Hi.
Alba19 said:
Hi,
I have a curve, impedance magnitude vs frequency. i would like to convert this curve to electrical conductivity vs time. how can i do that?
Impedence is the complex number that captures the magnitude and phase of the
quotient of the complex numbers of a sinusoidal voltage over a sinusoidal current. The units of impedence are Ohms.
$$ v(t) = A \cos( \omega t) \,\, V \rightarrow A \angle{0} \,\, V \\
i(t) = B \cos(\omega t + \phi) \,\, A \rightarrow B \angle{\phi} \,\, A \\
Z_{L} = \dfrac{ A \angle{0} }{ B \angle{\phi} } \,\, \Omega = \dfrac{A}{B} \angle{ - \phi } \,\, \Omega
$$
The inverse of this complex number, is
admittance. Admittance has the same units as conductance: Siemens $$\,\, S $$
You can convert an impedence to an admittance by taking its reciprocal:
$$ \dfrac{1}{Z_{L} } = \dfrac{1 \angle{0} }{ \dfrac{A}{B} \angle{ - \phi } \,\, \Omega} = \dfrac{B}{A}\angle{ \phi } \,\, S = Y_{L} $$
The admittance is simply the
quotient of the complex current over voltage.
The units of
admittance are Siemens, same as conductance. You can convert the impedence response by taking the reciprocal of the impedence function. But you cannot convert an impedence to a conductance, just that its
reciprocal has the same units as conductance. You can them compare the
magnitudes.
For example, the impedence of a capacitor can be proven to be:
$$ -j \cdot \dfrac{1}{ \omega C} \,\, \Omega $$ Where $$ \dfrac{1}{ \omega C } \,\, \Omega $$ is the
capacitive reactance.
The reciprocal of
capacitive and inductive reactance is called capacitive susceptance $$\dfrac{1}{\dfrac{1}{ \omega C } \Omega } = \omega C \,\, S $$ and inductive susceptance respectively, they have units of Siemens.
As an example, take a series RLC circuit:
$$ R + j( \omega L - \dfrac{1}{\omega C} ) \,\, (\Omega) = Z(\omega) $$
The
magnitude response is:
$$ | Z(\omega)| \,\, \Omega = \sqrt{ {R^2 + ( \omega L - \dfrac{1}{\omega C} )^2} } \,\, (\Omega) $$
Taking the inverse of this:
$$ \boxed{|Y(\omega)| S = \dfrac{1}{| Z(\omega)| (\Omega) } = \dfrac{1}{\sqrt{ {R^2 + ( \omega L - \dfrac{1}{\omega C} )^2} } \,\, (\Omega) }} $$
You can convert an impedence function of angular velocity to an admittance function of angular velocity, as stated in my earlier post. To convert it to the time domain is not possible.
The complex numbers that represent current and voltage can indeed be converted to the time domain by multiplying them with the exponential that was factored out previously:
