Generalized s-plane electrical impedance – Fusion of Engineering, Control, Coding, Machine Learning, and Science

In this tutorial, we explain the generalized s-plane electrical impedance. This impedance is also called as the complex impedance in the s Laplace domain. We also derive expressions for the electrical impedance of the resistor, capacitor, and inductor electrical components. The YouTube tutorial is given below.

Definition of Impedance

Consider the figure shown below. An electrical component is represented by the block EC. Let $i(t)$ be the current flowing through the component, and let $v(t)$ be the voltage across the electrical component terminals.

Next, let us introduce the following notation.

Let $I(s)$ be the Laplace transform of the current $i(t)$ . That is:

(1) $\begin{align*}\mathcal{L}\{ i(t) \} = I(s)\end{align*}$

where $\mathcal{L}\{ \}$ is the Laplace transform operator.

Let $V(s)$ be the Laplace transform of the current $v(t)$ . That is:

(2) $\begin{align*}\mathcal{L}\{ v(t) \} = V(s)\end{align*}$

Definition of the impedance $Z(s)$ : The (complex) impedance $Z(s)$ of a two-terminal electrical component is defined as the ratio of $V(s)$ and $I(s)$ under the assumption that the initial conditions for the Laplace transforms are zero. That is,

(3) $\begin{align*}Z(s)=\frac{V(s)}{I(s)}\end{align*}$

The complex impedance is shown in the figure below. Note that the current and voltage signals are represented in the Laplace s domain.

Impedance of an inductor component

To derive the impedance of an inductor component, we need to start from the basic equation governing the behavior of an inductor

(4) $\begin{align*}v(t)=L\frac{\text{d}i}{\text{d}t}\end{align*}$

where $L$ is inductance. By applying the Laplace transform to this equation, we obtain

(5) $\begin{align*}\mathcal{L} \{ v(t) \}=\mathcal{L} \{ L\frac{\text{d}i}{\text{d}t} \} = L \mathcal{L} \{ \frac{\text{d}i}{\text{d}t} \}\end{align*}$