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What you always wanted to know about
PHASORS and IMPEDANCES
but were afraid to ask
We have learned that the voltage
across an inductor is proportional to the time derivative of the current through the inducto
There is, however, one very important special case when differential equations can be replaced by complex analysis. This i
s
V(t) = Vp cos (w t + q v)
for voltages and
i(t) = ip cos (w t +q i)
By definition,
any complex number Z is represented as
Z = Re{Z} + j Im{Z}
(rectangular form),
or
Z = M (cos a + j sin a)
(polar form)
where j = sqrt(1), Re{Z} is the real part and Im{Z} is the imaginary part of the complex number Z, M is its magnitude, and a is
t
h
exp (j a) = cos a + j sin a
It follows from here that the polar form of any complex number can be represented as
Z = M exp (j a)
and our functions can be rewritten as real parts of complex numbers:
V(t) = Re{Vp exp[j (w t + q v)]}
and
i(t) = Re{ip exp[j (w t +q i)]}
Let us rewrite our functions once again:
V(t) = Re{Vp exp (jq v) exp (jwt)} = Re{V exp (jw t )}
and
i(t) = Re{ip exp(jqi) exp (jwt)} = Re{i exp (jw t )}
where
V =
Vp exp (jq v)
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 Spring '07
 SZILAGYI
 Electrical Engineering, Impedance, Volt

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