Applications of Complex Numbers
Electronics:
Direct Current (DC) : The opposition to the flow of electricity for a stable DC circuit, i.e. a flashlight, can be
described by just the circuit’s resistance.
Resistance is the opposition that a circuit has to a steady flow of electrons = R.
Voltage
Resistance = Amperage
for a DC circuit
Alternating Current (AC): AC circuits, like a household outlet, have a continuously changing voltage. This causes
AC circuits to have an opposition to the flow of electricity, called the impedance, caused by both the resistance and
the reactance.
Reactance is the opposition that a circuit has to a change in the flow of electrons = X.
Impedance is the total opposition that a circuit has to a flow of electrons = Z = R + jX = R − i X .
where j = -i = - -1 .
For AC circuits, it’s now impedance =
phasors.
voltage
amperage
where voltage and amperage are now imaginary numbers called
voltage = V = V ⎡⎣ cos ( ωt + φV ) + j sin ( ωt + φV ) ⎤⎦ = V e
Amperage = I = I ⎡⎣cos ( ωt + φI ) + j sin ( ωt + φI ) ⎤⎦ = I e
t = time
ω = 2π (frequency)
φV and φI are the phase shifts
e ≈ 2.71828...
j ( ωt +φV )
j ( ωt +φI )
x + yi = x 2 + y 2
Even though it might not look like it, using complex numbers in this application actually makes the math a lot
easier than not using complex numbers.