WAVE-BASED TRANSIENT ANALYSIS USING BLOCK NEWTON-JACOBI

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1 WAVE-BASED TRANSIENT ANALYSIS USING BLOCK NEWTON-JACOBI Muhammad Kabir McGill Uiversity Departmet of Electrical ad Computer Egieerig Motreal, QC H3A 2A7 Carlos Christofferse Lakehead Uiversity Departmet of Electrical Egieerig Thuder Bay, ON P7B 5E1 ABSTRACT A recetly itroduced wave-based trasiet aalysis uses relaxatio ad thus does ot require large matrix decompositios at each oliear iteratio. The use of waves results i guarateed covergece for ay liear passive circuit ad some types of oliear circuits, but the covergece rate ca ot be cotrolled. I this work, the wave-based trasiet aalysis is re-formulated usig a block Newto-Jacobi approach ad the covergece properties of the origial ad ew formulatios are compared with the simulatio of two microwave circuits. Idex Terms trasiet aalysis, microwave circuits, wave digital filters, state variables, Newto-Jacobi, scatterig matrix. 1. INTRODUCTION Recetly a ew wave-based trasiet aalysis aimed to microwave circuits was itroduced i Referece [1]. This trasiet aalysis formulatio is based o fixed-poit iteratios of waves at the ports of oliear devices. Ulike Newto s method, fixed-poit (or relaxatio) methods do ot require a matrix decompositio of a large Jacobia matrix at each oliear iteratio. Also fixed-poit methods exploit the latecy of the circuit by decouplig it ito smaller pieces ad solvig each piece idepedetly. Thus they are more suitable to be implemeted i parallel computer systems. Differet relaxatio methods have bee previously proposed for circuit simulatio [5]. The greatest advatage of the approach preseted i [1] compared to the traditioal fixed-poit iteratio approaches usig voltages ad currets is that the power delivered to oliear devices at ay iteratio is bouded [2]. Covergece is guarateed for ay liear passive circuit ad some types of oliear circuits. However the covergece rate is i geeral slower tha with Newto method. Trasiet simulatio of circuits usig waves have bee proposed i a few works (Ref. [4] is oe example) i the The authors would like to thak the Natioal Research Coucil of Caada (NSERC) for supportig this work. cotext of wave digital filters (WDF) [3]. WDF are discrete structures that mimic a aalog referece circuit, iitially employed to implemet digital filters but they ca be applied to model ay circuit. Circuit simulatio usig relaxatio i the WDF cotext was explored i Refereces [4] ad [2]. The approach i [2] allows oliear devices to be modelled usig voltages ad currets, ad thus it is more suitable for implemetatio i a circuit simulator. I Ref. [1], the method proposed i [2] was further developed with parameterized oliear device models, implemeted i the freeda TM [6] simulator ad tested with a wider variety of circuits. This paper proposes for the first time to use a block Newto- Jacobi approach to accelerate the covergece of the oliear equatios that arise i the trasiet aalysis aimed to microwave circuits preseted i [1]. Other methods to solve these equatios exist but the Newto-Jacobi approach is attractive because its implemetatio requires oly mior modificatios to the origial fixed-poit approach ad shares the fixed-poit-method advatage of exploitig the latecy of the circuit. The covergece properties of the ew ad the origial approach are compared i this work. Relevat equatios ad the iteratio scheme for Wave-Trasiet aalysis is preseted i Sectio 2. Sectio 3 describes the proposed Newto-Jacobi approach. It is show i this sectio that the proposed approach requires few additioal computatios compared to the origial fixed-poit approach. Simulatio results for two microwave circuits are preseted ad discussed i Sectio FORMULATION OF WAVE-BASED TRANSIENT ANALYSIS Equatios are formulated followig the state-variable approach [7] used i freeda TM. The circuit is partitioed i sources, liear ad oliear parts (Fig. 1). For each oliear elemet ports are defied with oe termial take as the referece. The liear etwork is assumed to be passive. The oliear subetwork is described by the followig

2 Thus voltage ad curret vectors correspodig to all oliear device ports are related to the power wave vectors (a ad b) as follows: v NL = D(a+b), (4) i NL = D 1 (a b), (5) where D is a diagoal matrix with the square roots of correspodig referece port resistaces i the mai diagoal. The relaxatio method is based o propagatig reflectios of waves betwee the liear ad oliear subetworks. Assume a iitial vector of reflected waves (b (k) ) is kow, where (k) deotes the iteratio umber. The correspodig waves set by the liear etwork (a (k+1) ) ca be calculated by replacig Eqs. (4) ad (5) i Eq. (3), parametric equatios [7]: ( v NL (t) = v i NL (t) = i Fig. 1. Network partitio x(t), dx dt,..., dm x ) dt m,x D(t) ) ( x(t), dx dt,..., dm x dt m,x D(t) wherev NL (t), i NL (t) are vectors of voltages ad currets at the ports of the oliear etwork, x(t) is a vector of state variables ad x D (t) is a vector of time-delayed state variables. All vectors i Eqs. (1) ad (2) have the same size equal to the umber of ports of the oliear etwork ( s ). Applyig umerical itegratio ad discrete covolutio o the stadard MNA equatio ad employig coectivity iformatio betwee the liear ad oliear subetwork the followig error fuctio is obtaied (the full derivatio ca be foud i [1]) (1) (2) s sv, M sv i NL (x ) v NL (x ) = 0, (3) where s sv, is a vector that accouts for the sources ad the previous history of the etwork. s sv, depeds oly o quatities kow at theth time step. TheM sv matrix is costat ad represets the liear etwork. The size of the algebraic system of oliear equatios (3) is s s. For each oliear port, a arbitrary referece resistace, R j with j the port umber, is chose. The icidet ad reflected power waves at Port j (a j ad b j, respectively) are defied as follows [3], a j = v j +R j i j 2 R j, b j = v j R j i j 2 R j, where v j ad i j are the istataeous values of the voltage ad curret at the port. The total voltage (v j ) ad curret (i j ) at Portj ca be expressed as v j = R j (a j +b j ), i j = (a j b j ) Rj. with a (k+1) = Sb (k) +a 0,, (6) S = [M sv D 1 +D] 1 [M sv D 1 D], (7) a 0, = [M sv D 1 +D] 1 s sv,, (8) here,sis the scatterig matrix of the liear etwork ada 0, is the cotributio of sources ad previous history to the icidet waves. 3. BLOCK NEWTON-JACOBI METHOD The proposed block Newto-Jacobi method is based o splittig the scatterig matrix, S ito a sum of a block-diagoal matrix (S B ) ad aother matrix with zeros i the diagoal blocks (S O ) [2]: S = S B +S O. (9) Each block i S B is associated to oe oliear device, i.e. the rows ad colums of each block correspod to the ports of oe oliear device. The S O matrix represets the coupligs betwee oliear devices. The basic idea of the approach is to iclude the correspodig block of S B i the equatios for each oliear device i the followig way: v NL (x ) = D(a S B b +b ), (10) i NL (x ) = D 1 (a S B b b ). (11) The cotributio of S O is cosidered i the mai relaxatio loop. Thus Eq. (6) is modified as follows: a (k+1) = S O b (k) +a 0, (12) where (k) deotes the relaxatio iteratio umber. Note that if there is o couplig betwee oliear devices (i.e., S O is zero) the Eqs. (10) ad (11) produce the exact solutio of the system. If the couplig is ot very strog it is expected

3 that the solutio produced by these equatios is a better approximatio tha the solutio obtaied by plai relaxatio. Newto s method is used to solve Eqs. (10) ad (11). By rearragig them, the error fuctios, Φ v ( ) ad Φ i ( ) are obtaied: Φ v (x,b ) = D 1 v NL (x ) a b (I S B ) (13) Φ i (x,b ) = Di NL (x ) a +b (I+S B ) (14) where, I is the idetity matrix. The umber of equatios ad ukows is2 s. The ukows are vectorsb adx. The a vector cotais the waves icidet to the oliear devices ad is calculated from Eq. (12). Note that this system of equatios is decoupled for each oliear device, thus the Jacobia matrices arisig from it are block-diagoal with small diagoal blocks, typically o more tha 6 6 elemets. The Jacobia matrix is as follows: J = [ D 1 J V (I S B ) DJ I +(I+S B ) ], (15) where, J V = v NL / x ad J I = i NL / x. These are the block-diagoal Jacobia matrices that are explicitly available i freeda TM. The updates from each Newto iteratio are calculated as follows: x (ν+1) = [ (I S B )D 1 ] 1 J V +(I+S B )DJ I { } [I+S B ]Φ (ν) i [I S B ]Φ v (ν), (16) b (ν+1) = [I S B ] 1( Φ (ν) i DJ I x (ν+1)), (17) were(ν) is the Newto iteratio umber, x ad b are the updates of state variables ad reflected waves at each Newto iteratio. Note that Newto method is used oly to approximate the solutio for oe iteratio. Most of the time oly a few iteratios (ofte oly oe) are required to reach a acceptable tolerace, as will be show i Sectio 4. Oe additioal matrix decompositio ([I S B ] 1 ) that was ot required with plai relaxatio is required for calculatig b with the Newto-Jacobi approach. But this matrix is small ad the decompositio is eeded oly oce for the whole simulatio, as log as the time step is kept costat ad therefore itroduces very little overhead. The effect of solvig Eqs. (10) ad (11) usig Newto method ca be summarized as a oliear vector fuctio G( ), b = G(a ). (18) The x vector is ot icluded here sice x is oly required at the oliear device level (Eqs. (16) ad (17)) ad is ot required i the mai iteratio loop. Thus the Newto-Jacobi scheme ca be expressed as a fixed-poit scheme by combiig Eqs. (12) ad (18): b (k+1) = G(S O b (k) +a 0,). (19) I additio a vector extrapolatio method called Miimum Polyomial Extrapolatio (MPE) [8] is applied to the vector sequece geerated by Eq. (19) to accelerate the covergece rate. The theory for MPE is covered i Ref. [1]. The algorithm for the aalysis is provided i Alg. 1. Here, K is Algorithm 1 Newto-Jacobi Wave-Based Trasiet Aalysis calculate M sv, calculate S usig Eq. (7), iitial guess is set to the results i previous step, update s sv,, calculate a 0, usig Eq. (8), calculate a usig Eq. (12). apply Eq. (16) ad Eq. (17) to obtai the approximatio of b. util(error l tol l ) update b for K iteratios, the extrapolate. util(error tol) icrease time, t = t+h. util(t < t ed ) the umber of vectors to extrapolate, tol is the predefied tolerace, h is the time step size ad t ed is the simulatio ed time. All those are set as simulatio parameters. tol l is the predefied tolerace for oliear iteratio (tol l >> tol). 4. SIMULATION RESULTS The proposed block Newto-Jacobi wave-based trasiet aalysis was implemeted i the freeda TM simulator i two aalysis types: WaveTra2 ad WaveTra2-E. WaveTra2 uses LU decompositio i Eq. (17) at each iteratio. WaveTra2-E is more efficietly implemeted ad uses a matrix-vector multiplicatio algorithm that cosiders the particular structure of S O i Eq. (12) ad Cholesky decompositio i Eq. (17) that is performed just oce. The performace of the proposed approach is compared with the origial (plai relaxatio) wavebased trasiet aalysis, type: WaveTra [1], ad the statevariable trasiet aalysis, type: Tra2 which uses the formulatio of Eq. (3) [7]. Results from Tra2 are assumed to be correct, as these aalysis has bee previously verified agaist measuremets ad other circuit simulators [9,10]. The performace of the proposed method for differet circuits is summarized i Table 1. All simulatios use fixed time step ad tol = 10 8 adtol l = 0.5. Circuit 1, show i Fig. 2, is a oliear trasmissio lie, or solito lie [10], composed of 47 diodes ad 48 lossy trasmissio lies. Circuit 2 is show i Fig. 3 ad is composed of 5 MMIC LNA [9] each coected to a solito lie (Circuit 1) at the output stage, each LNA fed with a differet iput frequecy. The size of the modified odal admittace matrix (MNAM) i each circuit are give i the table to give a idea of the problem size. Values i the table are average per time step except Newto iteratio, this is the average value per

4 Table 1. Summary of simulatio results Circuit Solito (Fig. 2) Multi-MMIC (Fig. 3) NL ports ( s) MNAM Size Ref. Port Res. (Ω) I. power level (dbm) Time Steps Iter. (WaveTra) 5 13 Iter. (WaveTra2) 4 5 Newto Iter. (WaveTra2) WaveTra Sim. T. (s) WaveTra2 Sim. T. (s) WaveTra2-E Sim. T. (s) Tra2 Sim. T. (s) Fig. 2. Solito-lie schematic Voltage (V) Voltage (V) WaveTra2-18 Tra Time (s) Fig. 4. Output voltage of Solito Lie WaveTra2 Tra Time (s) Fig. 5. Voltage at Output 5 of the Multi MMIC-Solito-lie Fig. 3. Multi MMIC-Solito-lie schematic relaxatio iteratio. The referece resistace used for each simulatio is show i the table. The same value of referece resistace is used for all ports as used i Ref. [1]. I the curret implemetatio this value is set maually as a simulatio parameter. The Iput Power Level row shows the AC power level of the iput sources referred to 50 Ω. The Iteratios rows, (WaveTra ad WaveTra2) compare the umber of iteratios eeded for WaveTra ad WaveTra2. The last four rows compare the total ruig time usig WaveTra, WaveTra2, WaveTra2-E, ad Tra2. All simulated output voltages are compared with that from Tra2 ad two zoom i plots are preseted i Figs. 4 ad 5. As expected the results from both simulatio methods agree. For all simulatios o average oly oe Newto iteratio is ecessary at the oliear device level. Thus the computatioal cost of usig Newto s method to obtai reflected waves from oliear device models formulated i terms of voltages ad currets is ot too expesive. Followig the same tred show i Ref. [1], WaveTra is slower tha Tra2 for the smaller circuit (Circuit

5 1) but as the circuit size icreases the wave method becomes more efficiet as it does ot require large matrix decompositios after S has bee calculated. It ca also be observed i Table 1 that WaveTra2-E is always faster tha WaveTra ad eve the less efficietly implemeted WaveTra2 is faster for Circuit CONCLUSIONS AND DISCUSSION A modificatio of a ovel trasiet aalysis approach was preseted i this paper. The proposed wave-based trasiet aalysis solves the oliear equatios usig a block Newto- Jacobi algorithm. As with the origial wave-based trasiet aalysis, o matrix decompositio of a large Jacobia matrix is required at each oliear iteratio. Thus the proposed method is also attractive to be implemeted i parallel computer systems. It was show that the additio of Newto-Jacobi requires oly a few extra computatios compared to the origial fixedpoit approach. The simulatio results preseted here show that the Newto-Jacobi-based method coverges faster ad is more efficiet for some circuits. Although these results are ecouragig some issues remai to be addressed. For example the choice of Newto- Jacobi blocks used i the curret implemetatio may ot be optimal for some circuit cofiguratios. I circuits where two (or more) oliear devices are coected i parallel the couplig betwee them is strog ad thus the two devices should be cosidered i the same block for optimum performace. It is also worth otig that although the block Newto-Jacobi method geerally improves the covergece rate, some of the covergece properties of the origial relaxatio approach are lost. Oe of the ideas that could address this issue is to implemet a adaptive algorithm that uses the Newto-Jacobi approach by default ad switches to plai relaxatio whe covergece is poor. Aother issue is the optimum selectio of the referece resistace. The referece resistace has some effect o the covergece rate ad the optimum value may be differet for each port ad also depeds o the operatig poit. I the curret implemetatio this resistace is the same for all ports ad maually set, but i the future a better approach should be ivestigated. [3] A. Fettweis, Wave Digital Filters: Theory ad Practice, IEEE Proceedigs, Vol. 74, pp , [4] T. Felderhoff, Jacobi s method for massive parallel wave digital filter algorithm, Proc. of the IEEE Coferece o Acoustics, Speech ad Sigal Processig, pp , Atlata, May [5] A. R. Newto ad A. L. Sagiovai-Vicetelli, Relaxatio-based electrical simulatio, IEEE Tras. o Electro Devices, Vol. ED-30, pp , [6] F. P. Hart, N. Kriplai, S. Luiya, C. E. Christofferse ad M. B. Steer, Streamlied Circuit Device Model Developmet with Freeda ad ADOL-C, 4th It. Cof. o Automatic Differetiatio, Chicago, USA, July [7] C. E. Christofferse Global modelig of oliear microwave circuits, Ph. D. Dissertatio, North Carolia State Uiversity, December [8] D. A. Smith, W. F. Ford, A. Sidi, Extrapolatio methods for vector sequeces, SIAM Review, Vol. 29, No. 2, pp , [9] S. Luiya, et al., Compact Electrothermal Modelig of a X-bad MMIC, 2006 IEEE It. Microwave Symp. Digest, Jue 2006, pp [10] C. E. Christofferse, et al., State variable-based trasiet aalysis usig covolutio, IEEE Tras. o Microwave Theory ad Tech., Vol. 47, Jue 1999, pp REFERENCES [1] M. Kabir, C. Christofferse, N. Kriplai, Trasiet Simulatio Based o State Variables ad Waves,, i It. Joural of RF ad Microwave Computer-Aided Egieerig, to appear. [2] C. Christofferse, Trasiet Aalysis of Noliear Circuits Based o Waves,, Mathematics i Idustry Scietific Computig i Electrical Egieerig (SCEE 2008), Spriger, pp