Math Bridge: Watcher Noise and Standing Energy

← Back to Context-Aware Energy Management
Math BridgeEnergy & PowerStruggle-friendly runway

Why does a watcher consume energy even when nothing changes?

Start at the thermal-noise floor, then keep the watcher's standing current visible in the policy ledger.

Battery Bruno, the energy and power guideBattery Bruno guides
The one targetPrice the physical floor beneath an always-listening context policy.
The chapter case290 K, a 200 kHz wake channel, and a 50 µA watcher on 3.3 V.
What it buys youA policy comparison that includes noise bandwidth and standing energy.

A field team faces an unresolved physical question: Why does a watcher consume energy even when nothing changes? They must answer it before changing bandwidth on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is bandwidth. The middle card applies this page's relationship. The green card is noise density. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Bandwidth changes noise density An input card leads through the page relationship to the noise density result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Integrating the same watts-per-hertz floor over more hertz admits more noise. A real circuit may also change current with bandwidth, but that needs measured device data.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for bandwidth is 200.

  2. 2

    Name the relationship. N0=kT=1.380649x10⁻²³x290=4.00x10⁻²¹ W/Hz Pn=N0B=4.00x10⁻²¹x200,000=8.01x10⁻¹⁶ W=-121.0 dBm Pwatch=3.3x50 uA=165 uW Eday=165 uWx24 h=3.96 mWh

  3. 3

    Substitute the chapter fixture. Set bandwidth to 200. The page ledger gives noise density as 4.00e-21 W/Hz.

  4. 4

    Read the result. Keep W/Hz beside the value. Use it only inside the technical boundary on this page.

Predict, then change bandwidth

Try Predict the direction of noise density. Move one control, calculate, then check your prediction.

200
Chapter baseline
Noise density

Observe Integrating the same watts-per-hertz floor over more hertz admits more noise. A real circuit may also change current with bandwidth, but that needs measured device data. Reset the control to 200 and compare noise density.

Explain Only bandwidth moves here. The other chapter fixtures remain fixed.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only bandwidth moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only bandwidth. It replays the chapter fixture; it is not a field prediction.

1. Start with the physical story

A quiet resistor still contains random thermal motion. A watcher must separate a trigger from that floor, so its amplifier needs bandwidth, gain, and standing bias current. Firmware can avoid needless sensing, but it cannot make an awake analogue front end cost zero.

Battery Bruno: Keep noise power and watcher power separate. One limits what can be heard; the other enters the energy budget.

2. Name every algebra move

1

Find noise densityMultiply Boltzmann's constant k by temperature T.

2

Choose bandwidthConvert kilohertz to hertz.

3

Integrate the floorMultiply kT by bandwidth B.

4

Use a logarithmic scaleConvert watts to dBm with 10 log10(P/1 mW).

5

Price the watcherMultiply rail voltage by standing current.

6

Extend through a dayMultiply watcher power by 24 hours.

3. Reproduce the chapter case

N0=kT=1.380649×10⁻²³×290=4.00×10⁻²¹ W/Hz
Pn=N0B=4.00×10⁻²¹×200,000=8.01×10⁻¹⁶ W=−121.0 dBm
Pwatch=3.3×50 µA=165 µW
Eday=165 µW×24 h=3.96 mWh

A 1 MHz channel contains five times the noise power of a 200 kHz channel, which is 7 dB more. The 50 µA watcher cost does not change when this bandwidth slider moves because the simple ledger holds circuit bias fixed.

4. Try one real input

TryWiden the channel and predict the noise-floor change before reading it.

Bandwidth
Noise density
Density in dBm/Hz
Noise power
Noise floor
Watcher power
Watcher energy/day

ObserveNoise power rises with bandwidth while density stays fixed. The watcher energy remains 3.96 mWh/day because the model does not invent a bias-current change.

ExplainIntegrating the same watts-per-hertz floor over more hertz admits more noise. A real circuit may also change current with bandwidth, but that needs measured device data.

Technical boundaries.

This is a thermal-floor and standing-energy ledger, not a receiver design.

Noise figure
The front end's added noise is not included.
Interference
Thermal noise does not represent nearby radios or impulsive sensor noise.
Bias model
The 50 µA watcher stays fixed while bandwidth changes.

Correct, not complete: this proves the floor and its energy cost, not trigger reliability in the field.

5. Use the result in the design

Measure watcher current, choose the narrowest bandwidth that still meets event latency, and compare its daily energy with the sensing or radio work it avoids.

6. Record the evidence state

Keep temperature, bandwidth, required sensitivity, noise figure, watcher current, rail voltage, trigger latency, miss rate, and avoided wake energy.

7. Check yourself

Why does five times the bandwidth add about 7 dB?
Answer: 10 log10(5) is about 7 dB.
Does -174 dBm/Hz describe the whole channel?
Answer: No. It is density; multiply by bandwidth before converting the channel total.
Why keep 50 µA separate from kTB?
Answer: kTB is available noise power, while 50 µA is the chosen circuit's standing supply current.
Honesty boundary.

The arithmetic uses the chapter's 290 K, 200 kHz, and catalog-typical 50 µA case.

Noise figure
The front end's added noise is not included.
Interference
Thermal noise does not represent nearby radios or impulsive sensor noise.
Bias model
The 50 µA watcher stays fixed while bandwidth changes.

Correct, not complete: this proves the floor and its energy cost, not trigger reliability in the field.

Return to Context-Aware Energy Management with the maths unpacked →