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Capacitor Behavior in DC Circuits

Charging and discharging through a resistor, over time — the RC time constant, the 63% rule, and why a capacitor's sense of time governs everything from timers to power-supply smoothing.

Beginner+Low Risk24 min read

What You Will Learn

  • You will learn how a capacitor charges and discharges through a resistor over time, following a smooth exponential curve rather than a straight line.
  • You will learn the time constant τ = R × C — what it means, how to calculate it, and why ohms times farads gives seconds.
  • You will learn the rules of thumb that make RC behavior usable without calculus: about 63% charged after one time constant, essentially finished after five.
  • You will learn how RC timing explains real circuits — timers, power-supply smoothing — and what it means for a capacitor that has lost capacitance.

What You Will Be Able To Do

  • You will be able to calculate the time constant τ = R × C for a resistor–capacitor circuit, with correct units.
  • You will be able to estimate how far a capacitor has charged or discharged after a given number of time constants using the 63% / 37% / 5τ rules.
  • You will be able to explain why a larger resistor or capacitor makes charging slower.
  • You will be able to connect RC behavior to real circuits and recognize how a capacitor that has lost capacitance misbehaves.

Required Tools

No physical tools required. This is a conceptual section.

Section Overview

The last section said a charged capacitor blocks steady DC. This one fills in the crucial word steady — because getting to "charged" takes time, and that time is where capacitors earn their keep. Connect a capacitor to a DC supply through a resistor and its voltage doesn't jump instantly; it rises along a smooth curve, fast at first and slowing as it fills, governed by a single number: the time constant, τ = R × C. This section teaches how a capacitor charges and discharges over time, what the time constant means, the handful of percentages that let you reason about it without calculus, and why this "sense of time" is behind timers, power-supply smoothing, and a whole class of repair symptoms.

Why This Matters

A capacitor's behavior in DC is almost entirely about timing. It's why a power-supply smoothing capacitor can hold the voltage up between the peaks of the incoming power, why a timer circuit blinks at the rate it does, and why a delay or a fade happens over the interval it does. For a repair technician, RC timing is also diagnostic: a smoothing capacitor that has lost capacitance can no longer hold the voltage between peaks, so the supply ripples and the device misbehaves — one of the most common failures in all of electronics. Understanding the charge and discharge curve turns "the capacitor is bad" from a guess into a prediction you can reason about and check.

Required Prerequisites

No consumables required. The exercises are pen-and-paper; an optional hands-on demonstration uses a single low-voltage battery, a resistor, and a capacitor.

  • Optional: a single battery (AA or 9 V), a large resistor (tens to hundreds of kΩ), a large capacitor (say 100–1000 µF electrolytic — mind its polarity), and a multimeter (from Section 1.7)
  • With a large enough R and C, the charge and discharge happen slowly enough — seconds — to watch the voltage climb and fall on the meter in real time

Everything here uses a single low-voltage battery. Observe the electrolytic capacitor's polarity, and remember from Section 3.1 that even a modest charged capacitor should be treated with respect — though the small values here store little energy.

Real-World Applications

Every blinking LED timer, every "hold the button for two seconds" delay, and every slow fade is an RC circuit choosing a time constant. Power supplies lean on capacitors to smooth: the rectified voltage from the mains arrives as a series of humps, and a large capacitor charges up on each hump and discharges gently between them, holding the output nearly steady — its time constant is deliberately made long compared to the gap between humps. On the repair bench, RC timing is a diagnostic lens: a smoothing capacitor that has dried out and lost capacitance has a shorter time constant, so it sags further between peaks, and the resulting ripple shows up as hum, flicker, resets, or instability. Recognizing that pattern is often the whole diagnosis.

Common Challenges

  • Expecting charging to be linear. A capacitor does not charge at a steady rate — it charges quickly at first, then slower and slower as it fills, along an exponential curve. The voltage never quite reaches the supply, though after a few time constants it's close enough to call full.
  • Fumbling the units of τ = R × C. The time constant comes out in seconds only when resistance is in ohms and capacitance is in farads. A capacitance in µF must be converted to farads first, or the answer is off by a factor of a million.
  • Thinking the time constant is the "time to charge." One time constant isn't full charge — it's about 63% of the way. Full (for practical purposes) takes about five time constants. Confusing the two is the most common RC mistake.

Safety Notes

Risk Level: Low. This section is analysis plus an optional slow-charge demonstration on a single low-voltage battery. No mains, no high-energy sources.

Professional Tips Before Starting

  • When a symptom is about timing, smoothing, or holding a voltage — a timer that drifted, a supply that hums, a delay gone wrong — reach for τ = R × C first. The time constant is almost always what changed.
  • Convert capacitance to farads before you multiply. µF → farads means shifting the decimal six places; skipping it is a million-fold error that makes every time-constant answer nonsense.
  • In a real circuit the "R" setting the discharge time is often the load the capacitor feeds, not a discrete resistor. Identify what is actually setting the time constant before you reason about it.

Charging and Discharging Over Time

Charging: Fast at First, Then Slower

Connect an uncharged capacitor to a DC supply through a resistor and, at the very first instant, the capacitor is empty (0 V) so the full supply voltage appears across the resistor. By Ohm's law that drives the largest charging current of the whole process. As charge piles onto the capacitor its voltage rises, which leaves less voltage across the resistor, which means less current, which means the capacitor fills more slowly. The result is a smooth exponential curve: the voltage climbs steeply at first and then ever more gently, approaching the supply voltage but — strictly — never quite reaching it.

This is the approach to the fully-charged, DC-blocking state you met in Section 3.1. That "a charged capacitor blocks DC" fact is simply where this curve ends up: once the capacitor voltage equals the supply, there's no voltage left across the resistor, so the current falls to zero. Section 3.1 described the destination; this section describes the journey.

Discharging: The Mirror Image

Now take a charged capacitor and let it discharge through a resistor. The same logic runs in reverse: at first the capacitor's full voltage drives a large current through the resistor, but as it empties its voltage drops, the current falls, and the voltage decays along a mirror-image exponential curve — steep at first, then leveling off toward zero. Charging fills up along a rising curve; discharging empties along a falling one, and both are governed by the same time constant.

The Time Constant: τ = R × C

The single number that sets the pace of both curves is the time constant, written with the Greek letter tau (τ):

τ = R × C

with R in ohms, C in farads, and τ in seconds. (Ohms times farads really does work out to seconds — the units are built to make this true.) The time constant is the natural clock of the circuit: a bigger resistor slows the current that charges the capacitor, and a bigger capacitor needs more charge to reach a given voltage — so increasing either one lengthens τ and makes everything happen more slowly.

A worked example. Take R = 10 kΩ and C = 100 µF. First convert: 100 µF = 0.0001 F. Then:

τ = R × C = 10,000 Ω × 0.0001 F = 1 second

So this circuit's natural timescale is one second. Change to a 100 µF capacitor with a 100 kΩ resistor and τ becomes 10 seconds; drop to a 1 µF capacitor with a 1 kΩ resistor and τ = 1000 × 0.000001 = 0.001 s, a millisecond. Same rule, wildly different speeds — which is exactly how designers pick a timing.

The Rules of Thumb: 63%, 37%, and Five

You don't need calculus to use RC circuits, just a few percentages tied to multiples of τ. For charging (rising from 0 toward the supply V), and discharging (falling from V toward 0), the capacitor voltage passes these milestones:

| Time elapsed | Charging (% of supply) | Discharging (% of start) | |---|---|---| | 1τ | ~63% | ~37% | | 2τ | ~86% | ~14% | | 3τ | ~95% | ~5% | | 4τ | ~98% | ~2% | | 5τ | ~99% (call it full) | ~1% (call it empty) |

The two numbers worth memorizing: after one time constant a charging capacitor reaches about 63% of the supply (and a discharging one falls to about 37% of where it started), and after about five time constants it's essentially done — over 99% charged, or under 1% remaining. So for our τ = 1 s example on a 10 V supply: after 1 second the capacitor is at about 6.3 V, and after about 5 seconds it's within a hair of the full 10 V. Discharging from 10 V, after 1 second it's dropped to about 3.7 V.

Notice these percentages don't depend on the actual R, C, or voltage — only on how many time constants have passed. That's what makes them so useful: work out τ once, and the whole curve is just "63% after one τ, done after five."

Common Mistakes

  • Leaving capacitance in µF. τ = R × C needs farads. 100 µF is 0.0001 F, not 100. Forgetting the conversion is a million-fold error.
  • Treating 1τ as fully charged. One time constant is ~63%, not 100%. Practical "full" is about 5τ.
  • Assuming linear charging. The curve is exponential — most of the change happens early, then it crawls. A capacitor gets to 63% in the first τ but takes four more τ to cover the rest.
  • Ignoring where the resistance comes from. In a real supply the "R" that sets discharge might be the load itself, not a discrete resistor — the principle is the same, but you have to identify what's actually setting the time constant.

Troubleshooting Guidance

RC timing gives you a way to reason about a huge class of failures. When a timing circuit runs too fast or too slow, the time constant has changed — usually a capacitor that has drifted or lost value (its capacitance fell, so τ = RC shrank and everything speeds up). When a power supply hums, flickers, or resets under load, suspect a smoothing capacitor that has lost capacitance: with a shorter time constant it discharges too far between peaks, so the output ripples instead of holding steady. You can often confirm the pattern by reasoning about the expected time constant versus the observed behavior — and, later, by measuring. The key insight from this section is that a capacitor's job in DC is fundamentally about how long it takes, so symptoms that are about timing, smoothing, or holding a voltage point straight at the capacitors.

Verification & Testing Methods

Check your understanding before moving on:

  • [ ] Explain why a capacitor charges quickly at first and then more slowly, rather than at a steady rate.
  • [ ] Calculate τ = R × C for a given resistor and capacitor, converting µF to farads correctly.
  • [ ] State how far a capacitor has charged after one time constant and after five, using the 63% and ~99% rules.
  • [ ] Explain how a smoothing capacitor that has lost capacitance causes ripple, in terms of a shortened time constant.

Then try the practice exercises below — pen-and-paper, with an optional slow-charge demonstration.

Practice Exercises

  1. Calculate the time constant (10 minutes, pen and paper). Find τ for each: (a) R = 10 kΩ, C = 100 µF; (b) R = 1 kΩ, C = 1 µF; (c) R = 470 kΩ, C = 10 µF. Remember to convert µF to farads first, and give each answer in seconds (or ms).
  2. Read the curve (10 minutes, pen and paper). A 10 kΩ resistor charges a 100 µF capacitor from a 12 V supply. State τ. Then, using the rules of thumb, estimate the capacitor voltage after 1τ, after 3τ, and after 5τ. About how long until it's essentially fully charged?
  3. Discharge reasoning (5 minutes, pen and paper). The same capacitor, charged to 12 V, is discharged through the same 10 kΩ resistor. Estimate its voltage after 1τ and after 5τ, and explain why the discharge curve is the mirror image of the charge curve.
  4. Watch it happen (5 minutes, optional hardware). Build a slow RC circuit (for example 100 kΩ and 100 µF, τ = 10 s) from a single battery and watch the voltage climb on a multimeter as it charges and fall as it discharges. Note how it moves fast at first and then crawls — the exponential curve, live.

These core ideas — the exponential charge/discharge curves, τ = R × C, and the 63% / 37% / 5τ rules — are tested in the Chapter Quiz at the end of this chapter, where a score of 80% is required to continue.

Key Takeaways

  • A capacitor charges and discharges through a resistor along a smooth exponential curve — fast at first, then slower — not at a constant rate.
  • The time constant τ = R × C (ohms × farads = seconds) sets the pace; a larger resistor or capacitor makes charging and discharging slower.
  • After one time constant a charging capacitor reaches about 63% of the supply (a discharging one falls to about 37% of its start); after about five time constants it's essentially done (over 99% charged, under 1% remaining).
  • These percentages depend only on the number of time constants elapsed, not on the specific R, C, or voltage.
  • Charging is the approach to the fully-charged, DC-blocking state from Section 3.1 — once charged, no more current flows.
  • A capacitor that has lost capacitance has a shorter time constant, which is why timers drift and smoothing capacitors let ripple through — a very common repair symptom.

Skills Learned

  • You can now calculate the time constant τ = R × C with correct unit conversion.
  • You can now estimate charge and discharge progress with the 63% / 37% / 5τ rules of thumb.
  • You can now explain why a bigger resistor or capacitor slows the charge, and why charging is exponential rather than linear.
  • You can now connect RC timing to real circuits — timers and power-supply smoothing.
  • You can now recognize how a capacitor that has lost capacitance misbehaves, in terms of a shortened time constant.

Glossary Additions

  • time constant — the product τ = R × C (in seconds) that sets how fast a capacitor charges or discharges through a resistor; after one time constant a charging capacitor reaches about 63% of the supply voltage, and after about five it is essentially fully charged.
  • RC circuit — a circuit of a resistor and a capacitor together, whose charging and discharging behavior over time is governed by the time constant τ = R × C.
  • exponential curve — the smooth, ever-slowing curve a capacitor's voltage follows while charging or discharging: rapid change at first, then progressively slower, approaching its final value without a constant rate.

Suggested Next Sections

Must read next:

  • Capacitor Behavior in AC Circuits — what happens when the voltage is always changing: the capacitor never settles, and its opposition to AC (reactance) depends on frequency, the basis of filters.

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