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Inductor Behavior in Circuits

The inductor over time and across frequency — its L/R time constant and its rising reactance, the point-for-point mirror of the capacitor, with current lagging where the capacitor's led.

IntermediateLow Risk26 min read

What You Will Learn

  • You will learn how the current in an inductor rises and falls over time, governed by the L/R time constant — the mirror of the capacitor's RC constant, but with current in place of voltage.
  • You will learn what inductive reactance is, the relationship XL = 2πfL, and why it rises with frequency — the exact opposite of a capacitor's reactance.
  • You will learn why an inductor passes DC and impedes AC, dissipates no power, and makes the current lag the voltage rather than lead it.
  • You will learn how these behaviors make an inductor a choke and filter, and how a shorted-turn fault changes them.

What You Will Be Able To Do

  • You will be able to calculate the inductive time constant τ = L ÷ R and use the 63% / 5τ rules to describe how the current builds and decays.
  • You will be able to calculate inductive reactance XL = 2πfL and predict how it scales with frequency, contrasting it with a capacitor.
  • You will be able to explain why an inductor passes DC, impedes AC, dissipates no power, and makes current lag voltage.
  • You will be able to describe how an inductor filters as a choke and how a shorted-turn fault degrades that behavior.

Required Tools

No physical tools required. This is a conceptual section.

Section Overview

You met the inductor as a component in the last section; now you'll see how it behaves — over time in DC, and across frequency in AC. Both stories are the point-for-point mirror of the capacitor. In DC, an inductor's current builds up and dies away along an exponential curve set by a time constant, exactly like a capacitor's voltage did — but the constant is τ = L ÷ R (division, not the capacitor's multiplication) and it governs current, not voltage. In AC, an inductor's opposition is inductive reactance, XL = 2πfL, which rises with frequency — the exact opposite of a capacitor's reactance, which falls. Along the way the inductor dissipates no power, and its current lags the voltage where a capacitor's led. This section is where the capacitor–inductor duality becomes a complete, working toolkit.

Why This Matters

Nearly every use of an inductor comes down to one of these two behaviors. Its time constant sets how fast current can build in a switch-mode power supply, a relay, or a solenoid — and how violently it kicks back when that current is interrupted. Its rising reactance is what makes a choke block high-frequency noise while passing DC power, the mirror image of a decoupling capacitor shunting noise to ground. For a repair technician, knowing that an inductor passes DC and impedes AC — and that a shorted-turn fault lowers its inductance, shifting both its time constant and its reactance — turns a class of confusing filter and power-supply symptoms into something you can reason about. And because it's the capacitor's mirror, everything you already learned about RC circuits and capacitive reactance transfers directly.

Required Prerequisites

No consumables required. The exercises are pen-and-paper calculations; an optional demonstration uses a signal source and a coil, but nothing is used up.

  • Optional: an inductor or two (a few mH ferrite choke works well), a resistor, a signal/function generator or a phone-driven AC source, and a multimeter with an AC range
  • The demonstrations — watching current build through a coil, or how much of a signal a choke passes at different frequencies — are instructive but entirely optional; the section stands on the calculations

Everything here is low-voltage signal-level work. Reiterating the Section 3.4 hazard: do not build a circuit that switches a coil's current on and off yet — that is where the inductive-kick spike lives, and it is saved for the later hands-on chapter.

Real-World Applications

The inductor's two behaviors are everywhere in power and signal work. A switch-mode power supply relies on the inductor's L/R current build-up to store and transfer energy each switching cycle; a snubber or flyback diode exists precisely because interrupting that current makes a spike. A choke — an inductor whose job is its rising reactance — sits in series with a power line and blocks high-frequency noise while passing the DC or low-frequency power straight through, the exact complement of a capacitor shunting noise to ground. Inductors pair with capacitors in the filters and tuned circuits of the next section. And when a filter or supply misbehaves in a frequency-dependent way, an inductor whose turns have partially shorted — lowering its inductance and therefore its reactance — is a real, if less common, culprit.

Common Challenges

  • Flipping the time-constant formula. The capacitor's constant was τ = R × C; the inductor's is τ = L ÷ R — division. A bigger resistor makes an inductor's current settle faster (shorter τ), the opposite of a capacitor. It's easy to carry the multiplication over by mistake.
  • Getting the reactance direction backwards. Capacitive reactance falls as frequency rises; inductive reactance rises. An inductor impedes high frequencies more, a capacitor less — mirror images. Mixing them up inverts every filter you analyze.
  • Confusing which quantity lags. In an inductor the current lags the voltage; in a capacitor the current leads. Both are 90° in the ideal case, but in opposite directions.

Safety Notes

Risk Level: Low. This section is analysis and optional low-voltage signal-level demonstration. No mains, no high-energy sources.

Professional Tips Before Starting

  • Carry the duality as a two-column mental table: capacitor resists voltage change / τ = RC / Xc falls with f / current leads; inductor resists current change / τ = L÷R / XL rises with f / current lags. Every inductor question is a capacitor question with the columns swapped.
  • Reason at the extremes first, as with the capacitor. At DC an inductor is a plain wire (XL = 0); at very high frequency it's nearly an open (XL huge). A capacitor is exactly the reverse. Most filter behavior is those two limits and the transition between.
  • When a power supply or filter fault is frequency-dependent, remember an inductor can be the cause as well as a capacitor — a choke or transformer winding with shorted turns has lost inductance, so its reactance and time constant have shifted.

Two Behaviors: Building Current and Rising Reactance

Current Builds Over Time: τ = L ÷ R

Apply a DC voltage to an inductor in series with a resistor and the current does not jump to its final value — the inductor opposes the change, so the current rises gradually along an exponential curve, just as a capacitor's voltage did. The final current is set by Ohm's law on the resistor, I_final = V ÷ R, and the pace of the rise is set by the inductive time constant:

τ = L ÷ R

with L in henries, R in ohms, and τ in seconds. Note the division: this is the mirror of the capacitor's τ = R × C, and it behaves oppositely with respect to R — a larger resistor makes an inductor's current settle faster (smaller τ), whereas a larger resistor made a capacitor charge slower.

A worked example. Take L = 10 mH (0.01 H) and R = 100 Ω:

τ = L ÷ R = 0.01 ÷ 100 = 0.0001 s = 0.1 ms

So the current reaches most of its final value in a fraction of a millisecond. With a much larger L = 1 H and R = 1 kΩ, τ = 1 ÷ 1000 = 0.001 s = 1 ms — ten times slower than the first example, because the hundred-times-larger inductance outweighs the ten-times-larger resistance. The same rules of thumb from the capacitor's time constant apply, now to current: after one time constant the current has reached about 63% of its final value; after about five time constants it's essentially there (over 99%). De-energizing mirrors it — remove the drive and the current decays with the same τ, falling to about 37% after one time constant. And this is the quantitative face of the inductive kick from Section 3.4: force the current to change fast (interrupt it, making the effective τ tiny) and the inductor answers with a large voltage.

Opposition That Rises With Frequency: XL = 2πfL

In AC, an inductor's opposition is inductive reactance, written XL, measured in ohms like a capacitor's — but with the frequency dependence turned upside down:

XL = 2πfL

where f is in hertz, L in henries, and XL in ohms. Because XL is proportional to frequency (not inversely proportional, as Xc was), it rises as frequency increases. That single difference flips the whole story:

  • At DC, f = 0, so XL = 0 — the inductor is just a coil of wire and passes DC freely. (A capacitor was the opposite: infinite reactance at DC, blocking it.)
  • As frequency rises, XL grows, so the inductor impedes high-frequency AC more and more. (A capacitor was again the opposite: its reactance shrank, passing highs more easily.)

A worked example. A 10 mH inductor (0.01 H) at 1 kHz:

XL = 2πfL = 2π × 1000 × 0.01 ≈ 62.8 ohms

Now raise the frequency ten times, to 10 kHz:

XL = 2π × 10,000 × 0.01 ≈ 628 ohms

Ten times the frequency gives ten times the reactance — the direct opposite of the capacitor, whose reactance dropped tenfold for the same change. And like capacitive reactance, XL dissipates no power: the inductor stores energy in its magnetic field for part of each cycle and returns it for the rest, so an ideal inductor consumes no net power even while limiting the current.

Phase: Current Lags the Voltage

The inductor also mirrors the capacitor's phase behavior — in the opposite direction. In a resistor, current and voltage move together. In an inductor, the current lags the voltage: because the coil resists the current changing, the current is always playing catch-up to the voltage driving it, arriving a quarter-cycle behind. This is the exact opposite of the capacitor, where the current led the voltage by a quarter-cycle. You don't need the trigonometry — the intuition is enough: a capacitor's current runs ahead (it responds to how fast voltage changes), while an inductor's current runs behind (it resists changing at all). In an ideal inductor the lag is 90°.

The Payoff: Chokes and Filters

Put the rising reactance to work and you get the inductor's signature job: the choke. A choke is an inductor placed in series with a line so that its reactance blocks what you don't want while passing what you do. Because XL is small at low frequency and large at high frequency, a choke in a power line passes the DC or low-frequency power almost untouched (XL near zero) while presenting a large opposition to high-frequency noise (XL large) — choking the noise off before it travels further. This is the mirror of the decoupling capacitor from Section 3.3, which did the same job by the opposite route: shunting high-frequency noise to ground (low Xc) rather than blocking it in series. Series inductor or shunt capacitor, the two are complementary tools for the same goal, and real filters often use both.

The repair connection mirrors the capacitor's too. An inductor whose turns have partially shorted has less inductance than it should, so both its time constant (τ = L ÷ R) and its reactance (XL = 2πfL) drop. A choke with shorted turns presents too little reactance and lets noise through; a switch-mode supply's inductor with shorted turns stores less energy and charges too fast. Frequency-dependent or timing-related symptoms in a circuit that contains inductors can trace back to an inductor whose inductance has drifted, exactly as they can to a capacitor whose capacitance has drifted.

Common Mistakes

  • Using τ = L × R instead of L ÷ R. The inductor's time constant is L divided by R. A larger resistor speeds an inductor up and slows a capacitor down — opposite effects.
  • Reversing the reactance trend. XL rises with frequency; Xc falls. An inductor blocks highs, a capacitor passes them.
  • Forgetting XL = 0 at DC. An inductor passes DC freely (it's just a coil); it does not block DC the way a capacitor does.
  • Swapping the phase direction. Inductor current lags voltage; capacitor current leads. Getting the direction backwards inverts the analysis.

Troubleshooting Guidance

The inductor's behaviors give you two more diagnostic levers, both mirrors of the capacitor's. When a filter or power supply misbehaves in a frequency-dependent way, consider not only the capacitors but any inductors or transformers: a choke that passes noise it should block, or a supply inductor that behaves as if it stores too little energy, points toward lost inductance — most often shorted turns, which lower L and therefore lower both XL and the L/R time constant. When a timing behavior in an inductive circuit is wrong, reason about τ = L ÷ R: has the inductance dropped, or the series resistance changed? And keep the duality in mind while diagnosing — if a symptom looks like a capacitor problem but the frequency dependence runs the wrong way (worse at high frequency where a failing capacitor would be worse at low, or vice versa), an inductor may be the real culprit. Detailed measurement comes later; the reasoning is available now.

Verification & Testing Methods

Check your understanding before moving on:

  • [ ] Calculate the inductive time constant τ = L ÷ R and state how the current behaves after one and after five time constants.
  • [ ] Explain why a larger resistor makes an inductor settle faster but made a capacitor charge slower.
  • [ ] Calculate inductive reactance XL = 2πfL and state how it changes when the frequency doubles, contrasting it with a capacitor.
  • [ ] Explain why an inductor passes DC, impedes AC, dissipates no power, and makes current lag voltage.

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

Practice Exercises

  1. Time constant (10 minutes, pen and paper). Find τ = L ÷ R for each: (a) L = 47 mH, R = 220 Ω; (b) L = 2.2 H, R = 4.7 kΩ; (c) L = 100 µH, R = 10 Ω. Convert to henries first and give each answer in seconds (or ms/µs). For (a), also state roughly how long until the current is essentially at its final value (about 5τ).
  2. Reactance and its direction (10 minutes, pen and paper). For a 22 mH inductor, find XL at 500 Hz and at 5 kHz. State the pattern in one sentence, and contrast it explicitly with what a capacitor's reactance would do over the same frequency change.
  3. The two extremes (5 minutes, reasoning). For any inductor, state its reactance at DC and describe its reactance as frequency becomes very high. Explain how these two limits make an inductor "pass DC, impede AC," and how that is the reverse of a capacitor.
  4. Choke reasoning (10 minutes, reasoning). A choke is placed in series with a DC power line to keep high-frequency noise out of a sensitive circuit. Explain, using XL = 2πfL, why it passes the DC power but blocks the noise — and why a choke with several shorted turns would let more noise through.

These core ideas — the L/R time constant, inductive reactance rising with frequency, no power dissipation, current lagging voltage, and chokes — are tested in the Chapter Quiz at the end of this chapter, where a score of 80% is required to continue.

Key Takeaways

  • An inductor's current builds and decays exponentially with the inductive time constant τ = L ÷ R (division, the mirror of the capacitor's τ = R × C), governing current where the capacitor governed voltage; a larger R makes an inductor settle faster.
  • The 63% / 37% / 5τ rules from the capacitor apply here to current: about 63% of final after one time constant, essentially complete after about five.
  • Inductive reactance XL = 2πfL rises with frequency — the exact opposite of a capacitor's reactance, which falls: an inductor passes DC (XL = 0) and impedes AC more at higher frequency.
  • Like capacitive reactance, XL dissipates no power — energy is stored in the magnetic field and returned each cycle.
  • In an inductor the current lags the voltage (by 90° when ideal), the opposite of a capacitor's current, which leads.
  • Rising reactance makes an inductor a choke — a series component that blocks high-frequency noise while passing DC/low-frequency power, the mirror of a shunt capacitor; a shorted-turn fault lowers L, dropping both XL and the time constant and degrading the filtering.

Skills Learned

  • You can now calculate the inductive time constant τ = L ÷ R and describe the current's rise and decay with the 63% / 5τ rules.
  • You can now calculate inductive reactance XL = 2πfL and predict its frequency scaling, contrasting it with a capacitor.
  • You can now explain why an inductor passes DC, impedes AC, dissipates no power, and makes current lag voltage.
  • You can now describe how a choke filters by frequency and how a shorted-turn fault degrades it.
  • You can now apply the full capacitor–inductor duality as a single mental model with the columns swapped.

Glossary Additions

  • inductive reactance — an inductor's opposition to alternating current, XL = 2πfL, in ohms; it is zero at DC (passing DC) and rises as frequency increases (impeding AC more at higher frequency), proportional to both frequency and inductance — the mirror of capacitive reactance, which falls with frequency.
  • choke — an inductor placed in series with a line to block unwanted higher-frequency current (via its rising reactance) while passing DC and low-frequency current; the series-inductor counterpart to a shunt (decoupling) capacitor.

Suggested Next Sections

Must read next:

  • LC Circuits and Resonance — put a capacitor and an inductor together and their opposite reactances cancel at one special frequency, producing resonance — the basis of tuning, oscillators, and filters that pick out a single frequency.

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