Broadband accuracy

The proposed T-model has been extensively verified by comparison with measurement in terms of S-parameter (S11, S22), L(ω), Re(Zin(ω)), and Q(ω) over broad bandwidth up to 20 GHz. The scalability is validated by various geometries with split of coil numbers, N=2.5, 3.5, 4.5, and 5.5, and width, W=3, 9, 15, 30μm for N=1.5.

Broadband accuracy is justified by good match with measurement in terms of the mentioned key performance parameters. Figure 3.23 (a)-(d) and figure 3.24 (a)-(d) indicate the comparison for magnitude of S21 (Mag (S21)) between the T-model and

measurement.

Figure 3.23 Comparison of S21 (magnitude) between T-model simulation and measurement for spiral inductors. Coil numbers (a) N=1.5, (b) N=2.5, (c) N=3.5, (d) N=4.5

Figure 3.24 Comparison of S21 (magnitude) between T-model simulation and measurement for spiral inductors. Width for N=1.5 (a) W=3μm, (b) W=9μm, (c)

W=15μm, (d) W=30μm

Excellent match is achieved for all coil numbers before resonance and agreement of curvature in maintained beyond resonance, which happened at fSR << 20GHz for larger coil numbers (N=4.5 and 5.5). We will explain above condition in the next section. Figure 3.25 (a)-(d) and figure 3.26(a)-(d) show the good agreement in terms of phase (S21) in which precise match of resonance frequency (fSR) is demonstrated for all coil numbers and width for N=1.5.

0 2 4 6 8 10 12 14 16 18 20

Figure 3.25 Comparison of S21 (phase) between T-model simulation and measurement for spiral inductors. Coil numbers (a) N=1.5, (b) N=2.5, (c) N=3.5, (d) N=4.5

0 5 10 15 20

Figure 3.26 Comparison of S21 (phase) between T-model simulation and measurement for spiral inductors. Width for N=1.5 (a) W=3μm, (b) W=9μm, (c) W=15μm, (d) W=30μm

Figure 3.27 and 3.28 reveals the exact match of Mag(S11) for smaller coils (N=2.5, 3.5) and various width for N=1.5 over full frequency range up to 20GHz, which is well beyond resonance for larger coils (N=4.5, 5.5) with fSR =6.9, 5.1 GHz, i.e., far below 20GHz, it happened to be a common issue suffered by EM simulation. Fortunately, this deviation didn’t make effect on the accuracy of L(ω), Re(Zin(ω)), and Q(ω) beyond resonance. Moreover, figure 3.29 and 3.30 confirms the model accuracy in terms of phase (S11) over broadband beyond resonance for each coil number and width for N=1.5

More extensive verification has been done by comparison of four key performance parameters for spiral inductors, i.e., L(ω), Re(Zin(ω)), Q(ω), and fSR. L(ω) is the imaginary part of input impedance Zin(ω), i.e.,

ω ω L ( ) = I ( _m Z _in ( )) ω

represents the real part of Zin(ω).

Figure 3.27 Comparison of S11 (magnitude) between T-model simulation and measurement for spiral inductors. Coil numbers (a) N=1.5, (b) N=2.5, (c) N=3.5, (d) N=4.5

Figure 3.28 Comparison of S11 (magnitude) between T-model simulation and measurement for spiral inductors. Width for N=1.5 (a) W=3μm, (b) W=9μm, (c) W=15μm, (d) W=30μm

Figure 3.29 Comparison of S11 (phase) between T-model simulation and measurement for spiral inductors. Coil numbers (a) N=1.5, (b) N=2.5, (c) N=3.5, (d) N=4.5

0 5 10 15 20

Figure 3.30 Comparison of S11 (phase) between T-model simulation and measurement for spiral inductors. Width for N=1.5 (a) W=3μm, (b) W=9μm, (c) W=15μm, (d) W=30μm

In this chapter, Q(ω) is the quality factor defined by

I _m ( Z _in ( )) / ω R Z _e ( _in ( ) ω )

All three parameters are frequency dependent that is critically related to the spiral conductor loss and Si substrate loss. In fact, accurate simulation to predict L(ω), Re(Zin(ω)) and Q(ω) is the major goal of inductor models for physical element design.

In our research, the proposed T-model can provide very good match with the measurement for the three parameters. Figure 3.31 and 3.32 illustrates the excellent fit to the measured L(ω) by T-model for all spiral inductors operating up 20GHz. The transition from inductive to capacitive model evoked by increasing frequency beyond fSR is accurately reproduced by the model. Regarding Re(Zin(ω)), good match between the T-model and measurement are shown in figure 3.33 and 3.34. The T-model can exactly capture the full band behavior of Re(Zin(ω)) even beyond resonance such as

the dramatic increase prior to resonance, peak at resonance, and then sharp drop after the peak. Moreover, Q(ω) is also good match. Q(ω) is for the primary concern for inductor design and the first key parameter governing RFIC performance such as power, gain, and noise figure, etc. Figure 3.35 and 3.36 reveals the excellent match with the measured Q(ω) over the broad bandwidth of 20GHz. Self-resonance frequency fSR is a key parameter accompanying with Q(ω) to quantify the useful bandwidth. In T-model, fSR can be accurately predicted by both full equivalent circuit simulation and analytical model of closed form given by equation 3.10

1 2

Figure 3.31 Comparison of L(ω) between T-model simulation and measurement for spiral inductors. Coil numbers (a) N=1.5, (b) N=2.5, (c) N=3.5, (d) N=4.5

0 5 10 15 20

Figure 3.32Comparison of L(ω) between T-model simulation and measurement for spiral inductors. Width for N=1.5 (a) W=3μm, (b) W=9μm, (c) W=15μm, (d) W=30μm

Figure 3.33 Comparison of Re(Zin(ω)) between T-model simulation and measurement

Figure 3.34 Comparison of Re(Zin(ω)) between T-model simulation and measurement for spiral inductors. Width for N=1.5 (a) W=3μm, (b) W=9μm, (c) W=15μm, (d) W=30μm

0 5 10 15 20

Figure 3.35 Comparison of Q(ω) between T-model simulation and measurement for spiral inductors. Coil numbers: N=2.5, 3.5, 4.5,5.5

0 5 10 15 20

Figure 3.36 Comparison of Q(ω) between T-model simulation and measurement for

spiral inductors. Width for N=1.5 (a) W=3μm, (b) W=9μm, (c) W=15μm, (d) W=30μm

According to equation 3.10, the analytical model is readily derived under appropriate approximation. The details of model equation derivation can be referred to the Appendix. The major approximation made by removing Lsub and Rloss (i.e. neglect eddy current effect.) was justified by impedance analysis and equivalent circuit simulation. Figure 3.37 presents Q(ω) calculated by reduced T-model without Lsub and Rloss and the comparison with original T-model with Lsub and Rloss.

0 5 10 15 20

-5 0 5 10 15 20

N=2.5 N=3.5

N=4.5 N=5.5

T-model, L

=0 T-model, w/i L

Mea

Q

frequency (GHz)

Figure 3.37 Comparison of Q(ω) and self-resonance frequency fSR corresponding to Q=0 among T-model, reduced T-model (Lsub = Rloss =0) and measurement for spiral inductors with various coil numbers.

N f

, Measured

f

,ADS equ. ckt sim

f

, Analytical model

f

, T-model simulation L

=0 L

=0

2.5 16.4 16.43 16.474

17.15

3.5 10.2 10.214 10.362

10.92

4.5 6.9 6.944 7.18

7.64

5.5 5.1 5.071 5.326

5.73

Table

Comparison of self-resonance (fSR) among measurement, simulation by original T-model and reduced T-model, and calculation by analytical model

The major difference is revealed in higher frequency region beyond the peak Q but the intercept point corresponding to Q=0. In next table, we will lists the exact values of fSR for comparison among measurement, simulation by original T-model and reduced T-model, and calculation by analytical model of equation 3.10.According to above table, the good agreement to each other in terms of deviation below 0.2 GHz justifies the approximation for reduced T-model and derived analytical model for fSR

The accuracy of fSR calculated by the equivalent circuit simulation and analytical model is further validated by good match with the measured result shown in figure 3.38 (a)

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 0

5 10 15 20 25

Measured

ADS equ. ckt sim Analytical model

f SR (G Hz)

Coil Number (a)

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 0

5 10 15 20 25 30 35 40

Analytical equation,Cp=0 ADS simulation,Cp=0 Analytical equation,Cox=0 ADS simulation,Cox=0 Analytical equation,Csub=0 ADS simulation,Csub=0

Measured

f SR (G Hz)

Coil Number

(b)

Figure 3.38 (a) Self-resonance frequency fSR of on –chip spiral inductors with various coil numbers, N=2.5, 3.5, 4.5, 5.5 (a) comparison between measurement, ADS

simulation, and analytical model. (b) Cp, Cox, and Csub effect on fSR calculated by ADS simulation and analytical model. Comparison with measured fSR to indicate the fSR

increase contributed by eliminating the parasitic capacitances, Cp, Cox, and Csub

respectively.

Regarding the parasitic capacitance effect on fSR as mentioned previously, Figure 3.38 (b) indicates the Cp, Cox, and Csub effect on fSR predicted by ADS simulation using full equivalent circuit and analytical model given by equation 3.10. The results from ADS simulation and analytical model show very good consistency. We see that the elimination of Cp or Csub can help to increase fSR by around 15~20% corresponding to N = 2.5, 3.5, 4.5, 5.5 while the elimination of Cox can dramatically boost fSR by more than 100%, i.e., more than double the existing performance for all coil numbers. The prediction from our T-model suggests that Cox plays a dominant role in determining fSR and spiral inductor on package is a potential solution to minimize Cox and achieve maximum fSR