Phoebe guidesA field team faces an unresolved physical question: What loop-detector ring-down reveals They must answer it before changing inductive loop quality factor 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 inductive loop quality factor. The middle card applies this page's relationship. The green card is ring-down time. 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.
Derive the baseline in four named moves
- 1
Name the input. The chapter baseline for inductive loop quality factor is 10.
- 2
Name the relationship. R=2πfL/Q; τ=Q/(2πf); P=I²R
- 3
Substitute the chapter fixture. Set inductive loop quality factor to 10. The page ledger gives ring-down time as 39.8 us.
- 4
Read the result. Keep us beside the value. Use it only inside the technical boundary on this page.
Predict, then change inductive loop quality factor
Try Predict the direction of ring-down time. Move one control, calculate, then check your prediction.
Observe Higher Q means less effective resistance and longer decay; frequency and inductance are fixed in this comparison. Reset the control to 10 and compare ring-down time.
Explain Only inductive loop quality factor moves here. The other chapter fixtures remain fixed.
Check yourself
What should you do before trusting a moved-control result?
What does this small model leave out?
1. Eddy currents cost real energy
A vehicle changes the loop's inductance, but induced currents also dissipate energy. We represent those losses with an effective series resistance R.
2. Use Q to expose the loss resistance
Make R the subjectR=ωL/Q.
Use the RL decayτ=L/R.
Substitute Rτ=Q/ω=Q/(2πf).
3. Count decay in cycles
Multiply the time constant by frequency to express it as oscillation cycles.
The tuned capacitor comes from f=1/(2π√LC), rearranged as C=1/[(2πf)²L].
4. Try the quality factor
TryMove Q while holding the chapter's 40 kHz loop and catalog-typical L and current fixed.
ObserveAt Q=10, R=2.51 Ω, τ=39.8 µs=1.59 cycles, C≈158 nF, and P≈251 µW.
ExplainHigher Q means less effective resistance and longer decay; frequency and inductance are fixed in this comparison.
The series-R tank is a small-signal lumped model.
- coupling geometry
- Needs separate evidence
- oscillator drive
- Needs separate evidence
- pavement
- Needs separate evidence
- cable losses
- Needs separate evidence
- temperature
- Needs separate evidence
- vehicle position/material
- Needs separate evidence
- cross-talk
- Needs separate evidence
- measurement bandwidth
- Needs separate evidence
- a ground-truth baseline
- Needs separate evidence
Use field evidence or a deeper model before release.
5. Work resistance and power
That power is the electrical cost of the tank's loss channel at the stated current.
6. Work the decay and tuning
Ring-down and resonance shift are related to the same tank, but they are independent evidence channels.
7. Check yourself
What happens to ring-down time when Q doubles at fixed frequency?
What effective loss resistance corresponds to Q=10 here?
Does a frequency shift alone prove vehicle class?
These are the chapter inputs, worked results, and named teaching assumptions.
- 40.000 kHz baseline and frequency-shift context come from the chapter
- Frequency, sample rate, or event rate
- 100 µH
- Inductance value
- Q=10
- Named physical or model constant
- 10 mA RMS are explicitly catalog-typical values used by its worked box
- Named teaching assumption
The 158 nF, 2.51 Ω, 251 µW, 39.8 µs, and 1.59-cycle results follow from those values. This is not a universal loop-detector classifier.