3 m to 10 m Limit Conversion: When the 20·log Rule Fails

Radiated emission limits are written for a fixed distance, but labs measure wherever the chamber allows: 3 m in a compact chamber, 10 m on an open-area test site, sometimes 1 m on the bench. The usual way to move a level from one distance to another is the 20·log rule. It is exact for a small source in the far field and wrong, sometimes by many decibels, close to the source.

The 20·log rule

In the far field the electric field strength falls in inverse proportion to distance, so a level measured or specified at distance d₁ converts to distance d₂ as L₂ = L₁ + n · log₁₀(d₁ / d₂), where L₁ and L₂ are field strengths in dBµV/m, d₁ and d₂ are distances from the EUT in metres, and n is the distance exponent in dB per decade. For a far-field source n = 20, which gives the familiar correction of 20 · log₁₀(10 / 3) = 10.46 dB between 3 m and 10 m.

Worked example: a 100 MHz reading at 3 m

A product measured at 3 m shows 38.0 dBµV/m (quasi-peak) at 100 MHz. Converted to 10 m: L₁₀ = 38.0 + 20 · log₁₀(3 / 10) = 38.0 − 10.46 = 27.5 dBµV/m. Against the CISPR 32 Class B limit at 10 m of 30 dBµV/m (30 to 230 MHz) the margin is 2.5 dB. At 100 MHz the wavelength is 3.00 m and the reactive near-field boundary λ/2π is 0.48 m, so 3 m is more than six times that boundary and the rule is trustworthy for a small source.

CISPR 32 also gives Class B limits for 3 m: 40 dBµV/m from 30 to 230 MHz and 47 dBµV/m from 230 to 1000 MHz, exactly 10 dB above the 10 m values. The rounded 10 dB is 0.46 dB smaller than the 20·log correction, so the same reading has 2.0 dB margin against the 3 m limit but 2.5 dB after conversion to 10 m. Use the limit table the standard gives for your distance; do not convert the 10 m limit yourself and expect exactly the same verdict.

Why the rule fails in the near field

Close to a small source the field has extra terms that fall faster than 1/r. For a short electric dipole (a short cable carrying common-mode current behaves like one), the broadside field magnitude is E ∝ (1/r) · √((1 − 1/(kr)²)² + (1/(kr))²). For a small magnetic loop (a PCB current loop), the electric field is E ∝ (1/r) · √(1 + 1/(kr)²). Here k = 2π/λ is the wavenumber in rad/m and r is the distance in metres. When kr ≫ 1 both reduce to 1/r and n = 20. When kr ≪ 1, inside the reactive boundary r = λ/2π, the dipole's E-field falls as 1/r³ (n = 60) and the loop's as 1/r² (n = 40).

The table applies these free-space expressions to a 3 m to 10 m conversion. Each drop column is the real change 20 · log₁₀(E₃ / E₁₀) predicted by the model; the last column shows how far the rule's 10.46 dB is from it.

Free-space model of a small source, broadside, no ground plane. Error = 10.46 dB − model drop. A positive error means the rule predicts a 10 m level that is too low (optimistic); a negative error means it predicts one that is too high (conservative).

Two things stand out. Below about 10 MHz the 20·log rule badly underestimates the drop, which is conservative when you extrapolate from a close measurement to a farther limit distance. From about 17 MHz upward, however, the dipole's drop is smaller than the 1/r value (the largest gap in this model, about 1.1 dB, is near 23 MHz). At 30 MHz the rule overestimates the drop by 0.87 dB, so a 3 m reading converted to 10 m comes out slightly optimistic. Repeat the first example at 30 MHz instead of 100 MHz: the rule gives 38.0 − 10.46 = 27.5 dBµV/m, but for a cable-like source the model gives 38.0 − 9.59 = 28.4 dBµV/m, so the real margin to 30 dBµV/m is 1.6 dB rather than 2.5 dB.

Below 30 MHz: 40 dB per decade is not safe either

FCC Part 15 (§15.31(f)(2)) allows measurements below 30 MHz at a distance closer than specified, with extrapolation by 40 dB per decade or by a factor found from measurements at two or more distances. The 40 dB figure is the near-field slope of a loop, but it holds only while both distances are inside λ/2π. At 10 MHz (λ/2π = 4.77 m) a 3 m to 30 m conversion crosses into the far field. The loop model gives a real drop of 25.37 dB, not 40 dB. A 3 m reading of 45.0 dBµV/m therefore becomes 5.0 dBµV/m at 30 m by the 40 dB rule, while the model predicts 19.6 dBµV/m: the rule is 14.6 dB optimistic.

Measuring at two distances helps, but only if they cover the slope you need. With the same loop at 10 MHz, readings at 3 m and 6 m differ by 9.37 dB, which gives n = 9.37 / log₁₀(2) = 31.13 dB per decade. Applied from 3 m to 30 m this predicts a 31.13 dB drop, still 5.8 dB more than the 25.37 dB from the model, because the slope keeps flattening toward 20 dB per decade beyond λ/2π. Put the farther measurement as close to the target distance as the site allows.

Other ways the conversion goes wrong

• Large EUT, distance set to the front face. If the main radiator is a cable bundle 1 m behind the face nearest the antenna, the real ratio is (10 + 1) / (3 + 1), an 8.79 dB drop instead of 10.46 dB. A 3 m reading converted with the rule is then 1.67 dB optimistic; with a 0.5 m offset the error is 0.92 dB.
• The source is not a point. The far-field boundary 2D²/λ for an EUT whose largest dimension is D = 1 m is 2.00 m at 300 MHz but 6.67 m at 1 GHz, so above about 450 MHz a 1 m EUT at 3 m is not in its far-field region and the measured maximum depends on direction and position.
• Ground plane and height scan. The free-space model ignores the floor reflection of a semi-anechoic chamber or open-area test site. The 1 m to 4 m height scan captures different combinations of direct and reflected waves at 3 m and at 10 m, so the ratio of the maximised readings can differ from the free-space value, most at the low end of the band.

Practical rules

• Above about 100 MHz with a small EUT, 20 · log₁₀(d₁ / d₂) is within about 0.1 dB of the free-space model; the site, not the formula, dominates the error.
• Between 30 and 50 MHz, keep about 1 dB in reserve when converting a 3 m reading to 10 m; a cable-like source can be up to 0.87 dB worse than the rule predicts.
• Below 30 MHz, compute λ/2π first. If both distances are inside it, n = 40 (loop) or n = 60 (dipole) is reasonable; if the conversion crosses it, n lies roughly between 20 and those values and a fixed 40 dB per decade can be more than 10 dB optimistic.
• In the test-distance calculator, set the exponent n for the case you are in and enter the distance to the radiating part, not only the distance to the front face.