Inductors, capacitors, impedance

Why it matters

Resistors are not the only things in AC circuits. Motor windings, transformers, fluorescent ballasts and long cables behave like inductors; power-factor correction banks, filters and motor start circuits use capacitors. Both oppose AC current in a way that changes with frequency and shifts the timing between voltage and current. That opposition is called reactanceThe opposition of an inductor or capacitor to AC, in ohms, which changes with frequency., and combined with resistance it is impedanceThe total opposition to AC from resistance and reactance combined; in series |Z| = √(R² + (XL − XC)²).. Without it you cannot size a capacitorTwo conductors separated by an insulator that store energy in an electric field and resist changes in voltage., predict a motor's current or understand filters.

Capacitors and inductors in one paragraph each

A capacitor is two conductors separated by an insulator. It stores energy in an electric field and resists changes in voltage. Its value is capacitance, C, in farads (F); practical values are microfarads (µF, 10⁻⁶ F) or nanofarads (nF, 10⁻⁹ F). A capacitor blocks steady DC once charged but passes AC, more easily at higher frequencies.

An inductorA coil of wire that stores energy in a magnetic field and resists changes in current. is a coil of wire, often around an iron core. It stores energy in a magnetic field and resists changes in current. Its value is inductance, L, in henries (H); practical values are millihenries (mH). An inductor passes DC easily (just the wire resistance) but opposes AC, more strongly at higher frequencies.

Reactance

Reactance, X, is measured in ohms like resistance:

Inductive reactance:   XL = 2π × f × L
Capacitive reactance:  XC = 1 / (2π × f × C)

• XL rises with frequency. Double the frequency, double XL.
• XC falls with frequency. Double the frequency, halve XC.

Example 1. A 100 mH coil at 60 Hz:

XL = 2π × 60 × 0.1 = 37.7 Ω

Example 2. A 10 µF capacitor at 60 Hz:

XC = 1 / (2π × 60 × 0.000010) = 265 Ω

At 600 Hz the same capacitor is 26.5 Ω and the same coil is 377 Ω.

Phase shift

In a pure resistor, voltage and current peak together. In a pure inductor, current lags voltage by 90°. In a pure capacitor, current leads voltage by 90°. A memory aid used in many trade schools is "ELI the ICE man": in an inductor (L), voltage E comes before current I; in a capacitor (C), current I comes before E.

Impedance

Because reactance is 90° out of step with resistance, you cannot simply add ohms. For a resistor, inductor and capacitor in series, the net reactance is X = XL − XC, and the impedance magnitude is found like the hypotenuse of a right triangle:

|Z| = √( R² + (XL − XC)² )
phase angle θ = arctan( (XL − XC) / R )

A positive angle means the circuit is inductive (current lags); negative means capacitive (current leads). Ohm's law works with impedance: I = V / |Z|.

Example 3. A series circuit with R = 50 Ω, L = 100 mH and C = 10 µF on 120 V, 60 Hz:

XL = 37.7 Ω,  XC = 265.3 Ω
X  = 37.7 − 265.3 = −227.6 Ω   (capacitive)
|Z| = √(50² + 227.6²) = 233.0 Ω
θ  = arctan(−227.6 / 50) = −77.6°
I  = 120 / 233.0 = 0.515 A

Resonance

When XL equals XC, they cancel and the impedance is just R. This happens at the resonant frequencyThe frequency at which XL equals XC, so they cancel and the impedance equals R.:

f0 = 1 / (2π × √(L × C))

For 100 mH and 10 µF, f0 ≈ 159 Hz. Resonance is used in radio tuning and filters, and it is also something power engineers avoid, because it can cause unexpectedly high voltages or currents.

Engineers often write impedance as a complex number, Z = R + jX, where j is the imaginary unit. Complex numbers make phase bookkeeping automatic; you will use them in the power and electronics course.

Common mistakes

• Forgetting unit prefixes. 10 µF is 0.000010 F; 100 mH is 0.1 H.
• Adding R and X directly. 50 Ω + 227.6 Ω is wrong; use the square-root formula.
• Mixing up which reactance rises with frequency. Inductors oppose fast change in current, so XL grows with f.
• Treating a charged capacitor as safe. Large capacitors in power supplies and motor circuits can hold a dangerous charge after power is removed. Only trained, supervised workers discharge and verify them.

Recap

• XL = 2πfL rises with frequency; XC = 1/(2πfC) falls with frequency.
• Inductor current lags voltage; capacitor current leads.
• Series impedance |Z| = √(R² + (XL − XC)²); then I = V / |Z|.

Try it

Open the Do tab and write the capacitive reactance formula in a Python impedance calculator.