How do you calculate the dimensions of a microstrip patch antenna?

The width is W = c/(2f₀)·√(2/(εr+1)). The length is L = c/(2f₀√εeff) − 2ΔL, where εeff is the effective permittivity of the patch strip and ΔL the open-end extension (Hammerstad). This calculator computes εeff with a 2D finite-element solver and then refines L and the feed in a network model so the input is matched at f₀.

Where should the inset feed or coaxial probe be placed?

The input resistance falls from the edge resistance Rin at the radiating edge to zero at the centre, roughly as Rin·cos²(πy₀/L). The feed sits where this equals the system impedance, usually 50 Ω. Substrate loss and the probe inductance move the best point, which the network refinement accounts for.

How does a truncated-corner patch produce circular polarization?

Cutting two opposite corners of a square patch splits its two orthogonal modes in frequency. When the split is about 1/Q, a single feed excites both modes 90° apart at the design frequency, giving right- or left-hand circular polarization depending on which diagonal is cut. The calculator sizes the cut, the square and the feed for a low axial ratio at f₀.

How accurate is the calculator?

The fast network model is typically within a few percent in resonant frequency. A 3D FDTD full-wave solve of the actual copper, ground plane and substrate can be run in the browser to check it, and one click corrects the model and the dimensions to agree with the full-wave result.

Microstrip Patch Antenna Calculator – Design, Simulate & Optimize

Step 1

Port solver

A 2D finite-element solve of the microstrip cross-section gives Z₀ and ε_eff for every width. Feed lines and transformers come from this library; the wide patch strip gets its own FEM solves in step 2.

Substrate & frequency

Laminate preset · typical datasheet design values, sorted by εr; check your supplier's data

Dielectric constant εr

Substrate height

Copper thickness

µm

Loss tangent

Conductivity

MS/m

Center frequency

GHz

Only need the feed line? Use the microstrip calculator.

Substrate changes need a new port solve. Center frequency changes apply immediately.

Cross-section field

Line library FEM points against the closed-form curve

2D FEMHammerstad–Jensen

Validation width needed for each impedance

Method. Quasi-static FEM with linear triangles on a graded mesh, half-structure by symmetry, shielded box at 14 h, direct banded Cholesky solve. It runs once with the dielectric and once in air to get C and C₀, so Z₀ = 1/(c√(C·C₀)) and ε_eff = C/C₀. Frequency dispersion (Kirschning–Jansen), conductor loss and dielectric loss are added on top.

Step 3

Analyze

The fast network model gives every curve on this page in milliseconds. A 3D full-wave solve of the actual copper checks it: its S11 and impedance appear on the same charts, and one click corrects the network model and the design to agree with it. The dashed vertical line marks f₀; dashed horizontal lines are your goals.

Sweep points

Input return loss S11, dB

Input impedance at the port, Ω

3D FDTD

Full-wave check

A 3D finite-difference time-domain (FDTD) solve of Maxwell's equations on the actual copper: patch, notches, feed, substrate, ground plane and open air above, with absorbing boundaries and a 50 Ω lumped port. Fringing, notch coupling, radiation and surface waves come out of the fields instead of a model.

Mesh

Engine

Checking…

Solve
Correct
Verify

Solve region

Copper (PEC sheet)50 Ω lumped portAbsorbing boundary (CPML)

Field under the copper |E_z| in the substrate, log scale

Run the solve to watch the pulse fill the patch cavity. The TM₀₁₀ mode shows as two bright radiating edges with a null across the middle.

Comparison

Port voltage against time

The patch rings at its resonance long after the pulse has passed. The run stops as soon as that ringing is predictable, and the rest of the decay is added analytically.

Correction details

Fits corrections so the network model reproduces this full-wave S11: an extra open-end length on each radiating edge, a shift of the effective feed point (inset and probe feeds) and, if you ask for it, a scale on the radiation conductance. Applying them switches steps 2–5 to the corrected model and re-tunes your current design (patch length and the feed's match dimension) for a match at f₀, keeping every other dimension. The card at the top of this page then compares the design before and after. The buttons above guide you through solve → correct → verify, or Tune automatically does all of it, up to three rounds, until your goals are met. Corrections are only offered from Normal or Fine mesh solves: the coarse mesh puts the resonance lower than finer meshes and changes the depth of the match, and fitting to it would build that grid error into the dimensions. Even the Normal mesh sits about 1–2% below the mesh-converged resonance in our tests, so check a final design on the Fine mesh.

Also fit the radiation conductance. S11 cannot separate radiated from dissipated power, so this changes the predicted gain and efficiency without the full-wave run confirming them. Leave it off unless the bandwidth disagrees.

Use the corrected model in steps 2–5

More actions

For when you want to step outside the guided flow, for example to correct again although the goals are met, or from a coarse-mesh solve.

Correction rounds every full-wave solve of this feed type

Method. Yee-grid FDTD in 3D on the finite board: a zero-thickness perfectly conducting ground sheet the size of the ground plane, the substrate only over the board, zero-thickness copper on top, lossy dielectric (σ from tan δ at f₀), and air on every side, with cells uniform through the substrate and growing 25% per layer above and below, and convolutional PML on all six faces. The cell size divides L exactly so the resonant length never snaps (for a circularly polarized patch the square side, with truncated corners as staircase triangles whose area matches the drawn cut); widths, gap and inset depth snap to the grid and the network curve is computed for the snapped values. Microstrip feeds use a distributed 50 Ω lumped port at the board edge, calibrated with a straight line on the same mesh; the probe feed is a single-cell 50 Ω column from ground to patch, which the network models as a pin of radius 0.135 cell. A Gaussian-modulated pulse, wider than the sweep so it is short, drives the port and the reflected wave is Fourier transformed into S11 over the sweep. Once the pulse has passed, the ringing of the patch is fitted with a sum of decaying modes (matrix pencil); when that model predicts the latest part of the signal to within 1%, the run stops and the rest of the decay is added analytically to every spectrum, which typically halves the run. During the run the tangential electric and magnetic fields on a closed box around the board are Fourier transformed at seven frequencies (21 for a circularly polarized patch, for the axial-ratio curve), sampled every few dozen steps since the pulse has no content above about twice f₀; as equivalent electric and magnetic surface currents they give the far field over the full sphere, front and back, and with the port power they give directivity, radiation efficiency, realized gain and front-to-back ratio. It runs on the GPU through WebGPU when the browser offers it, otherwise on the CPU in a background thread. Limits: copper loss and thickness are not included, and the ground sheet has no thickness.

Network model, full-wave once solved

Radiation and losses

Pattern, gain and efficiency over the full sphere, front and back of the board. The network model adds an edge-diffraction estimate for the back side; once the full-wave check has solved this design, its far field from a closed box around the board is laid over the network curves.

3D radiation pattern at f₀

Radiation pattern at f₀ realized gain, dBi

Smith chart S11 locus, band in accent

Gain against frequency broadside, dBi

Where the power goes at f₀ share of available power

Touchstone export

One-port S11 over the sweep in Touchstone 1.1 format (GHz, magnitude/angle, referenced to the system impedance). Paste into a .s1p file for ADS, AWR, scikit-rf or HFSS.

Model. The patch is a wide microstrip line between two radiating slots at its open ends. Each slot carries the conductance G₁ (space wave) plus a surface-wave conductance, and the susceptance of its fringing field, Y_c·tan(βΔL); the slots are coupled through the mutual conductance G₁₂. Line sections use the FEM-anchored impedance and ε_eff with Kirschning–Jansen dispersion, conductor loss and dielectric loss. The pattern is that of two in-phase magnetic line currents over an infinite ground, weighted by the solved slot voltages, so it shows no back lobe or cross-polarization; a finite ground plane adds radiation behind the board and a little ripple in front.

Step 4

Optimize

Set the goals and the dimensions the optimizer may change, then minimize the goal cost. A cost below zero means every enabled goal is met; a small margin term keeps improving the design past that point.

Goals penalty = weight × (violation / scale)²

Return loss is checked at f₀. Resonance is the frequency of the S11 minimum; bandwidth is the span around it where S11 stays below −10 dB, so the bandwidth goal sets how wide a band the patch must cover. Gain and efficiency are at f₀.

Design variables

Step is how far the optimizer moves that dimension on its first iteration, and the grid every candidate is rounded to. All dimensions stay on a 0.01 mm grid regardless, so every value has at most two decimals; set the step to 0 to let the optimizer use any 0.01 mm value.

Parameter sweep try one dimension over a range, compare, pick one

Dimension

From

Values

Then fine-tune the other free dimensions around it

Which dimensions matter optional screening before you optimize ▾

Method

Trajectories

Rank by

Morris samples r random paths through the variable space, changing one variable at a time. μ* is the mean absolute effect; σ shows interaction or nonlinearity.

Effect ranking

Interaction map Morris μ* against σ

Points far right matter most. Points high above the diagonal interact with other variables or act nonlinearly.

Fix below%

Optimizer

Algorithm

Max evaluations

Convergence best cost so far

Start against best

About this microstrip patch antenna calculator

This free calculator from NextAntennas designs a microstrip patch antenna from the substrate and the design frequency, then lets you analyze, verify and optimize it in the browser. It covers the rectangular patch for linear polarization and the truncated-corner square patch for circular polarization (RHCP or LHCP), each with an inset microstrip feed, an edge feed with a quarter-wave transformer, or a coaxial probe.

How to use it

Port solver: pick a laminate such as FR-4 or Rogers RO4003C; a 2D finite-element solver finds the 50 Ω feed-line width and effective permittivity.
Patch synthesis: closed-form patch width and length, open-end extension, inset depth or probe position, then a network refinement for a match at f₀. For circular polarization it sizes the square, the corner cut and the feed for a low axial ratio.
Analyze: S11 and return loss, input impedance, Smith chart, bandwidth, gain, radiation efficiency, axial ratio, polar and 3D radiation patterns, and a power budget. A 3D FDTD full-wave solve checks the design and corrects the model.
Optimize: set goals, sweep a dimension, or run Nelder–Mead or differential evolution; dimensions are rounded to a 0.01 mm fabrication grid.
Yield: Monte Carlo analysis of etching, permittivity, thickness and probe-position tolerances.

Patch antenna design equations

Patch width W = c/(2f₀)·√(2/(εr+1)); length L = c/(2f₀√ε_eff) − 2ΔL with the Hammerstad open-end extension ΔL. The input resistance at the radiating edge is R_in = 1/(2(G₁ + G₁₂)) from the slot and mutual conductances, and an inset or probe at distance y₀ sees about R_in·cos²(πy₀/L). A quarter-wave transformer has Z_t = √(Z₀R_in). For a truncated-corner circularly polarized patch the corner cut c splits the two modes by about κc²/a², chosen equal to 1/Q.

Frequently asked questions

Which substrate should I use? FR-4 is inexpensive but lossy (tan δ ≈ 0.02), so a small 2.4 GHz patch on 1.6 mm FR-4 reaches only about 40–50% efficiency; low-loss laminates such as RO4003C give higher gain.

Why is the S11 dip of a circularly polarized patch not at the design frequency? The two modes make the input response broad; the frequency that matters is where the axial ratio is lowest, and the calculator places that at f₀.

Do I need a GPU? No. Everything runs on the CPU, but the 3D full-wave solve is much faster with WebGPU in Chrome or Edge.

More RF tools: microstrip line calculator · all NextAntennas calculators.

Next step — design your feed line The World's Best Online Microstrip Calculator Calculate your 50 Ω feed line width, εeff, and guided wavelength for any substrate — instantly, free, no sign-up.

Engineering Theory

Theory Behind the Equations

How the calculator works: closed-form design equations for the starting point, a 2D finite-element (FEM) solve of the microstrip cross-section, a transmission-line network model of the patch, and a 3D finite-difference time-domain (FDTD) solve to verify and correct it.

1. What Is a Microstrip Patch Antenna?

A microstrip patch antenna is a metal patch printed on one face of a dielectric substrate, with a ground plane on the other face. This calculator designs three feeds: an inset microstrip feed, where the line is recessed into the patch through two notches; an edge feed with a quarter-wave transformer; and a coaxial probe soldered to the patch from below. It also designs the truncated-corner square patch, which radiates circular polarization from a single feed.

The patch radiates from the fringing fields at its two radiating edges, the edges perpendicular to the feed direction. The fields at the other two edges cancel in the broadside direction, so the rectangular patch is a linearly polarized, broadside radiator. Gain is typically 6–8 dBi on low-loss laminates and 2–4 dBi on FR-4, where dielectric loss takes much of the power.

The calculator works in three layers. Closed-form equations give the starting dimensions. A 2D finite-element (FEM) solve of the microstrip cross-section gives the impedance and effective permittivity of the feed line and the patch strip, and a transmission-line network model built on it gives S11, impedance, gain and patterns in milliseconds. A 3D finite-difference time-domain (FDTD) solve of the actual copper, substrate and ground plane then checks the design and corrects the network model.

2. Patch Width W

The width is chosen for efficient radiation from the transmission-line model:

Patch width

W = \frac{c}{2 f_{0}} \sqrt{\frac{2}{ε_{r} + 1}}

c = speed of light · f₀ = design frequency · εr = substrate relative permittivity

W comes out a little larger than the half wavelength in the substrate. A wider patch has more bandwidth and a lower edge resistance, but beyond about λ₀ it can excite higher-order modes. You can override W in step 2. For the circularly polarized patch the shape is square, so W = L = a.

3. Effective Permittivity ε_eff — FEM Solve of the Cross-Section

The field of a microstrip line lies partly in the substrate and partly in the air above it, so the line behaves as if it were filled with a uniform medium of effective permittivity ε_eff. The classic closed form (Hammerstad) is:

Closed form (Hammerstad)

ε_{eff} = \frac{ε_{r} + 1}{2} + \frac{ε_{r} - 1}{2} {(1 + \frac{12 h}{W})}^{- \frac{1}{2}}

This calculator does not stop there. Step 1 solves Laplace’s equation on the cross-section with the finite-element method, on a triangular mesh refined at the strip edge, once with the substrate and once with air. The two capacitances per unit length give the quasi-static impedance and effective permittivity exactly for that geometry:

FEM capacitance method

ε_{eff} = \frac{C}{C_{air}}

Z_{0} = \frac{1}{c \sqrt{C C_{air}}}

C = capacitance per unit length with the substrate · C_air = the same line in air

The patch itself is a very wide strip, so step 2 runs extra FEM solves at the patch width and uses them to correct the Hammerstad–Jensen expressions. The Kirschning–Jansen formula then adds dispersion, the slow rise of ε_eff with frequency.

4. Length Extension ΔL — Fringing at the Radiating Edges

The fringing field at each open edge makes the patch electrically longer than the copper. Hammerstad’s open-end formula gives the extension per edge:

Open-end extension (Hammerstad)

Δ L = 0.412 h \frac{(ε_{eff} + 0.3) (\frac{W}{h} + 0.264)}{(ε_{eff} - 0.258) (\frac{W}{h} + 0.8)}

Added at each radiating edge; the electrical length is L + 2ΔL

In the network model the extension appears as the susceptance of each edge, B = Y_c·tan(βΔL), where Y_c and β are the admittance and propagation constant of the patch strip. After a full-wave solve, the calculator fits a correction to ΔL so the network model matches the FDTD resonance.

5. Resonant Patch Length L

The patch resonates in the TM₀₁₀ mode when its electrical length is half a guided wavelength:

Resonant length

L = \frac{c}{2 f_{0} \sqrt{ε_{eff}}} - 2 Δ L

This equation sets the operating frequency: a patch that is too long resonates low, and one that is too short resonates high. The closed form is the starting point. The network refinement then adjusts L, together with the feed, so the input is matched exactly at f₀ including dispersion, loss, the notches and the probe inductance.

6. Slot Conductances G₁, G₁₂ and the Edge Resistance R_in

Each radiating edge is modelled as a slot of width W carrying a magnetic current. The calculator integrates the slot field numerically instead of using the narrow- or wide-patch approximations:

Self conductance

G_{1} = \frac{1}{120 π^{2}} \int_{0}^{π} {[\frac{\sin (\frac{k_{0} W}{2} \cos θ)}{\cos θ}]}^{2} \sin^{3} θ d θ

The two slots are half a wavelength apart and couple through the mutual conductance, which the old rule of thumb R_in = 1/(2G₁) leaves out. It is not small: for a typical patch G₁₂ is more than half of G₁.

Mutual conductance

G_{12} = \frac{1}{120 π^{2}} \int_{0}^{π} {[\frac{\sin (\frac{k_{0} W}{2} \cos θ)}{\cos θ}]}^{2} J_{0} (k_{0} L \sin θ) \sin^{3} θ d θ

J₀ = Bessel function of the first kind, order zero · k₀ = 2π/λ₀

Part of the power leaves as surface waves in the substrate rather than as radiation. The calculator adds this as a surface-wave conductance from the space-wave efficiency of a horizontal dipole on a grounded slab (Jackson and Alexopoulos):

Space-wave efficiency

e_{sw} = \frac{1}{1 + \frac{3 π}{4} k_{0} h \frac{{(1 - \frac{1}{ε_{r}})}^{3}}{c_{1}}}

c_{1} = 1 - \frac{1}{ε_{r}} + \frac{2}{5 {ε_{r}}^{2}}

The lossless edge resistance is then:

Edge resistance

R_{in} = \frac{1}{2 (\frac{G_{1}}{e_{sw}} + G_{12})}

Typically 200–300 Ω. Conductor and dielectric loss in the patch lower it further, which the network model includes.

7. The Network Model, the Inset Depth y₀ and the Other Feeds

The network model treats the patch as a lossy transmission line of length L between the two slot admittances G + jB, coupled through G₁₂. The line sections use the FEM impedance and ε_eff with dispersion, conductor loss and dielectric loss (tan δ). Solving the network gives the input impedance and the reflection coefficient at every frequency:

Reflection coefficient

Γ = \frac{Z_{in} - Z_{0}}{Z_{in} + Z_{0}}

S_{11} = 20 \log_{10} | Γ |

Inside the patch the TM₀₁₀ voltage follows a cosine along L, so recessing the feed by y₀ lowers the resistance roughly as:

Inset feed

R_{in} (y_{0}) \approx R_{in} \cos^{2} (\frac{π y_{0}}{L})

y_{0} = \frac{L}{π} \arccos \sqrt{\frac{Z_{0}}{R_{in}}}

This gives the starting depth. The network model then sets y₀ exactly, including the narrower patch beside the notches.

The other two feeds follow the same idea. The quarter-wave transformer matches the edge resistance with a line of impedance Z_t = √(Z₀R_in). The coaxial probe is placed at y₀ in the same way, but its pin adds a series inductance that the refinement offsets by shortening the patch slightly:

Probe reactance

X_{p} = \frac{η_{0} k_{0} h}{2 π} [\ln (\frac{2}{k_{0} \sqrt{ε_{r}} r}) - γ]

r = pin radius · η₀ ≈ 377 Ω · γ ≈ 0.5772 (Euler’s constant)

8. Notch Gap g

The notches beside an inset feed line keep the line from coupling into the patch edge. A narrow gap is enough: the calculator uses about one third of the feed-line width (at least 0.1 mm) by default, typically 0.5–1 mm. Wider gaps remove copper next to the feed and shift the match. The network model treats the part of the patch beside the notches as a narrower line of width W − w_f − 2g, so the inset depth it reports already accounts for the gap you choose.

9. Circular Polarization with Truncated Corners

A square patch has two orthogonal modes at the same frequency. Cutting two opposite corners with legs c shortens the mode polarized along that diagonal and splits the pair in frequency:

Mode split

\frac{Δ f}{f} \approx κ \frac{c^{2}}{a^{2}} \approx \frac{1}{Q}

c = \frac{a}{\sqrt{κ Q}}

a = side of the square · Q = quality factor of one mode · κ ≈ 2; a full-wave solve fits κ

With the split equal to 1/Q, the feed excites both modes equally and they sit 45° either side of resonance at f₀, 90° apart in phase, so the field rotates. The feed sees the average of the two mode impedances, Z_F = ½(Z₁ + Z₂), so it is placed where one mode alone would show 2Z₀. The sense, right-hand (RHCP) or left-hand (LHCP), depends on which diagonal is cut. The calculator splits the far field into its circular components and reports the axial ratio:

Circular components and axial ratio

E_{R} = \frac{E_{θ} + j E_{φ}}{\sqrt{2}}

E_{L} = \frac{E_{θ} - j E_{φ}}{\sqrt{2}}

AR = \frac{| E_{R} | + | E_{L} |}{| | E_{R} | - | E_{L} | |}

AR = 0 dB is perfect circular polarization; below 3 dB is the usual design target

10. Radiation Pattern, Directivity and Gain

The far field is the sum of the two slot fields weighted by the solved slot voltages. On a finite ground plane, Uniform Theory of Diffraction (UTD) terms at the board edges add the radiation behind the board. Integrating the radiation intensity U over the full sphere gives the directivity, and the power budget gives the realized gain:

Directivity and realized gain

D = \frac{4 π U_{\max}}{P_{rad}}

G_{realized} = D e_{rad} (1 - {| Γ |}^{2})

e_rad = radiated power / accepted power, after conductor, dielectric and surface-wave loss

11. Full-Wave Verification — 3D FDTD

The network model is fast but approximate, so step 3 can solve Maxwell’s equations directly with the finite-difference time-domain (FDTD) method on a 3D Yee grid. The grid contains the actual copper, the lossy substrate only under the board, the finite ground plane and air on every side, ending in a convolutional perfectly matched layer (CPML) that absorbs outgoing waves. The fields leap-frog in time:

Yee update

H^{n + \frac{1}{2}} = H^{n - \frac{1}{2}} - \frac{Δ t}{μ} \nabla\times E^{n}

E^{n + 1} = E^{n} + \frac{Δ t}{ε} (\nabla\times H^{n + \frac{1}{2}} - σ E)

σ = 2πf₀ε tan δ represents the dielectric loss

Courant stability limit

Δ t \leq \frac{1}{c \sqrt{\frac{1}{{Δ x}^{2}} + \frac{1}{{Δ y}^{2}} + \frac{1}{{Δ z}^{2}}}}

A Gaussian pulse drives a 50 Ω lumped port; Fourier transforming the port voltage and current gives S11 across the whole band in one run. A closed box of tangential fields around the board gives the far field over the full sphere through the equivalence principle, with surface currents J = n̂ × H and M = −n̂ × E. That yields directivity, efficiency, realized gain, front-to-back ratio and, for circular polarization, the axial ratio. The solve runs on the GPU through WebGPU where the browser supports it, otherwise on the CPU. One click then fits the network model to the FDTD result (edge extension, feed-point shift and, for circular polarization, κ) and re-tunes the dimensions.

12. Validity and Limitations

The network model is accurate to a few percent in resonant frequency for electrically thin substrates (h below about 0.05λ₀) and a single dominant mode, and it includes mutual slot coupling, surface waves, conductor and dielectric loss, probe reactance, the inset notches and an edge-diffraction estimate for a finite ground plane. The FDTD solve removes most of the remaining modelling error, but note:

Copper is modelled as a zero-thickness perfect conductor in FDTD; conductor loss appears only in the network model.
FDTD results depend on the mesh: zero-thickness copper edges make the patch behave slightly longer than drawn, so the resonance converges from below as the cells shrink. In our tests the Normal mesh sat about 1–2% below the mesh-converged resonance and the Fine mesh about 1%; the coarse mesh is a quick preview only. Check a final design on the Fine mesh.
The probe is a single-cell wire in FDTD, and connector and solder effects are not modelled.
Laminate εr and tan δ vary with frequency and supplier; use the datasheet values for your board.

Treat the results as a strong starting point and verify the built antenna by measurement before production.

References

Balanis, C.A., Antenna Theory: Analysis and Design, 3rd ed., Wiley, 2005, ch. 14.
Pozar, D.M., Microwave Engineering, 4th ed., Wiley, 2011.
Garg, R., Bhartia, P., Bahl, I. & Ittipiboon, A., Microstrip Antenna Design Handbook, Artech House, 2001.
Hammerstad, E.O., “Equations for Microstrip Circuit Design,” Proc. 5th European Microwave Conf., 1975.
Hammerstad, E. & Jensen, Ø., “Accurate Models for Microstrip Computer-Aided Design,” IEEE MTT-S Int. Microwave Symp. Digest, 1980.
Kirschning, M. & Jansen, R.H., “Accurate Model for Effective Dielectric Constant of Microstrip with Validity up to Millimetre-Wave Frequencies,” Electronics Letters, vol. 18, 1982.
Jackson, D.R. & Alexopoulos, N.G., “Simple Approximate Formulas for Input Resistance, Bandwidth, and Efficiency of a Resonant Rectangular Patch,” IEEE Trans. Antennas Propag., vol. 39, no. 3, 1991.
Pozar, D.M., “Input Impedance and Mutual Coupling of Rectangular Microstrip Antennas,” IEEE Trans. Antennas Propag., vol. 30, no. 6, 1982.
Kouyoumjian, R.G. & Pathak, P.H., “A Uniform Geometrical Theory of Diffraction for an Edge in a Perfectly Conducting Surface,” Proc. IEEE, vol. 62, no. 11, 1974.
Yee, K.S., “Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media,” IEEE Trans. Antennas Propag., vol. 14, no. 3, 1966.
Taflove, A. & Hagness, S.C., Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed., Artech House, 2005.
Roden, J.A. & Gedney, S.D., “Convolution PML (CPML): An Efficient FDTD Implementation of the CFS-PML for Arbitrary Media,” Microwave Opt. Technol. Lett., vol. 27, no. 5, 2000.
Jin, J.-M., The Finite Element Method in Electromagnetics, 3rd ed., Wiley, 2014.