# Impedance

Electrical impedance is a complex-valued measure of the opposition that a circuit presents to a time-varying electric current when a voltage is applied.  Combined with the Laplace transform it allows us to use algebra to calculate electrical networks.

## Capacitor impedance

Capacitors store energy in an electric field.  The amount of charge $$q$$ stored in a capacitor is linearly proportional to the voltage $$u$$ over the capacitor. [MIT]
$$q(t) = Cu(t)\label{eq:c_equiv}$$
where $$C$$ is a constant called capacitance.  The SI unit for capacitance is Farad with values typically range from from 2.2 pF to 470 μF. Electrical charge $$q(t)$$

The electrical charge $$q(t)$$ is a function of the current accumulated over time, assuming that there is no initial charge.
$$q(t)=\int\limits_{0}^{t} \! i(\tau) \, \mathrm{d}\tau\label{eq:c_integral}$$
Combining equation $$\eqref{eq:c_equiv}$$ and $$\eqref{eq:c_integral}$$ and solving for the current and taking the derivative on both sides, gives the current as a function of the voltage
\begin{align} \int\limits_{0}^{t} \! i(\tau) \, \mathrm{d}\tau &= C\,u(t) \\ \Rightarrow i(t) &= C\frac{\mathrm{d}u(t) }{\mathrm{d}t} \end{align}
The Laplace transform allows us to use algebra in complex frequency domain, instead of working with differential equations.  Using the Laplace transform of the first derivative
\left. \begin{align} I(s) &=\mathfrak{L}i(t) =\mathfrak{L}\left\{ C\frac{\mathrm{d}u(t)}{\mathrm{d}t} \right\} =C\mathfrak{L}\left\{ \frac{\mathrm{d}}{\mathrm{d}t}u(t) \right\}\nonumber\\ \mathfrak{L}\left\{\tfrac{\mathrm{d}}{\mathrm{d}t}f(t)\right\}&= -f(0^-)+s\mathfrak{L}\left\{f(t)\right\}\nonumber\\ f(0^-)&=0\nonumber \end{align} \right\} \Rightarrow I(s)=s\,C\,U(s) \label{eq:c_laplace}
where capital letters are used to indicate complex domain variables, such as $$I(s)=\mathfrak{L}i(t)$$ and $$U(s)=\mathfrak{L}u(t)$$.

Solving $$\eqref{eq:c_laplace}$$ for the complex impedance $$Z_{C}(s) \equiv\frac{U(s)}{I(s)}$$ we get
$$\shaded{Z_{C}(s) = \frac{1}{sC}}\label{eq:c_impedance}$$
The plot below shows the magnitude of the capacitor impedance $$Z_C$$ as a function of the frequency where $$s=\sigma+j\omega$$. $$Z_C(\omega)$$

## Inductor impedance

An inductor stores energy in a magnetic field.  The magnetic flux $$\phi$$ in the inductor is linearly proportional to the current $$i$$ through the inductor.  The role played by the inductor in this magnetic case is analogous to that of a capacitor in the electric case.
$$\phi(t) = L i(t)\label{eq:l_equiv}$$
where $$L$$ is a constant called inductance.  The SI unit for inductance is Henry with values typically range from from 0.1 µH to 1 mH.

According to Faraday’s law of induction [MIT], an inductor opposes changes in current by developing a voltage $$\varepsilon=-u$$ proportional to the negative of the rate of change of magnetic flux $$\phi$$.
$$\varepsilon(t) = -u(t) = -\frac{\mathrm{d}\phi(t) }{\mathrm{d}t}\label{eq:l_emf}$$ $$i(t)$$

Combining equation $$\eqref{eq:l_equiv}$$ and $$\eqref{eq:l_emf}$$ yields
\begin{align} u(t) = -\varepsilon(t) = +\frac{\mathrm{d}\phi(t) }{\mathrm{d}t} = L \frac{\mathrm{d}i(t) }{\mathrm{d}t} \end{align}
Once more, the Laplace transform allows us to use algebra in complex frequency domain, instead of working with differential equations.  Using the Laplace transform of the first derivative
\left. \begin{align} U(s) &=\mathfrak{L}u(t) =\mathfrak{L}\left\{ L\tfrac{\mathrm{d}i(t)}{\mathrm{d}t} \right\} =L\mathfrak{L}\left\{ \tfrac{\mathrm{d}}{\mathrm{d}t}i(t) \right\}\nonumber\\ \mathfrak{L}\left\{\tfrac{\mathrm{d}}{\mathrm{d}t}f(t)\right\}&= -f(0^-)+s\mathfrak{L}\left\{f(t)\right\}\nonumber\\ f(0^-)&=0\nonumber \end{align} \right\}\Rightarrow U(s)=s\,L\,I(s) \label{eq:l_laplace}
where capital letters are used to indicate complex domain variables, such as $$I(s)=\mathfrak{L}i(t)$$ and $$U(s)=\mathfrak{L}u(t)$$.

Solving $$\eqref{eq:l_laplace}$$ for the complex impedance $$Z_L(s) \equiv\frac{U(s)}{I(s)}$$ we get
$$\shaded{Z_L(s) = sL}\label{eq:c_inductance}$$
The plot below shows the magnitude of the inductor impedance $$Z_L$$ as a function of the frequency $$Z_L(\omega)$$

## Resistor impedance

For completeness we show the resistor impedance $$Z_r$$ as a function of the frequency
$$\shaded{Z_r(s) = R}\label{eq:r_inductance}$$
The plot below shows the magnitude of the resistor impedance $$Z_r$$ as a function of the frequency $$Z_R(\omega)$$

## Overview

Electric quantities
name symbol unit impedance
Voltage $$u(t)$$ Volt
Current $$i(t)$$ Ampere
Resistance $$R$$ $$u(t)=R\cdot i(t)$$ $$Z_r=R$$
Capacitance $$C$$ $$u(t)=\frac{1}{C}\int_0^t i(\tau)\,\mathrm{d}\tau$$ $$Z_c=\frac{1}{j\omega C}$$
Inductance $$L$$ $$u(t)=L\frac{\mathrm{d}i(t)}{\mathrm{d}t}$$ $$Z_l=j\omega L$$
$$\sum u(t)=0$$ $$\sum i(t)=0$$