Combined EM field/linear and nonlinear circuit simulation using the FDTD method

of linear and nonlinear sub-circuits)

Q.Chen V.F. Fusco

Indexing terms: Geometrical decomposition, Mixed geonieiry prohltws, FD TD method

Abstract: A review of several techniques related to the combination of geometrical decomposition of mixed geometry problems coupled with the inclusion of single and multiple linear and nonlinear circuit elements is addressed for use with the FDTD method. As a demonstrator of the algorithmic aspects related to this problem, a slot loop antenna will be analysed. Several variants of the antenna are described, the simplest variation consisting of a slot loop antenna fed by a CPW line connected via a slotline transition. More elaborate variations include techniques for the inclusion of lumped linear and/or nonlinear elements into the FDTD code. The inclusion of three terminal lumped active devices is demonstrated in order to describe the algorithmic details of the slot loop operated as an active antenna. Where possible, experimental and theoretical predictions are compared in order to validate the methodologies adopted.

1 Introduction

The next generation of CAD simulators for use by high frequency electronic circuit designers will need to incor- porate the detailed interaction between lumped devices, operated either in linear or in nonlinear mode, and dis- tributed circuitry. Thus, techniques for the integrated analysis of mixed geometry EM field problems (to accommodate distributed circuit layout), including lin- ear and nonlinear devices, need to be assessed. In the work presented here mixed circuit theoretical and elec- tromagnetic field simulation, methodologies involving finite difference time domain (FDTD) and diakoptics techniques are described.

The rationale adopted for following this approach is that linear and nonlinear device parameter estimation are established methods for two and three terminal pas- sive and active device characterisation, with many equivalent circuit models for circuit elements available to designers [l]. Thus for the foreseeable future, linear and nonlinear equivalent circuits are likely to remain

0 IEE, 1998 IEE Proceedings online no. 19981617 Paper first received 22nd May and in revised form 25th September 1997 The authors are with High Frequency Electronics Laboratory, Depart- ment of Electrical and Electronic Engineering, The Queen’s University of Belfast, Ashby Building, Stranmillis Road, Belfast, BT9 5AH, UK

-

an attractive option for inclusion in mixed EM simula- tors. In addition, the general utility of the FDTD approach as a robust methodology for dealing with general EM field problems is indicated [2, 31. The inclu- sion of appropriate network theoretical considerations should allow these diverse areas of study to be incorpo- rated into a single mutually co-operating analysis envi- ronment, as in Fig. 1.

nonlinear EM problem

region from large linear area

for nonlinear

(time-domain integration network reassembly lumped elements

. curved slotline

. CPW short end

slotline-CPW balun

Ro=4.25mm

Fig. 2 Diukol,tics-ha,~ed slot loop mitennu unalysis

2 concepts

It is often difficult or impossible to compute the elec- tromagnetic field behaviour around a computationally large, complex geometric shape such as might occur in a MMIC circuit. For example, the slot loop antenna in Fig. 2 contains mixed geometry. This antenna part is best described in a cylindrical co-ordinate system, while

FDTD diakoptics using network theoretic

185 TEE Pum-Microw. Antennas Propug., Vol 145, No. 2, April 1998

the CPW feed line and matching structure relates to a rectangular Cartesian co-ordinate system. Dismantling the structure into smaller sub-circuits and then analys- ing these independently using specialist FDTD algo- rithms followed by subsequent reassembly allows the overall characteristic behaviour of the system to be evaluated [4].

‘2

a b C

Fig. 3 U i v = 6 3 5 p n , s = 381pm, d = 3810 e , = 4.699mm, t2 = 5.65” substrate: cr = 10.2, t = 635pii 0 A = 63.5,uni c At = 0 . 1 ~ ~

Multiport excitation FDTD analysis of slotline-CP Wjunction

The reassembly process is based on conventional net- work theory. The slotline to CPW junctions is a four port network, as in Fig. 3a. By exploiting the symmet- rical nature between ports 3 and 4 the problem reduces to a three port, as in the Appendix (Section 8.1); multi- port excitation can then be used to analyse the struc- ture. Excitation of port 1 exploits the perfect magnetic wall condition that exists along the line of symmetry (Fig. 3h). At port 3 dual symmetry exists (a perfect magnetic wall and a perfect electric wall) as in Fig. 3c. In essence this is exactly the same as two identical sources exciting both slotline ports. On applying these techniques, Fourier analysis of the reflected Gaussian voltage responses obtained for the network allows the S-parameters for the junction to be found subject to appropriate impedance scaling, in this case

Having obtained the 5’-parameters of the CPW-slot- line junction using the idea of multiport excitation, the cylindrical slotline, analysed in cylindrical co-ordinate FDTD [5], can be added to the CPW-slotline junction at port 3 as reflection coefficient r3

( 2 ) r - -3/3910t*(2.rrR-W-2G-2t2)

3 - - e

where Prior is the phase constant of the curved slotline, R is the radius of the slot loop, W is the coplanar waveguide signal conductor width, G is the coplanar waveguide ground plane gap width and t2 is the length of the slotline arm. The termination condition at port 2 for optimum coupling into the slotline arms (port 3) can now be obtained in order to ensure a matched input on port 1,

( 3 ) 311

Sll’22 - s12s21 r2 =

This reflection coefficient equation, when connected to port 2, leaves only the length to the CPW short circuit e for optimum match to be computed as

(4) $ e n d - $l?2 f 2nll i =

2Pcpw

where (bend is the phase of the CPW short circuit, h2 is the phase of the required r2 and pcpw is the phase con- stant of the uniform CPW. Here, (bend and pcpw are

186

themselves computed using a Cartesian co-ordinate FDTD programme. In this way the design of the CPW feed slot loop antenna is optimised on a circuit theoret- ical basis without recourse to iterative FDTD studies

Fig. 4 shows the predicted and measured amplitude responses for the circuit. All of the FDTD calculations were obtained using full 3D simulations with modified absorbing boundary conditions [7]. The radiated far- field patterns of the antenna can, if required. be formed from the near-field transformation [8, 91.

161.

0

m -10 2 : -20

a, ._

8 5; -30 .- - a, - e ? -40

-50 I 0 5 10 15 I

frequency,GHz Fig.4 frequency 6.3 GHz) refection coejficient of slotline ring antenna

~ predicted _ ~ _ _ measured

Predicted (working frequency 6.7GHz) and measirred (working

3 nonlinear devices

Inclusion of single lumped element linear and

In this Section, a method for including linear and non- linear lumped elements within a 3D FDTD environ- ment is described. The method proposed analyses lumped elements of any size. The conventional approach of using a discretised conduction current term in Maxwell’s equation [lo-141 makes it difficult to explicitly express the relationship between the lumped current and the FDTD electric field component. Here we choose to modify the media parameters associated with cells into which the lumped element is added. The method adopted for single elements is to incorporate the functional I- V characteristic equation, which can be nonlinear, in such a way that the governing Maxwell’s equations remain valid.

Fig.5 A lumped element connected across a slotline

Consider Fig. 5, where a lumped element is con- nected across a slotline of width 6 A, thickness 12 A and E, = 10.3, where A = 1.25“. Lumped current is expressed through the media conductivity (r and per- mittivity E in a parametric fashion as Kunz and Lueb- bers demonstrated for one cell [2], but now over more than one cell. Thus the formulation in eqn. 5 can be

IEE PYOC -Microw Anlennas Pwpug I Vol 145, No 2, Apurl 1998

obtained [15]. The effective media parameters oerr and .qff can then be calculated from the values of resistance andlor capacitance; hence a lumped element of any size can be analysed. The iteration formula governing the available x-directed current flow for a lumped element placed across the slotline is therefore the same as that used for all other grid elements, but with modifications to ocff Eqn. 5:

The Appendix (Section 8.2) shows the equivalent rela- tionship between this formulation and that normally adopted by the lumped current approach when applied over one cell only.

As a demonstration of the method, a Schottky diode is used as the mixing element in a slotline. The diode Z- V relationship is written as

7,

Here V, is the voltage across the slot, and is obtained by integration of the principal x-directed slotline elec- tric field. Hence the effective conductance of the diode can be found as

where Io = le - 14A, T = 300K, W = 2 A, L = 6 A, H = 1 A, A = 1.25".

I I I

5 10 15 20 25 time,ns

-6b

Fig. 6 MDS ( j j ,= 200MHz, f2 = 300 MHz)

~ this paper MDS [20]

Diode voltage compurison between present simulution und thut o f '

~~~~

Consideration of eqns. 5 and 8 indicates the stabilis- ing self-limiting mechanism inherent in the formula- tion. If the signal voltage is large, then a high E, field is generated. This gives a large V,, resulting in a large conductivity which will subsequently result in a small voltage. For a diode, if the voltage across it is large then a small voltage change gives rise to a large current difference. In this case the xlf-limiting mechanism might not be convergent and a standard root solver can be used (E, field convergence normally occurs after two or three iterations). If the lumped element is inductive, the total inductance can be distributed across each cell

equally, over the space the inductor occupies, and con- ventional FDTD discretised equations applied. When a diode is placed across the slotline, and multitone exci- tation used, mixing action is observed to occur at the diode, as in Fig. 6 [ 161. In this case the input signal is a combination of two different tones: 300 and 200MHz, respectively, and any reflected signal from the mixer is allowed to penetrate into the source and be absorbed there. The overall computation domain used in this paper is 56A x 170A x 50A. The total number of time steps is 12000, each with a duration of 2 . 2 5 ~ ~ .

4 Circuit with multiple linear/nonlinear elements

In this Section we illustrate how time domain diakop- tics and comprehensive device equivalent circuits, in conjunction with a numerical EM field simulator can be used to analyse an active FET slot loop antenna, to provide a complete hybrid circuitifield simulator capa- ble of providing quantitative nonlinear circuit predic- tion.

In this problem, as shown in Fig. 7 [17], the loop (forming the oscillator feedback loop and oscillator res- onant load) occupies a much larger space than the active device. Therefore it would be numerically and computationally expensive to map the complete struc- ture into the FDTD computation domain. Instead we utilise a similar concept to that used in Section 2, but this time we decompose it in the time domain rather than in the frequency domain.

'FET / \

/ Fig. 7 untennu w = 0.635mm, r, = 6mm, g = 0.2541nm, A = 6 3 . 5 ~ , At = 0.1 ps substrate: E, = 10.8, H = 0.635"

Conjigurution und diukoptics unu1ysi.s of' an uctive slot-ring

= 6.635"

Here, since the reverse transmission coefficient of the active device is much smaller than the forward trans- mission coefficient, clockwise signal flow will dominate the behaviour of the active antenna. Thus the analysis consists of three parts: a curved section of slotline (including the left and right parts of the slotline shown

allow for DC bias); and the FET. First the curved slotline passive elements are ana-

lysed over a wide frequency range using Gaussian exci- tation in cylindrical co-ordinates [5]. Then the impulse

in Fig. 7); R gapped section of slotline discontinuity (to

187 IEE Proc.-Mlcuow. Antennas Pvupug., Vol. 145, No. 2, April 1998

response of the loop is obtained as

i? = T ( T l + 7-21 - t f e t - ( g a p (10) where y is the propagation constant, e is the total length of the uniform curved slotline, and is the transmission parameter for the slotline gap discontinu- ity. From this we calculate the time domain response of the composite feedback loop for any given excitation (in this case the FET output signal). Interconnection of the nonlinear FET model, consisting of a collection of linear and nonlinear circuit ports, has now to be made to the distributed feedback loop. Once this has been achieved, the time domain output signal from the FET is convolved with H(t) to give the FET input excitation ~ F E T - ~

V F E T - ~ ~ ( ~ ) = H ( T ) V F E T - ~ ~ ~ ( ~ - 7 ) d ~

(11) .ioi

+ Initial Excitation If required, single elements representing additional

parasites or components can also be included in the FDTD; as in Section 3. Due to the resonant character- istic of the structure only harmonic frequencies exist, and only a few iterations of eqn. 11 are required to establish the steady-state oscillation condition. Thus the time domain diakoptics strategy adopted here ena- bles full-wave EM simulation of the entire circuit including a nonlinear three terminal active device.

The active nonlinear device is described in the form of a time-discretised matrix in the Appendix (Section 8.3):

such that the FET (an NE72089A) terminal currents become known. The lumped currents representing the active three terminal device in Maxwell’s equations are used as an interface between the FDTD simulation and the time domain device model [18].

Before any large signal AC analysis of the structure occurs, a DC analysis is first completed. The Zds charac- teristic of the FET is made to be a realistic nonlinear function of VgY and Vds according to the Curtice Cubic relationship 1191.

A”Y” = X” (12)

2

1 ?

$ 0

-1

-2

Fig.8 Predicted (4.52GHz) and Insert (measured as 4 49GHz) response of active slot loop antenna

Based on this method a quantitative prediction of the actual active antenna behaviour is obtained, as in Fig. 8. A 56A x 80A x 50A computation domain is used for the nonlinear FET region analysis and the compu- tation time is roughly 8 hours on a Sun Sparc Station 5 computer. It was found that during oscillator start up, the method described here successfully accounted for dispersive effects in the slot feedback line, FET source,

188

load amplitude and frequency dependence, all of which induce pulling effects on the oscillator during the oscil- lator start up period. The hybrid strategy adopted here predicted an oscillating frequency of 4.52GHz (c.f. 4.49GHz measured).

5 Conclusions

This paper gives a review of several techniques of value for enhancing the behaviour of numerical EM field solving simulators for use in nonlinear circuitifield hybrid simulation.

The first method, based on domain decomposition of numerically derived scattering parameters, allows the inclusion of network theory so that optimised solutions can be obtained without recourse to iterative numerical computation. The second method illustrates how single linear and nonlinear lumped elements can be integrated into existing FDTD code without major code modifica- tion. The final method shows how systems theory and circuit theory can be adapted for interfacing with exist- ing FDTD code. In all cases multiple source excitation is permitted. Comparisons of theoretical, experimental and simulated prediction show good agreement, thereby establishing the validity of the methods employed.

6 Acknowledgments

This work was supported by EPSRC under contract numbers GRIJ40188 and GRiK75682.

7 References

1 EESOF INC.: ‘EEsof model libraries: User’s guide’. May 1991 2 KUNZ, K.S., and LUEBBERS, R.J.: ‘The finite difference time

domain method for electromagnetics’ (CRC Press, 1993) 3 TAFLOVE, A.: ‘Computational electrodynamics: The finite-dif-

ference time-domain method’ (Artech House, 1995) 4 KRON, G.: ‘Diakoptics-The piecewise solution of large scale sys-

tems’ (Mcdonald Press, 1963) 5 CHEN, Q., and FUSCO, V.F.: ‘Three dimensional cylindrical

coordinate finite-difference time-domain analysis of curved slot- line’. Proceedings of IEE 2nd International Conference on Com- putation in electromagnetics, University of Nottingham, UK, April 1994, pp. 323-326 CHEN, Q,, and FUSCO, V.F.: ‘FDTD diakoptics design of a slot-loop antenna excited by a coplanar waveguide’.Proceedings of 25th European Microwave conference, Bologna, Italy, Sept. 1995, pp. 2149-2155 CHEN, Q., and FUSCO, V.F.: ‘Three dimensional finite-differ- ence time-domain slotline analysis on a limited memory personal computer’, IEEE Trans. Microwave Theory Tech., 1995, 43, (2), pp. 358-362 YEE, K.S., INGHAM, D., and SCHLAGER, K.: ‘Time domain extrapolation to the far field based on FDTD calculations’, ZEEE Trans. Antennas Propag., 1991, 39, (3), pp. 411413

9 LUEBBERS, R.J., KUNZ, K.S., SCHNEIDER, M., and HUNSBERGER. F.: ‘A finite difference time domain near zone

6

7

8

to far zone transformation’, lEEE Trans. Antennas Propag., 1991, 39, (4), pp. 429-433

10 SUI, W., CHRISTENSEN, D.A., and DURNEY, C.H.: ‘Extend- ing the two-dimensional FDTD method to hybrid electromagnetic systems with active and Dassive lummd elements’. fEEE Trans. k i c row. Thcory Tech., 1952, 40, (4), pp. 724-730

11 TOLAND, B., LIN, J., HOUSHMAND, B., and ITOH, T.: ‘FDTD analysis of an active antenna’, ZEEE Microw. Cuid. Wave Lett., 1993, 3, (11), pp. 423-425

12 LUEBBERS, R., BEGGS, J. , and CHAMBERLIN, K.: ‘Finite difference time-domain calculation of transients in antennas with nonlinear loads’, IEEE Trans. Antennas Propag., 1993, 41, (5) , pp. 566-573

13 PIKET-MAY, M., TAFLOVE, A., and BARON, J.: ‘FD-TD modelline of digital sienna1 urouacration in 3-D circuits with Das- sive and -active Toads’,”ZEEE f r a i l . Microw. Theory Tech., 1994, 42, (8), pp. 1514-1523

14 CIAMPOLINI, P., MEZZANOTTE, P., ROSELLI, L., and SORRENTINO, R.: ‘Accurate and efficient circuit simulation with lumped-element FDTD technique’, ZEEE Trans. Microw. Theory Tech., 1996, 44, (12), pp. 2207-2215

IEE Proc.-Microrv. Antennas Propag., Vol. 145, No. 2, April 1998

15 CHEN, Q., and FUSCO, V.F.: 'A new algorithm for analysing lumped parameter elements using 3D FDTD method',Proceedings of 25th European Microwave conference, Bologna, Italy, Sept. 1995, pp. 410413

16 CHEN, Q., and FUSCO, V.F.: 'Microwave mixer analysis using 3-D FDTD'. Proceedings of IEE 3rd international conference on Computation in electromagnetics, University of Bath, UK), April 1996, pp. 155-158

17 HO, C.H.. FAN. L.. and CHANG. K.: 'New FET active slotline ring antenna', Electr'on. Lett., 1993,' 29, (6), pp. 521-522

18 CHEN, Q., and FUSCO, V.F.: 'Hybrid FDTD large signal mod- eling of three terminal active devices', IEEE Trans. Microw. The- ory Tech., 1997, 45, @), pp. 1267-1270

19 PENGELLY, R.S.: 'Microwave field-effect transistors - Theory, design and applications' (John Wiley & Sons Inc., New York, 1986), Chapter 8

Design Systems', May 1997, Release 7.1 20 HEWLETT-PACKARD COMPANY: 'HP R F and Microwave

8 Appendixes

8.1 FDTD slotline to coplanar waveguide junction modelling We define the simplified 3-port voltage S-parameters as S[:] and the 4-port ones as S[;41 so that

v,, = sp;1v,, + spvz, + 2sylv,3

vr3 = spv,l + s ~ ~ v v z 2 + s.y' + SF'Vi3

vr, = sL1 ~ 3 1 v,, + sK?",31vz2 + 2syvz3

(13) Exciting the structure at port 1 and applying absorbing boundary conditions at port 2 and port 3, i.e. VL1 f 0, VI2 = VL3 = 0 (Fig. 3b), we obtain

sl";"~ = s~~ = sp;~ = s,, [ U 4 1 = VTl/V,l

sp~ = [ U 3 1 = [ U 4 1 - [ U 4 1 = SI, SI, - s,, vr2/v,1 [ U 3 1 = - s ~ ~ = st;] = [ U 4 1 = -sy

s3.3 s4,

= -st;' = vr3/vz1 (14) If we excite port 3 and apply the absorbing boundary conditions at ports 1 and 2, i.e. V I 3 # 0, VI, = VI2 = 0 (Fig. 3c), we obtain

s p ~ ~ = - s ~ ~ = 2 s [ w 4 ~ = [ ~ 4 1 = -2sk~ 13 2s,4

= -2SY' = v,,/v13 s;';"] = s 3 3 [ U 4 1 + SEI = SEI + SEI vr3/vz3

(15)

8.2 Lumped circuit field equivalencies Consider, for example, if the lumped element is a capacitor with capacitance C, then the lumped current I,, is

cax At

I,, = -(E," - q - 1 )

Substituting Zl, into the Maxwell equation, and rear- ranging it, we obtain for the x-directed E-field

2 & ( k ) + &] - o a t

2 [e (k ) + s] +oat E,"( i , j ,k) = [ E,"-l(i, j , k )

2 a t t -

2 k ( k ) + +oat H," ( i , j , k ) - H,n( i , j - 1 ) k )

AY

] (17) H,"(i , j ,k - 1) - H," ( i , j ) k )

a2 + Comparing the above equation with eqn. 5 it can be seen that &(k) + C Ax/(Ay Az) in eqn. 17 corresponds to

in eqn. 5 . For the one cell case [2], Eejf = C Ax/(Ay Az). In the lumped current method [lo-141, since the lumped element is assumed to be dimensionless, the media permittivity &(k) is added to the C Ax/(Ay Az) term. When compared with the latter term, ~ ( k ) is very small and can be neglected. A similar result can be obtained for the conductivity parameter oeff for a resis- tive lumped element.

CF 'I

2 IT2 -"2

ICOUT

"3

Fig.9 Prrckuged FET model

8.3 Time-domain large signal FET modelling Fig. 9 shows the MESFET model used to represent the packaged NE72089A FET where CF, CIN and C O u T represent packaging capacitors. To include the FET model in the FDTD active slot-ring antenna analysis, the model itself has to be analysed in the time domain, i.e. terminal currents ZT1, I , and Zn are to be com- puted under certain input voltage Vl, V, and V,. The currents through the package capacitors GIN, C, and CouT are easy to obtain since the terminal voltage Vi, V . and V, are known, thus the problem of computing Z,, , In and In is transformed to that of computing ZI4, Zz5 and Z36.

Nine independent equations in the nine unknowns, Y n = { v t , v;, v i , i r 4 , ii5, i76, ice<, icDsn, iccDnT, can be established at each time step H :

Any" = X" (18) where An (a 9 x 9 square matrix) and X n (a nine ele- ment vector) can be computed from the model parame- ters and previous time-step solutions.

IEE Proc.-Microw. Antennus Propug., Vol. 145, No. 2, April 1998