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Complex Numbers and AC Circuits
We pretend that there is an (imaginary) number,
i
, which, multiplied by itself, equals &1:
1
−
≡
i
(1)
A socalled
complex
number,
z
=
x
+
iy
, has both, a real part (Re(
z
) =
x
) and an imaginary
part (Im(
z
) =
y
). The complex
conjugate
z
* of
z
one obtains by flipping the sign of all
terms with an
i
in them, i.e.,
z
* =
x
&
iy
.
Leonhard Euler (1707 & 1783) discovered the relation, which relates complex numbers to
the (periodic) trigonometric functions (and which is the reason why we are doing what
we are doing!):
α
sin
cos
⋅
+
=
i
e
i
(2)
Consider the voltage
U(t)
that appears somewhere in some AC circuit. This is a periodic
function, so it makes sense to try to write this as
t
i
e
U
t
U
ω
⋅
=
0
)
(
(3)
This is a complex number, but instead of specifying its real and imaginary parts, we
specify its
amplitude
U
0
and its
phase
ω
t
. Here,
ω
is the angular frequency, which is
related to the frequency
f
by
ω
= 2
π
f
.
When we have any complex number
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 Spring '11
 Urheim

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