A pure sine wave driven into the -3 dB cut-off frequency of an oscilloscope, amplifier, or front-end filter emerges at 70.7% of its true amplitude, because 20·log10(0.707) ≈ -3 dB [S1][S5]. That is not a defect: it is the literal definition of the -3 dB bandwidth spec every instrument datasheet quotes, and it imposes a ~29.3% amplitude error the moment the signal frequency touches the rated bandwidth [S1][S6].
For sine-wave work the practical fix is to keep the highest signal frequency well below the instrument's -3 dB point: the 3x rule (signal ≤ BW/3) holds amplitude error to roughly 5%, while the 5x rule commonly cited by oscilloscope vendors pushes the error under ±2% [S1][S5]. The same instrument's rise-time spec is then RT = 0.35/BW for an oscilloscope below 1 GHz with a Gaussian response, so a 200 MHz scope implies a 1.75 ns minimum observable edge [S1].
Why 70.7% at -3 dB is a definition, not a calibration bug
The -3 dB point is the frequency at which a sinusoidal input is attenuated to 1/√2 ≈ 0.7071 of its true amplitude, the convention used across oscilloscope, vector network analyser, and filter datasheets [S5]. On a dB scale that maps exactly to -3.0103 dB, which is why the spec is shortened to "minus 3 dB" [S1][S5]. For a Gaussian-response front end, the measured error at the rated bandwidth sits at about 30%, so a 100 MHz scope measuring a 100 MHz, 1 Vpp sine will read roughly 0.707 Vpp [S1].
The same rule applies in any low-pass stage: a 1 GHz sine fed through a system whose -3 dB bandwidth is 1 GHz will be displayed with an amplitude near 70% of its true value, the textbook behaviour of a single-pole RC or Gaussian response [S3]. Designers who ignore this convention end up either trusting under-readings or building in margin they do not actually have, a recurring trap in measurement and test instrument selection.
3x vs 5x bandwidth rule: where the error numbers come from
For a first-order Gaussian response, amplitude error falls off roughly as 1 - exp(-0.5·(f/BW)²) when expressed in linear units, and a faster curve for higher-order responses. Working that through: at f/BW = 0.5, error ≈ 11.7%; at f/BW = 0.333 (3x rule), error ≈ 5.4%; at f/BW = 0.2 (5x rule), error ≈ 2.0% [S1][S5]. Tektronix publishes the 5x rule as the threshold for less than ±2% sine amplitude error, the level most general-purpose measurements need [S5].
OWON's bandwidth primer states the same relationship from a different angle: if 3% measurement error is the target, the input frequency must sit "much lower than the oscilloscope bandwidth," and the rule-of-thumb cited in vendor literature is required BW = highest signal frequency × 5 [S1]. Farnell's primer rounds the 70.7% point to "approximately -30% amplitude error" at the -3 dB frequency, the same number engineers see on the screen [S6]. A short comparison:
Bandwidth multiplier vs sine amplitude error (Gaussian response, f = highest signal frequency):
· f = BW (1x): about -29.3% amplitude error, the -3 dB point itself [S1][S5]
· f = BW/2 (2x rule): about -11.7% error, still significant for compliance work
· f = BW/3 (3x rule): about -5.4% error, the entry-level target for general sine measurements [S1]
· f = BW/5 (5x rule): under ±2% error, the vendor-recommended target for accurate sine reproduction [S1][S5]
For harmonics-rich signals the multiplier must be applied to the highest significant harmonic, not just the fundamental, which is where the 3x rule starts to collide with real edges and clocks.
Why the 3x rule matters for edges, harmonics, and digital signals
A square or pulse waveform is a sum of sine harmonics at 1·f, 3·f, 5·f, 7·f..., with amplitudes 2/(nπ) for the n-th harmonic, so the 3rd harmonic of a 100 MHz square is at 300 MHz with amplitude 2/(3π) ≈ 0.21 of the fundamental [S4]. To preserve the 3rd harmonic with ≤5% loss the scope must have a -3 dB point above 300 MHz, which is the 3x rule applied to the highest harmonic you care about, not the fundamental [S4].
For step and pulse edges the same physics shows up as rise time. The standard relation f3dB ≈ 0.35 / tr is exact for a single-pole RC low-pass and a reasonable approximation for a Gaussian-response scope [S1][S3]. Inverting it, a scope with 100 MHz of bandwidth will pass a step with about 3.5 ns of rise time, a 200 MHz scope passes 1.75 ns, and a 1 GHz scope passes 0.35 ns [S1]. The same trade-off shows up in the K-factor, or rise-time-bandwidth product: K = 0.32 is a true Gaussian response, K = 0.35 is the "essentially Gaussian" design Tektronix uses for minimum overshoot, and K = 0.45 trades rise time for about 5% overshoot on the step [S3].
Where the 3x and 5x rules break down
The 3x rule only holds for the sine at one specific frequency; harmonic-laden waveforms need the multiplier applied to the highest significant harmonic, often the 5th or 7th, which is why RF and pulse work routinely demand the 5x rule or stricter [S4][S5]. Bandwidth-limit filters built into scopes (for example a 20 MHz low-pass used to reject broadcast RF) will distort a 27 MHz crystal oscillator waveform even though the crystal frequency is "below bandwidth," because the 27 MHz sits in the filter's stop band and the harmonic structure collapses [S1].
Slew-rate-limited stages add a separate ceiling. Two waveforms can share an identical rise time but differ by a factor of five in slew rate, and the one with the higher slew rate carries higher-frequency content and therefore needs more system bandwidth to pass cleanly [S2]. The Signal Integrity Journal piece makes the point explicitly: bandwidth is "the frequency range of a signal's significant spectral components," and "significant" depends on the application, not on a single -3 dB number [S2]. A related trap lives in accuracy and tolerance stacks: a 3% amplitude error from bandwidth derating plus a 1% probe error plus a 1% ADC error compounds well past 5%, the same compound-accuracy pattern that hits process measurements and instrumentation.
Applying the rules in a real selection
A 50 MHz sine to be measured to ±2% needs a scope with at least 250 MHz of bandwidth under the 5x rule, or at least 150 MHz if ±5% from a 3x rule is acceptable [S1][S5]. A 100 MHz clock with 2 ns rise time needs the highest significant harmonic covered; a 5th-harmonic target at 500 MHz, multiplied by 5x, points to a 2.5 GHz scope, and a 3rd-harmonic-only target at 300 MHz needs a 1.5 GHz scope at the 5x multiplier or a 900 MHz scope at the 3x multiplier [S4][S5].
The procedure in three steps, all citable: (1) find the highest frequency component you must preserve, fundamental for sine, 3rd-7th harmonic for digital, 0.35/tr for unknown rise time [S1][S3][S4]; (2) multiply by 3 for a ~5% target or by 5 for a ±2% target [S1][S5]; (3) confirm the front-end response type, since true Gaussian (K = 0.32) and "essentially Gaussian" (K = 0.35) overshoot less than higher-order flat responses at the same bandwidth [S3]. For bench work in lighting and electronic instrumentation and similar wide-frequency-range test setups, the same 3x/5x decision sets whether harmonics, ringing, or switching edges survive the measurement chain.
Failure modes engineers actually hit
Three failure patterns recur when the bandwidth rule is ignored. First, amplitude under-read at the rated bandwidth: a 100 MHz, 1 Vpp sine on a 100 MHz scope reads ~0.707 Vpp, which looks like a DUT problem but is the front-end [S1][S5]. Second, lost harmonic content on pulses: a clock whose 5th harmonic at 500 MHz sits above the scope's -3 dB point shows rounded edges, reduced amplitude, and jitter that does not exist on the real signal [S4][S5]. Third, bandwidth-limit-filter artefacts: enabling a 20 MHz limit to clean up a low-frequency measurement will crush a 27 MHz crystal, and the resulting waveform is not a property of the device under test [S1].
Two guard rails keep these out of reports. Always state the scope bandwidth, the response type (Gaussian vs flat), and the chosen multiplier (3x or 5x) next to any amplitude or rise-time number, so the reader can back out the derating [S1][S3]. And treat the K-factor, K = 0.35 for "essentially Gaussian" scopes, as a known overshoot source: if the DUT has sharp edges, expect up to ~5% overshoot on K = 0.45 systems versus near-zero on K = 0.35 systems [S3]. For adjacent analogue domains, the same error stack shows up in flow measurement selection, where derating the bandwidth-equivalent dynamic response changes the published accuracy by a similar percentage.
Trackable next signal: watch for vendor literature to keep publishing the 5x rule as the headline "accurate amplitude" multiplier while the 3x rule spreads into education-grade material as the entry threshold for general sine work, and watch for scope firmware to expose the response type (Gaussian vs flat) on screen so the K-factor decision is auditable, not assumed.