Current-Based Method
Solving for the Current Instead of the Field
The defining choice of a current-based method is the primary unknown. Rather than computing the electric and magnetic fields everywhere in space, the formulation treats the induced current density J on conductors (and the equivalent polarization current in dielectrics) as the quantity to solve for. Once that current is known, every other observable, including the radiated field, input impedance, near-field distribution, and S-parameters, follows from a straightforward radiation integral. This is a direct application of integral-equation theory: the unknown current is buried inside an integral operator, and the operator is inverted numerically after discretization.
The mechanism that makes this possible is the dyadic Green's function, which already satisfies Maxwell's equations and the Sommerfeld radiation condition in the background medium. Because the radiation condition is built into the kernel, the solver never meshes the open air around the structure and never needs an absorbing boundary or PML layer. That property is the reason current-based methods dominate open-region problems. The cost appears elsewhere: the Green's function couples every part of the structure to every other part, producing a fully populated matrix where field-based methods would give a sparse one.
The discretization workflow is consistent across implementations. The current is expanded as a weighted sum of N basis functions, the relevant boundary condition (tangential field continuity on a conductor) is enforced by projecting the residual onto N test functions, and the result is a linear system Z I = V. The N-by-N matrix Z is the generalized impedance matrix, V is the excitation vector, and the solution vector I holds the basis-function coefficients. Galerkin testing, in which the test functions equal the basis functions, keeps Z symmetric and is the most common scheme in production tools.
The Governing Integral Equation
For a perfect electric conductor, enforcing zero tangential total field gives the Electric Field Integral Equation (EFIE), which is the most widely used current-based statement. The Magnetic Field Integral Equation (MFIE) and the Combined Field Integral Equation (CFIE) are alternatives that suppress the spurious interior-resonance breakdown of the EFIE on closed bodies.
[Z]N×N {I}N = {V}N, I(r) ≈ ∑n=1N In fn(r)
Electric Field Integral Equation (PEC):
Einctan = jωμ ∫S G̅(r,r′)·Js(r′) dS′ |tan
Matrix element (Galerkin):
Zmn = ⟨fm, L(fn)⟩, Vm = ⟨fm, Einc⟩
PEEC partial inductance:
Lp,ij = (μ0 / 4π) (1 / aiaj) ∫Vi∫Vj (1 / Rij) dVi dVj
Where fn = basis function, G̅ = dyadic Green's function, Js = surface current, L = integro-differential operator, ⟨·,·⟩ = inner product (testing), Rij = distance between current cells. Storage of [Z] scales as N2; direct solve as N3.
Current-Based vs. Field-Based Solvers
| Method | Primary unknown | What is meshed | Matrix | Open-region cost | Best application |
|---|---|---|---|---|---|
| MoM (EFIE) | Surface current Js | Conductor surfaces | Dense, N×N | Excellent (no air mesh) | Antennas, planar circuits |
| PEEC | Branch / cell currents | Conductor volume | Dense R, L, C, G | Good (SPICE-coupled) | Interconnect, EMI |
| MLFMA | Surface current Js | Conductor surfaces | Implicit, N log N | Excellent (electrically large) | Radar cross-section |
| FEM (field) | E or H field | Full 3D volume + PML | Sparse | Poor (volumetric mesh) | Enclosed 3D structures |
| FDTD (field) | E, H in time | Full 3D grid + ABC | Explicit time-march | Poor (grid fills space) | Broadband, nonlinear |
Frequently Asked Questions
How does a current-based method differ from a field-based solver like FEM or FDTD?
A current-based method meshes only the conductors and dielectric interfaces and solves an integral equation for the induced current; the Green's function already enforces the radiation condition, so open air needs no mesh and no PML. Field-based FEM and FDTD mesh the entire volume including free space. The current approach therefore wins on open-region problems like antennas, scaling with surface area, but pays with a dense matrix instead of a sparse one.
Why does the Method of Moments produce a dense impedance matrix?
Every basis function radiates through the Green's function, which has infinite spatial support, so it couples to every test function and fills all N×N entries of Z. Direct LU factorization then costs order N3 operations and order N2 memory; a 10,000-unknown model needs roughly 1.6 GB in complex double precision. Fast algorithms like MLFMA and adaptive cross approximation drop the matrix-vector product to order N log N for million-unknown runs.
What basis functions are used to expand the unknown current?
Wires use piecewise-sinusoidal or triangular rooftop functions. Arbitrary metal surfaces use the Rao-Wilton-Glisson (RWG) basis over adjacent triangle pairs, which enforces normal-current continuity across the shared edge. Dielectric volumes use SWG tetrahedral functions. Typical accuracy needs 10 to 20 unknowns per wavelength, and Galerkin testing keeps the matrix symmetric.