Impedance

Recall the voltage drop of a resistor, capacitor, and inductor.

Even though these formulas have a phase shift, it is possible to express them in a form similar to Ohm's law, if we use complex numbers.

Recall Euler's Formula:

this means that

so the current can be expressed as

and then we will write current as a complex quantity:

What does this mean?

  1. Each value of gives a point in the complex plane plane, on the circle of radius
  2. We can picture as a point travelling around the circle
  3. The actual current is given by the imaginary part of the complex one, so we only consider the coordinate.
Abstract

This quantity is called Impedance.

Recall . We just have to replace with everywhere.

For Resistors

Resistors and Resistance

Law


where is voltage, is current, and is resistance.

Remark

This is the equation of a line, where is the slope ()

Ohm's law remains unchanged after our impedance calculations. becomes , and since ,

Law

For Capacitors

Capacitors and Capacitance

Recall . This becomes

(see Euler's Formula)

Recall

Law

For Inductors

Inductors and Inductance

Recall . This becomes

(see Euler's Formula)

Recall

Law