Since integrodifferential equations are complicated to solve, we introduce complex numbers instead as a device for easier calculation.
So, let the complex root be known as j:
Then, the wave equation for alternating current i and voltage u are (with ω being the angular frequency, t being the time and φ_u, φ_i being the phases):
Impedance
The impedance Z is a resistance-like number, just as a complex number.
Resistor
Or with the complex impedance Z (a complex number):
where:
Capacitor
When we let u and i be:
or nicer:
where:
Inductor coil
Induction and magnetic flux Φ:
then the self-induced voltage is:
When we let U and I be:
Then:
or nicer:
where: