| Long before computers, engineers derived equivalent circuits of important mathematical equations, such as Maxwell's equations, for use as analogue computers or for solution by direct methods. However, with the advent of computers, these direct methods gave way to new and faster iterative techniques for solving larger problems. In the process, use of such equivalent circuit representations also faded away. In this work a numerical technique has been developed, using an equivalent circuit that is essentially the same as one developed by Gabriel Kron over 50 years ago, to solve Maxwell's equations in the frequency domain using nonuniform rectangular cells. This three-dimensional equivalent-circuit formulation creates a linear system of complex equations of the form Ax = b, where matrix A is complex, symmetric, and very sparse. This non-Hermitian system is solved iteratively using the complex symmetric quasi-minimal residual method. The resulting equivalent-circuit method merges field and circuit theory making it ideal for circuit-model active devices and hybrid systems containing lumped elements and propagating structures. The method also utilizes sources from circuit theory (nonideal and ideal voltage and current sources) and from field theory (applied electric and magnetic field sources). Convergence rates for frequency-domain methods are dependent only on the properties of the system being solved, i.e., matrix conditioning. This provides potential advantages for this method in the lower and middle frequency ranges over the finite-difference time-domain method which requires more time to achieve steady-state as the frequency is lowered. However, this finite-difference frequency-domain method does have the disadvantages associated with solving a set of matrix equations. The equivalent-circuit method, which reduces to circuit theory at low frequencies, has been validated from dc to microwave frequencies. This method was used for extensive calculations of electric fields induced near metal implants by switched-gradient magnetic fields. The results are presented and discussed in detail. The greatest E-field concentration occurred near the ends of the cut insulated wire, where the E field was 196.7 times greater than in the absence of the wire. These data were used by Reilly et al. at Johns Hopkins University to assess possible nerve stimulation. Results for high-frequency test cases using absorbing boundary conditions are also presented. |
