Chapter 3 Broadband and Scalable On-chip Inductor
4.1 Symmetric inductor design and fabrication
4.1.4 Comparison with conventional symmetric inductors
The conventional symmetric inductors is realized by joining groups of coupled metal lines from one side of an axis of symmetry to the other using a number of cross-over and cross-under connections. This connection method may cause port 1 and port 2 to be not equivalent, especially for S11 and S22. This style of winding was generally applied to monolithic transformers for coupling both primary and secondary coils but this kind of layout suffers some inconvenience for metal connection from port 3 to other components. The fully symmetric inductor of new layout design was proposed and fabricated to solve the mentioned weakness. Verification will be done by checking the equivalence of S11 and S22 , particularly the phase portion. The improved symmetry is one of major reasons why we adopted this new structure. (refer to figure 4.4). One major advantage provided by the new symmetric inductor is the easy extension from 2-port to 3-port scheme for applications of differentially driven circuit design. The inductor quality factor (Q) can be significantly enhanced through
operation mode of differential excitation as compared to the conventional single end scheme.
For a symmetric inductor designed for differential excitation, the voltage and current at port -1 and port-2 are 180o out of phase. The symmetric inductor excited differentially can realize a substantially higher Q. It has been recognized that the quality factor an inductor is higher when driven differentially than subject to single-ended excitation. The mechanism responsible for Q enhancement is due to the reduced substrate loss under balanced excitation. A natural field of applications of such kind of inductors is the tank circuit of differential VCOs where the center tap is used for biasing and the capacitances are connected across the symmetric terminals.
Previous work reported that use of differentially driven symmetric inductor can reduce the total VCO area by 35% compared to a design using two conventional spiral inductors. an additional benefit is the improved electrical performance due to the increased tank impedance caused by higher Q. Accordingly, a larger output voltage swing and a reduced phased noise can be achieved.
4.2 Symmetric inductor model development
For computer-aided design (CAD) purposes, a lumped-element equivalent or SPICE-compatible model is needed to predict the large signal performance of an RF circuit correctly. The lump element sub-circuit in the scalable RF model is developed.
In this section, a new symmetric inductor model has been developed to accurately simulate the broadband characteristics of on-Si-chip symmetrical inductors, up to 20GHz. Good match with the measured S-parameter, L(ω), Re(Zin(ω)), and Q(ω) proves the proposed 2T-model. Besides, in order to quantify the improvement in Q factor of the differentially driven symmetric design, we also built differential model in
2T-model. The broadband feature and scalability have been justified by good match with a linear function of inner radius for all model parameters employed in the RLC network. A parameter extraction flow is established through equivalent circuit analysis to enable automatic parameter extraction and optimization. In next section, we will discuss them in detail.
4.2.1 Model parameter extraction flow
In this section, we extend the parameter extraction flow from previously T-model for spiral inductors to 2T model for symmetric inductors. All the unknown R, L, C parameters will be extracted from the analytical equations derived through equivalent circuit analysis as shown in figure 4.37. The analytical equations are composed of Z-parameters and Y-parameters listed in the first block of extraction flow illustrated in figure 4.38, which can be easily transformed from the measured S-parameters after appropriate de-embedding developed for this fully symmetric inductor. Under the condition that the number of unknown elements is larger than the number of equations, approximation valid under very low or very high frequency is generally made to remove some unknown elements and extract the remaining ones as the first step.
Then we can extract the others at the second step and go for optimization. According to the necessary approximation, the extracted R,L,C parameters in the first procedure of flow (figure 4.38) are generally not exactly the correct solutions but all unknown parameters must be given the initial guess for further optimization through best fitting to the measured S-parameters, L(ω), Re(Zin(ω)), and Q(ω).
(a)
(b)
(c)
Figure 4.5 2T model for fully taper symmetric inductor (a) equivalent circuit schematics (b) intermediate stage (c) final stage of block diagram for circuit analysis
Figure 4.6. 2T-model parameter derivation formulas and extraction flow chart.
As a result, all the physical elements composing the model can be extracted through the flow shown in figure 4.38. Attributed to the symmetric nature, all the elements appearing at both sides are assumed equal to simplify the problem, i.e.
Ls=Ls1=Ls2, Rs=Rs1=Rs2, Lsk=Lsk1=Lsk2, Rsk=Rsk1=Rsk2 and Cox=Cox1=Cox2=Cox3. At the first step, skin effect incorporated elements Rsk and Lsk are neglected, then Rs and Ls
representing the physical inductor under very low frequency can be extracted. After extraction of Ls and Rs, the measured Y-parameters (Y21) and extracted (Ls, Rs) are adopted to extract Lsk and Rsk. The ideal quality factor free from conductor and substrate losses, denoted as Qs is given by Qs= L s/Rs. After that, Rp and Cox can be extracted by close forms as a function of (Ls, Rs, Qs) and measured Ydiff=1/Zdiff
(equivalent impedance of differential mode). Rp is a new element introduced in our 2T-model and it represents conductor loss and Q degradation before resonance (ω<ωSR). Cox is one major element to determine self-resonance frequency (ωSR) and can be derived by the equation shown in the flow chart. Then, Csub ,Rsub and Rloss can be extracted easily from Zsub under very low frequency and the assumption of Rsub= Rloss for initial guess. To the end, Lsub is extracted under very high frequency for initial guess and then (Rsub, Rloss, Lsub, Csub) are iterated to obtain the optimized parameters.
In table I, the initial guess and optimized value of each parameter corresponding to different radiuses are listed. Herein, the substrate parameters such as Rsub, Lsub, and Rloss reveal obviously bigger error in the initial guess. The error generally came from the assumption and approximation to simplify the equations for extraction. However, few iterations were required to reach optimization.
R=30 Initial guess Optimize Error (%)
L
s( nH ) 0.302 0.249 -21.285%
R
s( Ω) 0.458 0.507 9.665%
L
sk( nH ) 0.1 0.127 21.260%
R
sk( Ω) 0.34 0.22 -54.545%
C
ox( fF ) 15.11 15.04 -0.465%
C
sub( fF ) 14.94 12.682 -17.805%
R
sub(Ω) 78.865 394.625 80.015%
R
p( Ω ) 757 773.59 2.145%
L
sub(nH) 0.028 0.101 72.277%
R
loss(Ω) 78.865 76.664 -2.871%
R=60 Initial guess Optimize Error (%)
L
s( nH ) 0.568 0.569 0.176%
R
s( Ω ) 0.52 0.62 16.129%
L
sk( nH ) 0.127 0.111 -14.414%
R
sk( Ω ) 0.413 0.324 -27.469%
C
ox( fF ) 31.09 34.7 10.403%
C
sub( fF ) 30.8 35.458 13.137%
R
sub(Ω) 41.875 188.635 77.801%
R
p( Ω ) 828 829.973 0.238%
L
sub(nH) 0.0898 0.151 40.530%
R
loss(Ω) 41.875 60.8 31.127%
R=90 Initial guess Optimize Error (%)
L
s( nH ) 0.975 0.912 -6.908%
R
s( Ω ) 0.7 0.797 12.171%
L
sk( nH ) 0.08 0.083 3.614%
R
sk( Ω ) 0.51 0.437 -16.705%
C
ox( fF ) 43.86 53.326 17.751%
C
sub( fF ) 45.3 52.55 13.796%
R
sub(Ω) 24.445 125.98 80.596%
R
p( Ω ) 913 902.212 -1.196%
L
sub(nH) 0.12 0.206 41.748%
R
loss(Ω) 24.445 33.665 27.387%
Table I Comparison with initial guess and optimize and error percentage
4.2.2 Broadband accuracy
The 2T-model has been verified by comparison with measurement in terms of S-parameters (S11, S21), Zdut1, Zdut2, L(ω), Re(Zin(ω)), and Q(ω) over frequency up to 20 GHz. For differential excitation, the 2T-model has also been verified by comparison with measurement in terms of Sd(ω), Ld(ω), Re(Zd(ω)),and Qd(ω). Figure 4.39 (a) ~ (d) indicate S11 and S21 in terms of magnitude and phase from measurement and simulation by 2T model. Good agreement is achieved between measurement and 2T-model simulation for all S-paramters. Figure 4.40 presents good match in respect of L(ω) and Re(Zin(ω)). It is even better match achieved as compared to 3D EM simulation by Ansoft HFSS (figure 4.23 ~ 4.29). Fig.4.40 (a) illustrates excellent fit to the measured L(ω) by 2T model for all symmetric inductors of various R operating up
to 20 GHz. The transition from inductive to capacitive mode evoked by increasing frequency beyond fSR is accurately reproduced by 2T model. Regarding Re(Zin(ω)), pretty good match between 2T model and measurement is shown in Fig.4.40 (b). 2T model can exactly capture the full band behavior of Re(Zin(ω)) even beyond resonance such as dramatic increase prior to resonance, peak at resonance, and then sharp drop after the peak. Moreover, the 2T-mdoel has a new feature created to simulate single-end and differential modes’ performance by a unified model parameters. In this way, it can reduce circuit simulation time. Figure 4.41 (a) ~ (d) exhibits good fit to measurement in terms of Sd, Ld, and Re(Zd) corresponding to differential excitation mode. Comparison between the single-end excitation in figure 4.40 and differential mode in figure 4.41 reveals obviously higher fSR for differential mode with delayed impedance sign change from inductive to capacitive mode. Figure 4.42(a)~(d) indicate the symmetric inductor coil impedance, Zdut1 and Zdut2 extracted from measurement after justified new de-embedding and calculated by 2T model using the optimized parameters. Good match is achieved between measurement and 2T model for both Zdut1 and Zdut2 in terms of real and imaginary parts for R=30 and 60 m over full range of frequency up to 20GHz. As for the largest inductor with R=90 m, good fit is maintained for Re(Z dut1) and Re(Zdut2) over 20GHz bandwidth but visibly larger deviation is identified for Im(Zdut1) and Im(Zdut2) at higher frequency, above 15GHz. Quality factors corresponding to single-end and differential modes defined as Q(ω) and Qd(ω) are two of most important parameters for symmetric inductors in circuit applications. Figure 4.43 shows good match with the measured Q(ω) and Qd(ω) by 2T model over broadband of 20 GHz. The good fit to the peak Q and capture of full band behavior for various R suggests the advantage of our 2T-model compared to the existing π-model or 2π-model. Self-resonance frequencies fSR are key parameters accompanying with Q(ω) and Qd(ω) to quantify the useful
bandwidth. In 2T-model, fSR can be accurately predicted by full equivalent circuit simulation. The extension verification proves the broadband accuracy of our 2T model and validate its applications for RF circuit simulation and design in which symmetric inductors will be adopted with single-end or differential configurations.
0 5 10 15 20
Figure 4.7. Comparison of 2T-model and measurement for R=30, 60, 90 μm (a) Mag (S11) (b) Phase (S11) (c) Mag (S21) (d) Phase (S21)
Figure 4.8. Comparison of 2T-model and measurement under single-ended excitation for R=30, 60, 90 m (a) L (ω) (b) Re(Zin(ω))
0 5 10 15 20
Figure 4.9. Comparison of 2T-model and measurement under differential excitation for R=30, 60, 90 μm (a) Re (Sd) (b) Im (Sd) (c) Ld (ω) (d) Re(Zd(ω))
Figure 4.10. Comparison of 2T-model and measurement for R=30, 60, 90 μm (a) Re(Zdut1(ω)) (b) Im (Zdut1(ω)) (c) Re(Zdut2(ω)) (d) Im (Zdut2(ω))
0 5 10 15 20 -5
0 5 10 15 20 25 30 35
40 mea_R30
mea_Qd_R30 mea_R60 mea_Qd_R60 mea_R90 mea_Qd_R90 2T-model 2T-diff-model
Q
frequency, f (GHz)
Figure 4.11. Comparison of Q(ω) between 2T-model simulation and measurement for fully taper symmetry inductor with various radiuses: R=30, 60, 90 μm
4.2.3 Model Scalability
Besides the broadband accuracy, another important feature realized by this 2T model is the good scalability w.r.t. geometry for all model parameters. Figure 4.44 and figure 4.45 present good match with a linear function of inner radius (R) for each model parameter in the symmetric spiral coils’ RLC network. Figure 4.46 reveal good fit with a linear function of R for substrate network involved model parameters, Csub, 1/Rsub, Lsub, and Rloss. The promisingly good scalability proven for full set model parameters suggests that this 2T model is useful in simulation for inductor layout optimization and design. The nature of easy link with standard circuit simulator makes
this 2T model useful in circuit element tuning and optimization for RF circuit design. It has emphasized the need for a powerful symmetry inductor to satisfy current circuit design trends. The development of an accurate and scalable equivalent circuit model for the symmetrical inductors named as 2T model has been demonstrated. The accuracy and continuity of both symmetrical inductance and quality factor for this scalable inductor model is closely examined and satisfyingly good agreement between the simulated and measured device characteristics has been realized. This 2T model can facilitate RF circuit design such as VCO, LNA, and mixer of differential circuit topology in which symmetric inductor become the key passive element to be adopted.
Cox1 (fF) Cox2 (fF)
Radius, R
(a)
Figure 4.12. 2T-model RLC network parameters versus inner radius, fully taper symmetry coil’s RLC network parameters (a) Ls1,2 (b) Rs1,2 (c) Cp1,2 (d) Cox1,2,3
30 40 50 60 70 80 90
Lsk =0.128+1.5E-4 R-7.22222E-6 R2 Lsk1, Lsk2
Taper moedl
Lsk1 (nH) Lsk2 (nH)
Radius, R
(
μm)
Rp1 (Ohm) Rp2 (Ohm)
Radius, R
Figure 4.13. 2T-model RLC network parameters versus inner radius, fully taper symmetry coil’s RLC network parameters (a) Lsk1,2 (b) Rsk1,2 (c) Rp1,2
Figure 4.14. 2T-model RLC network parameters versus inner radius, lossy substrate RLC network parameters (a) C (b) 1/R (c) L (d) R
Chapter 5 Future work
Simple T-model and 2T model of broadband accuracy for spiral inductors and fully symmetric inductors on a silicon substrate have been presented. The proposed model containing a combination of RLC networks has been developed to accurately simulate on-chip inductors operating up to 20 GHz. Verification with measurement data from various structures has validated the proposed models. Our models show excellent agreement with measured data over the entire frequency range of interest.
All model parameters are validated with good scalability with varying inductor geometries.
Moreover, substrate effect pertaining to on-chip is an important issue of major concern. Through EM simulation by Ansoft HFSS, they demonstrate that energy dissipation, which degrades Q, occurs predominately in the bulk silicon substrates of semiconducting property. Regarding the substrate effect, all model parameters manifest themselves the physical property associated with varying substrate resistivities. As a result, physics-based model parameters enable the developed models applicable for three operation modes (eddy current, slow wave, and TEM modes) under varying substrate resistivities.
In our research work, we design a new inductor structure, which can effectively reduce substrate parasitic effect through differentially driven operation. We name this new structure a fully taper symmetric inductor. A new de-embedding method has been developed accordingly to fit the new inductor structure of fully symmetric feature. We find that a symmetric inductor which is excited differentially can realize a substantial improvement in both Q factor and component bandwidth. Differentially excited
inductors can effectively reduce substrate parasitics that was demonstrated through both simulation and measurement. This leads to higher Q factor than for a signle-end excitation when fabricated in silicon technology. This improvement in Q factor can translate directly into lower phase noise and greater output signal swing for the oscillator, especially for the higher frequencies applications. In addition, the proposed and fabricated new symmetric inductor can ensure port-1 equal to port-2. Then, it will help to improve the RF circuit performance considerably such as power consumption.
A comprehensive extraction flow has been established through equivalent circuit analysis to enable automatic model parameter extraction and optimization. Moreover, our model can be easily implemented in SPICE-compatible simulator to improve accuracy in circuit simulation.
The future work emerging through this study can be summarized as follows. The first part is the implementation of T-model and 2T-model in Spice compatible circuit simulators to verify model accuracy in the design of RF circuits such as VCO, LNA, and mixer, etc. The second one is the further investigation of substrate resistivity effect subject to different inductor geometries such as metal strip width and space, coil radius, coil number, and coil shape, etc. Moreover, simple spiral of single coil set and symmetric inductor of two spiral coil sets will be covered. The third part is the extension of 2-port de-embedding method for spiral and symmetric inductors to 3-port de-embedding method for differential inductors or transformers. The last one is the extension of 2T model from application in fully symmetric inductor to that appropriate for stack symmetric inductors of elevated broadband and quality factor (higher fSR and Q).
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