CR · AC steady state

Impedance: Resistance and Reactance

In AC, inductors and capacitors impede current too, but in a different way from a resistor. Learn to bundle that impeding together with resistance into one complex impedance.

Resistance and reactance add at a right angle

Raise the reactance X and the impedance arrow tilts up from the resistance axis. Find where the reactance equals the resistance and the arrow stands at exactly 45 degrees.

Inductive reactance X = ωLX = 4 Ω

Impedance magnitude and phase

|Z| = 10.8 Ω  ·  θ = 22°

|Z| = √(R² + X²) = √(100 + 16)

Resistive

Reactance impedes from 90 degrees away

A resistor keeps voltage and current in step and throws energy away as heat. An inductor is different: it opposes any change in current and makes the voltage lead the current by 90 degrees. So an inductor’s impeding (its reactance) does not point along the resistance but stands at a right angle in the complex plane.

A right angle means Pythagoras

Resistance R lies on the horizontal axis, reactance X on the vertical. Because they are at a right angle, the magnitude of the combined impedance is not a plain sum but the hypotenuse, |Z| = √(R²+X²). How much the impedance impedes current (its magnitude) and how much it makes the voltage lead (its phase) both live in this one triangle.

The phase is set by the ratio of the legs

In the impedance triangle the phase angle is the vertical leg over the horizontal one, θ = arctan(X/R). When the reactance is small it is almost pure resistance, so the phase is near zero; when the reactance equals the resistance the two legs are the same length and the phase is exactly 45 degrees. For a capacitor the reactance points downward, the phase goes negative, and the current leads the voltage.

ObserveZ = R + jX

Real part is resistance, imaginary part reactance.

Choose|Z| = ?

A right angle, so combine as a hypotenuse.

Fill inθ = arctan(?)

Vertical leg over horizontal leg.

On your ownX = R → θ = ?

Equal legs give 45 degrees.

Back to the first screen

As you raised the reactance X, the impedance arrow tilted up from the resistance axis, and the instant X equaled the resistance R it stopped at exactly 45 degrees. The length of that arrow was the magnitude |Z|, its tilt the phase θ by which the voltage leads the current. One complex number carried both the size of the impeding and the phase at once.

The impedance Z = R + jX writes, as a complex number, how much a component impedes current in sinusoidal steady state. The real part resistance R keeps voltage and current in step; the imaginary part reactance X (ωL for an inductor, −1/ωC for a capacitor) sets them 90 degrees apart. The magnitude |Z| = √(R²+X²) says how much current is impeded, the phase θ = arctan(X/R) says how far the voltage leads the current.