Automatic Extraction of Analog Circuit Macromodels. Rohan gatra Advisor: I~rof. Pileggi

Transcription

1 Automatic Extraction of Analog Circuit Macromodels Rohan gatra 2005 Advisor: I~rof. Pileggi

2 Automatic Extraction of Analog Circuit Macromodels by Rohan Batra A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Electrical and Computer Engineering Carnegie Mellon University Advisor: Prof. Larry Pileggi Prof. Patrick Yue

3 Abstract Verifying a complete analog system via trar~sistor-level simulation is an extremely difficult process and can often become infeasible due to the limitation of simulation capacity. A similar difficulty is encountered when high-level design analysis is performed for the whole system. For these reasons, compact macromodels of analog blocks are desired which can be substituted in place of the actual ~ransistor-level netlist to speedup the simulation with sufficiently high accuracy. The NORM algorithm that was proposed in [1] utilizes the Volterra-Series to represent nonlinear transfer functions and employs projection-based techniques to significantly reduce the size of the nonlinear system equations, thereby generating compact representations for analog and RF circuits. This algorithm can be applied to time-invariant as well as time-varying "weakly" nonlinear circuits. Som extensions have been described in [2][3]. In the first part of this work, we describe a complete methodology which extends the NORM algorithm to generate nonlinear reduced-order macromodels directly from transistor-level netlists [4].Even though the NORM algorithm works extremely well for "weakly" nonlinear circuits, due to the limitations of the Volterra-Series [5], it cannot be extended to more "strongly" nonlinear circuits like Phase-Locked loops. Phase-locked loops are used in applications such as frequency synthesis and clock and data recovery. The nature of these applications requires high accuracy for predicting the PLL nonlinearities and dynamics during the design process, which thereby requires prohibitively expensive transistor-level simulation. In the second part of this report we propose a compact behavioral model for voltage-controlled oscillators, which are the key components of PLLs, and the most difficult to macromodel due to their dynamic, large-scale nonlinear behavior. Unlike previous works in this area, this approach follows a systematic modeling of nonlinear dynamics in a topology independent manner. We demonstrate that our model can reduce the system-level simulation runtime significantly without sacrificing the required accuracy.

4 Part - I : Automatic extraction of [nacro-models of weakly nonlinear circuits Introduction To generate analog macromodels that can be used in commercial simulation environments, the circuits under consideration must be characterized and then modeled based on industry-standard device models in these environments. We developed a complete methodology which extends the NORM algorithm [1] to generate nonlinear reduced-order macromodels directly from transistor-level netlists. The reduced nonlinear macromodels will capture the nonlinear characteristics of corresponding circuit blocks, such as IIP3, THD and gain compression, in a compact form while maintaining an accuracy comparable to commercial simulators such as SpectreP~ ~ and HSPICE. The purpose of developing compact nonlinear analog macromodels is two-fold. Firstly, macromodels can facilitate efficient system-level design exploration by allowir~g designers to effectively "re-use" the macromodels from their previous designs to predict the system-level behavior. High-level decisions and tradeoff analyses can be made efficiently by evaluating system specifications through the use of a library of "reduced-order" macromodels corresponding to a variety of circuit topologies and configurations. Secondly, compact component macromodels also facilitate the full-system verification which is otherwise intractable.

5 Background Volterra Series provide an elegant way to characterize weakly nonlinear systems in terms of nonlinear transfer functions. For a circuit with input u(t), the response x(t) can be expressed as the sum of responses at different orders: x(t) = ~ x. (t) (1) where, x. (t) is the n-th order response. More generally~ we can use Volterra kernels to capture both nonlinearities and dynamics by convolution [5]: n=l x(t): ~... ~h. t...,r.)u(t- 1)... z~(t-~:. )~. 9~:. (2) where, h. (rl... r.) is the nth order Volterra kernel. The frequency domain transform of the nth order Volterra kernel denoted by H. (s 1... s,,) is generally referred to as the nth order nonlinear transfer function. These nonlinear transfer functions are independent of the input and fully describe the weakly nonlinear behavior of the circuit. In order to apply the Volterra nonlinear transfer function for a SIMO weakly nonlinear system we can consider its standard MNA formulation: f(x(t))+ -~(q(x(t)) = bu(t), y(t) (3) For a circuit with a time-invariant operating condition given by x = x0, the first order linear transfer function is given by: (G~ + sc~ )Ht (s) (4) The symmetrized second order nonlinear transfer function is determined by [ 1 ]: (5) where k = s~ +sz and f~r equation (4) and (5): 4

6 1 ~ 0 1 ~ 0 G, = ~x,. (f)... C, = ~0--~ (q).~=~o H 1 (s~) H l (S 2) = ~(H~ (s~)@ H~ z) + H~ H~ (s~ ) When a nonlinear circuit has a large input excitation; however, it causes the operating points of the active devices to change with time. For example, in a ~xer, the operating condition with respect to ~ signal is dete~ned by a large LO signal rather than a fixed operating point. This requires the analysis of a small-signal excitation over a large periodic operating condition. Therefore, we can apply a time-va~ing formulation of the Volte~a transfer functions H. (t, s~, s~... s. ) which can be formulated si~lar to the fime-inv~iant case [9][2] [3]. Volte~a based nonline~ descriptions, however, often increase drastically with problem size, thereby maeng them ineffective when used directly. Therefore, we instead apply the projection based nonlinear reduced order method (NORM) proposed in [1] to reduce the model size. The algorithm computes a projection matrix by explicit]iy considering moment-~tching of nonline~ transfer functions. For example, if we expand the first-order transfer function H~ (s) at the origin: H~(s) = ~ s~g~,~, (6) k=0 where M1, k is a kth order moment for the first-order transfer function. Now, expanding the second-order nonlinear transfer function H 2 (S1, 2 )attheorigin (0,0 ~ k k=0 l=0 l k-i (7) where, M2,k, l is a kth order moment of the second-order transfer function. The actual expression for Mz,~,,i can be obtained by first substituting (6) into (5) and expanding w.r.t k 1 + sz.this procedure can be also applied to obtain the moments of the third order transfer functions. In NORM, a projection matrix is built such that the reduced order model will match certain number

7 of transfer function moments. It has also been demonstrated that multi-point expansion based approach produces much more compact models than the single-point expansion.

8 Overall Macromodeling Flow We outline the complete flow for the generation of reduced-order models from transistor-level netlists in Fig. 1. First, we simulate the transistor-level netlist in a commercial simulator such as SPICE to determine a proper operating condition for the circuit. In the case of a time-invariant circuit, a fixed DC operating point will be computed. Otherwise, a large-signal time-varying operating point will be computed for a time-varying circuit such as a mixer. We model the nonlinearities for each transistor in the circuit as a third-order polynomial. We simulate each transistor in the circuit multiple times, varying the bias-voltage for its terminals to generate accurate data-points for fitting the polynomial. We then construct the full Volterra-based model of the circuit and generate the reduced-order model of the circuit using NORM[ 1]. Spice simulation Determine op point Collect data-points Fit nonlinearities Order Model Reduced-order model [~alx+a2x2 +a3x3 +...[ Figure. 1. Extraction of reduced-order model

9 Extraction of Volterra Parameters The nonlinear modeling techniques outlined in Section 2 depend on extracting the parameters of the Volterra model accurately. In Volterra series, a nonlinearity is represented as a power series expansion around a bias point. To illustrate, let us consider a nonlinear device characteristic f(x) expanded about a bias point x0: where, f (x) = f o)+a~ (x- ~ 0 )+a 2 (x - 2 o) +... (8) ai = x~ ~ ~7"x~ 1 0 i f (x) Many different small-signal models for MOS transistors exist, and most sophisticated models include substrate coupling effects and transcapacitances[6][7]. Spice models like BSIM3 not only represent physical effects but also include many numerical parameters which further increase the complexity of the model equations. It is infeasible to find the coefficients of the equation given in (8) by finding the higher-order derivatives from the model equations in BSIM3 and other models. Instead, we employ least-mean-square error (LMSE) fitting techniques to find the coefficients [6]. We will show how the nonlinear parameters for the drain current of a MOS transistor are extracted. We model the drain current Ids as a third-degree polynomial with respect to the drain, source and gate voltages. For simplicity, we have used the body terminal as the reference voltage although other possibilities can be easily accommodated. The equation includes individual voltage terms as well as cross-terms. Compared to ordinary hand-analysis equations we model not only the first-order nonlinearities but also the second and third-order nonlinearities as smallsignal quantities around the bias point: Ia~ = las ~ + gav,~ + gsv~ + g~v~ + ga~vav~ +.. (9) gssvs V 3 "t-* "~gddsl)dvs... gsss s

10 where, I d, o = bias current value at operating point = small-signal voltage at terminal x = {d,g,s} first-order coefficient for voltage at terminal x gxy = second-order coefficient for cross-product of voltages at terminals x and y g xyz third-order coefficient for cross-product of voltages at terminals x, y and z Therefore, there are 3 first-order terms, 6 second-order terms and 10 third-order terms in the equation. It is not possible to get the second and third-order terms directly from transistor-level. simulation so we have formulated an efficient way to get these terms and model the nonlinearity accurately. We extract the first-order model parameters from Hspice simulation[8]. For a timeinvariant circuit, we perform a DC operating point analysis to obtain the bias current value and the first-order coefficients. For a time-varying circuit, we perform a single-tone transient analysis for a sufficient settling time and then sample a single time-period of the settled response to obtain time-varying operating point; for the circuit. We then perform a DC operating-point analysis at each of these points to get the first order coefficients ga, gs and gg. We can express these coefficients in terms of the more commonly used small-signal coefficients G,~, G,~. and Gm~,, : gg =Gin, gd =Gds and gs =-(Gds +Gin +Gmbs) (11) For both the time-invariant and time-varying cases, the bias voltages for each transistor are perturbed by small-amounts to obtain data-points for fitting the second and third-order coefficients in the appropriate fitting range represented by the bounding box shown in Fig. 2. The

11 figure shows the drain current as a function of Vds and Vg s./t is possible to measure the current lds by perturbing the Vd~, and Vg~ slightly around each bias-point to obtain many different points. From (9): (Iris i --ldso)= gdvdi + gsvsi + ggvg i +gdsvdivsi +.. (10) where, the subscript i denotes the i-th data-point and (las i - Ia~ o) is called the "residue" To solve the coefficients of the RHS in equation (10) we write the powers and cross-terms v d, vs and v s for n sampling points into matrix Y, the corresponding coefficients into the vector p and the residue (Ia, ~ - I,ts0) into matrix R: Vdl Y = Vd2 3 Vsl Vgl V~I Vd~Vgl... Vsl 3 Vs2 Vg2 V~t2 Vd2Vg2... Vs2 V dn 3 V stt V gtl 12 ~rt ~ dn l)gn... llsn P=[ga g,~ gg ga,~... gsss] and R=[I, I~... I,,] r I,, =Ia~,,-Ia,. o (11) We have to fit the coefficients of (10) such that the error for each of the n data-points around the operating point is minimized. The aggregate error for the ith data-point is denoted by e i. We have to minimize the error e = [e~ e 2... e,] r : Yp-R =e (12) The LMSE algorithm estimates p by minimizing the sum of squared errors: F = ete = (~ p-- R)~ (Yp- (13) This leads to the optimal solution: p = (Y~Y)-~.(YrR) (14) 10

12 In order to guarantee a good fit for the nonlinearities we ensure that the fitting range for the data is correct [6] (Fig. 2). The range must be large enough to fit the nonlinearities accurately but it should not attempt to cover the effects outside the signal-swing range. For example, if the gate of a transistor has an expected signal swing of ±10mV, the fitting range for vg of this transistor should be limited by the signal swing. It is also imperative to select enough data-points to fit the nonlinear parameters accurately. Ids 1 Vgs Vds Figure. 2. Effective fitting range for Volterra parameters In some cases, the :fitted results might still cause large relative errors for certain points in the data. The fit may be improved by using a weighted-least squares method instead of the conventional method. In this case, it is important to select individual weights for each equation: 14~(Idsi--Id~o) g,~vaiwi gsvsiwi + +o..+ gdsvdivsiwi. +. (15) such that the effective residue wi(ia~ i -las o) for each data-point is in the same range. This weighting scheme gives each individual data-point the same importance as far as the fitting process is concerned. This aids in reducing the error fbr each point and gives a better fit for the entire range of data-points. First, we can perfbrm a LMSE fit on the data to get an initial estimate 11

13 of the points where the error e i may be large. We select the appropriate weight w i for each datapoint and scale the residue accordingly. After performing the weighted least-squares fit using (16) and (17) we can look at the error e i again and modify the weights if we are still not satisfied with the results. This is done for a few iterations till we can no longer improve the results. For the weighted least-squares method we introduce another matrix, W, which is the diagonal matrix of the individual weights. We have to minimize the weighted least squares error function: F = (Yp - R)7"W(Yp - (16) where Y,p and R have been defined earlier. This can be solved to get the nonlinear parameters in (9): p = (YT"WY)-I. (YrWR) (17) 12

14 Results The methodology presented in the previous sections has been demonstrated on a doublebalanced mixer and an opamp. The macromodels generated using this approach are compared with detailed transistor-level simulation of the circuits with HSPICE. A Double-Balanced Mixer Figure. 3. A Double-Balanced Mixer A double-balanced mixer (Fig. 3) is modeled as a time-varying weakly nonlinear system with respect to the RF input. The LO frequency is set at I[GHz, and we calculate the time-varying operating point of the circuit by setting the RF input voltage to zero and using transient analysis in Hspice to sample a single time-period of the settled response. The third order nonlinearities are modeled around this time-varying operating point using numerical fitting techniques outlined in Section 4. The fitted second and third order coefficents are used to generate a Volterra-based full model for the circuit. A single-tone RF input is used to verify the model results with the transient simulation results. The third-order harmonic of the RF input frequency down-converted with respect to the LO frequency is compared between the model and the simulation results. The second order nonlinearities should ideally be zero except for numerical noise, by design. We performed 13

15 transient analysis for the circuit in Hspice followed by an accurate Fourier Transform of the output time-domain waveform to verify the results. The RF input frequency is varied from 300MHz to 1200MHz. The maximum error in the full model as compared to Hspice simulation for the first-order results is less than 2% for all frequencies. The maximum error in the third-order results is less than 10% for third-order for all frequencies. The results in Fig. 4 have been normalized with respect to the RF input amplitude. -4 I~ I O "1200 Fr equen c~" (Mhz) [D Hspic e Sin ulation [] Our rrodel ] Figure. 4. Third-order harmonic (down-converted) for different input frequencies Once we have the Volterra Series based full model, it is possible to measure the third-order response at more useful harmonics also. For example, Fig. 5 shows the plot for the third-order transfer functionh3(t.,j2nf~,j2zf2,j2rf3)where,3oomhz<_f, f2<12oomhz are the RF inputs and f3 = 1GHz is the LO frequency. The full model has 1350 rime-sampled circuit unknowns which is reduced to approximately 14 circuit variables using the NORM method, while still capturing the dominant response of the circuit. The relative modeling error between the full and reducedorder model for the first-order results is less than 0.01%. Fig. 6 shows that the relative percentage error between the full-model and reduced-order model for the third-order results is less than 6% for all cases. 14

16 2 1/V ~ 400 ~ 300 ~ 200 ~ 2 Frequency (Hz) Figure. 5. Third-order transfer function for mixer(full) x ~ I Frequency (Hz) Figure. 6. Relative modeling error for third-order transfer function 15

17 An Operational Amplifier A two-stage Operational Amplifier topology is shown in Fig. 7. The closed-loop AC response of this circuit is shown in Fig. 8. Using the extraction method described in Section 4, it is possible to match the AC (first order response) of the circuit accurately to about % compared with Hspice simulation. For this circuit, second order nonlinearities are more important than the thirdorder nonlinearities since they are much higher in magnitude. The opamp is modeled as a timeinvariant system and is linearized at the DC bias point to fit the second and third order coefficients for each transistor in the circuit. The second-order distortion for a single-tone input is shown in Fig. 9. We compared the Hspice simulation results for input frequencies ranging from 1 MHz to 100 MHz with our model results. The relative error between the full-model and the simu]iation results is less than 10% for all input frequencies. The number of state-variables for the circuit reduced from 22 in the full-model to about 5 in the reduced order model. The comparison of the full and reduced order model results shows that there is less than 0.01% error for both first and second-order responses. Vdd Figure. 7. A two-stage opamp 16

18 ~... T... ~... ~... ~ Frequency (Hz) x 1 Figure. 8. Closed-loop AC response of the opamp 10 Hspice Simulation Full Model Reduced Order Model "/ Frequency (Hz) 18 O Figure. 9. Second order distortion as a function of frequency 17

19 Part - II A Behavioral-Level Approach for Nonlinear Dynamic Modeling of Voltage- Introduction Controlled Oscillators Phase-locked loops (PLLs) have found their way in many important applications, such frequency synthesis and clock/data recovery. The performance validation of a PLL at the system level generally requires transistor-level simulation with tight accuracy control. This accuracy requirement, however., combined with the the nonlinearities and dynamics inherent in a complex PLL, lead to prohibitively expensive simulation runtime to evaluate key performance specifications such as acquisition time, capture/lock range and phase noise. In many cases, a full SPICE simulation of the complete PLL can even become infeasible based on the meantime to failure for the computer. For this reason, compact macromodels of the key PLL components are desired to be substituted in place of the actual transistor-level models without sacrificing any of the required accuracy. A key component of the PLL (Fig. 10) is the voltage-controlled oscillator (VCO), which has dynamic large-scale nonlinear behavior that substantially impacts the overall performance of the PLL. Input ~ +..I Phase Loop Detector Filter t~ t._~ Voltage, Controlled Oscillator Output Divider Fig. lo.block diagram of a PLL. Referring to the block diagram of a PLL in Fig. 10, the phase detector and loop filter can be readily modeled using behavioral-level or circuit-level models that can capture the non-idealities, including the weak nonlinearities. However, the nature of the VCO makes it a very challenging 18

20 modeling problem. The output frequency/phase of the VCO is a nonlinear function of the input control voltage (Fig. 11). Moreover~ the output of the VCO is designed to be very sensitive slight variations in the input control voltage, which makes the dynamic response of the VCO very important. Phase (frequency)-domain models of the VCO have been used in the past [11][12][17] to speed-up the simulation, and significant progress has been made in analyzing the behavior of these oscillators, especially in terms of their plhase noise [181119][20][21][22]. The static nonlinear behavior can be captured easily by sweeping the input control voltage and measuring the output frequency [ 11] [ 12], but modeling the dynamics is much more challenging [13][14][15][16]. Most attempts for modeling the dynamics are very simplistic and do not guarantee sufficient accuracy. In this report we propose a systematic approach to accurately capture both the static nonlinear behavior and nonlinear dynamics of a VCO using a behavioral model [ 10]. Importantly, our approach is topology independent and does not require specific information regarding the characteristics of the VCO that is being modeled. 19

21 VCO model For simulation efficiency and generality, we propose a behavior modeling approach to relate the VCO input voltage directly to the output frequency. To capture both the static and dynamic nonlinear behavior in the model, we propose a second-order differential equation template is used to describe the VCO:.? = g(f,j:,u,~) (18) where f, ) and j~ are the output frequency, and its first and second order derivatives, respectively; u and ~ are the input control voltage and its first order derivative, g(.) is nonlinear mapping function. In our model, g(.) is a polynomial inf, ~, j~, u and ~ : g(x1,x2,x3,x4) = ~ ajx~ +~.~ aijxix j +E aijkxixjxk +"" (19) j i,j i,j,k where at, % and ai~ are the coefficients to be determined in the model. Note that the model template in (19) is completely generic, and not tied to any specific circuit topology or VCO architectures. In the following sections we will outline a two-step simulation based characterization process to extract the model parameters which capture both the static and dynamic behavior of a VCO. 20

22 Static nonlinear modeling Prior to modeling the dynamics of a VCO, our first step is to preserve the nonlinear static voltage-to-frequency behavior. As shown in Fig. 11, we want to ensure that in the steady-state our model will output a frequency which closely follows that of the original circuit. Output Frequency H.~_ Input Control Voltage Fig. ll.static nonlinear behavior of a VCO. The model proposed in (19) contains both static terms (e.g. terms containing only f, u ) and their cross-terms and dynamic terms which contain the higher-order derivatives of these terms. In order to ensure that the model is "correct" in the steady-state, we need to consider a reduced static nonlinear behavior by excluding the derivative terms in (20): where j = 1,2,x I = u, x 2 = f. ~ajxj +Eaijxixj +~_~aijkxixjxk (20) j i,j i,j,k In order to fit the coefficients of this template, we need to employ different control voltages at the VCO input and measure the output frequency at those control voltages as shown in Fig. 11. This can be implemented easily in most commercial simulators. For a particular input voltage, the frequency of the output waveform can be obtained using the time-difference At between adjacent zero-crossing times on the waveform (Fig. 12). The output frequency can be calculated 21

23 as f =1/2nSt. In SpectreRF, for example, this can be done using the "frequency" command [13]. The data obtained from this procedure can be used to fit the coefficients in the static model of (20). Vout Fig. 12. Static nonlinear behavior of a VCO. 22

24 Dynamic nonlinear modeling In addition to the static behavior, we capture the nonlinear VCO dynamics in our behavioral model by determining the remaining coefficients in equations (21) and (22). We apply multiple sinusoidal voltage inputs to the VCO, and use the simulation data in a least square fitting procedure to fit these coefficients. For simplicity, consider applying a single-tone sinusoidal input which can be rewritten in the complex exponential forrn: u = v 0 + v s cos(g0t) (21) U : V 0 q- -~ --~-. ~ vs (ejo~ -t-e ) (22) The output signal (instantaneous oscillation frequency) will contain a frequency component which is at the same frequency of the input-voltage tone. However, due to circuit nonlinearities, the output signal will also contain some harmonics of the input tone (possibly including a DC term). By using circuit simulation and post-simulation processing, we express the output in terms of first k harmonics of the input tone: For our experiments in this paper we chose K = 3. Given the input and output expressions in (22) and (23), their time derivatives can be readily computed, l~or instance, the first derivative of the output is given as K je = Z Jka)o~keJk~ (24) k=-k Substituting the expressions of u, f and their derivatives into (21) and equating the coefficients M harmonic components at the both side of equation, we have obtain the following set of equations: F(co)~ = ~(09) (25) 23

25 where ~ = [a~,az,a~,a4,...,au,...,a~,...~ is the vector of model parameters, F(co) is the coefficient matrix given by - F(a)) = f12 (co) (26) oo f,~ (0~) and g(@=[b (@,bt(~o),b2(@,...,bm(w)~is the constant right l~and side Note that there are M equations in (26), one for each harmonic. Each entry in the coefficient matrix F(@ represents contribution of the corresponding model parameter to a particular harmonic. For instance, f~ (~o) determines the contribution of model parameter auto the DC component Both the coefficient matrix and the right hand vector are functions of input voltage frequency ~o. It should be noted that some of model parameters may have been determined in the preceding static modeling step The parameter values are substituted into (26) to determine the remaining parameters. We determine the model parameters via optimization by applying N test inputs and setting up a least square fitting problem as follows: _b((on Although we only described how to set up a least square fitting by using single-tone input data, it is also possible to incllude multi-tone simulation data in (27). However, our experiments have shown that using single-tone test data is generally safficient. (27) 24

26 Implementation Since our generic model template abstracts the VCO behavior in terms of the input voltage output and the oscillation frequency as well as its distortions, a post-simulation processing step is needed to convert the actual signal waveforms into quantities used in the model. This processing step is described as follows. For a particular training input such as the one shown in Fig. 13 (a), we obtain the output frequency vs. time relationship from the output voltage-domain waveform using the zero crossing time or by using other techniques [14]. Since the VCO oscillation frequency is much higher than the voltage input, the instantaneous oscillation frequency can be obtained by averaging the zero crossing times over several consecutive periods. Once the output frequency vs. time relationship is obtained (Fig. 13(c)), an FFT is applied to identify various frequency components in the output signal (which is also a frequency signal). As such, the output frequency signal is expressed in the form of (24), as shown in Fig. 13(d). In other words, the above processing step translates the actual response w~tveforms to the behavioral level representations used in our model. f V four w 2w 3w W Fig. 13. ((a)-(d) in clockwise direction) Simulation data extraction. 25

27 Results We used the LC oscillator type VCO shown in Fig. 14 for testing our methodology, the specifications for which are shown in Table. 1. We fitted the static nonlinear curve using the simulation data extrac~led from SpectreRF, followed by :fitting the VCO dynamics model template, and then compared the results with SpectreRF simulations. We further expressed our macromodel in Verilog-A. -+- Fig. 14. Static nonlinear behavior of a VCO. Technology Power Supply Center Frequency Tuning Range Power TSMC 0.25 um 2.5 V 2.38 Ghz Ghz 3 mw Consumption VCO gain 100 Mhz/V Table.1. VCO specifications 26.

28 Static modeling The data-points for fitting the static nonlinearities are obtained by sweeping the input control voltage of the LC oscillator and measuring the frequency of the output waveform at each dc control voltage. The nonlinear relationship between the output frequency and input control voltage is fitted by a template containing only 3 rd order terms as well as a template containing only 4 th order terms: allu + a21u2 + a31u3 + a41bt4 f4th (28) -- 1 u3-t- a23bt + a33b/2 d- a43 The two static models are compared with the simulation data in Fig. 15. The relative modeling errors are shown in Fig. 16. As can be seen in the figure, both models are very accurate, with the largest error less than 0.06% for a wide range of the control voltage. x rd order LS fit -- 4th order LS fit ~ ~ N 2.4 ~ _ Vctrl(V) I I I 2.5 Fig. 15. Static nonlinearity fitted with 3rd order and 4th order templates. 27

29 rd order static model --- 4th order static model ~ 0.04 ~0.03 g Vctrl(V) Fig. 16. Percentage error in frequency for 3 rd order and 4 th order static models. For this reason, we chose a 3 rd order static mode~ without sacrificing too much accuracy with respect to the actual simulation results. 28

30 Dynamic modeling The datapoints for fitting the coefficients corresponding to the dynamics are obtained by applying multiple single-tone simulations at a fixed bias point, small amplitude (50-200mV) and frequencies much smaller than the operating frequency of the VCO. Following this, we extracted the changing-frequency vs. time characteristics for each of the datapoints using the zero-crossing times. For example, we applied sinusoids with amplitude of 200mV and input frequencies ranging from 1 to 10 MHz at a dc bias point of 1.4V at the input of the VCO. We used the resulting data to fit the VCO model, and applied test inputs to verify the results with Spectre simulation. Fig. 17 shows the comparison between the original frequency waveform extracted from SpectreRF simulation for a test input sinusoid biased at 1.6V with a 50mV amplitude and 50MHz frequency. The frequency waveform re-constructed using the first 3 input harmonics, the output frequency reported by the model containing static terms only, and the output frequency reported by the dynamic model are shown in the figure. One can observe the error incurred with the static model, and the accurate matching of the 3-harmonic mode[ as compared to the transistor-level simulation. We further incorporated the static and dynamic behavioral template in Verilog-A. The comparison between the simulation results obtained from the transistor level SpectrRF simulation and the Verilog-A m,odel simulation results for a single-tone input t (Fig 18) and multiple tone inputs ~ (Fig 19) shows that the dynamic model is extremely accurate while offering 3 orders magnitude speedup as compared with the transistor-level simulation. Most importantly, this speedup is even more significant when considered in the context of a much larger simulation (e.g. the complete PLL) which is forced to take small timesteps at all nodes to accommodate the nonlinear dynamic behavior of the VCO. 29

31 x ~ ~ ~ ,4 original - re-constructed using 3 harmonics static + dynamics model static model ~ O.. ~ ~ ~ ~ ~ ~ L Time (sec) Test input : V = *sin(2*pi*(10 Mhz)*t) x -710 Fig. 17. Comparison of VCO static and dynamic model with SpectreRF simulation. x spectrerf simulation... vedlog-a dynamic model vedlog-a static model "330 5~ ]]me(ns) Test input : V = sin(2*pi*(10Mhz)*t) Fig. 18. Comparison of Verilog-A model results with SpectreRF simulation: Single-tone test 3O

32 Test input : V = sin(2*p~*(10Mhz)*t) Test input" V = sin(2*pi*(TMhz)*t) sin(2*pi*(10Mhz)*t) x Verilog-A dynamic model -- SpectreRF simulation t 2.33 = "rime(ns) Test input : V = sin(2*pi*(7Mhz)*t) sin(2*pi*(10Mhz)*t) 1000 Fig. 19. Comparison of Verilog-A model results with SpectreRF simulation: Multi-tone test 31

33 Conclusions In the first part of this work, we presented a methodology for generating analog circuit macromodels of weakly nonlinear circuits from the transistor-level netlists. This methodology can be applied to a broad range of time-invariant and time-varying weakly nonlinear circuits. The macromodels generated using this methodology are characterized using efficient numerical fitting of simulation data and model order reduction techniques Our experimental results have shown that the macromodels offer significant decrease in model size with comparable accuracy to full transistor- level simulation in Hspice. In the second part of the work, we proposed a novel behavioral VCO model which is capable of capturing both the static and dynamic behaviors. Our model relates the input voltage directly with the output frequency leading to much improved efficiency in simulation. The model configuration is completely generic and can be applied to multiple VCO architectures. Our experiments have shown that the model is sufficiently accurate tbr a LC oscillator designed in TSMC 0.25um technology. This model has been implemented in Verilog-A and can be readily adopted for system-level simulation of PLLs. 32

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