Impedance, Reactance and Ohm's Law in AC

The impedance is the complex relationship between voltage and current phasors in a linear circuit at a given frequency.

In a direct current resistive circuit under steady-state conditions, we work with resistance ().

But in Alternating Current (AC), the coil and the capacitor also oppose the current flow, not through friction, but through reaction to changes.

• The coil opposes changes in its magnetic field.
• The capacitor opposes changes in its voltage.

We call this “dynamic resistance” that depends on frequency reactance ().

The complex combination of resistance and reactance forms the impedance ().

Reactance ()

Reactance is also measured in Ohms (Ω), but it has a curious property: it changes with the wave speed ().

Inductive Reactance ()

This is the opposition presented by a coil.

• If the frequency increases (): The coil opposes much more. The changes are too fast for it.
• In DC under steady-state conditions: and an ideal coil is equivalent to a short circuit.

Capacitive Reactance ()

This is the opposition presented by a capacitor.

• If the frequency increases (): The capacitor allows current to pass easily (it charges and discharges very quickly). The opposition decreases.
• In DC (): Dividing by zero gives infinity (∞). The capacitor is an open circuit (it blocks DC).

Complex Summation and Phasors

Suppose you have a 3 Ω resistor and a coil with a reactance of 4 Ω in series. What is the total opposition?

Why can’t they be added? Because, as we saw in the previous article, Resistance acts “now” and the Coil acts “90 degrees later”. They are not in the same time direction.

To add these quantities, we use phasors, complex representations of sinusoids that share the same frequency.

The Real and Imaginary Axes

We can represent this on a complex plane:

1. X-axis (Real): Here lives the Resistance ().
2. Y-axis (Imaginary): Here lives the Reactance ().

• Upward (+90º): The Coil ().
• Downward (-90º): The Capacitor ().

The Impedance Triangle

The complex impedance of a series RLC circuit is:

Its magnitude is the hypotenuse of the triangle:

Applying the good old Pythagorean Theorem:

And the phase angle () between voltage and current will be:

Solved Example: If we have and .

The total opposition is 5Ω, not 7Ω.

Ohm’s Law with Phasors

Now that we have , we can update the most famous equation in electricity to work with AC:

This formula ALWAYS works, both in circuits with a single light bulb and in industrial motors.

• If the circuit is purely resistive, .
• If it is mixed, is the hypotenuse.

Impedance allows us to calculate the amplitude and phase of voltage and current. When a phase shift exists, we must distinguish between active, reactive, and apparent power.