A quarter-wave monopole on an infinite ground is the textbook case: the image fills in the missing half of a half-wave dipole, input impedance is about half of 73 ohm, and the pattern is half of the dipole figure-eight. Real boards and vehicle roofs are finite. Here is what changes and how to size the ground without a full-wave run every time.

1. Ideal numbers first
At 915 MHz, lambda/4 is 82 mm in free space. With a thin wire over infinite PEC, Zin is about 36 + j21 ohm at exact lambda/4; shortening to about 0.24 lambda cancels the inductive part and lands near 36 ohm real. A series or shunt L-match then brings it to 50 ohm if you need it.

2. Finite circular ground: the radial that matters
For a circular disk of radius a, the monopole looks into a radial transmission line. Measured and published curves (e.g. Weiner, Balanis summaries) show:
- a = 0.25 lambda: resonance rises a few percent in frequency, R_in drops toward 20 to 25 ohm, and elevation beam peaks near 30 to 40 deg instead of the horizon.
- a = 0.5 lambda: R_in is close to the infinite-ground value, but the pattern still has a null fill and a lobe tilt of order 10 to 20 deg.
- a >= lambda: pattern and Zin are within a couple of dB / ohms of the infinite case for most practical work.

Rule of thumb I use for sub-GHz IoT: aim for at least 0.4 lambda radius of continuous metal under the feed, or equivalent area in a rectangular plane. Smaller than 0.25 lambda and you are really building a top-loaded short dipole, not a monopole.

3. Rectangular PCB grounds
A phone-sized board at 2.4 GHz (lambda/4 ~ 31 mm) with the antenna on the short edge behaves like a PIFA or IFA more than a monopole: the board itself is the other half of the radiator. Expect:
- Strong dependence on how the user hand loads the ground.
- Resonance that moves when you add shields, batteries or display flexes.
- Return current that prefers the nearest ground edge, so keep a clear keep-out and do not route noisy clocks under the antenna clearance.

4. Radials instead of a solid disk
Four to eight wires of length ~0.25 lambda, sloping down 30 to 45 deg, raise R_in toward 50 ohm and lift the beam slightly. For a mast mount this is cheaper than a disk. Keep the feed coax common-mode choked (ferrite or sleeve) or the feedline becomes another radial and skews the pattern.

5. Quick bench checklist
- Sweep S11 with the intended ground only (no bench metal within ~1 lambda).
- Note resonant frequency vs the infinite-ground target; if it is high, lengthen the whip or enlarge the ground.
- Check |S11| with a hand or metal plate under the board; more than 50 MHz of shift means the design is ground-size limited.
- For vehicle roofs, treat the roof as "large" only above about 400 MHz; below that, door seams and roof curvature still move the nulls.

Image theory remains the starting point, but the finite-ground correction is where most "my monopole will not match" tickets actually live.