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lesson · 12min · Lesson 33 of 37

RC and RL circuits: time constants

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01Atoms, Electrons & Electric Charge
  1. ReadWhat is electricity?: atomic model
  2. ReadElectric charge and the coulomb
  3. ReadConductors, insulators, and semiconductors
  4. ReadConventional current vs electron flow
  5. quizModule quiz
02Voltage, Current & Resistance
  1. ReadPotential difference: the driving force
  2. ReadCurrent: the rate of charge flow
  3. ReadResistance and resistivity
  4. ReadOhm's Law: derivation and examples
  5. exerciseWorked examples: Ohm's Law problems
03Series & Parallel Circuits
  1. ReadSeries circuits: characteristics and rules
  2. ReadParallel circuits: characteristics and rules
  3. ReadCombined series-parallel networks
  4. ReadVoltage dividers and current dividers
  5. exerciseCircuit analysis practice
04Kirchhoff's Laws
  1. ReadKCL: Kirchhoff's Current Law
  2. ReadKVL: Kirchhoff's Voltage Law
  3. ReadMesh and nodal analysis
  4. exerciseKirchhoff's law problems set
05Power & Energy
  1. ReadElectrical power: watts and horsepower
  2. ReadEnergy: kilowatt-hours and joules
  3. ReadEfficiency and power loss in cables
  4. quizPower quiz
06Alternating Current Fundamentals
  1. ReadAC vs DC: why AC won
  2. ReadSinusoidal waveforms: peak, RMS, average
  3. ReadFrequency and period
  4. ReadPhase relationships
  5. ReadAC circuit analysis introduction
07Capacitors & Inductors
  1. ReadCapacitor construction and capacitance
  2. ReadCapacitors in AC circuits: reactance
  3. ReadInductor construction and inductance
  4. ReadInductors in AC circuits: reactance
  5. ReadRC and RL circuits: time constants
08Measurement & Instruments
  1. ReadMultimeters: AC/DC voltage and current
  2. ReadClamp meters and measuring live current
  3. ReadOscilloscopes: reading waveforms
  4. quizFinal assessment
Lesson · 12min
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PurposeCore theoryWorked exampleKnowledge checkSources

In this lesson

PurposeCore theoryWorked exampleKnowledge checkSources
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ElectraCore lesson handout

RC and RL circuits: time constants

Electrical Fundamentals · Lesson 33 · Beginner

BeginnerReview: professional review pending

Purpose

Use RC and RL time constants to predict first-order charging and current transients.

Before you beginCapacitance · Inductance · Exponential idea

Learning objectives

  • Calculate τRC and τRL
  • Use 63.2% and 36.8% landmarks
  • Estimate five-time-constant settling
  • Respect initial continuity conditions

First-order circuits do not jump directly between steady states: capacitor voltage and inductor current change exponentially.

FIRST-ORDER TRANSIENTSFIRST-ORDER TRANSIENTS τ = RC or L/Rstored energy and time-dependent response

Core theory

For RC, τ=RC; a charging capacitor reaches 63.2% of final voltage after one τ.

For RL, τ=L/R; current reaches 63.2% of final value after one τ.

After about 5τ, remaining error is below 1%, though never mathematically zero.

Terms, symbols, and units
TermMeaningSymbolUnit
Time constantCharacteristic exponential response timeτs
Initial conditionStored voltage or current at switchingNot applicableNot applicable
Steady stateLong-time limiting conditionNot applicableNot applicable
τRC=RC; τRL=L/R

R is the resistance seen by the storage element.

s
Worked exampleA 10 kΩ resistor charges a 100 µF capacitor from 0 V toward 12 V.

Assumptions: Ideal source; No other loading.

  1. Constant: τ=10,000×100e−6=1.0 s.
  2. One tau: Vc=0.632×12=7.58 V at 1 s.
  3. Settling: About 5 s reaches over 99% of final.

τ=1.0 s; Vc≈7.58 V at 1 s; near-settled after about 5 s.

Reasonableness check: Voltage changes quickly first and slows near 12 V.

Common mistakes
  • Using µF without conversion
  • Assuming 1τ means fully charged
  • Using total resistance without finding what the element sees

Where this appears in practice

Time constants govern delays, debounce, filters, coil release, soft starts, and transient measurements.

SafetyStored capacitor energy and inductive current can persist or create transients after switching. Apply verified discharge and suppression methods.
Local code checkThe component relationships are universal. Product ratings, discharge provisions, protective measures, permitted work, and test procedures must follow current local rules and manufacturer data.

Knowledge check

Approximately what fraction of final value is reached after one charging time constant?

63.2%. The remaining error is e⁻¹≈36.8%.

Answer: 63.2%. The remaining error is e⁻¹≈36.8%.

Practical exercise

Find τ and 5τ for 47 kΩ with 10 µF, then sketch the charging landmarks.

Summary

  • RC uses RC; RL uses L/R
  • One τ reaches 63.2%
  • Five τ is a practical settling estimate

Sources and review

  • University Physics Volume 2: Capacitance, Inductance, and AC Circuits: OpenStax; Current web edition; Physics reference
  • The International System of Units (SI Brochure): BIPM; 9th edition, updated 2026; International

Editorial review date: 2026-08-21. Professional electrical review is pending.

Educational material for learning and preliminary checks. Verify current local requirements and exact equipment instructions. This lesson does not replace competent professional work.

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