1. Introduction

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.

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).

3. Extraction methodology

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.

3.1 Series branch extraction

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:

$ $$\begin{equation} \label{eq1} R_{\rm x} =\frac{R_{\rm hf} L_1^2 \omega ^2+R_{\rm hf}^2 R_1 +R_{\rm hf} R_1^2 }{L_1^2 \omega ^2+(R_{\rm hf} +R_1 )^2}, \end{equation}$$ $

(1)

$ $$\begin{equation} \label{eq2} L_{\rm x} =\frac{R_{\rm hf}^2 L_1 }{L_1^2 \omega ^2+(R_{\rm hf} +R_1 )^2}+L_{\rm hf} . \end{equation}$$ $

(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:

$ $$\begin{equation} \label{eq3} \frac{\omega ^2}{R_{\rm x}}=\frac{L_1^2 \omega ^2+(R_{\rm hf} +R_1 )^2}{R_{\rm hf} L_1^2 }, \end{equation}$$ $

(3)

$ $$\begin{equation} \label{eq4} \frac{\omega ^2}{R_{\rm x}}L_{\rm x} =\frac{L_{\rm hf} (L_1^2 \omega ^2+(R_{\rm hf} +R_1 )^2)+R_{\rm hf}^2 L_1 }{R_{\rm hf} L_1^2 }. \end{equation}$$ $

(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.

3.2 Parallel branch extraction

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:

$ $$\begin{equation} \label{eq5} \frac{C_{\rm x} }{G_{\rm x} }\omega ^2=\frac{C_{\rm si} R_{\rm si} (C_{\rm si} +C_{\rm ox} )}{C_{\rm ox} }\omega ^2+\frac{1}{R_{\rm si} C_{\rm ox} }, \end{equation}$$ $

(5)

$ $$\begin{equation} \label{eq6} \frac{\omega ^2}{G_{\rm x}}=\frac{R_{\rm si}(C_{\rm si} +C_{\rm ox} )^2}{C_{\rm ox}^2 }\omega ^2+\frac{1}{R_{\rm si} C_{\rm ox}^2 }, \end{equation}$$ $

(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).

$ $$\begin{equation} \label{eq7} R_{\rm sg2} ={\rm real}(Z_{\rm parallel} ), \end{equation}$$ $

(7)

$ $$\begin{equation} \label{eq8} \omega \cdot{\rm imag}(Z_{\rm parallel} )=L_{\rm sg2} \omega ^2-\frac{1}{C_{\rm sg2} }, \end{equation}$$ $

(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.

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].

$S$-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.

5. Conclusion

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.