Extending the two-dimensional FDTD method to hybrid electromagnetic systems with active and passive lumped elements

This paper extends the finite-difference time-domain (FDTD) method to include distributed electromagnetic systems with lumped elements (a hybrid system) and voltage and current sources. FDTD equations that include nonlinear elements like diodes and transistors are derived. Calculation of driving-point impedance is described. Comparison of FDTD calculated results with analytical results for several two-dimensional transmission-line configurations illustrate the accuracy of the method. FDTD results for a transistor model and a diode are compared with SPICE calculations. The extended FDTD method should prove useful in the design and analysis of complicated distributed systems with various active, passive, linear and nonlinear lumped electrical components.

724 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 40, NO. 4, APRIL 1992 Extending the Two-Dimensional FDTD Method to Hybrid Electromagnetic Systems with Active and Passive Lumped Elements Wenquan Sui, Douglas A. Christensen, Member, IEEE, and Carl H. Dumey, Senior Member, IEEE Abstract-This paper extends the finite-difference time-do- main (FDTD) method to include distributed electromagnetic systems with lumped elements (a hybrid system) and voltage and current sources. FDTD equations that include nonlinear elements like diodes and transistors are derived. Calculation of driving-point impedance is described. Comparison of FDTD calculated results with analytical results for several two-dimen­sional transmission-line configurations illustrate the accuracy of the method. FDTD results for a transistor model and a diode are compared with SPICE calculations. The extended FDTD method should prove useful in the design and analysis of com­plicated distributed systems with various active, passive, linear and nonlinear lumped electrical components. I. Introduction THE FDTD method, first presented by Yee in 1966 [1], numerically solves Maxwell's equations in the time domain on a spatial grid. Because of its computa­tional efficiency, accuracy and direct physical interpreta­tions, the FDTD method has become increasingly popular for computations of electromagnetic wave propagation and scattering problems and electromagnetic biological effects [2]-[6], including the FDTD analysis of microwave cir­cuits [7], [8]. Previous analytical techniques for hybrid circuits (dis­tributed electromagnetic systems with lumped elements) have used equivalent circuits or have combined different numerical methods to analyze lumped-element or semi­conductor devices in a distributed system [9]-[14]. For example, for a nonuniform transmission line loaded with lumped elements equivalent transformations or equivalent circuits have been employed [11]-[13]. The transmission line matrix (TLM) method [15], [16], which simulates the wave propagation by an equivalent circuit based on Huy- gen's principle, is another popular numerical method. Comparisons between the FDTD and TLM methods are reported in [7], [17] and [18], with the general conclusion that the two methods, although based upon different mod­eling philosophies, are similar in several respects, and can be considered to be complementary; both utilize meshes and a time-domain approach. Voelker and Lomax [14] Manuscript received April 5, 1991; revised November 11, 1991. The authors are with the Department of Electrical Engineering, Univer­sity of Utah, Merrill Engineering Building, Salt Lake City, UT 84112. IEEE Log Number 9106044. combined the TLM method with the FDTD method to gain the advantages of both methods for a semiconductor prob­lem including both lumped elements and nonlinear de­vices. The TLM method has also been employed in a mi­crowave field simulator formulated to be particularly user- friendly [19], similar to microwave circuit CAD packages which are being developed to meet the increasing demand for all-purpose CAD tools [19]-[21]. This paper extends the two-dimensional finite-differ- ence time-domain (FDTD) numerical analysis technique to include hybrid systems, such as microwave circuits with lumped capacitors, inductors and diodes mixed with strip lines or radiating elements [22]-[25], and discrete sources with antennas such as hyperthermia applicators for cancer therapy [26]. The formulation includes calculation of driving-point impedance, which is often needed for impedance matching. A brief review of the conventional FDTD method is followed by a derivation of the relationships between the E and H fields and the voltages and currents that are used to extend the general equations to include lumped ele­ments inside the grid. Both R, L, and C lumped elements and discrete voltage and current sources with internal impedance are modeled; by specifying the appropriate I-V characteristics of the sources and elements, both pas­sive and active, linear and nonlinear circuit elements, can be treated. Several test cases are analyzed by this method and the results compared with analytical solutions or the results of SPICE to verify the validity of the technique. To demonstrate the capability of the method, diodes and transistors connected to transmission lines are analyzed and the results from the extended FDTD method are com­pared with those from SPICE. The transmission line con­figurations are chosen for the comparisons because ana­lytic solutions for these cases are easily obtained, but the method developed here applies to more complex config­urations as well. Some considerations of the extended method are then discussed. Finally, prospective applica­tions and future development of the method are discussed. II. FDTD Formulations with Lumped Elements Since the FDTD method has been discussed exten­sively in the literature [27], the two-dimensional FDTD iteration equations are stated (but not derived), and used 0018-9480/92S03.00 © 1992 IEEE SUI el al.: EXTENDING TWO-DIMENSIONAL FDTD METHOD TO HYBRID ELECTROMAGNETIC SYSTEMS 725 as the basis for extending the method to include lumped elements. The extension is based upon the derivation of the FDTD equations from the integral form of Maxwell's equations. A. Conventional FDTD Method The standard FDTD equations may be derived by in­tegrating Maxwell's equations: E ■ dl i dH s dt dS (1) dE Jc ■ dS + e - • dS (2) s Js at appropriately on a Yee cell to obtain the iteration equa­tions: zjn + I _ Tin . ^ ttrn _l_ Tpn j?n j?n \ n zi.j - nZi,j f f, ' Xl,j + I T ZyiJ - £,xjJ - Ej yj + ]j) fidl (3) rn + 1 _ tLxi+\,j ~ e It rn + 1 yi + 1J e a It + 2 • (Hzi+uj e o It " 2 T7n c'xi + 1 ,j di e a 5t + 2 HnZi+ ij - i) (4) e a 8t + 2 j?n E* yi + 1J i + 81 € a It + 2 ~ Hnzi+ , ,j) - Eyi+lj 61 Fig. 1. Two-dimensional FDTD unit cell (Yee cell) at position Eyi+2j (ij) cell (i+lj) cell (a) Position of lunged element 1 j F xi+l j+l -t i Eyij t 1 Hzij * Vyi+lj Eyi+lj • Hri+1j C Eyi+2j .4 1 (5) In (3)-(5), hi is the length of a side of the square cell of the grid, 6 is the permittivity and a the conductivity, bt is the time interval of each iteration time step which is de­termined by 8t = w~ (6) 1 cmax where F > V2 for stability in the two-dimensional prob­lem [2] and cmax is the maximum speed of the electro­magnetic wave inside the grid. Note that to derive (4) and (5) from (2), the E term in the current density Jc has been taken as a forward time average, i.e., J" = a(En +1 + En)/2, to avoid an instability problem in the FDTD it­eration equation when the conductivity of the medium (a) becomes very large. The locations and orientations of the E and H field com­ponents are shown for a typical two-dimensional unit cell of the grid (the Yee cell) in Fig. 1. We have used Mur's first-order absorbing boundary conditions [28] at the outer edges of the grid, which have been shown to provide rea­sonable accuracy in previous investigations [8], [26]. B. The Extended FDTD Method Lumped elements may be accounted for in Maxwell's equations by starting with the interpretation that in (2), the right-hand side is the total integrated current density Eyi+2j (i j) cell (i+lj) cell (C) Fig. 2. Interpretation of a lumped element in an FDTD cell pair, (a) Illus­tration of the integral contour and field components, (b) Position of the lumped element and the related voltage, (c) Example for modeling a volt­age source with internal resistor. through a surface, which is the sum of the conduction cur­rent \SJC ■ dS and the displacement current e\s(dE/dt) • dS. This total current is related by (2) to the H field in­tegrated over the corresponding contour. An example of such a contour and surface for the finite-difference for­mulation is shown in Fig. 2(a). The contour abcda is in the x-z plane. In this two-dimensional formulation, the length ab in the z direction is arbitrary, since all functions do not vary with z, and since the currents in the x and y directions are defined as currents per unit z length. For the contour shown in Fig. 2(a), the total current on the right-hand side of (2) is the integral of the current densi­ties over the area of the surface between the two adjacent //-field components, shown as a shaded surface com­prised of 5, and S, +, in the figure. The conduction current per unit length is thus given by726 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 40, NO. 4, APRIL 1992 61 2 81 Icy ai,jEyi+lJ 0 + <7;+ \jEyj + \j °ij + G,+ 1, ME. yi + Kj- (7) Note that the conduction current is proportional to the av­erage conductivity times the electric field. Similarly, the displacement current per unit length is given by Id\ 51 dt (8) The currents in (7) and (8) are currents per unit length in z, with units of A/m. Using the classical definition of potential difference of point 1 with respect to point 2, Vn = - j E ■ dl (9) we define Vyi+lJ = 8lEyi+Uj (10) with the polarity as shown in Fig. 2(b). With this defini­tion, (7) and (8) can be written as (11) ?cy Syi + 1, j r yi + I ,j j „ dVyi+1; ldy {~yi+\,j ^ where Syi + 1 ,j (aiJ + ai + Ij) Cyi + 1 ,j (£i,j + 1 j) (12) (13) (14) are the average conductance per unit length and average capacitance per unit length, respectively. Similar rela­tions can be written for currents in the x direction. With these relations, (2) can be extended to include lumped elements by adding Ih the current through the lumped element, to the right-hand side of (2). For the ex­ample shown in Fig. 2(a), this results in Hzu H71 + 1 ,j ~ Iy - I/y + /,■„ + /, dy (15) J?n + 1 C'xi+ 1J e a 8t " 2 e a bt + 2 J7n c'xi + 1 ,j dl e a It + 2 • fn * (f?n /T J x yE'xi + 1 ^xi + 1J9 M „ 11 + - m+x,j - eo bt + 2 (16a) ^, bt 2 Fn. . - --------Fn t-* vi 4- 1 1 vi dl yi + 1, j yi + 1, / e ff e ct dt 2 bt 2 . j7n ~ ^ . . . j7 ^ \ Jy \E,yj + Etyj + \J, , EL yj + | j ) + 7 Sr + 2 (16b) where 7/y is the current per unit length through the lumped element and Icy and Idy are the conduction and displace­ment currents per unit length as given by (11) and (12). The lumped element is assumed to be two-dimensional (no variation in the z direction) and thin enough in the x direction that it can be considered lumped, as indicated in Fig. 2(b). When a lumped element whose current/voltage rela­tionships are given by the general forms Iix = fx(Vx) and hy = fy (K) is connected between cells (i, j) and (i + 1, j) in Fig. 2, the finite-difference expressions (4) and (5) for those two cells become: where, depending upon the functions fx(:) and fy(-) de­termined by the I-V relations of the lumped element(s) to be modeled, the above equations correspond to the finite- difference approximations to linear, integral, differential equations or their combination. Equations (16) are general expressions for modeling any lumped element or combination of lumped elements within an FDTD cell pair. Since there is no limitation on the I-V relations of the lumped element(s), i.e., the func­tions/( K) in (16), it appears to be possible to model any component, linear or nonlinear, in the FDTD grid. Note that the lumped element is connected in "paral­lel" with the cell while the original displacement and con­duction current of the cell remains; i.e., the lumped ele­ment does not replace the cell contents, since the current calculated by (15) is the total current, conduction current and displacement current of the medium plus the lumped- element current, flowing through the cross section en­closed by the contour abcda in Fig. 2. In other words, the lumped element is treated as "sizeless" compared with the cell size, which requires, of course, that the lumped element is physically small compared to a cell. Equations (16) include two components of the function /(•), allowing for the possibility that the current in the lumped element has two components, namely in the x and y directions. Often one direction of the current dominates for cells connected with lumped elements. C. Modeling of R, L, C Elements The lumped resistor, inductor or capacitor can be mod­eled easily by substituting into (16) the appropriate I-V relations for each component. In finite-difference terms, these relations take the following forms (for y-directedSUI et at.: EXTENDING TWO-DIMENSIONAL FDTD METHOD TO HYBRID ELECTROMAGNETIC SYSTEMS 727 current) for the resistor, inductor, and capacitor, respec­tively: Sl(En/+\j + E?yi + ]J) - (17a) in In lyi + 1 ,j Jn - *yi + 1 ,j mm 2 L CS[ 2bt 2 R Eyi+\j + 2 S Ekyi+Uj ) (17b) *= i /rn + 1 rrt - 1 \ (C-yi+lJ ~ Zyi+\,j) (17c) where R, L, and C are given per unit length, and zero values are assumed for both inductor current and capaci­tor voltage at time zero. Forward time averages have been taken in the above equations to be consistent with the treatment of the con­ductance current in Section II-A. Also, if no time aver­aging is used for the currents, the forward-difference form of the difference equations (16) will lead to instabilities for certain values of R (we have found that a backward- difference form must be implemented to avoid instabilities if no time averaging is used). D. Modeling of Discrete Sources When a discrete source such as a voltage source Vs with internal series resistance Rs is included in the y direction in the cells, as shown in Fig. 2(c), the I-V relation of the branch connected with voltage source and resistor can be written as Vv = IyRs - In Sl{E" +1 or in finite-difference form yi+\.j + Eyj+l,j) + 2V" 2R„ (18) A similar current/voltage equation can be derived for a current source Is with an internal parallel conductance Gs. Then substituting (18) or the corresponding equation for the current source into the FDTD iteration equation (16b) for Ey will give, respectively, 1 _ _ J_ H_ St 2 2R, SIR, En i-* v/ n + 1 yi+Uj e a St + 2 2 R _ r?n | Gyi+lJ e a I It + 2 + 2^ (HndJ - Hnzi+i,j) e ct 1 St + 2 + 2R, SI or j?n + 1 _ Zyi+XJ "" 6 51 e a St + 2 + 9* E- a yi+iJ r 1 s SI e a Gs - + - + - St 2 2 rn, - Hnzi+tJ) e a G, - _j_ _ ---- 5 / 2 2 (19b) SI (a) Transistor model (19a) (b) Fig. 3. Modeling of a transistor in the FDTD grid, (a) Simplified equiva­lent circuit of the transistor: there are n cells, n = 1, 2, • • • , between the base and collector cells, as shown by the dashed line, which modifies the fringing field effect, (b) Equivalent circuit for comparisons with the SPICE calculation: the transistor (equivalent model enclosed by dashed box) is connected in an infinitely long transmission line driven by a voltage source and loaded by a resistor. When modeling a voltage source whose internal series resistor Rs equals zero, corresponding to an ideal voltage source, the electric field is given directly from the source voltage with no need for other terms; in this case (19a) should be replaced by (10) with Vy = -Vs. When Rs goes to infinity, the voltage source branch becomes an open circuit, and (19a) correctly reduces to (5). Similarly, when modeling a current source, if the parallel conductance Gs goes to zero, corresponding to an ideal current source, then (19b) reduces to (16b) with/(-) = /*. E. Modeling of Active and Nonlinear Devices An active device can also be modeled by a combination of lumped elements, including dependent sources, in one or more cells inside the grid. As an example, a transistor can be modeled by the simplified equivalent circuit shown in Fig. 3(a). The base and collector of the transistor are modeled at two edges of separate cells; the collector cur­rent is driven by the base-current-controlled current source. To account for the effect of fringing fields be­tween the base and collector in the transistor model, the distance between the base and collector may be chosen to be a varying number of cells as shown in Fig. 3(a). For more advanced modeling, lumped-element capacitors may be added in parallel to any branch. Also, a nonlinear diode model may be included in the FDTD calculations by using the diode equation: /, = l0(e<y'/kT - 1) (20) where k is Boltzmann's constant, q is the charge of an electron, T is temperature in Kelvin, and Vt is the voltage across the diode (which is related to E by (10)). F. Calculation of Driving-Point Impedance For any distribution system or hybrid system, the driv- ing-point impedance Z at the source can be calculated728 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 40, NO. 4, APRIL 1992 when the system reaches the steady state by using the fol­lowing equation: where VQ and 70 are the magnitudes of the driving-point voltage and current, and <t>{ and <j>2 are their phases, for the cell that contains the excitation source, as calculated from (10) and (15). III. Comparisons of FDTD with Transmission-Line Theory and Spice To verify the method, we compared FDTD calculations with analytical solutions or results from SPICE for a two- ended parallel-plate transmission line with various lumped elements connected either at the source or as a load. This general configuration is chosen because analytical solu­tions for it are available, and because it can include the effects of multiple reflections and interference between the incident and reflected waves. Unwanted radiation into free space, which would be included in the FDTD results but which would unduly complicate the analytical analysis, are eliminated by selecting a closed transmission system. Of the many calculations we have done, we present four that typify the agreement between FDTD results and an­alytical or SPICE results. The following parameters were used in the FDTD calculations: cell size was 1 cm square; all excitation signals were sinusoidal waves, V5 (t) = V0 sin (ut) or Is (t) = Ia sin (ut), where V0 and 10 are the magnitudes of the excitations; and the frequency was 200 MHz. The separation distance between the two transmis­sion- line plates was 2 cm (two cells) and the line was air­spaced, so the characteristic impedance of the transmis­sion line was Z0 = 7.5347 Q and only the electromagnetic TEM mode was present. The time increment used for all FDTD results was 8t = 16.667 ps, according to (6) with F = 2. The infinitely long transmission line was simu­lated by placing Mur's first order absorbing boundary conditions [28] at both ends of a 200-cell long transmis­sion line. Both the FDTD numerical computations and the SPICE simulations were done on an HP 9000 computer system. The derivations of the analytical solutions for the test models can be found in [29]. Fig. 4 shows the transmission line configuration used for several of the comparison tests. When the load Zt in Fig. 4 is an inductor, the current through the load, as cal­culated by both FDTD and the analytical solution, is shown in Fig. 5. The agreement between the two methods is very good. The effect on the resulting load current from reflections between the source and load can be clearly seen; for this case the source resistance is zero. Fig. 6 shows the excellent agreement between the FDTD calculation and SPICE calculated currents in Zt when Z, is a diode. For certain combinations of parame­ters which cause the voltage across the diode to become large, the exponential term in the diode I-V relationship wil cause instabilities in the FDTD results, apparently due Fig. 4. One of the configurations for comparison calculations: infinite transmission line loaded by a lumped element Z, (which can be R, L, C or diode). Fig. 5. Load current through L, in Fig. 4 (inductor load) as calculated by the FDTD method (solid line) and analytical solution (dashed line) with parameters K0 = 1 V, Rs = 0 fi, Lt = 50 pH, and / = 0.4 m. Fig. 6. Current through a diode load (Fig. 4) as calculated by the FDTD method (solid line) and SPICE simulation (dashed line) for the parameters V0 = 1 V, Rs = 0.5 Q, / = 0.4 m, T = 300 K, and I0 = 10 14 A. The two lines are nearly coincident. to the extremely large currents involved. In such high- current cases, the exponential relationship (20) between current and voltage is no longer physically meaningful, and a modified linear I-V relationship would probably avoid the instabilities, although we have not pursued this modification. Fig. 7 shows the FDTD results compared to SPICE cal­culations for the current through a load resistor in the col­lector/ transmission-line circuit of the transistor configu­ration shown in Fig. 3(b). The two techniques give very similar results for the chosen set of model parameters. When the base resistance is significantly increased above the particular value chosen, however, the difference be­tween the results of the two methods increases, probably due to an increase in the base voltage and consequent in-SUI et al.: EXTENDING TWO-DIMENSIONAL FDTD METHOD TO HYBRID ELECTROMAGNETIC SYSTEMS 729 Fig. 7. Current through a load resistor in the collector circuit of a transis- tor-transmission line model (Fig. 3(b)) as calculated by the FDTD method (solid line) and SPICE simulation (dashed line) for the parameters V0 = 0.1 V, Rs = 20 Q, R, = 2.5 12, 0 = 50, rh = 240 fi, r,. = 10 kfi, I, = 0.37 m and l2 = 0.4 m. Also, there are three cells between the base and collector in the FDTD model. crease in the magnitude of the fringing fields between the base and the collector; these fields are included in the FDTD analysis but not in the SPICE simulation. In a sim­ilar fashion, when the base resistance becomes very small (less than 100 fl), the FDTD technique exhibits large in­stabilities, perhaps due to positive feedback via the fring­ing fields between the collector and base. We have placed an unphysical three-cell gap between the base and collec­tor for the FDTD model in order to minimize the effect of the fringing fields, as indicated in Fig. 3(a), but electric field values in the region of the gap from the FDTD re­sults show that coupling still exists. Studies using more complex models, including shielding and variable gap spacing, are needed to determine more clearly the role of fringing fields in our transistor model. A parallel-plate transmission-line model excited by a voltage source with complex R-L-C impedance was also analyzed by the FDTD method. The results using (21) give a driving-point impedance value of 3.76 fl, which is very close to the theoretical value of Z0/2 = 3.7673 fl (the factor of 2 being due to the double-ended nature of the transmission line). IV. Discussion of Results Calculations presented in last section show that the time-domain FDTD results are close to the analytical or SPICE results; in fact most of the curves fit each other within \% relative error. Although our examples show only source or load currents, currents and voltages at other locations have also been checked and found to be accurate to approximately the same degree. Other configurations were tested using the method, and excellent agreement with analytical and/or SPICE results was obtained. These results illustrate the potential usefulness of the extended FDTD method for hybrid system analysis. One interesting phenomenon we have observed during this study is the occurrence of small "early signals," i.e., the arrival of low-amplitude waves traveling at speeds faster than the actual speed of the electromagnetic waves in the transmission line. The magnitudes of these early field components are very small compared to the eventual signal strengths, and therefore they have little effect on the final results. One possible explanation of this phenom­enon is that, since the selection of 8t in FDTD must sat­isfy the stability requirement given by (6), the electro­magnetic values in the numerical iterations travel twice as fast as in the real wave (when F in (6) equals 2). Because of the half-cell offset between the E and H field components for each cell as shown in Fig. 1, the positions for calculating the analytical source current in the trans­mission line must also be offset by one half cell from the source position (z = 0). Ignoring the half-cell offset causes significantly larger differences between the FDTD and the analytic results. This is consistent with the interpretation in Fig. 2(b) in which the current through the surface is related to the two magnetic field components at the center of the nearby cells. By the nature of the calculation of the current through each cell pair (Fig. 2(b)) in the FDTD technique, these currents include both the lumped-element current and the displacement current through the medium contained in the cells (and conduction current if the medium is lossy). Therefore, the displacement current due to the cell's me­dium must be subtracted from the total FDTD current to obtain a comparison with the analytic solution (which gives the current only through the lumped element). This displacement current is usually small, and becomes an ap­preciable factor only for very large impedance loads. In this paper, the perfectly conducting metal walls of the transmission line are modeled by cells with very high conductivity in the FDTD grid. Sheen et al [8] simulated the perfect conductor by setting the tangential electric field components to zero and assuming the conductor to be of zero thickness. Combining the lumped-element modeling technique in this paper with Sheen's method for metal modeling may be a way to handle configurations where the thickness of the conductor layer is very small (like a microstrip line or thin film) without having to make the cell size correspondingly small. In principle, there is no upper frequency bound for FDTD calculations as long as the cell size is kept small enough compared with the wavelength (usually one-tenth of the wavelength) to give adequate spatial resolution. The FDTD method has been used in the analysis of optical waveguides operating with frequencies higher than 10l4Hz [30]. From the lumped-element point of view, lumped elements have been modeled in X-band [24], [31] and re­cently up to 18 GHz [21]. V. Conclusion We have developed a method for analyzing hybrid elec­tromagnetic systems by extending the FDTD equations, and have verified its accuracy for several test cases. The extended FDTD method should prove useful in the design and analysis of complicated systems with various electri­ 730 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 40, NO. 4, APRIL 1992 cal components such as active, passive, linear and nonlin­ear lumped elements. The straightforward extension of this technique to three­dimensional systems is anticipated. It also seems possible to extract the 5-parameters of the electromagnetic system by this method, or reciprocally, to analyze a system using its S-parameter descriptions. Implementing more compli­cated device models or equivalent circuits should give more accurate simulations of real systems. For example, noise sources might be included in the equivalent FDTD circuit to simulate random or thermal noise in the system to determine its noise parameters. References [1] K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. An­tennas Propagat., vol. 14, no. 5, pp. 302-307, 1966. [2] A. Taflove and M. E. Brodwin, "Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Max­well's equations," IEEE Trans. Microwave Theory Tech., vol. 23, no. 8, pp. 623-630, 1975. [3] D. M. Sullivan, D. T. Borup, and O. P. Gandhi, "Use of the finite- difference time-domain method in calculating EM absorption in hu­man tissues," IEEE Trans. Biomed. Eng., vol. 34, no. 2, pp. 148- 157, 1987. [4] J. G. Maloney et al., "Accurate computation of the radiation from simple antennas using the finite-difference time-domain method," IEEE Trans. Antennas Propagat., vol. 38, no. 7, pp. 1059-1068, 1990. [5] T. Shibata et al., "Analysis of microstrip circuits using three-dimen­sional full-wave electromagnetic field analysis in the time domain," IEEE Trans. Microwave Theory Tech., vol. 36, no. 6, pp. 1064-1070, 1988. [6] J-Y, Chen, and O. M. Gandhi, "Current induced in an anatomically based model of a human for exposure to vertically polarized EMP," IEEE Trans. Microwave Theory Tech., vol. 39, no. 1, pp. 31-39, 1991. [7] W. K. Gwarek, "Analysis of arbitrarily shaped two-dimensional mi­crowave circuits by finite-difference time-domain method," IEEE Trans. Microwave Theory Tech., vol. 36, no. 4, pp. 738-744, 1988. [8] D. M. Sheen et al., "Application of the three-dimensional finite-dif- ference time-domain method to the analysis of planar microstrip cir­cuits," IEEE Trans. Microwave Theory Tech., vol. 38, no. 7, pp. 849-857, 1990. [9] K. Kobayashi et al., "Equivalent transformations for mixed-lumped and multiconductor coupled circuits," IEEE Trans. Microwave The­ory Tech., vol. 30, no. 7, pp. 1034-1041, 1982. [10] D. Winklestein et al., "Simulation of arbitrary transmission line net­works with nonlinear terminations," IEEE Trans. Circuits Syst., vol. 38, no. 4, pp. 418-422, 1991. [11] K. Kobayashi et al., "Kuroda's identity for mixed lumped and dis­tributed circuits and their application to nonuniform transmission lines," IEEE Trans. Microwave Theory Tech., vol. 29, no. 2, pp. 81-86, 1981. [12] I. Endo et al., "Design of transformerless quasi-broad-band match­ing networks for lumped complex loads using nonuniform transmis­sion lines," IEEE Trans. Microwave Theory Tech., vol. 36, no. 4, pp. 629-634, 1988. [13] Y. Nemoto et al., "Equivalent transformations for the mixed lumped Richards section and distributed transmission line," IEEE Trans. Mi­crowave Theory Tech., vol. 36, no. 4, pp. 635-641, 1988. [14] R. H. Voelker and R. J. Lomax, "A finite-difference transmission line matrix method incorporating a nonline device model," IEEE Trans. Microwave Theory Tech., vol. 38, no. 3, pp. 302-312, 1990. [15] P. B. Johns and R. L. Beurle, "Numerical solution of 2-dimensional scattering problems using transmission-line matrix,' Proc. Inst. Elec. Eng., vol. 118, no. 9, pp. 1203-1208, 1971. [16] W. J. R. Hoefer, "The transmission-line matrix method-theory and applications," IEEE Trans. Microwave Theory Tech., vol. 33, no. 10, pp. 882-893, 1985. [17] P. B. Johns, "On the relationship between TLM and finite-difference methods for Maxwell's equations," IEEE Trans. Microwave Theory Tech., vol. 35, no. 1, pp. 60-61, 1987. [18] W. K. Gwarek and P. B. Johns, "Comments on ‘On the relationship between TLM and finite-difference methods for Maxwell's equa­tions," IEEE Trans. Microwave Theory Tech., vol. 35, no. 9, pp. 872-873, 1987. [19] P. P. M. So et al., "A two-dimensional transmission line matrix mi­crowave field simulator using new concepts and procedures," IEEE Trans. Microwave Theory Tech., vol. 37, no. 12, pp. 1877-1884, 1989. [20] R. H. Jansen et al., "A comprehensive CAD approach to the design of MMIC's up to MM-wave frequency," IEEE Trans. Microwave Theory Tech., vol. 36, no. 2, pp. 208-219, 1988. [21] E. Pettenpaul et al., "CAD models of lumped elements on GaAS up to 18GHz," IEEE Trans. Microwave Theory Tech., vol. 36, no. 2, pp. 294-304, 1988. [22] R. J. Hwu et al., "Array concepts for solid-state and vacuum mi­croelectronics millimeter-wave generation," IEEE Trans. Electron Devices, vol. 36, no. 11, pp. 2645-2650, 1989. [23] C. F. Jou et al., "Millimeter-wave diode-grid frequency doubler," IEEE Trans. Microwave Theory Tech., vol. 36, no. 11, pp. 1507­1514, 1988. [24] M. Caulton et al., "Status of lumped elements in microwave inte­grated circuits-Present and future," IEEE Trans. Microwave Theory Tech., vol. 19, no. 7, pp. 588-599, 1971. [25] R. Levy, "A new class of distributed prototype filters with applica­tions to mixed lumped/distributed component design," IEEE Trans. Microwave Theory Tech., vol. 18, no. 12, pp. 1064-1071, 1970. [26] J. A. Shaw, C. H. Dumey, and D. A. Christensen, "Computer-aided design of two-dimensional electric-type hyperthermia applicators us­ing the finite-difference time-domain method," IEEE Trans. Biomed. Eng., vol. 38, no. 9, pp. 861-870, 1991. [27] A. Taflove, "The FDTD method for numerical modeling of EM wave interactions with arbitrary structures," in Electromagnetics Re­search, Part 2: Progress in Electromagnetics Research, J. A. Kong Ed., Elsevier, New York, 1990, ch. 8. [28] G. Mur, "Absorbing boundary conditions for the finite-difference ap­proximation of the time-domain electromagnetic-field equations," IEEE Trans. Electromagn. Compat., vol. 23, no. 4, pp. 377-382, 1981. [29] W. Sui, "An extended finite difference time domain method for hy­brid electromagnetic systems with active and passive lumped ele­ments," Master's thesis, Electrical Engineering Department, Uni­versity of Utah, June 1991. [30] D. Christensen et al., "Evanescent-wave coupling of fluorescence into guided modes: FDTD analysis," SPIE, vol 1172, Chemical, Biochemical, and Environmental Sensors, pp. 70-74, 1989. [31] C. S. Aitchison et al., "Lumped-circuit elements at microwave fre­quencies," IEEE Trans. Microwave Theory Tech., vol. 19, no. 12, pp. 928-937, 1971. Wenquan Sui, photograph and biography not available at the time of pub­lication. Douglas A. Christensen (M'69), photograph and biography not available at the time of publication. Carl H. Durney (S'60-M'64-SM'80), photograph and biography not available at the time of publication.