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An equivalent circuit model for on-chip inductors with gradual changed structure.

1. Introduction

With the rapid development of wireless technology, RF ICs have played a vital role in wireless circuits. On-chip spiral inductors are widely used in RF IC design as system-on-chip solution, such as filters, low-noise amplifiers, and voltage-controlled oscillators. To increase the Q factor, inductors with changed metal width or space structure have been presented [1, 2]. In [1], a single gradually changed structure with fixed space but gradually reduced metal width from outside to inside has been proposed. The Q factor of a 20-nH inductor with this structure is up to 60% better than the result of a single strip-width inductor at 3.5GHz. In [2], the sum of metal width and space of inductor's each coil is fixed while the ratio of the metal width to space is gradually reduced from outside to inside. With this design, the Q factor of a 7-nH inductor on high-resistivity silicon is 23.5% higher than the conventional inductor with fixed metal width and space at 2.1GHz. All of these show that the gradually changed structure inductor is attracting for RF ICs.

Various models of spiral inductors on silicon substrate have been reported in resent years [3-4]. The model in [3] can well modify the eddy-current loss in the silicon substrate, while the T-model in [4] has been proposed to accurately simulate the broadband characteristics of spiral inductors. As mentioned above, gradually changed inductors can gain good performance, but the models of these inductors have not been reported. Therefore, a lumped element model applied to gradually changed inductor is necessary.

Comprehensively considering the achievements of skin and proximity effects and substrate coupling effect, a model of on-chip spiral inductor with gradually changed structure has been discussed. The model in this paper shows the performance of the inductor with gradually changed structure by taking ohmic losses and magnetic field distribution into account. The experimental results show that the model is both suitable for the fixed inductor and the gradually changed inductor. So it's very useful for the on-chip inductor simulation and optimization design.

2. Proposed Inductor Model

In [2], the paper presents a novel gradually changed structure. The sum of the metal width and space is fixed while the ratio of the metal width to space is gradually reduced from outside to inside. It is shown in Figure 1.

[FIGURE 1 OMITTED]

The Structure parameters of the inductor include number of turns (N), inner opening diameter ([D.sub.in]), outer opening diameter ([D.sub.out]), metal line width of the nth turn ([W.sub.n]), conductor inter-turn space ([S.sub.n]) and two metal layers (M1 and M2).

According to skin and proximity effects, current does not flow uniformly in wiring and resistance increases with increasing frequency. Inductance decreases when the current flowing in wiring becomes less uniform with increasing frequency [5, 6]. So the addition of a combination of resistance Ro and inductor Lo in parallel has been added in the model to simulate the increase in resistance and the decrease in inductance due to the skin and proximity effects [7].

It has been reported that in order to represent the lateral substrate coupling, [R.sub.sub] and [C.sub.sub] are introduced in the inductor model [8]. A parallel combination of [R.sub.sub] and [C.sub.sub] is placed under the silicon dioxide layer, similar to the equivalent-circuit model for substrate coupling [9] and on-chip interconnects [10].

Comprehensively covering the research achievements in inductor models such as the skin and proximity effect and substrate coupling effect, the model for gradually changed spiral inductor is presented in Figure 2. According to this model, the series resistance [R.sub.s] and the series inductance [L.sub.s] of equivalent circuit can be extracted as (2)

[R.sub.s] = [R.sub.1] + [R.sub.0] [[omega].sup.2][L.sup.2.sub.0]/[R.sup.2.sub.0] + [[omega].sup.2][L.sup.2.sub.0] (1)

[L.sub.s] = [L.sub.1] + [L.sub.0] [R.sup.2.sub.0]/[R.sup.2.sub.0] + [[omega].sup.2][L.sup.2.sub.0]

When [omega]) [right arrow] 0.[R.sub.DC] = [R.sub.l], [L.sub.DC] = [L.sub.1] + [L.sub.0]; When [omega] [right arrow] [infinity]. [R.sub.HF] = [R.sub.1] + [R.sub.0], [L.sub.RF] = [L.sub.1].

[R.sub.DC] and [L.sub.DC] are series resistance and inductance in low frequency. [R.sub.HF] and [L.sub.HF] are series resistance and inductance in high frequency. It means that the proposed model can represent the frequency characteristics of spiral inductors. The series resistance [R.sub.s] will increase and the series inductance [L.sub.s] will decrease with increasing frequency.

[FIGURE 2 OMITTED]

Compared with fixed structure inductor with the same outer opening diameter and number of turns, gradually changed structure inductor has wider metal strip in outer turns. Ohmic losses are inversely proportional to the metal strip width and a wide metal strip width is expected to decrease the ohmic losses. So [R.sub.s] of gradually changed structure inductor is smaller than that of fixed structure inductor. In the other case, magnetically induced losses mainly influence the center of the inductor. Increasing the distance between inner turns can decrease magnetically induced losses. Due to using wider distance in the inner turns, [L.sub.s] of the gradually changed structure inductor is smaller than that of the fixed structure inductor. Hence, besides validating the model's availability, we also pay much attention on the validity of [R.sub.s] and [L.sub.s] in inductors with different geometrical configuration.

3. Model validation and Discussion

To verify the validity and accuracy of the model above, square spiral inductors with various geometrical configuration have been designed first by HFSS and then fabricated on high-resistivity silicon substrate ([rho] = [10.sup.3] [ohm] x cm) in the following processes. Firstly, Ti/Au metals approximately 0.6[micro]m are electroplated and patterned to form the underpass of the inductors. Secondly, a PECVD Si[O.sub.2] layer about 0.8[micro]m is deposited for isolation. Subsequently, 1.5[micro]m thick Ti/Au layer is evaporated and electroplated for patterning spiral coil of inductor. The physical dimensions of the inductors are summarized in Table 1. The parameters of the inductor include number of turns (N), outer opening diameter ([D.sub.out]), metal width (W), conductor inter-turn space (S). Q1, Q4, Q7 are the conventional inductor with fixed metal width and space. Q2, Q5, Q8 are the inductors with fixed space and gradually reduced metal width from outside to inside, we call them single gradually changed inductors. And Q3, Q6, Q9 are gradually changed structure inductors with the sum of metal width and space of each coil fixed while the ratio of the metal width to space is gradually reduced from outside to inside.

Measurements are carried out at frequencies ranging from 100MHz to 10GHz by E8363B network analyzer and Cascade on-wafer probe. Accurate measurements for the inductors alone can be obtained, by measuring S parameters of both the device under test (DUT), probe pads and ground planes (PAD), and subtracting the effects of PAD from DUT. After inductors have been measured, the equivalent circuit parameters for the proposed model can be extracted.

Each parameter in equivalent circuit model should be extracted accurately by using gradient algorithm in ADS,. [R.sub.si] and [C.sub.si] in two branches should be in the same value. Considering the asymmetry of the two ports, [C.sub.ox1] and [C.sub.ox2] should be different. The Q factor, the equivalent series resistance [R.sub.s] and the equivalent series inductance [L.sub.s] are added to the aim function. For the spiral inductor always works under the self-resonance frequency, the equivalent circuit model should meet the practical requirement accurately under the self-resonance frequency. According to the data measured, all parameters in Figure 2 model have been extracted and shown in Table 2.

It is shown that the model extracted parameters in Table 2 can be well fitted the inductor parameters and the changes of the extracted parameters are consistent with the change of geometrical configuration. Parts of validated results are listed below.

Figure 3 shows the model validation results for Inductor Q7 and Q9. Figure 3a gives the Q factor comparison. As being seen, Q factor of the proposed model shows the good agreement with not only the conventional structure Q7 but also the gradually changed structure Q9. Q factor of Q9 is higher than that of Q7, which is also identical with the design. Figure 3b and 3c show the equivalent series resistance [R.sub.s] and series inductance [L.sub.s] curves for Q7 and Q9. In Figure 3b and 3c, [R.sub.s] and [L.sub.s] of Q9 are smaller than the [R.sub.s] and [L.sub.s] of Q7, which consists with the analysis above. Figure 4 shows the Q factor comparison of experimental results and model simulations for Inductor Q6 and Q9, Q6 with 3.5 turns and Q9 with 4.5 turns. It is found that the proposed model shows good agreement with the gradually changed structure of different numbers of turns.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

4. Conclusion

A simple wide-band inductor has been presented considering the skin, proximity effects and substrate coupling. A combination of resistance Ro and inductor Lo in parallel in the proposed model model the skin and proximity effects with increasing frequency. A parallel combination of [R.sub.sub] and [C.sub.sub] captures the substrate coupling. The experimental data indicates that the model can be used in both conventional and gradual changed inductor and the model extracted parameters are consistent with the physical effects in different geometrical configuration. These all show the model's value in inductor design.

5. Acknowledgment

This work is supported by Natural Science Foundation of China (No. 60676047, 60606010), Foundation of Shanghai Science&Technology Committee (04QMX1419) and Shanghai-Applied Materials Research and Development Fund (No.0522).

References

[1] Lopez, Villegas, J. M, Samitier, J., Cane, C. (2000). Improvement of the quality factor of RF integrated inductors by layout optimization. IEEE Transactions on Microwave Theory and Techniques, 48(1) 76-83.

[2] Liu, J., Shi, Y. L., Wen, X. Z., Chen, D. W. (2008). On-chip spiral inductor with novel gradually changed structure. Microwave and Optical Technology Letters, 50(8) 2210-2213.

[3] Melendy, D., Francis, P., Pichler, C., Hwang, K., Srinivasan, G., Weisshaar, A. (2002). A new wideband compact model for spiral inductors in RFICs. IEEE Electron Device Letter, 23 (5) 273-275.

[4] Guo, J. C., Tan, T. Y. (2006). A broadband and scalable model for on-chip inductors incorporating substrate and conductor loss effects, IEEE Trans. Electron Devices, 53 (3) 413-421.

[5] Kuhn, W. B., Ibrahim, N. M. (2001). Analysis of Current Crowding Effects in Multitrun Spiral Inductors. IEEE Transactions on Microwave Theory and Techniques, 49(1).

[6] Ooi, B. L., Xu, D. X., Kooi, P. S., Lin, F. J. (2002). An Improved Prediction of Series Resistance in Spiral Inductor Modeling With Eddy-Current Effect. IEEE Transaction on Microwave Theory and Techniques, 50,2202-2206.

[7] Watson, A.C., Melendy, D., Francis, P., Kyuwoon Hwang, Weisshaar, A. (2004). A comprehensive compact-modeling methodology for spiral inductors in silicon-based RFICs. IEEE Transactions on Microwave Theory and Techniques, 52 (3) 849-857.

[8] Joonho Gil, Hyungcheol Shin, (2003). A Simple Wide-Band On-Chip Inductor Model for Silicon-Based RF ICs. IEEE Transaction on Microwave Theory and Techniques, 51, 2023-2028.

[9] Jin, W., Eo, Y., Shim, J. I., Eisenstadt, W. R., Park, M. Y., Yu, H. K. (2001). Silicon substrate coupling noise modeling, analysis, and experimental verification for mixed signal integrated circuit design. IEEE MTT-S Int. Microwave Symp. Dig:1727-1730.

[10] Zheng, J., Hahm, Y.-C., Tripathi, V. K., Weisshaar, A. (2000). CAD-oriented equivalent-circuit modeling of on-chip interconnects on lossy silicon substrate. IEEE Trans. Microwave Theory Tech., 48, 1443-1451.

Xi Li (1), Zheng Ren (2), Yanling Shi (1)

(1) East China Normal University Shanghai 200241 People's Republic of China

(2) Shanghai IC Research and Development Center Shanghai 201203 People's Republic of China {ximeir@hotmail.com, alancatrz@icrd.com.cn, ylshi@ee.ecnu.edu.cn}
Table 1. Various physical dimensions of the inductors

Sample   [D.sub.out]   N     W + S
Number   ([micro]m)          ([micro]m)

Q1       400           3.5   30
Q2
Q3

Q4                     3.5   40
Q5
Q6

Q7                     4.5   30
Q8
Q9

Sample   W([micro]m)      S([micro]m)   Inductor Type
Number

Q1       15               15            conventional
Q2       24.23.20.15      10            single gradually changed
Q3       24,23,20,15      6,7,10,15     gradually changed

Q4       20               20            conventional
Q5       32.30.27.20      10            single gradually changed
Q6       32.30.27.20      8.10.13.20    gradually changed

Q7       15               15            conventional
Q8       25.24.23.20.15   10            single gradually changed
Q9       25.24.23.20.15   5.6.7.10.15   gradually changed

Table 2. Model Parameters Extracted from Measurement Data

     [L.sub.1]   [L.sub.0]   [R.sub.1]   [R.sub.0]   [C.sub.ox1]
       (nH)        (nH)       ([ohm])     ([ohm])       (fF)

Q1     5.83        0.28        5.11        4.27         126.5
Q2     5.44        0.23        4.51        3.67         151.4
Q3     4.93        0.16        3.71        3.07         134.7
Q4     6.13        0.26        5.41        4.17         148.5
Q5     5.92        0.23        4.81        3.86         152.8
Q6     5.58        0.16        4.38        2.86         142.5
Q7     4.43        0.23        4.11        3.86         138.6
Q8     4.53        0.18        4.05        3.57         155.8
Q9     3.93        0.16        3.15        2.56         136.5

     [C.sub.ox2]   [R.sub.si1]   [R.sub.si2]   [C.sub.si1]
        (fF)         ([ohm])       ([ohm])        (fF)

Q1      128.3         887.1         887.1         183.1
Q2      153.5         864.4         864.4         191.2
Q3      136.9         877.6         877.6         188.6
Q4      149.8         874.3         874.3         173.2
Q5      153.4         854.1         854.1         183.1
Q6      143.8         861.2         861.2          191
Q7      139.7         897.8         897.8         173.1
Q8      155.8         879.3         879.3         181.5
Q9      137.1         885.1         885.1         158.2

     [C.sub.si2]   [C.sub.s]   [R.sub.sub]   [C.sub.sub]
        (fF)         (fF)        ([ohm])        (fF)

Q1      183.1        30.1         753.9         9.77
Q2      191.2        37.2         717.7         15.6
Q3      188.6        35.6         733.9         12.7
Q4      173.2        36.1         736.8         20.8
Q5      183.1        41.6         706.3         30.4
Q6       191         53.1         726.4         25.7
Q7      173.1        32.1         683.5         35.7
Q8      181.5        38.4         668.9         41.3
Q9      158.2        36.5         679.2         37.7
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Author:Li, Xi; Ren, Zheng; Shi, Yanling
Publication:Journal of Digital Information Management
Article Type:Report
Date:Apr 1, 2012
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