The rectangular microstrip patch is usually the first antenna people design, and also the one that most often comes back from the board house a few percent off frequency. Here is the transmission-line-model workflow I use, with a worked example and the places where the model lies to you.

1. Pick the substrate first
Bandwidth and efficiency are set mostly by substrate thickness and permittivity, not by clever geometry. Thicker and lower-Dk means wider bandwidth and higher radiation efficiency, at the cost of a larger patch and more surface-wave excitation if you go too thick (rule of thumb: keep h below about 0.05 lambda0 for a simple patch).

Use the design Dk the laminate vendor recommends for antenna and circuit work at your frequency, which is often not the headline "process" Dk. For the example below I use a hydrocarbon-ceramic laminate with design Dk 3.55, h = 1.524 mm, target f0 = 2.45 GHz.

2. Width
W = c / (2 f0) * sqrt(2 / (er + 1))
W = 0.1224 m / 2 * sqrt(2 / 4.55) = 40.6 mm

W mainly controls radiation resistance and efficiency; it is not critical for frequency. Avoid making W much larger than this or higher-order modes start to appear.

3. Effective permittivity
eeff = (er + 1)/2 + (er - 1)/2 * (1 + 12 h/W)^(-1/2) = 3.33

4. Fringing extension and length
The fields fringe past each radiating edge, so the patch looks electrically longer than it is:
dL = 0.412 h (eeff + 0.3)(W/h + 0.264) / ((eeff - 0.258)(W/h + 0.8)) = 0.73 mm
L = c / (2 f0 sqrt(eeff)) - 2 dL = 32.1 mm

For comparison, the same 2.45 GHz patch on 1.524 mm FR-4 (er about 4.4) comes out near 37.2 x 28.8 mm, but with noticeably lower efficiency because of the loss tangent.

5. Bandwidth expectations
A classic empirical estimate for VSWR < 2 on a thin substrate is
BW = 3.77 (er - 1)/er^2 * (W/L) * (h/lambda0)
which gives about 1.2 % here (roughly 30 MHz). Full-wave results for this geometry usually land somewhere in the 1 to 2 % range. Either way it is narrow, and that has a direct consequence:

A change of +/-0.1 in Dk moves the resonance of this patch by roughly 33 MHz, about the same as the whole bandwidth. Dk tolerance, copper thickness and etch tolerance are the reason first prototypes miss. Budget for a trim iteration, or design with a thicker substrate so the bandwidth swallows the tolerance.

6. Feeding with an inset
At the radiating edge the input resistance is high. With the simple single-slot conductance
G1 = W / (120 lambda0) * (1 - (k0 h)^2 / 24)
you get Rin(edge) = 1/(2 G1), about 180 ohm here. Ignoring the mutual conductance between the two radiating slots makes this approximate (typical real edge values are 150 to 300 ohm), so treat what follows as a starting point.

The resistance falls toward the centre roughly as
Rin(y0) = Rin(edge) * cos^2(pi y0 / L)
For 50 ohm: y0 = (L/pi) * arccos(sqrt(50/180)) = about 10.4 mm.

The 50 ohm microstrip on this 1.524 mm board is about 3.4 mm wide. Make the inset notch gap roughly equal to the feed width on each side to start, then sweep y0 and the gap in an EM solver. The notch also shifts resonance slightly downward, so re-tune L after the feed is set.

Alternatives: a coaxial probe feed (no notch, probe position sets impedance, probe inductance limits thick substrates), a quarter-wave transformer from the edge, or aperture coupling when you need more bandwidth and a clean feed layer.

7. Ground plane and pattern
A finite ground plane changes resonance, back radiation and gain. A common minimum is the patch outline plus about 6h on each side (about 9 mm here), but pattern and front-to-back ratio keep improving up to roughly a quarter wavelength of margin. Expect around 6 to 7 dBi directivity for a single patch on a substrate like this, with realized gain a little lower after conductor, dielectric and mismatch loss.

8. Circular polarization options
- Nearly square patch fed on the diagonal, or truncated corners: single feed, narrow axial-ratio bandwidth.
- Square patch with two feeds 90 degrees apart through a hybrid: wider axial-ratio bandwidth, more board area.

Checklist before ordering boards
- Simulate with the real laminate design Dk, loss tangent and copper roughness.
- Include the connector launch and feed line in the model.
- Plan a measurement: S11 on a VNA with the cable routed away from the radiating edges, ideally with a ferrite or choke on the cable to stop it becoming part of the antenna.
- Leave room to trim L by a fraction of a millimetre on the first spin.

If you have measured-versus-simulated data for patches on other laminates, please post the offsets you saw.