Analysis of Microstrip Circuits Using a Finite-Difference Time-Domain Method M.G. BANCIU and R. RAMER School of Electrical Engineering and Telecommunications University of New South Wales Sydney 5 NSW AUSTRALIA Abstract The Finite-Difference Time-Domain method provides analysis with a high degree of accuracy and versatility of high frequency (RF, microwave etc.) components and circuits. The paper presents aspects of application of this method to microstrip circuits often used in Monolithic Microwave Integrated Circuits (MMIC) with direct application for wireless and satellite communications. Key-Words: - microstrip circuits, time domain methods, full wave analysis. Introduction The miniaturization of the high frequency integrated circuits requires more accurate design methods to control all the effects such as packaging effects, cross coupling, surface waves effects mostly neglected by some commercial computer aided design (CAD) software packages. To go beyond the model limits of components accepted by a program as Touchstone TM [] or similar one needs to apply a full wave analysis method. The Finite-Difference Time- Domain (FDTD) method distinguish itself from the class of full wave analysis methods because its accuracy and versatility. Some aspects of application of FDTD method to the microwave microstrip circuits are discussed in this paper. Foundations of FDTD. The FDTD cell Early finite-difference methods applied to Maxwell equations as an attempt to describe the electromagnetic field propagation leaded to spurious (nonphysical) solutions. The FDTD method selects from all four Maxwell equations the two curl equations and solves them numerically by finite differences in time domain. The grid points where the electric and magnetic fields are calculated alternates in space forming the FDTD cell. Each component of the electric field is surrounded by four components of the magnetic field and each component of the magnetic field has as first order neighbors four electric field components []... Source excitation The field excitation was chosen as a Gaussian pulse E ( t) = E z z e ( t T ) T where T is the pulse half width in time domain and T =4T. The maximum frequency from the Gaussian spectrum we can rely on is f = max. The stability of T the method is assured by Courant condition t = kc x + y + z
where k is Courant factor and c is the speed of light. A common excitation for microstrip circuits contains a voltage source below microstrip. The signal needs to propagate a certain distance along the microstrip line to let the transient modes to vanish and to reach their true modal nature. In order to minimize the computational domain, first it is simulated the pulse propagation along a simple input microstrip line. After propagating a certain distance, when the signal got the correct transversal profile the pulse is copied at the input microstrip line on structure to be analyzed. The same input signal can be used for several simulations for various planar structures when the same substrate and the same FDTD grid are preserved..3 Absorbing boundary conditions (ABC) The computational domain is always finite therefore some conditions at its boundaries need to be applied. Sometimes, very seldom, the analyzed structure has symmetry planes and the boundary conditions can be expressed as magnetic walls or perfect magnetic conductors (PMC). The metallic enclosure (housing), ground plane etc. can be considered as perfect electric conductors (PEC). Besides PMC and PEC, a very important case is the absorbing boundary conditions (ABC) when the pulse should suffer a minimum spurious reflection at the boundary of the computational domain. This paper demonstrates that the FDTD analysis of microstrip circuits depends drastically on the ABC quality. We will show that a poor ABC as st order Mur s ABC [3] limits the method accuracy. Nevertheless we show in the next section that Bérenger s Perfectly Matched Layer (PML) [4-5] can be extended to microstrip circuits increasing the method precision and maximum frequency bandwidth for accurate results. The PML technique decomposes each field projections in two and the wave incident to PML is attenuated via electric (σ) and magnetic (σ * ) conductivity. The extremely small reflection is satisfied by the impedance matching condition perpendicularly to the PML layer. 3. FDTD applied to microstrip circuits FDTD method allows the pulse propagation simulation in the microstrip circuit. In Figs., is presented the E z component of the electric field just underneath the dielectric-air interface. TS=55 Fig.. The pulse is guided by the microstrip line along Oy axis Fig.. The pulse changed direction after a mittered corner and reached a wider microstrip line (lower impedance)
3.. Dispersion effects The fundamental propagation mode for the microstrip line it is considered as approximating the Transversal Electric and Magnetic (TEM) mode when the fields are oscillating only in a perpendicular plane on the direction of propagation. For an ideal TEM mode, not including the material effects, the pulse should suffer no dispersion. In practice the microstrip properties are considered not depending on frequency for a frequency bandwidth up to GHz. However, the fundamental mode for the microstrip line it is considered as quasi-tem only and different models for dispersion effects were proposed. [6] In Fig. 3 it is presented the Ez field just underneath a straight microstrip line versus time at different locations. The distances from the source plane to measuring points are given in y units. The substrate was alumina (Al O 3 ) with dielectric constant ε r =9.98, thickness h=.635mm. The 5Ω line width was w5=.635mm. The FDTD grid has x= y=.55 mm and z=.7mm relat. units).8.6.4..8.6.4. y= y=3 y=5 y=9 y=3 -. 4 6 8 Time steps Fig. 3 The pulse is measured versus time at different locations along the microstrip. The distance y from the source is given in y units and the time in t units. ε r =9.98, h=.635mm, t=.7ps, w5=.635mm x= y=.55mm, z=.7mm, T =4T, T=38 t The incident Gaussian pulse has T=38 t and T =4T. Fig. 3 shows that the dispersion of the quasi-tem propagating mode distorts the propagating pulse. 3. Linear low impedance microstrip resonators The time domain data provided by FDTD can be Fourier transformed in frequency domain. The ratio between the transmitted and reflected pulses at all the circuits ports over the incident pulse provide the scattering S parameters over the frequency bandwidth. In this paper there were analyzed reciprocal two-ports therefore only the complex values of S (transmission) and S (reflection) could fully describe the circuits. A class of microstrip circuits examined with FDTD was containing the low impedance (larger strip width) linear resonators directly coupled. (db) and S (db) - - -3-4 -5-6 5 5 5 3 Fig 4. S (closer to. db) and S data in decibells versus frequency (GHz) obtained with FDTD program (solid line) and TOUCHSTONE (dashed line) for a linear microstrip resonator. ε r =3.38, h=.8mm. w5=.87mm, w reson = 4.7, l reson = 5.54mm.
Fig. 4 presents the results for a low impedance linear resonator on a substrate with dielectric constant ε r =3.38, and thickness h=.8mm. The 5 ohms line with is w5=.87mm, resonator line width w reson =4.7 and the resonator length l reson =5.54mm. First order Mur s ABC were used as boundary conditions at input and output ports. Touchstone can provide accurate S parameters and it was used to validate our FDTD software. Fig. 4 shows a good accord between FDTD results and Touchstone output up to 7 GHz. The FDTD accuracy is limited mainly by the poor ABC. The st order Mur s ABC assumes a non-dispersive mode propagating perpendicularly to the boundary plane. The dispersive effects increase with frequency therefore these ABC give spurious numerical reflections at higher frequency. To improve the FDTD accuracy the perfect matched layer (PML) was implemented as ABC. PML is not restricted to non-dispersive modes and gives very good results even for higher dispersive microstrip lines on a higher dielectric constant substrate as shown by the S parameters versus frequency plots in Fig. 5. The substrate, 5Ω line, FDTD grid, time step and source were chosen identical to those in Fig. 3. The resonator line width is was w reson =.68mm, and the resonator length was l reson =.6mm. 3.3 Gap end coupled linear resonator This kind of resonator has a higher quality factor (Q-factor) requiring a more time step to complete the FDTD simulation. The S parameters plots shown in Fig. 6 were calculated with a number of time steps TS=6384. A less number of time steps do not decrease very much the accuracy but a small ripple starts to be noticed in the frequency data. The parameters were chosen as in Fig. 3. The resonator line width is w 5 and the length is l reson =mm. The coupling gap spacing is s=.3mm. (db) and S(dB) - - -3-4 -5-6 5 5 5 3 Fig. 5. S (dashed line) and S (continuous line) parameters for low impedance linear microstrip resonator on Al O 3 substrate. The substrate, 5Ω line, FDTD grid, time step and source were chosen identical to those in Fig. 3. w reson =.68mm, l reson =.6mm. (db) and S (db) - - -3-4 -5-6 -7 5 5 5 3 Fig. 6. S (continuous line) and S (dotted line) versus frequency for a linear gap end coupled resonator. Parameters as in Fig. 3. l reson =mm, s=.3mm. 4 Conclusion A Finite-Difference Time-Domain method for the analysis of the microwave microstrip circuits was successfully implemented. In the time domain the method allows the simulation of a Gaussian pulse propagation through a microstrip circuit. The dispersion effects of the quasi-tem propagation mode were revealed after the
pulse propagation along a linear microstrip line. Using a discrete Fourier Transform (DFT) the S parameters plots versus frequency can be obtained. In the paper there were analyzed two-ports especially microstrip linear resonators to be able to compare the FDTD results with results given by other commercial software. However the FDTD method is not limited to a finite set of microwave modeled components. The analysis of linear low impedance resonators revealed the importance of a good quality absorbing boundary conditions (ABC). Perfect Matched Layer (PML) was proven to provide a higher degree of accuracy than st order Mur s ABC. PML applied to FDTD for microstrip circuits is numerical stable even for a large number of time steps required by a high quality factor (Q factor) as for the end gap coupled linear resonators. [6] R.L. Veghte, C.A. Balanis, Dispersion of Transient Signals in Microstrip Transmission Lines, IEEE Trans. Microwave Theory Tech., Vol. MTT- 34, No., 986, pp. 47-436 References: [] HP-Eesof Microwave & RF Circuit Design Circuit Element Catalog, Hewlett R -Packard, March 994 [] A. Taflove, Computational Electrodynamics The Finite-Difference Time-domain Method, Artech, 995 [3] G. Mur, Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations, IEEE Trans. Electromagn. Compat., Vol. EMV-3,, Nov. 98, pp. 377-38 [4] J.-P. Bérenger, Perfectly Matched Layer for the FDTD Solution of Wave- Structure Interaction Problems, IEEE Trans. Microwave Theory Tech., Vol. MTT-44, 996, pp. -7 [5] J.-P. Bérenger, Improved PML for the FDTD Solution of Wave-Structure Interaction Problems, IEEE Trans. Microwave Theory Tech., Vol. MTT- 45, 997, pp. 466-473