Editing Laplace transform (section)

=== Complex impedance of a capacitor ===
In the theory of [[electrical circuit]]s, the current flow in a [[capacitor]] is proportional to the capacitance and rate of change in the electrical potential (with equations as for the [[International System of Units|SI]] unit system). Symbolically, this is expressed by the differential equation
<math display=block>i = C { dv \over dt} ,</math>
where {{math|''C''}} is the capacitance of the capacitor, {{math|1=''i'' = ''i''(''t'')}} is the [[electric current]] through the capacitor as a function of time, and {{math|1=''v'' = ''v''(''t'')}} is the [[electrostatic potential|voltage]] across the terminals of the capacitor, also as a function of time.

Taking the Laplace transform of this equation, we obtain
<math display=block>I(s) = C(s V(s) - V_0),</math>
where
<math display=block>\begin{align}
  I(s) &= \mathcal{L} \{ i(t) \},\\
  V(s) &= \mathcal{L} \{ v(t) \},
\end{align}</math>
and
<math display=block>V_0 = v(0). </math>

Solving for {{math|''V''(''s'')}} we have
<math display=block>V(s) = { I(s) \over sC } + { V_0 \over s }.</math>

The definition of the complex impedance {{math|''Z''}} (in [[ohm]]s) is the ratio of the complex voltage {{math|''V''}} divided by the complex current {{math|''I''}} while holding the initial state {{math|''V''<sub>0</sub>}} at zero:
<math display=block>Z(s) = \left. { V(s) \over I(s) } \right|_{V_0 = 0}.</math>

Using this definition and the previous equation, we find:
<math display=block>Z(s) = \frac{1}{sC}, </math>
which is the correct expression for the complex impedance of a capacitor. In addition, the Laplace transform has large applications in control theory.