Lesson Sources and induction · Induction and spikes
Inductance and RL time constants
Inductance (L) measures how strongly a coil opposes change of current — unit henry (H). In an RL circuit, current cannot step instantly; it rises and falls on an exponential curve. One time constant τ = L/R (seconds) marks the pace; after 5τ current is ~99% of final value — “essentially steady.” That timing governs relay pull-in, contactor chatter risk, and why coils need suppression when opened.
Resistive vs inductive turn-on
Close a switch on pure resistance (e.g., 10 Ω on 20 V): current immediately hits I = V/R = 2 A — Ohm’s law, no delay.
Replace resistor with a coil that still has 10 Ω wire resistance: same ultimate 2 A, but current ramps along an exponential — cannot jump to 2 A instantly because rising current builds expanding flux → Lenz EMF bucks the supply.
Why applied and induced voltages oppose on rise
As current tries to increase, expanding magnetic field cuts coil turns → induced voltage polarity opposes applied voltage — acts like extra “resistance” to change.
At first instant rate of change is maximum → induced voltage maximum → current increase slowest.
As current nears Ohm’s-law value, dI/dt falls → induced voltage falls → current approaches steady 2 A limited only by wire resistance.
The exponential curve — 63.2% per τ
Many natural processes follow exponential shape — RL rise, RC charge (later courses), even drying clothes.
Each time constant τ, current rises 63.2% of what remains to reach final value.
Example target 1.5 A, τ = 20 ms:
| After | Cumulative fraction of max |
|---|---|
| 1τ | 63.2% |
| 2τ | 86.5% |
| 3τ | 95.0% |
| 4τ | 98.1% |
| 5τ | 99.2% (~ “done”) |
After 5 time constants, treat current as steady for field work.
Defining inductance — the henry
1 henry = inductance when 1 A/s change of current produces 1 V of induced EMF.
Symbol L (honoring Lenz). Practical coils: often mH or µH.
Physical factors increasing L:
- More turns (N² effect in design formulas)
- Magnetic core (iron/silicon steel) vs air core — better flux path
- Larger core cross-section
- Tighter turn spacing (turns closer → more flux linkage)
Iron core boosts L but adds core losses at high frequency — why air-core or powdered iron in RF, steel laminations in power chokes.
RL time constant τ = L/R
One time constant:
τ = L / R
- L in henrys
- R in ohms (usually coil wire resistance plus series circuit R)
- τ in seconds
Example: L = 1.5 H, R = 6 Ω → τ = 0.25 s. To reach full Ohm’s-law current (~0.5 A on 3 V): 5 × 0.25 s = 1.25 s.
Buildup table (fraction of IMax):
| τ multiple | Current reached |
|---|---|
| 1τ | 0.632 × IMax |
| 2τ | 0.865 × IMax |
| 3τ | 0.950 × IMax |
| 4τ | 0.981 × IMax |
| 5τ | 0.992 × IMax |
Current decay when power removes
Opening the supply (with a complete path for decay) — current falls exponentially too. Each τ, current drops to 36.8% of what remained ( loses 63.2% of remainder).
| τ multiple | Remaining fraction of IMax |
|---|---|
| 1τ | 0.368 |
| 2τ | 0.135 |
| 3τ | 0.050 |
| 5τ | ~0.008 (~ done) |
Example: 4 A coil, after 2τ decay → 4 × 0.135 = 0.54 A still flowing.
Decay timing sets how long contact arc persists and how hot a snubber gets — next lesson.
Field math without fear
You rarely compute L from the core formula on a truck; you do use τ = L/R to estimate:
- How long until a contactor coil pulls in solidly
- How long until current is safe after open
- Why two coils with same resistance but different L behave differently
If doubling L doubles τ — same R, twice as long to steady state.
Field case
Situation. PLC output drives 24 V DC relay with 500 mA coil. Logic scans output every 10 ms; coil spec says 15 ms operate time. Chatter and missed aux contacts on fast pulse trains.
Analysis. Operate time scales with L/R. Undersized pulse width vs τ → coil never reaches 5τ holding current → armature buzzes.
Applied lesson. Match minimum energize time to coil τ and load; or pick relay with lower L or adequate core; verify with scope on coil current, not just voltage at terminals.
In the field
Symptom
Slow pull-in, chatter, or long drop-out
Where to look
Coil V rating, supply sag, series resistance, mechanical binding
Likely causes
- Low V during rise, high R in line, excessive L, pulse too short
What to measure
- Coil current waveform
- compute τ from L,R nameplate if given
What not to do
- Assume instant coil response on DC
Checklist
- I contrast instant R circuit vs exponential RL rise
- I define henry and factors that increase L
- I compute τ = L/R with correct units
- I use 5τ ≈ steady state for rise and decay
- I cite 63.2% buildup per τ
- I explain Lenz EMF during current increase