 8.


## 3.3. Materialwise Effect of Height, Radius, and Thickness of Shield

Inner radius of shield was kept constant in Section3.2 to analyze effects separately. Another set of analysis was designed for SRR structures with shield of different materials to analyze effects on resonant frequency and Q factor against additional dimension, that is, shield radius. Each analysis was designed to have three hundred fifteen solutions due to additional nine levels of shield radius. Range of parametric variations introduced in this analysis is given in Table 12.Table 12
Range of parametric variations.
Parameters
Range of variation
Minimum (mm)
Maximum (mm)
Increment (mm)
Height
24
30
1
Radius
22
30
1
Thickness
4
8
1Results of SRR structure with shield of different materials were obtained in a manner described in previous section. Data pertaining to these simulations were used for statistical analysis. Extreme values of parametric output with related shield dimensions are presented in Table13.Table 13
Extreme values related to shield dimensions.
Shield material
Parametric output
Shield dimensions (mm)
Height
Radius
Thickness
AL
f
0 (max) (GHz)
2.2184
24
23
4
f
0 (min) (GHz)
1.9322
29
30
5
Q factor (max)
3226.05
28
27
4
Q factor (min)
2129.90
29
30
4
BR
f
0 (max) (GHz)
2.2184
24
23
4
f
0 (min) (GHz)
1.9322
29
30
5
Q factor (max)
3151.12
28
27
4
Q factor (min)
2100.71
29
30
4
SS
f
0 (max) (GHz)
2.2183
24
23
4
f
0 (min) (GHz)
1.9322
29
30
5
Q factor (max)
2697.54
28
27
4
Q factor (min)
1799.00
24
29
5
CI
f
0 (max) (GHz)
2.2176
24
23
4
f
0 (min) (GHz)
1.7938
30
29
4
Q factor (max)
1388.51
30
29
5
Q factor (min)
830.21
24
23
5
SN
f
0 (max) (GHz)
2.2184
24
23
4
f
0 (min) (GHz)
1.9322
29
30
5
Q factor (max)
3090.23
28
27
4
Q factor (min)
2066.21
27
22
7Maximum resonant frequency for all SRR structures was found almost equal. Interestingly, this value was obtained for the same shield dimensions. This observation also showed that resonant frequency remains unaffected by shield material. Similar observation was made for minimum value of resonant frequency and maximum value ofQ factor except for structure with CI shield. Minimum Q factor was obtained for the same shield dimensions made of AL and BR. Highest Q factor was noted for structure with AL shield. An improvement of around 26 percent in Q factor was noted from original design along with slight increase in resonant frequency. Regression analysis for resonant frequency and Q factor was performed for full factorial DOE to analyze dependence of these factors on shield dimensions as shown in Tables 14 and 15, respectively, whereas main effects’ plot for resonant frequency and Q factor is shown in Figure 6.Table 14
Regression analysis: resonant frequency.
Factors
DF
SS
MS = SS/DF
F-value
P value
Regression
3
0.17150
0.057167
19.21
0.000
Height
1
0.13821
0.138208
46.45
0.000
Radius
1
0.00113
0.001125
0.38
0.539
Thickness
1
0.03217
0.032166
10.81
0.001
Error,E
311
0.92527
0.002975
Total
314
1.09677Table 15
Regression analysis:Q factor.
Factors
DF
SS
MS = SS/DF
F-value
P value
Regression
3
8852
2950.5
0.06
0.981
Height
1
6674
6674.5
0.14
0.713
Radius
1
562
561.8
0.01
0.915
Thickness
1
1615
1615.3
0.03
0.857
Error,E
311
15358691
49384.9
Total
314
15367542Figure 6
Main effects plot: resonant frequency andQ factor—AL shield.
(a)
Mean ofQ factor and resonant frequency versus height of shield
(b)
Mean ofQ factor and resonant frequency versus thickness of shield
(c)
Mean ofQ factor and resonant frequency versus radius of shieldTwo-level factorial regression equations were derived for full factorial DOE [23]. Equations obtained for resonant frequency and Q factor of AL shield are given as(7)f0=-0.22+0.0996×H+0.0909×R0+0.469×T-0.00383×H×R0-0.0195×H×T-0.0165×R0×T+0.000683×H×R0×TQ=-7677+421×H+333×R0+1866×T-13.6×H×R0-74.8×H×T-60.8×R0×T+2.47×H×R0×T.Regression analysis for SRRs enclosed in shields made of other materials was carried out. Two-level factorial regression equations for full factorial DOE were generated for each structure and are given below.Regression equations for SRR structure with BR shield are(8)f0=0.25+0.0829×H+0.0705×R0+0.403×T-0.00309×H×R0-0.0171×H×T-0.0134×R0×T+0.000571×H×R0×TQ=-6215+363×H+265×R0+1626×T-11.1×H×R0-65.7×H×T-50.6×R0×T+2.08×H×R0×T.Regression equations for SRR structure with SS shield are(9)f0=0.85+0.0597×H+0.0460×R0+0.341×T-0.00217×H×R0-0.0146×H×T-0.0109×R0×T+0.000473×H×R0×TQ=-3448+227×H+134×R0+1086×T-5.5×H×R0-43.9×H×T-32.8×R0×T+1.36×H×R0×T.Regression equations for SRR structure with CI shield are(10)f0=0.62+0.0767×H+0.0655×R0+0.364×T-0.00325×H×R0-0.0167×H×T-0.0135×R0×T+0.000619×H×R0×TQ=-18+34.3×H-19.4×R0+283×T+0.99×H×R0-12.0×H×T-9.5×R0×T+0.410×H×R0×T.Regression equations for SRR structure with SN shield are(11)f0=0.34+0.0810×H+0.0655×R0+0.394×T-0.00325×H×R0-0.0170×H×T-0.0131×R0×T+0.000567×H×R0×TQ=-6613+377×H-276×R0+1613×T-11.5×H×R0-65.7×H×T-50.4×R0×T+2.10×H×R0×T.Equations (7)–(11) present dependence of output parameters on shield dimensions. Values of resonant frequency and Q factor can be predicted for various structures with predesigned SRR as shown in Table 1.


## 4. Analysis

Effects of material properties of shield on output parameters of SRR structure were analyzed. It was observed that resonant frequency remained almost unaffected when shield material was changed. Maximum variation recorded in resonant frequency was less than 1 percent which shows independence of this parameter of shield material and dependence entirely upon dimensions of SRR structure. However,Q factor was found considerably affected around 65 percent with change in shield material. Regression analysis showed heavy dependence of this factor on material properties. Highest value of Q factor for SRR structure was found with AL shield, whereas it exhibited lowest value for structure with SN shield. Analysis of Q factor pertaining to SRR structures of various shield materials along with study of regression equations for the same parameter revealed that increase in relative permeability had adverse effect on this parameter although mass density and bulk conductivity have similar but less significant effects.Dependence of output parameters on variation in height and thickness of shield made of different materials was also analyzed. Inner radius of shield was kept constant to have systematic insight into output variations related to shield dimensions. Relatively larger variation (around 10 percent) was noted in resonant frequency for various geometries of SRR structure. This was expected as resonant frequency depends on shield dimensions as well. ForQ factor, it increased to around 70 percent. It was also observed that higher values of Q factor were obtained for shield of less thickness while lower values were noted for structure with thicker shield. Regression equations showed heavy reliance of Q factor on shield dimensions, whereas resonant frequency was found less affected due to dimensional variations. An increase of around 23 percent in Q factor was noted against previous analysis.Finally, inner radius of shield was also varied along with height and thickness of shield. For resonant frequency maximum variation was around 20 percent, whereas around 75 percent variation was observed inQ factor for various geometries. It was noted that in each case maximum value of both resonant frequency and Q factor was achieved for the same shield dimensions. However, lowest values were obtained for different shield dimensions. Regression equations showed heavy reliance of Q factor on shield dimensions, whereas resonant frequency was found less effected by dimensional variations. An overall increase of around 26 percent in Q factor was noted against first analysis.In all cases, SRR structure with AL shield provided highest value forQ factor as compared to its counterparts. Structures with BR, SS, and SN shields produced comparable results. However, structure with SN shield produced poor results. Main reason lies in relative permeability of material. Higher permeability material stores higher amount of energy, thereby absorbing energy from surrounding. As a result the amount of available energy around SRR reduces resulting in low Q factor. Analysis suggests that use of material for shield with high permeability is not recommended.Analysis presented in this work was performed with a view to develop an optimized SRR device/structure. Optimized device can sense small variation in composition of MUT. This information is gathered in the form of measured resonant frequency andQ factor. Sensing very small changes is vital in high resolution compositional analysis of dielectric materials in mixture/composite.


## 5. Conclusion

Designing of appropriate shield around resonator ensures stability and enhancement in output parameters of SRR structure. This work presents an analysis of factors affecting these parameters. Dependence of these parameters on shield parameters was shown through regression equations.While analyzing dependence over properties of shield material, it was observed that resonant frequency remained independent of shield material. Maximum variation in this parameter for SRR shield made of different material was 0.039 percent. However,Q factor was considerably affected with change in shield material. Variation up to 64.83 percent was observed for shields of different materials. Highest Q factor was noted for AL shield while it was lowest for CI shield. Regression equations provided dependence of output parameters over shield material properties.Study for height and thickness of shield showed that resonant frequency varied least for AL shields of different dimensions, that is, 0.1757 GHz, whereas it varied maximum for BR shields, that is, 0.1997 GHz. Maximum and minimum variation inQ factor was noted as 791.69 and 393.56 for AL and CI shields of different dimensions, respectively. Highest Q factor, that is, 3115.55, was observed for AL shield while the lowest, that is, 836.7, was noted for CI shield. Regression equations provided quantified dependence of output parameters over shield dimensions.When shield radius was included in study, it was observed that minimum variation in resonant frequency, that is, 0.2862 GHz, was noted for AL, BR, and SN shields of different dimensions. SS shield showed 0.2861 GHz variation. While maximum variation, that is, 0.4238 GHz, was noted for CI shield. Maximum variation inQ factor, that is, 1096.15, was noted for AL shield of different dimensions while lowest variation, that is, 558.3, was noted for CI shields. Highest Q factor, that is, 3226.05, was observed for AL shield while the lowest, that is, 830.21, was noted for CI shield. Regression equations quantified dependence in this case also.It was also shown that shield enclosing a predesigned SRR with appropriate dimensions can result in optimized output parameters. Moreover, SRR structure can be designed for desired value ofQ factor with corresponding resonant frequency with the help of data obtained for analysis.Optimized SRR device/structure could be used to improve performance of devices where such structure is used. For instance high selectivity can be achieved in filters and tuned amplifiers. Regression equations developed in this work provide an easy and effective tool to judge performance of device.



---

*Source: 4754192-2016-01-05.xml*