Abstract: Coplanar waveguides (CPW) are widely used in mm-wave circuits designs for their good performance. A novel unified model of various on chip CPWs for mm-wave application, together with corresponding direct parameter extraction methodologies, are proposed and investigated, where standard CPW, grounded CPW (GCPW) and CPW with slotted shield (SCPW) are included. Several kinds of influences of different structures are analyzed and considered into the model to explain the frequency-dependent per-unit-length L, C, R, and G parameters, among which the electromagnetic coupling for CPWs with large lower ground or shield is described by a new C-L-R series path in the parallel branch. The direct extraction procedures are established, which can ensure both accuracy and simplicity compared with other reported methods. Different CPWs are fabricated and measured on 90-nm CMOS processes with Short-Open-Load-Through (SOLT) de-embedding techniques. Excellent agreement between the model and the measured data for different CPWs is achieved up to 67 GHz.
In recent years, along with the improvement of the CMOS technology, CMOS circuits for millimeter-wave (mm-wave) applications have drawn attentions from majority, such as high speed wireless communication at 60 GHz[1-2], and automobile radar at 24 GHz and 77 GHz[3]. At mm-wave regime, circuit components suffer from significant parasitic effects, which strongly depend on the layout structure, leading to significant fluctuations of circuit performance. For example, conventional passive components, such as inductor and transformer, suffer from parasitic effects of current return path at mm-wave frequencies[4], leading to the uncertainty of modeling.
Transmission lines (TLs) with low loss and a high dielectric constant become attractive for attaining miniaturization and high performance in mm-wave systems. In contrast to conventionally passive components, TLs can be flexibly utilized as inductors, capacitors and for impedance transformation. A well-designed mm-wave circuit based on transmission lines minimizes (or even completely removes) the need for a parasitic extraction as they have definite a current return path and the parasitic effects are already taken into account in the line model.
Microstrip lines (MSs) and coplanar waveguides (CPWs) are the most commonly used TLs. Although MSs provide shielding from the lossy substrate, the gap between the signal wire and bottom metal is set by the technology, leading to poor circuit design flexibility. CPWs, on the other hand, have the freedom to adjust both the width and gap, and are easy to integrate in standard CMOS processes.
It is well known that electromagnetic (EM) simulation can provide the $S$-parameter of CPWs with good accuracy, but it is very time consuming to construct a sufficiently large model library; besides, such method only supports small-signal frequency domain analysis and suffers from poor convergence especially in periodic steady-state (PSS) simulations. There have been extensive works on constructing SPICE models of CPWs. For example, SPICE models given in Ref. [7] achieve relatively better accuracy with analytical extraction methodology. However, both the measurement and 3D EM simulation show that the per unit capacitance increases with frequency for transmission lines with a ground structure, which is quite different from the standard CPW[6, 7]. A unified model for CPWs has been co-developed by our group[8] and an LC series sub-circuit is proposed to model the phenomenon. The model parameters are extracted by optimization and some empirical formulas are adopted without considering the specific foundry processes. Besides, the model has been verified up to about 40 GHz due to the limitations of measurement equipment and the de-embedding method.
In this paper, we take a further step towards the practical application of CPWs, a new unified model for CPWs, GCPWs, and SCPWs is proposed in which a $C$-$L$-$R$ branch is introduced to describe the EM coupling to the lower metal for the CPWs with large lower ground or shield as shown in Section 2, and the corresponding direct extraction methodology is provided to acquire the parameters of the model in Section 3. The complete experimental results and verifications up to 67 GHz with short-load-open-through (SLOT) de-embedding techniques are presented and discussed in Section 4. Finally, this paper is concluded in Section 5.
2.
Proposed unified model for CPWs
Based on the quasi-TEM assumption, CPWs could be fully characterized by the Telegrapher's equation with RLGC parameters. In this work, $R_{\rm x}$, $L_{\rm x}$, $G_{\rm x}$, and $C_{\rm x}$ are adopted to describe frequency dependent per-unit-length (PUL) RLGC parameters. The CPWs are modeled by multiple cascaded blocks $B_{\rm i}$, in which equivalent circuits are adopted to capture the behaviors of $R_{\rm x}$, $L_{\rm x}$, $G_{\rm x}$, and $C_{\rm x}$, as shown in Fig. 1.
Figure
1.
Proposed unified equivalent circuit model for CPWs.
The series branch consists of an inductor $L_{\rm hf}$ in series with an R/L ladder composed of $R_{\rm hf}$, $R_{1}$ and $L_{1}$ to characterize the skin effect at high frequencies.
The parallel branch is comprised of three parts to describe three different distributed effects: (1) $C_{\rm sg1}$ characterizes the pure capacitance between the signal wire and ground for various types of CPWs, as shown in Fig. 2(a). (2) A $C$-$R$-$C$ network is incorporated here to describe the EM coupling to the lossy substrate, as shown in Fig. 2(b). (3) The $C$-$L$-$R$ series path describes the EM coupling to the lower metal, especially for CPWs with a large lower ground or shield, e.g. GCPW and SCPW, as shown in Fig. 2(c).
Figure
2.
Parallel branch of the CPWs model. (a) $C_{\rm sg1}$. (b) $C$-$R$-$C$ network. (c) $C$-$L$-$R$ series path.
Based on the quasi-TEM assumption, the series and parallel branches could be extracted independently. From the $S$ parameters, one could derive the ABCD parameters of the entire CPWs and subsequently the ABCD parameters of a single block by calculating the $n$-th root. Based on the ABCD parameters, the $Y$ and $Z$ parameters of the block are acquired, from which $R_{\rm x}$, $L_{\rm x}$, $G_{\rm x}$, $C_{\rm x}$ are obtained to extract the series and parallel branches, as shown in Fig. 3.
Figure
3.
Proposed extraction flow of the unified CPW model.
The series branch is comprised of a second-order $R/L$ ladder to describe the skin effect of the conductor at mm-wave frequencies. The equivalent resistance and inductance of the $R/L$ ladder could be derived as follows:
$
Rx=RhfL21ω2+R2hfR1+RhfR21L21ω2+(Rhf+R1)2,
$
(1)
$
Lx=R2hfL1L21ω2+(Rhf+R1)2+Lhf.
$
(2)
At high frequency, $R_{\rm hf} L_1^2 \omega ^2$$\gg$$R_{\rm hf}^2 R_1 +R_{\rm hf} R_1^2$, and the above equations can be reduced to:
$
ω2Rx=L21ω2+(Rhf+R1)2RhfL21,
$
(3)
$
ω2RxLx=Lhf(L21ω2+(Rhf+R1)2)+R2hfL1RhfL21.
$
(4)
Based on Eqs. (3) and (4), the four parameters $R_{\rm hf}$$, L_{\rm hf}$, $R_{1}$, and $L_{1}$ of the series branch could be extracted by using the least square method from the data. The four parameters in the series branches are all extracted so far. This method is verified by the measured data of a 400-$\mu $m-long CPW up to 67 GHz as shown in Fig. 4, where (a) (b) show the linear relation in Eqs. (3) and (4) and (c) (d) compare the model with the measured results.
Figure
4.
(color online) Serial extraction of a CPW, comparisons of (a) $\omega^{2}$/$R_{\rm x}$ versus $f^{2}$, (b) $L_{\rm x}$$\omega^{2}$/$R_{\rm x}$ versus $f^{2}$, (c) $R_{\rm x}$ versus $f$, (d) $L_{\rm x}$ versus $f$ between model and measured results.
For CPWs, $C_{\rm sg1}$ characterizes the capacitance between the signal wire and upper ground (Fig. 2(a)) which can be calculated by the Y1 formula given in Ref. [8] As part of the EM field is shielded by the below shielding metal while the other part may penetrate to the substrate, a weight factor $\eta$ is used here to describe the effect of incomplete shielding, namely, (1-$\eta)$$Y_{\rm parallel}$ being the admittance of the $C$-$R$-$C$ network and $\eta Y_{\rm parallel}$ being the admittance of the $C$-$L$-$R$ path.
For SCPW, $\eta$ is a nonlinear function of the metal filling ratio and can be expressed as [$f_{\rm w}$/($f_{\rm w}+f_{\rm s})]^{\alpha}$, where $f_{\rm w}$ and $f_{\rm s}$ are the width and spacing of shielding metal, $\alpha$ is the fitting parameter[8]. Therefore both the $C$-$R$-$C$ network and the $C$-$L$-$R$ path are included in the parallel branch extraction of the SCPW. The seven parameters in the parallel branch can be acquired by combining the following mathematics for the extraction of the CPW and the GCPW.
For a standard CPW, $\eta$ is set to zero, i.e. the $C$-$L$-$R$ series path could be omitted and the direct extraction of the $C$-$R$-$C$ network is given as follows:
$
CxGxω2=CsiRsi(Csi+Cox)Coxω2+1RsiCox,
$
(5)
$
ω2Gx=Rsi(Csi+Cox)2C2oxω2+1RsiC2ox,
$
(6)
where $G_{\rm x}$$=$ real($Y_{\rm parallel})$, $C_{\rm x}$$=$ imag($Y_{\rm parallel}$ -j$\omega C_{\rm sg1})$, and $C_{\rm ox}$, $C_{\rm si}$, and $R_{\rm si}$ can therefore be extracted.
For a GCPW, since the signal line is semi-enclosed by the ground metal, little electromagnetic field couples to the lossy substrate, therefore $\eta$ is set to 1 and the $C$-$R$-$C$ network can be omitted. The direct extraction of the $C$-$L$-$R$ path is given in Eqs. (7) and (8).
$
Rsg2=real(Zparallel),
$
(7)
$
ω⋅imag(Zparallel)=Lsg2ω2−1Csg2,
$
(8)
where $Z_{\rm parallel}=1/(Y_{\rm parallel}$ -j$\omega_{\rm sg1})$. Based on Eqs. (8) and (9), $R_{\rm sg2}$, $C_{\rm sg2}$, and $L_{\rm sg2}$ can be extracted. The model is verified by the measured results of a CPW and an SCPW up to 67 GHz as shown in Figs. 5 and 6.
Figure
5.
(color online) $C$-$R$-$C$ extraction of a CPW, comparisons of (a) $C_{\rm x}$$\omega^{2}$/$G_{\rm x}$ versus $f^{2}$, (b) $\omega^{2}$/$G_{\rm x}$ versus $f^{2}$, (c) $C_{\rm ox}$ versus $f$, (d) $G_{\rm x}$ versus $f$ between model and measured results.
Figure
6.
(color online) $C$-$L$-$R$ extraction of an SCPW, comparisons of (a) $\omega$$Z_{\rm parallel}$ versus $f^{2}$, (b) $C_{\rm x}$ versus $f$, (c) $G_{\rm x}$ versus $f$ between model and measured results.
Thus far, the direct extraction of all three kinds of CPWs is accomplished.
4.
Experimental validation and discussion
In order to verify the proposed unified CPW model and direct extraction methodologies, various CPWs structures, including a CPW and an SCPW with lengths of 400 $\mu $m are designed and fabricated in a 90 nm IBM CMOS process. Layout of CPWs and de-embedding structures are shown in Fig. 7. $S$-parameters are measured up to 67 GHz. Simplified SOLT de-embedding for identical and reciprocal structure, calculated by OPEN, SHORT, and LOAD test structures[10], has been applied for parasitic de-embedding to acquire the intrinsic CPWs data. As the coupling between the two GSG pads is significantly small due to the large distance, the OPEN and SHORT structures are therefore integrated as shown in Fig. 7. The $S$-parameter of LOAD structure is acquired through EM simulation (Ansoft HFSS V12) with measurement calibration. Proposed direct extraction methodologies are adopted to acquire the parameters of the model, which are summarized in Table 1. The measured per-unit-length $R_{\rm x}$, $L_{\rm x}$, $G_{\rm x}$, $C_{\rm x}$ parameters of different structures are compared with the proposed model in Figs. 8(a)-8(d). Excellent agreement between the measured and simulated results is obtained with maximal $C_{\rm x}$ and $L_{\rm x}$ errors less than 5% over the entire frequency range up to 67 GHz. When $f$$>$ 20 GHz, $\omega L_{\rm x}$$\gg$$R_{\rm x}$ and $\omega C_{\rm x}$$\gg$$G_{\rm x}$, the errors of $G_{\rm x}$ and $R_{\rm x}$ contribute little to the overall impedance and admittance of the series and parallel branches and can be neglected. It can be seen from Fig. 8(c) that the model can fit the increasing of $C_{\rm x}$ well which is impossible for the model in Ref. [7-8].
Figure
7.
Layout of (a) SCPW, (b) CPW, and (c) Open-Short.
Figure
8.
(color online) Comparisons between the experimental data (dot) and the modeled results (line) of per-unit-length (a) resistance, (b) inductance, (c) capacitance, and (d) conductance.
$S$-parameter comparisons between measurements and simulated results of 400 $\mu$m CPWs with 10 cascaded blocks are shown in Fig. 9. Excellent data agreement of the three structures is achieved over the entire frequency range up to 67 GHz.
Figure
9.
(color online) $S$-parameter comparisons between the experimental data (dot) and the modeled results (line). (a) Magnitude of $S_{11}$. (b) Phase of $S_{11}$. (c) Magnitude of $S_{21}$. (d) Phase of $S_{21}$.
A unified CPW, GCPW and SCPW model for mm-wave applications is proposed and the direct extraction procedure is established for the first time in this paper. A new $C$-$L$-$R$ series path, together with the conventional one, is introduced to describe the electromagnetic coupling for CPWs with large lower ground or shield, while other kinds of high frequency effects including substrate loss, and skin effect are considered to explain the frequency-dependent per-unit-length Lx, Cx, Rx, and Gx parameters. Direct extraction methodologies of the model are presented and verified by measurement on different CPW structures fabricated in 90nm IBM CMOS processes. Excellent agreement of RLGC parameters and $S$-parameter between the measured data and modeled result is obtained with maximal $C_{\rm x}$ and $L_{\rm x}$ errors of $ < $ 5% over the entire frequency range up to 67 GHz. The direct extraction methodologies ensure the feasibility and availability of scalable modeling of CPWs for circuit designer.
References
[1]
Emami S, Wiser R F, Ali E, et al. A 60 GHz CMOS phased-array transceiver pair for multi-Gb/s wireless communications. ISSCC Dig Tech Papers, 2011:164 http://ieeexplore.ieee.org/document/5746265/
Lee J, Li Y A, Hung M H. A fully-integrated 77-GHz FMCW radar transceiver in 65-nm CMOS technology. IEEE J Solid-State Circuits, 2010, 45(12):2746 doi: 10.1109/JSSC.2010.2075250
[4]
Wang H, Zhang L, Yang D, et al. Modeling of current-return-path effect on single-ended inductor in millimeter-wave regime. IEEE Electron Device Lett, 2011, 32(6):737 doi: 10.1109/LED.2011.2136312
[5]
Sayag A, Ritter D, Goren D. Compact modeling and comparative analysis of silicon-chip slow-wave transmission lines with slotted bottom metal ground planes. IEEE Trans Microw Theory Tech, 2009, 57(4):840 doi: 10.1109/TMTT.2009.2015041
[6]
Cho H Y, Yeh T J, Liu S, et al. High-performance slow-wave transmission lines with optimized slot-type floating shields. IEEE Trans Electron Devices, 2009, 56(8):1705 doi: 10.1109/TED.2009.2024034
[7]
Gevorgian S, Linn Cr L J P, Kollberg E L. CAD models for shielded multilayered CPW. IEEE Trans Microw Theory Tech, 1995, 43(4):772 doi: 10.1109/22.375223
[8]
Brinkhoff J, Koh K S S, Kang K, et al. Scalable transmission line and inductor models for CMOS millimeter-wave design. IEEE Trans Microw Theory Tech, 2008, 56(12):2954 doi: 10.1109/TMTT.2008.2007337
Tiemeijer L F, Havens R J, Jansman A B M, et al. Comparison of the 'pad-open-short' and 'open-short-load' de-embedding techniques for accurate on-wafer RF characterization of high-quality passives. IEEE Trans Microw Theory Tech, 2005, 53(2):723 doi: 10.1109/TMTT.2004.840621
[11]
Gao W, Yu Z. Scalable compact circuit model and synthesis for RF CMOS spiral inductors. IEEE Trans Microw Theory Tech, 2006, 54(3):1055 doi: 10.1109/TMTT.2005.864134
Fig. 1.
Proposed unified equivalent circuit model for CPWs.
Fig. 4.
(color online) Serial extraction of a CPW, comparisons of (a) $\omega^{2}$/$R_{\rm x}$ versus $f^{2}$, (b) $L_{\rm x}$$\omega^{2}$/$R_{\rm x}$ versus $f^{2}$, (c) $R_{\rm x}$ versus $f$, (d) $L_{\rm x}$ versus $f$ between model and measured results.
Fig. 5.
(color online) $C$-$R$-$C$ extraction of a CPW, comparisons of (a) $C_{\rm x}$$\omega^{2}$/$G_{\rm x}$ versus $f^{2}$, (b) $\omega^{2}$/$G_{\rm x}$ versus $f^{2}$, (c) $C_{\rm ox}$ versus $f$, (d) $G_{\rm x}$ versus $f$ between model and measured results.
Fig. 6.
(color online) $C$-$L$-$R$ extraction of an SCPW, comparisons of (a) $\omega$$Z_{\rm parallel}$ versus $f^{2}$, (b) $C_{\rm x}$ versus $f$, (c) $G_{\rm x}$ versus $f$ between model and measured results.
Fig. 8.
(color online) Comparisons between the experimental data (dot) and the modeled results (line) of per-unit-length (a) resistance, (b) inductance, (c) capacitance, and (d) conductance.
Fig. 9.
(color online) $S$-parameter comparisons between the experimental data (dot) and the modeled results (line). (a) Magnitude of $S_{11}$. (b) Phase of $S_{11}$. (c) Magnitude of $S_{21}$. (d) Phase of $S_{21}$.
Table 1.
The extracted parameters of the proposed model for various types of CPWS.
[1]
Emami S, Wiser R F, Ali E, et al. A 60 GHz CMOS phased-array transceiver pair for multi-Gb/s wireless communications. ISSCC Dig Tech Papers, 2011:164 http://ieeexplore.ieee.org/document/5746265/
Lee J, Li Y A, Hung M H. A fully-integrated 77-GHz FMCW radar transceiver in 65-nm CMOS technology. IEEE J Solid-State Circuits, 2010, 45(12):2746 doi: 10.1109/JSSC.2010.2075250
[4]
Wang H, Zhang L, Yang D, et al. Modeling of current-return-path effect on single-ended inductor in millimeter-wave regime. IEEE Electron Device Lett, 2011, 32(6):737 doi: 10.1109/LED.2011.2136312
[5]
Sayag A, Ritter D, Goren D. Compact modeling and comparative analysis of silicon-chip slow-wave transmission lines with slotted bottom metal ground planes. IEEE Trans Microw Theory Tech, 2009, 57(4):840 doi: 10.1109/TMTT.2009.2015041
[6]
Cho H Y, Yeh T J, Liu S, et al. High-performance slow-wave transmission lines with optimized slot-type floating shields. IEEE Trans Electron Devices, 2009, 56(8):1705 doi: 10.1109/TED.2009.2024034
[7]
Gevorgian S, Linn Cr L J P, Kollberg E L. CAD models for shielded multilayered CPW. IEEE Trans Microw Theory Tech, 1995, 43(4):772 doi: 10.1109/22.375223
[8]
Brinkhoff J, Koh K S S, Kang K, et al. Scalable transmission line and inductor models for CMOS millimeter-wave design. IEEE Trans Microw Theory Tech, 2008, 56(12):2954 doi: 10.1109/TMTT.2008.2007337
Tiemeijer L F, Havens R J, Jansman A B M, et al. Comparison of the 'pad-open-short' and 'open-short-load' de-embedding techniques for accurate on-wafer RF characterization of high-quality passives. IEEE Trans Microw Theory Tech, 2005, 53(2):723 doi: 10.1109/TMTT.2004.840621
[11]
Gao W, Yu Z. Scalable compact circuit model and synthesis for RF CMOS spiral inductors. IEEE Trans Microw Theory Tech, 2006, 54(3):1055 doi: 10.1109/TMTT.2005.864134
Song Min, Zheng Yaru, Lu Yongjun, Qu Yanling, Song Limin, et al.
Chinese Journal of Semiconductors , 2005, 26(12): 2407-2410.
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Jun Luo, Lei Zhang, Yan Wang. Modeling and parameter extraction of CMOS on-chip coplanar waveguides up to 67 GHz for mm-wave applications[J]. Journal of Semiconductors, 2013, 34(12): 125008. doi: 10.1088/1674-4926/34/12/125008
J Luo, L Zhang, Y Wang. Modeling and parameter extraction of CMOS on-chip coplanar waveguides up to 67 GHz for mm-wave applications[J]. J. Semicond., 2013, 34(12): 125008. doi: 10.1088/1674-4926/34/12/125008.
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Received: 26 May 2013Revised: Online:Published: 01 December 2013
Jun Luo, Lei Zhang, Yan Wang. Modeling and parameter extraction of CMOS on-chip coplanar waveguides up to 67 GHz for mm-wave applications[J]. Journal of Semiconductors, 2013, 34(12): 125008. doi: 10.1088/1674-4926/34/12/125008 ****J Luo, L Zhang, Y Wang. Modeling and parameter extraction of CMOS on-chip coplanar waveguides up to 67 GHz for mm-wave applications[J]. J. Semicond., 2013, 34(12): 125008. doi: 10.1088/1674-4926/34/12/125008.
Citation:
Jun Luo, Lei Zhang, Yan Wang. Modeling and parameter extraction of CMOS on-chip coplanar waveguides up to 67 GHz for mm-wave applications[J]. Journal of Semiconductors, 2013, 34(12): 125008. doi: 10.1088/1674-4926/34/12/125008
****
J Luo, L Zhang, Y Wang. Modeling and parameter extraction of CMOS on-chip coplanar waveguides up to 67 GHz for mm-wave applications[J]. J. Semicond., 2013, 34(12): 125008. doi: 10.1088/1674-4926/34/12/125008.
Jun Luo, Lei Zhang, Yan Wang. Modeling and parameter extraction of CMOS on-chip coplanar waveguides up to 67 GHz for mm-wave applications[J]. Journal of Semiconductors, 2013, 34(12): 125008. doi: 10.1088/1674-4926/34/12/125008 ****J Luo, L Zhang, Y Wang. Modeling and parameter extraction of CMOS on-chip coplanar waveguides up to 67 GHz for mm-wave applications[J]. J. Semicond., 2013, 34(12): 125008. doi: 10.1088/1674-4926/34/12/125008.
Citation:
Jun Luo, Lei Zhang, Yan Wang. Modeling and parameter extraction of CMOS on-chip coplanar waveguides up to 67 GHz for mm-wave applications[J]. Journal of Semiconductors, 2013, 34(12): 125008. doi: 10.1088/1674-4926/34/12/125008
****
J Luo, L Zhang, Y Wang. Modeling and parameter extraction of CMOS on-chip coplanar waveguides up to 67 GHz for mm-wave applications[J]. J. Semicond., 2013, 34(12): 125008. doi: 10.1088/1674-4926/34/12/125008.
Institute of Microelectronics, Tsinghua University, Beijing 100084, China
Funds:
the National Natural Science Foundation of China61204026
the National Natural Science Foundation of China61101001
the National Natural Science Foundation of China61176034
the State Key Development Program for Basic Research of China2010CB327404
the Tsinghua University Initiative Scientific Research Program
Project supported by the State Key Development Program for Basic Research of China (No. 2010CB327404), the National High Technology Research and Development Program of China (No. 2011AA010202), the National Science and Technology Major Project of China (No. 2012ZX03004004), the National Natural Science Foundation of China (Nos. 61176034, 61101001, 61204026), and the Tsinghua University Initiative Scientific Research Program
the National High Technology Research and Development Program of China2011AA010202
the National Science and Technology Major Project of China2012ZX03004004
Coplanar waveguides (CPW) are widely used in mm-wave circuits designs for their good performance. A novel unified model of various on chip CPWs for mm-wave application, together with corresponding direct parameter extraction methodologies, are proposed and investigated, where standard CPW, grounded CPW (GCPW) and CPW with slotted shield (SCPW) are included. Several kinds of influences of different structures are analyzed and considered into the model to explain the frequency-dependent per-unit-length L, C, R, and G parameters, among which the electromagnetic coupling for CPWs with large lower ground or shield is described by a new C-L-R series path in the parallel branch. The direct extraction procedures are established, which can ensure both accuracy and simplicity compared with other reported methods. Different CPWs are fabricated and measured on 90-nm CMOS processes with Short-Open-Load-Through (SOLT) de-embedding techniques. Excellent agreement between the model and the measured data for different CPWs is achieved up to 67 GHz.
In recent years, along with the improvement of the CMOS technology, CMOS circuits for millimeter-wave (mm-wave) applications have drawn attentions from majority, such as high speed wireless communication at 60 GHz[1-2], and automobile radar at 24 GHz and 77 GHz[3]. At mm-wave regime, circuit components suffer from significant parasitic effects, which strongly depend on the layout structure, leading to significant fluctuations of circuit performance. For example, conventional passive components, such as inductor and transformer, suffer from parasitic effects of current return path at mm-wave frequencies[4], leading to the uncertainty of modeling.
Transmission lines (TLs) with low loss and a high dielectric constant become attractive for attaining miniaturization and high performance in mm-wave systems. In contrast to conventionally passive components, TLs can be flexibly utilized as inductors, capacitors and for impedance transformation. A well-designed mm-wave circuit based on transmission lines minimizes (or even completely removes) the need for a parasitic extraction as they have definite a current return path and the parasitic effects are already taken into account in the line model.
Microstrip lines (MSs) and coplanar waveguides (CPWs) are the most commonly used TLs. Although MSs provide shielding from the lossy substrate, the gap between the signal wire and bottom metal is set by the technology, leading to poor circuit design flexibility. CPWs, on the other hand, have the freedom to adjust both the width and gap, and are easy to integrate in standard CMOS processes.
It is well known that electromagnetic (EM) simulation can provide the $S$-parameter of CPWs with good accuracy, but it is very time consuming to construct a sufficiently large model library; besides, such method only supports small-signal frequency domain analysis and suffers from poor convergence especially in periodic steady-state (PSS) simulations. There have been extensive works on constructing SPICE models of CPWs. For example, SPICE models given in Ref. [7] achieve relatively better accuracy with analytical extraction methodology. However, both the measurement and 3D EM simulation show that the per unit capacitance increases with frequency for transmission lines with a ground structure, which is quite different from the standard CPW[6, 7]. A unified model for CPWs has been co-developed by our group[8] and an LC series sub-circuit is proposed to model the phenomenon. The model parameters are extracted by optimization and some empirical formulas are adopted without considering the specific foundry processes. Besides, the model has been verified up to about 40 GHz due to the limitations of measurement equipment and the de-embedding method.
In this paper, we take a further step towards the practical application of CPWs, a new unified model for CPWs, GCPWs, and SCPWs is proposed in which a $C$-$L$-$R$ branch is introduced to describe the EM coupling to the lower metal for the CPWs with large lower ground or shield as shown in Section 2, and the corresponding direct extraction methodology is provided to acquire the parameters of the model in Section 3. The complete experimental results and verifications up to 67 GHz with short-load-open-through (SLOT) de-embedding techniques are presented and discussed in Section 4. Finally, this paper is concluded in Section 5.
2.
Proposed unified model for CPWs
Based on the quasi-TEM assumption, CPWs could be fully characterized by the Telegrapher's equation with RLGC parameters. In this work, $R_{\rm x}$, $L_{\rm x}$, $G_{\rm x}$, and $C_{\rm x}$ are adopted to describe frequency dependent per-unit-length (PUL) RLGC parameters. The CPWs are modeled by multiple cascaded blocks $B_{\rm i}$, in which equivalent circuits are adopted to capture the behaviors of $R_{\rm x}$, $L_{\rm x}$, $G_{\rm x}$, and $C_{\rm x}$, as shown in Fig. 1.
Figure
1.
Proposed unified equivalent circuit model for CPWs.
The series branch consists of an inductor $L_{\rm hf}$ in series with an R/L ladder composed of $R_{\rm hf}$, $R_{1}$ and $L_{1}$ to characterize the skin effect at high frequencies.
The parallel branch is comprised of three parts to describe three different distributed effects: (1) $C_{\rm sg1}$ characterizes the pure capacitance between the signal wire and ground for various types of CPWs, as shown in Fig. 2(a). (2) A $C$-$R$-$C$ network is incorporated here to describe the EM coupling to the lossy substrate, as shown in Fig. 2(b). (3) The $C$-$L$-$R$ series path describes the EM coupling to the lower metal, especially for CPWs with a large lower ground or shield, e.g. GCPW and SCPW, as shown in Fig. 2(c).
Figure
2.
Parallel branch of the CPWs model. (a) $C_{\rm sg1}$. (b) $C$-$R$-$C$ network. (c) $C$-$L$-$R$ series path.
Based on the quasi-TEM assumption, the series and parallel branches could be extracted independently. From the $S$ parameters, one could derive the ABCD parameters of the entire CPWs and subsequently the ABCD parameters of a single block by calculating the $n$-th root. Based on the ABCD parameters, the $Y$ and $Z$ parameters of the block are acquired, from which $R_{\rm x}$, $L_{\rm x}$, $G_{\rm x}$, $C_{\rm x}$ are obtained to extract the series and parallel branches, as shown in Fig. 3.
Figure
3.
Proposed extraction flow of the unified CPW model.
The series branch is comprised of a second-order $R/L$ ladder to describe the skin effect of the conductor at mm-wave frequencies. The equivalent resistance and inductance of the $R/L$ ladder could be derived as follows:
Based on Eqs. (3) and (4), the four parameters $R_{\rm hf}$$, L_{\rm hf}$, $R_{1}$, and $L_{1}$ of the series branch could be extracted by using the least square method from the data. The four parameters in the series branches are all extracted so far. This method is verified by the measured data of a 400-$\mu $m-long CPW up to 67 GHz as shown in Fig. 4, where (a) (b) show the linear relation in Eqs. (3) and (4) and (c) (d) compare the model with the measured results.
Figure
4.
(color online) Serial extraction of a CPW, comparisons of (a) $\omega^{2}$/$R_{\rm x}$ versus $f^{2}$, (b) $L_{\rm x}$$\omega^{2}$/$R_{\rm x}$ versus $f^{2}$, (c) $R_{\rm x}$ versus $f$, (d) $L_{\rm x}$ versus $f$ between model and measured results.
For CPWs, $C_{\rm sg1}$ characterizes the capacitance between the signal wire and upper ground (Fig. 2(a)) which can be calculated by the Y1 formula given in Ref. [8] As part of the EM field is shielded by the below shielding metal while the other part may penetrate to the substrate, a weight factor $\eta$ is used here to describe the effect of incomplete shielding, namely, (1-$\eta)$$Y_{\rm parallel}$ being the admittance of the $C$-$R$-$C$ network and $\eta Y_{\rm parallel}$ being the admittance of the $C$-$L$-$R$ path.
For SCPW, $\eta$ is a nonlinear function of the metal filling ratio and can be expressed as [$f_{\rm w}$/($f_{\rm w}+f_{\rm s})]^{\alpha}$, where $f_{\rm w}$ and $f_{\rm s}$ are the width and spacing of shielding metal, $\alpha$ is the fitting parameter[8]. Therefore both the $C$-$R$-$C$ network and the $C$-$L$-$R$ path are included in the parallel branch extraction of the SCPW. The seven parameters in the parallel branch can be acquired by combining the following mathematics for the extraction of the CPW and the GCPW.
For a standard CPW, $\eta$ is set to zero, i.e. the $C$-$L$-$R$ series path could be omitted and the direct extraction of the $C$-$R$-$C$ network is given as follows:
where $G_{\rm x}$$=$ real($Y_{\rm parallel})$, $C_{\rm x}$$=$ imag($Y_{\rm parallel}$ -j$\omega C_{\rm sg1})$, and $C_{\rm ox}$, $C_{\rm si}$, and $R_{\rm si}$ can therefore be extracted.
For a GCPW, since the signal line is semi-enclosed by the ground metal, little electromagnetic field couples to the lossy substrate, therefore $\eta$ is set to 1 and the $C$-$R$-$C$ network can be omitted. The direct extraction of the $C$-$L$-$R$ path is given in Eqs. (7) and (8).
where $Z_{\rm parallel}=1/(Y_{\rm parallel}$ -j$\omega_{\rm sg1})$. Based on Eqs. (8) and (9), $R_{\rm sg2}$, $C_{\rm sg2}$, and $L_{\rm sg2}$ can be extracted. The model is verified by the measured results of a CPW and an SCPW up to 67 GHz as shown in Figs. 5 and 6.
Figure
5.
(color online) $C$-$R$-$C$ extraction of a CPW, comparisons of (a) $C_{\rm x}$$\omega^{2}$/$G_{\rm x}$ versus $f^{2}$, (b) $\omega^{2}$/$G_{\rm x}$ versus $f^{2}$, (c) $C_{\rm ox}$ versus $f$, (d) $G_{\rm x}$ versus $f$ between model and measured results.
Figure
6.
(color online) $C$-$L$-$R$ extraction of an SCPW, comparisons of (a) $\omega$$Z_{\rm parallel}$ versus $f^{2}$, (b) $C_{\rm x}$ versus $f$, (c) $G_{\rm x}$ versus $f$ between model and measured results.
Thus far, the direct extraction of all three kinds of CPWs is accomplished.
4.
Experimental validation and discussion
In order to verify the proposed unified CPW model and direct extraction methodologies, various CPWs structures, including a CPW and an SCPW with lengths of 400 $\mu $m are designed and fabricated in a 90 nm IBM CMOS process. Layout of CPWs and de-embedding structures are shown in Fig. 7. $S$-parameters are measured up to 67 GHz. Simplified SOLT de-embedding for identical and reciprocal structure, calculated by OPEN, SHORT, and LOAD test structures[10], has been applied for parasitic de-embedding to acquire the intrinsic CPWs data. As the coupling between the two GSG pads is significantly small due to the large distance, the OPEN and SHORT structures are therefore integrated as shown in Fig. 7. The $S$-parameter of LOAD structure is acquired through EM simulation (Ansoft HFSS V12) with measurement calibration. Proposed direct extraction methodologies are adopted to acquire the parameters of the model, which are summarized in Table 1. The measured per-unit-length $R_{\rm x}$, $L_{\rm x}$, $G_{\rm x}$, $C_{\rm x}$ parameters of different structures are compared with the proposed model in Figs. 8(a)-8(d). Excellent agreement between the measured and simulated results is obtained with maximal $C_{\rm x}$ and $L_{\rm x}$ errors less than 5% over the entire frequency range up to 67 GHz. When $f$$>$ 20 GHz, $\omega L_{\rm x}$$\gg$$R_{\rm x}$ and $\omega C_{\rm x}$$\gg$$G_{\rm x}$, the errors of $G_{\rm x}$ and $R_{\rm x}$ contribute little to the overall impedance and admittance of the series and parallel branches and can be neglected. It can be seen from Fig. 8(c) that the model can fit the increasing of $C_{\rm x}$ well which is impossible for the model in Ref. [7-8].
Figure
7.
Layout of (a) SCPW, (b) CPW, and (c) Open-Short.
Figure
8.
(color online) Comparisons between the experimental data (dot) and the modeled results (line) of per-unit-length (a) resistance, (b) inductance, (c) capacitance, and (d) conductance.
$S$-parameter comparisons between measurements and simulated results of 400 $\mu$m CPWs with 10 cascaded blocks are shown in Fig. 9. Excellent data agreement of the three structures is achieved over the entire frequency range up to 67 GHz.
Figure
9.
(color online) $S$-parameter comparisons between the experimental data (dot) and the modeled results (line). (a) Magnitude of $S_{11}$. (b) Phase of $S_{11}$. (c) Magnitude of $S_{21}$. (d) Phase of $S_{21}$.
A unified CPW, GCPW and SCPW model for mm-wave applications is proposed and the direct extraction procedure is established for the first time in this paper. A new $C$-$L$-$R$ series path, together with the conventional one, is introduced to describe the electromagnetic coupling for CPWs with large lower ground or shield, while other kinds of high frequency effects including substrate loss, and skin effect are considered to explain the frequency-dependent per-unit-length Lx, Cx, Rx, and Gx parameters. Direct extraction methodologies of the model are presented and verified by measurement on different CPW structures fabricated in 90nm IBM CMOS processes. Excellent agreement of RLGC parameters and $S$-parameter between the measured data and modeled result is obtained with maximal $C_{\rm x}$ and $L_{\rm x}$ errors of $ < $ 5% over the entire frequency range up to 67 GHz. The direct extraction methodologies ensure the feasibility and availability of scalable modeling of CPWs for circuit designer.
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Jun Luo, Lei Zhang, Yan Wang. Modeling and parameter extraction of CMOS on-chip coplanar waveguides up to 67 GHz for mm-wave applications[J]. Journal of Semiconductors, 2013, 34(12): 125008. doi: 10.1088/1674-4926/34/12/125008 ****J Luo, L Zhang, Y Wang. Modeling and parameter extraction of CMOS on-chip coplanar waveguides up to 67 GHz for mm-wave applications[J]. J. Semicond., 2013, 34(12): 125008. doi: 10.1088/1674-4926/34/12/125008.
Jun Luo, Lei Zhang, Yan Wang. Modeling and parameter extraction of CMOS on-chip coplanar waveguides up to 67 GHz for mm-wave applications[J]. Journal of Semiconductors, 2013, 34(12): 125008. doi: 10.1088/1674-4926/34/12/125008
****
J Luo, L Zhang, Y Wang. Modeling and parameter extraction of CMOS on-chip coplanar waveguides up to 67 GHz for mm-wave applications[J]. J. Semicond., 2013, 34(12): 125008. doi: 10.1088/1674-4926/34/12/125008.
Figure Fig. 1. Proposed unified equivalent circuit model for CPWs.
Figure Fig. 2. Parallel branch of the CPWs model. (a) $C_{\rm sg1}$. (b) $C$-$R$-$C$ network. (c) $C$-$L$-$R$ series path.
Figure Fig. 3. Proposed extraction flow of the unified CPW model.
Figure Fig. 4. (color online) Serial extraction of a CPW, comparisons of (a) $\omega^{2}$/$R_{\rm x}$ versus $f^{2}$, (b) $L_{\rm x}$$\omega^{2}$/$R_{\rm x}$ versus $f^{2}$, (c) $R_{\rm x}$ versus $f$, (d) $L_{\rm x}$ versus $f$ between model and measured results.
Figure Fig. 5. (color online) $C$-$R$-$C$ extraction of a CPW, comparisons of (a) $C_{\rm x}$$\omega^{2}$/$G_{\rm x}$ versus $f^{2}$, (b) $\omega^{2}$/$G_{\rm x}$ versus $f^{2}$, (c) $C_{\rm ox}$ versus $f$, (d) $G_{\rm x}$ versus $f$ between model and measured results.
Figure Fig. 6. (color online) $C$-$L$-$R$ extraction of an SCPW, comparisons of (a) $\omega$$Z_{\rm parallel}$ versus $f^{2}$, (b) $C_{\rm x}$ versus $f$, (c) $G_{\rm x}$ versus $f$ between model and measured results.
Figure Fig. 7. Layout of (a) SCPW, (b) CPW, and (c) Open-Short.
Figure Fig. 8. (color online) Comparisons between the experimental data (dot) and the modeled results (line) of per-unit-length (a) resistance, (b) inductance, (c) capacitance, and (d) conductance.
Figure Fig. 9. (color online) $S$-parameter comparisons between the experimental data (dot) and the modeled results (line). (a) Magnitude of $S_{11}$. (b) Phase of $S_{11}$. (c) Magnitude of $S_{21}$. (d) Phase of $S_{21}$.