CR · AC steady state

Impedance: Resistance and Reactance

In AC circuits, inductors and capacitors also hold back current, but in a different way from resistors. This unit shows how to bundle that opposition together with resistance into a single complex impedance.

Resistance and reactance add at a right angle

In the triangle, the horizontal side is a fixed resistance, the vertical side is the reactance, and the arrow along the hypotenuse is the impedance. Raise the reactance and the arrow rises from the horizontal axis; find the setting where the reactance equals the resistance and the arrow points 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)

Far from 45 degrees

Reactance impedes from 90 degrees away

A resistor keeps voltage and current in step and throws energy away as heat. An inductor behaves differently: it fights every attempt to change its current, and as a result the voltage leads the current by 90 degrees. So an inductor's opposition, its reactance, does not point the same way as resistance but stands at a right angle to it in the complex plane.

A right angle means Pythagoras

Resistance R lies along the horizontal axis and reactance X along the vertical axis. Because the two are at right angles, the magnitude of the combined impedance is not a simple sum but the length of the hypotenuse, |Z| = √(R²+X²). This one triangle holds both the magnitude, which tells how much current is held back, and the phase, which tells how far the voltage is pushed ahead.

The phase is set by the ratio of the legs

In the impedance triangle the phase angle is set by the ratio of the vertical side to the horizontal side, θ = arctan(X/R). With a small reactance the element is almost purely resistive and the phase is near 0; when the reactance equals the resistance the two sides are equally long and the phase is exactly 45 degrees. The resistance in the plot is 10 Ω, so the 45-degree point comes at a reactance of 10 Ω. For a capacitor the reactance points downward, the phase turns negative, and the current leads the voltage.

ObserveZ = R + jX

The real part is resistance; the imaginary part is reactance.

Choose|Z| = ?

The two parts are at right angles, so they combine along the hypotenuse.

Fill inθ = arctan(?)

It is the arctangent of the vertical side over the horizontal side.

On your ownX = R → θ = ?

Equal sides give 45 degrees.

Back to the first screen

As you raised the reactance, the impedance arrow rose from the horizontal axis and reached exactly 45 degrees the moment the reactance equaled the resistance. The arrow's length was the impedance magnitude |Z|, and its slope was the phase θ by which the voltage leads the current. A single complex number carried both the size and the phase of the opposition.

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