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1
Methods of Electromagnetic Simulation
1.1 Introduction
An easy methodology for constructing diagrams (plates) showing the electric lines of force and equipotential surfaces of different charge distributions was presented more than one century ago [1]. This methodology was probably the first graphical method for performing electrostatic analysis. In fact, this technique has three formulations, two analytical (closed-form equations or the solution of simultaneous equations) and one graphical (the connection of curve intersection points). Since then, many techniques have been used to simulate all sorts of electromagnetic phenomena. These techniques can be separated into seven kinds: analytical, graphical, circuital, experimental, statistical, numerical, and those based on analogies [2].
Among the numerical techniques, the following ones are recognized: the finite-difference spatial-domain (FDSD) method [3]; the waveguide model (WGM), 1955 [4]; the generalized scattering matrix (GSM) technique, 1963 [5]; the method of moments (MoM), 1964 [6,7,8]; the method of lines (MoL), 1965 [9]; the finite-difference time-domain (FDTD) method, 1966 [10,11]; the mode matching method (MMM), 1967 [12]; the spectral domain approach (SDA), 1968 [13]; the finite-element (FE) method, 1968 [14]; the transmission line matrix (TLM) method, 1971 [15]; the integral-equation (IE) method, 1977 [16,17]; the finite-integration technique (FIT), 1977 [18]; the transverse resonance technique (TRT), 1984 [19]; and the generalized multipole technique (GMT), 1990 [20]. Many of these methods are presently being used in very powerful commercial simulation software programs.
All these techniques can be divided into two groups: domain methods and boundary methods [20]. In domain methods the region limited by the boundaries is discretized, and differential equations must be solved, whereas in boundary methods the boundaries themselves are discretized, reducing by one the size of the problem, and integral equations must be solved. Some of the methods belong to both categories.
As can be realized from previous paragraphs, a plethora of numerical analysis methods have been proposed to solve electromagnetic problems. Among these, the FDTD is a very good candidate to perform microstrip simulations due to its simplicity and excellent didactical properties. This method has the following characteristics:
• The algorithm is formulated with easy-to-solve differential equations instead of complicated integral equations.
• It simulates passive and active (linear and no linear) circuits.
• It analyzes planar circuits (microstrip, stripline, coplanar, etc.) and waveguide structures.
• The media parameters (ε, μ, and σ) are assigned to each individual cell allowing analysis of compounded structures with different kinds of conducting and dielectric materials.
• Although the method has a large numerical expense, it is very efficient because saves much memory storing the field distribution at one moment only, instead of working with bulky matrix equation systems.
• Typical time-domain pulses, like Gaussian, sinusoidal, or step, can be used as stimulus to obtain broad-band frequency responses via the discrete Fast Fourier Transform.
• When the telegrapher equations are used, the circuit terminations (matched, unmatched, open, or short) are themselves the boundary conditions, conferring to the model a natural or intrinsic feature.
• Use of circuit terminations as boundary conditions reduces the numerical error caused in the frequency-domain responses, by the highly sensitive Fourier Transform of the time-domain data when imperfect boundaries are employed.
• Only two approximations of concern are utilized, the physical segmentation of geometries with the consequent numerical discretization, and the consideration of thin substrates with the empirical calculation of fringing when a two-dimensional model is used.
All these characteristics confer to the FDTD method the quality of a well-structured and powerful simulation technique.
1.2 Two- and Three-Dimensional FDTD Models
The Maxwell partial differential equations describe wave propagation on regions consisting of a kind of dielectric or free-space, which requires artificial boundaries to limit, to manageable values, their physical dimensions and hence the computational space of analysis. On the contrary, the telegrapher equations describe wave propagation on transmission lines (confined or semiconfined regions) bounded by physical charges representing less computational effort. Then, the Maxwell equations can be discretized into 1-D, 2-D, or 3-D models, whereas the telegrapher equations may be only discretized into 1-D and 2-D models. Nonetheless, the results obtained from 2-D FDTD simulations of planar circuits, such as those carried out on a microstrip, can be good enough or even comparable to that obtained using 3-D simulations, making the use of the 3-D analysis sometimes unnecessary. A comprehensive study of the theory and techniques of 3-D models can be found in what is considered as the FDTD bible [11].
Whatever the case, the typical parameter assessed in FDTD analysis of microwave circuits, which is determined by the ratio of electric to magnetic fields or voltage to current waves, is the impedance. In general, however, in most radio- and high-frequency circuits, the parameter of interest is the reflection coefficient instead of the impedance. As a consequence, a direct transformation between immittance and reflection coefficient must be performed. Unfortunately, because of the discontinuities and due to the change of the reference plane introduced by the connectors (embedding), sometimes this transformation is not as direct as seems to be, as will be shown in Chapter 5.
To solve this problem, a turn away from the discontinuities or a connector’s de-embedding must be performed in one of two ways: by incrementing or decrementing the length of the transmission lines sections (augment or reduction of cells) or by using some of the simple transformations constituting the more general bilinear or Möbius transformation [21], which correct the deviations when the input impedance is transformed to input reflection coefficient in 2-D FDTD simulations of connectorized microstrip transmission line circuits. By using the latter, the divergences are sensibly corrected when transformations of translation, dilatation (expansion or compression), and sometimes reciprocation, rotation, and inversion [22,23] are used. However, as will be explained in Chapter 5, due to pedagogical motives, the former method is more suitable when pertinent. As examples, a straight transmission line, two impedance transformers, one synchronous and the other non-synchronous, a right-angle bend, a low-pass filter, and a two-stub four-port directional coupler, all of them constructed on microstrip technology and terminated in SMA (SubMiniature version A) connectors, are analyzed, simulated, and characterized (except the filter that was not constructed).
References
1. J. C. Maxwell, A Treatise on Electricity and Magnetism, Dover, New York, 1954. Two volumes.
2. S. R. H. Hoole, Computer-Aided Analysis and Design of Electromagnetic Devices, Elsevier, New York, 1989.
3. R. V. Southwell, Relaxation Methods in Engineering Science, Clarendon Press, Oxford, 1940.
4. A. A. Oliner, Equivalent circuits for discontinuity in balanced strip transmission line, IRE Trans. Microwave Theory Tech., vol. MTT-3, pp. 134–143, Mar. 1955.
5. R. Mittra and J. Pace, A new technique for solving a class of boundary value problems, Rep. 72, Antenna Laboratory, University of Illinois, Urbana, 1963.
6. L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, translated by D. E. Brown, Pergamon Press, Oxford, 1964, pp. 586–587.
7. R. F. Harrington, Field Computation by Moment Methods, Macmillan, New York, 1968.
8. M. N. O. Sadiku, Numerical Techniques in Electromagnetics, CRC Press, Boca Raton, FL, 1992.
9. O. A. Liskovets, The method of lines, Review, Diferr. Uravneniya, vol. 1, pp. 1662–1678, 1965.
10. K. S. Yee, Numerical solution of initial boundary-value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas Propagat., vol. AP-14, pp. 302–307, May 1966.
11. A. Taflove and S. C. Hagness, Computational Electrodynamics the Finite-Difference Time-Domain Method, Artech House, Norwood, MA, 2000.
12. A. Wexler, Solution of waveguide discontinuities by modal analysis, IEEE Trans. Microwave Theory Tech., vol. MTT-15, pp. 508–517, Sep. 1967.
13. E. Yamashita and R. Mi...
