03. Complex Numbers

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:

j:=√{-1}

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):

u:=Û⋅e^{j⋅(ω⋅t+φ_u)}
i:=Î⋅e^{j⋅(ω⋅t+φ_i)}

Impedance

The impedance Z is a resistance-like number, just as a complex number.

Resistor

u=R⋅i

Or with the complex impedance Z (a complex number):

u=Z⋅i

where:

Z:=R

Capacitor

Q=C⋅u
i=C⋅{\dot u}

When we let u and i be:

u:=Û⋅e^{φ_u}⋅e^{j⋅ω⋅t}
{\dot u}=Û⋅e^{φ_u}⋅j⋅ω⋅e^{j⋅ω⋅t}
i:=Î⋅e^{φ_i}⋅e^{j⋅ω⋅t}
Î⋅e^{φ_i}⋅e^{j⋅ω⋅t}=C⋅j⋅ω⋅Û⋅e^{φ_u}⋅e^{j⋅ω⋅t}
i=C⋅j⋅ω⋅u
÷{i}{C⋅j⋅ω}=u

or nicer:

u=Z⋅i

where:

Z:=÷{1}{j⋅ω⋅C}

Inductor coil

Induction and magnetic flux Φ:

L:=÷{dΦ}{di}

then the self-induced voltage is:

u=L⋅÷{di}{dt}

When we let U and I be:

u:=Û⋅e^{φ_u}⋅e^{j⋅ω⋅t}
i:=Î⋅e^{φ_i}⋅e^{j⋅ω⋅t}
{\dot i}=÷{di}{dt}=j⋅ω⋅Î⋅e^{φ_i}⋅e^{j⋅ω⋅t}

Then:

Û⋅e^{φ_u}⋅e^{j⋅ω⋅t}=L⋅j⋅ω⋅Î⋅e^{φ_i}⋅e^{j⋅ω⋅t}
u=j⋅ω⋅L⋅i

or nicer:

u=Z⋅i

where:

Z:=j⋅ω⋅L

Author: Danny (remove the ".nospam" to send)

Last modification on: Thu, 09 May 2013 .