Math Bridge: Relay Flyback and Coil Decay

A technician must decide whether ideal inductive spike is safe before changing current fall time on the real device. The result is unresolved until the rule and units are checked. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is current fall time. The middle card applies this page's rule. The green card is ideal inductive spike. 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 model keeps those stated values fixed and changes only current fall time, so the numeric fixture does not switch without explanation.

Walk the arrows. Slowing the same current fall reduces the ideal unclamped voltage spike.

Derive the baseline in four named moves

1
Name the input. The chapter baseline is 100 ns.
2
Name the relationship. spike = 0.020 H x 0.50 A x 1,000,000,000 / fall time
3
Substitute with units. 10,000,000 / 100 = 100,000 V
4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change current fall time

Try Predict the direction of spike = 0.020 H x 0.50 A x 1,000,000,000 / fall time. Test another current fall time, then compare ideal inductive spike.

Observe Slowing the same current fall reduces the ideal unclamped voltage spike. Reset current fall time to 100 and compare ideal inductive spike.

Explain Slowing the same current fall reduces the ideal unclamped voltage spike.

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 current fall time moves here. Field effects named in the technical boundary stay fixed.

1. A coil stores energy in current

An inductor resists a change in its current. Opening the switch removes the old path but does not erase the magnetic energy already stored.

Max: The circuit must provide somewhere safe for that current to decay.

2. Find the coil scales

Use Ohm's lawR=V/I=12/0.5=24 Ω.

Compute stored energyE=½LI²=2.50 mJ.

Compute safe decay timeτ=L/R=0.833 ms.

3. Force a fast current change

V=LΔI/Δt; fc=1/(2πτ); fknee≈0.35/trise

Making Δt tiny makes the ideal unclamped voltage huge. Slowing and clamping the edge protects the switch and reduces high-frequency energy.

4. Try the turn-off time

R=V/I; E=LI²/2; τ=L/R; Vspike=LI/toff; fknee=0.35/toff

TrySlow the ideal unclamped current fall from the chapter's 100 ns example.

ObserveAt 100 ns the ideal result is 100,000 V and a 3.50 MHz edge knee, while the diode-guided L/R decay has a 0.833 ms time constant and 191 Hz corner.

ExplainTurn-off time changes only the unclamped spike and edge knee. Coil resistance, stored energy, and the L/R decay remain tied to the fixed chapter hardware.

Technical boundaries.

The 100,000 V result is an ideal warning, not a prediction

parasitic capacitance: Needs separate evidence
arcing: Needs separate evidence
avalanche: Needs separate evidence
wiring: Needs separate evidence
snubbers: Needs separate evidence
diode drop: Needs separate evidence
coil resistance: Needs separate evidence
switch behaviour: Needs separate evidence
measurement bandwidth clamp or reshape the transient: Needs separate evidence

Use field evidence or a deeper model before release.

5. Read the diode's job

The flyback diode becomes the new current path. It lets current fall over the coil's resistance-scale time instead of demanding an impossible instantaneous stop.

6. Protect function and radio evidence

Choose diode voltage/current ratings and release-time behaviour, then measure the transient at the switch and check nearby radio performance under repeated switching.

7. Check yourself

How much energy does the coil store?

Answer: ½(0.020)(0.5²)=0.0025 J=2.50 mJ.

Why does a shorter turn-off time raise voltage?

Answer: V=LΔI/Δt; the same current change divided by less time needs more voltage.

Does the ideal 100,000 V appear unchanged in hardware?

Answer: No. Real parasitics and breakdown clamp and reshape it, often destructively.

Honesty boundary.

These are the chapter inputs, worked results, and named teaching assumptions.

12 V: Voltage or voltage-step value
0.5 A: Current or responsivity value
24 Ω: Resistance or impedance value
20 mH: Inductance value
2.50 mJ: Charge or energy value
0.833 ms: Time, interval, or service-life value
191 Hz: Frequency, sample rate, or event rate
100 ns: Time, interval, or service-life value
100,000 V: Voltage or voltage-step value
3.50 MHz: Frequency, sample rate, or event rate

They do not replace a rated protection design or oscilloscope measurement; Under the Hood keeps those requirements.