AC Circuit Design Calculator

AC RLC series circuit with source V, resistor R, inductor L, and capacitor C
Inductive reactance equals 2 pi times frequency times inductance

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Inductive Reactance

Inductive reactance increases with frequency because an inductor opposes changes in current. At DC (0 Hz), an ideal inductor has zero reactance. The result is measured in ohms.

X_L = 2πfL

Capacitive Reactance

Capacitive reactance falls with frequency because a capacitor passes higher-frequency current more easily. Together, resistance and reactance combine into impedance: Z = √(R² + X²).

X_C = 1 / (2πfC)

How It Works

Reactance is the opposition to AC current caused by inductors and capacitors. Unlike resistance, reactance changes with frequency. Inductive reactance (Xₗ = 2πfL) rises with frequency, while capacitive reactance (Xᴄ = 1/2πfC) falls. This calculator solves either equation for any unknown variable. The results are measured in ohms, the same unit as resistance. Together, resistance and reactance combine into impedance: Z = √(R² + X²).

Example Problem

A 10 mH inductor operates in a 60 Hz power circuit. What is its inductive reactance?

  1. Identify the knowns. Inductance L = 10 mH (a typical line-filter or smoothing choke), source frequency f = 60 Hz (standard North American mains).
  2. Identify what we're solving for. We want the inductive reactance Xₗ, the opposition the inductor presents to the 60 Hz AC current in ohms.
  3. Write the formula: Xₗ = 2πfL. Reactance scales linearly with frequency — at DC (0 Hz) an ideal inductor passes current freely, and at higher frequencies it pushes back harder.
  4. Convert to SI units before substituting. The formula needs L in henries, so L = 10 mH × (1 H / 1,000 mH) = 0.01 H.
  5. Substitute and simplify: Xₗ = 2π × 60 × 0.01 = 6.2832 × 0.6 = 3.7699 Ω.
  6. State the final result: the inductor presents **Xₗ ≈ 3.77 Ω** at 60 Hz. A 100 µF capacitor at the same frequency would show Xᴄ = 1/(2π × 60 × 0.0001) ≈ 26.53 Ω in the opposite direction on the impedance triangle.

When to Use Each Variable

  • Solve for Reactance — when you know the frequency and the component value (inductance or capacitance), e.g., finding the opposition a 10 mH inductor presents at 60 Hz.
  • Solve for Frequency — when you need the frequency that produces a specific reactance, e.g., tuning an LC filter to resonate at a target impedance.
  • Solve for Inductance — when you know the reactance and frequency and need to select an inductor, e.g., choosing a choke for a power supply filter.
  • Solve for Capacitance — when you know the reactance and frequency and need to select a capacitor, e.g., sizing a coupling capacitor for an audio crossover.

Key Concepts

Reactance is measured in ohms like resistance, but it shifts the phase between voltage and current rather than dissipating energy. Inductive reactance grows linearly with frequency, while capacitive reactance shrinks inversely. When both are present, they partially cancel: the net reactance is X = X_L − X_C, and at resonance (X_L = X_C) the circuit behaves as a pure resistance.

Applications

  • Power systems: sizing capacitor banks for power factor correction on industrial motors
  • Audio engineering: designing crossover networks that split signals between woofers and tweeters
  • RF design: tuning LC tank circuits to select a specific radio frequency
  • Signal filtering: calculating cutoff frequencies for low-pass and high-pass RC/RL filters

Common Mistakes

  • Forgetting to convert units — inductance must be in henries and capacitance in farads, not millihenries or microfarads, before plugging into the formula
  • Confusing reactance with impedance — reactance is the imaginary component only; impedance combines both resistance and reactance as Z = sqrt(R^2 + X^2)
  • Using DC resistance values for AC analysis — at non-zero frequency, inductors and capacitors contribute reactance that pure resistance calculations ignore

Frequently Asked Questions

What is the difference between reactance and impedance?

Reactance is the opposition to AC current from inductors or capacitors alone. Impedance combines reactance with resistance into a single value: Z = √(R² + X²). In a purely reactive circuit with no resistance, impedance equals the absolute value of the reactance.

Why does inductive reactance increase with frequency?

An inductor opposes changes in current. At higher frequencies the current changes direction more often, so the inductor resists more strongly. At DC (0 Hz), an ideal inductor has zero reactance and behaves like a short circuit.

How do you calculate capacitive reactance at 1 kHz?

Use Xᴄ = 1/(2πfC). For a 10 µF capacitor at 1,000 Hz: Xᴄ = 1/(2π × 1000 × 0.00001) ≈ 15.92 Ω. Larger capacitors or higher frequencies yield lower reactance.

What is power factor correction?

Industrial motors draw inductive current that lags behind the voltage, wasting energy. Adding capacitors reduces the net reactance, bringing current and voltage closer in phase. This lowers the apparent power demand and can reduce electricity costs by 10–30%.

At what frequency do inductive and capacitive reactance cancel?

When Xₗ = Xᴄ, the circuit is at resonance: 2πfL = 1/(2πfC), which gives f = 1/(2π√(LC)). At resonance the net reactance is zero and the circuit behaves as a pure resistance — this is the principle behind LC tank circuits and tuned filters.

Does reactance dissipate power like resistance?

No. Reactance stores and returns energy each AC cycle — inductors in their magnetic field, capacitors in their electric field. Only resistance dissipates real power as heat. That is why reactive components stay cool while still affecting current and phase angle.

How accurate is the ideal reactance formula in practice?

The formulas Xₗ = 2πfL and Xᴄ = 1/(2πfC) assume ideal lossless components. Real inductors have winding resistance and parasitic capacitance; real capacitors have equivalent series resistance (ESR) and lead inductance. These parasitics shift the effective impedance noticeably above ~100 kHz, so for RF work you should consult the component's full impedance curve rather than rely on the textbook reactance alone.

Reference:

Tipler, Paul A. 1995. Physics For Scientists and Engineers. Worth Publishers. 3rd ed.

Worked Examples

Power Electronics

What is the inductive reactance of a 100 mH choke on a 60 Hz mains line?

A 100 mH iron-core choke sits on a single-phase 60 Hz mains feed for harmonic filtering. Find the inductive reactance X_L the inductor presents to the line-frequency fundamental.

  • Knowns: L = 100 mH = 0.1 H, f = 60 Hz
  • Formula: X_L = 2π × f × L
  • X_L = 2π × 60 × 0.1

X_L ≈ 37.7 Ω

Reactance scales linearly with frequency, so the same choke presents about 628 Ω to a 1 kHz harmonic and roughly 6.3 kΩ to a 10 kHz switching-noise burst — useful for filtering high frequencies while passing the 60 Hz fundamental.

Radio Broadcasting

What capacitive reactance does a 100 pF tuning capacitor show at 1 MHz?

An AM tuning network uses a 100 pF variable capacitor in series with the antenna at a station frequency of 1.0 MHz (squarely in the AM broadcast band). Find X_C to size the matching network.

  • Knowns: f = 1 MHz = 1,000,000 Hz, C = 100 pF = 1 × 10⁻¹⁰ F
  • Formula: X_C = 1 / (2π × f × C)
  • X_C = 1 / (2π × 1,000,000 × 1 × 10⁻¹⁰)
  • X_C = 1 / (6.283 × 10⁻⁴)

X_C ≈ 1,592 Ω

Capacitive reactance falls as frequency rises, which is why the same cap presents only ~16 Ω at 100 MHz (FM band) — the reason variable-capacitance tuning becomes physically impractical above HF.

Audio Engineering

What inductor sizes an 8 Ω tweeter for a 4 kHz first-order high-pass crossover?

A two-way loudspeaker uses a first-order high-pass crossover at f = 4 kHz on an 8 Ω tweeter. The series inductor at this crossover frequency must present X_L equal to the driver impedance. Solve for L.

  • Knowns: X_L = 8 Ω, f = 4,000 Hz
  • Formula (solved for L): L = X_L / (2π × f)
  • L = 8 / (2π × 4,000)
  • L = 8 / 25,133

L ≈ 3.18 × 10⁻⁴ H ≈ 318 μH

Real loudspeaker crossovers are usually second-order or higher and account for the driver's non-flat impedance curve — a first-order pass-band calculation gives the correct ballpark inductance but underdamps the response near crossover.

Inductive & Capacitive Reactance Formulas

Two reactance equations cover the AC behavior of inductors and capacitors at a given frequency:

XL = 2π × f × LInductive reactance
XC = 1 / (2π × f × C)Capacitive reactance

Where:

  • XL — inductive reactance, in ohms (Ω)
  • XC — capacitive reactance, in ohms (Ω)
  • f — signal frequency, in hertz (Hz)
  • L — inductance, in henries (H)
  • C — capacitance, in farads (F)

These formulas describe ideal lossless components. Inductive reactance grows linearly with frequency, while capacitive reactance falls inversely with frequency — the opposite behaviors that make LC networks selective filters. When both elements appear in series, the net reactance is X = XL − XC, reaching zero at the resonant frequency f₀ = 1 / (2π√(LC)).

Reference: Tipler, Paul A. 1995. Physics For Scientists and Engineers. Worth Publishers. 3rd ed.

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