Developments of fundamental locally one-dimensional finite-difference time-domain methods for transmission lines and lumped elements

This thesis presents the developments of fundamental locally one-dimensional finite-difference time-domain (FLOD-FDTD) methods for transmission lines and lumped elements. Unlike explicit FDTD method, the FLOD-FDTD method can use time step larger than Courant-Friedrichs-Lewy (CFL) limit. It also achieves higher efficiency and simplicity with matrix-operator-free right-hand sides (RHS) compared with the conventional LOD-FDTD method. The updating equations of the FLOD-FDTD method incorporating resistors, inductors and capacitors are derived. In addition to the three commonly used lumped elements, explicit FDTD method and FLOD-FDTD method incorporated with memristor are also developed. Memristor is regarded as the fourth lumped element which links charge and flux. The update equations are derived based on Maxwell's equations and the physical model given by Hewlett-Packard (HP) lab. Numerical examples including transmission lines and various lumped elements are provided to show the trade-off between efficiency and accuracy of the proposed 3-D FLOD-FDTD method. Since the conventional LOD-FDTD method is of first-order accuracy only, second-order temporal-accurate scheme for 3-D FLOD-FDTD method is also introduced to improve the accuracy for LOD-FDTD method with three split matrices. The main iterations of the proposed scheme comprise only one more procedure in addition to the existing three procedures. Using proper (initial only) input and (often infrequent, field-point only) output processings, the second-order temporal accuracy is achieved and verified analytically. Stability analysis is provided to ascertain the unconditional stability of the proposed scheme and numerical results are demonstrated to justify its higher accuracy. Moreover, an alternative reversed second-order temporal-accurate FLOD-FDTD scheme is also proposed. Numerical examples such as reverberation chamber and transmission line circuit are demonstrated to show the higher accuracy of the proposed FLOD-FDTD method. Although full 3-D FDTD methods can cater for most general structures and 3-D FLOD-FDTD method can use time step larger than CFL limit for higher efficiency, the computational domain is still large and full 3-D methods are not well-suited to run on mobile devices such as smart phones/pads due to the requirement for large memory storage and long simulation time. To circumvent this problem, novel interconnected multi-one-dimensional (IM1-D) fundamental alternating-direction-implicit (FADI)- and FLOD-FDTD methods for transmission lines with interjunctions are introduced to further enhance the computational efficiency. The proposed methods are unconditionally stable and capable of treating multiple main transmission lines and stubs interconnected at various interjunctions using time step larger than CFL limit. Fundamental scheme-based IM1-D FADI- and FLOD-FDTD methods are both derived to enhance the efficiency with matrix-operator-free RHS. The methods involve one-step update procedure optimized for simulation of main transmission lines and stubs on mobile device. Using proper treatments at the interjunctions for various interconnection conditions, the electromagnetic fields in all interconnected main transmission lines and stubs can be updated cooperatively and efficiently to solve practical problems. A microstrip line loaded with stubs and a branch-line coupler are simulated to show the accuracy and efficiency of the proposed methods. To extend the applicability for handling the couplings between two transmission lines via gaps, a microstrip circuit with gaps is simulated using the proposed methods incorporated with equivalent circuit models involving capacitances. Real-time simulations of these numerical examples provide much intuitional insight for one to observe the electromagnetic waves propagation in time domain on computer or mobile device. Finally, a novel non-uniform time step (NUTS) FLOD-FDTD method for multiconductor transmission lines (MTLs) including lumped elements. Unlike conventional and the other LOD-FDTD methods, the NUTS scheme adopts different (non-uniform) time steps for different periods during simulation, in order to reduce the large errors caused by unconditionally stable FDTD methods with uniform time step larger than CFL limit. It is found that NUTS scheme is potentially unstable for ADI-FDTD method, but it is stable for LOD-FDTD method or its implementation based on fundamental scheme. The NUTS FLOD-FDTD method is useful to simulate MTLs with lumped elements in series and parallel connections. Moreover, the method based on multiple 1-D approach is also extended to incorporate lumped elements in cross connection to model near-field coupling between the MTLs. While the electric current density is commonly used for field-circuit coupling of parallel connected lumped elements, the magnetic current density will be utilized for field-circuit coupling of series and cross-connected lumped elements. Numerical results for MTLs with lumped elements in series, parallel and cross connections are provided to show the trade-off between efficiency and accuracy of the proposed NUTS FLOD-FDTD method. Compared with the FLOD-FDTD method with uniform time step, the NUTS FLOD-FDTD method may achieve higher accuracy by using smaller time step for certain periods of simulation.Doctor of Philosophy (EEE