Generalized s-plane electrical impedance

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 be the current flowing through the component, and let be the voltage across the electrical component terminals.

Next, let us introduce the following notation.

Let be the Laplace transform of the current . That is:

(1)  

where is the Laplace transform operator.

Let be the Laplace transform of the current . That is:

(2)  

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

(3)  

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)  

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

(5)  

Taking into account that the definition of the impedance assumes zero initial conditions, from the last equation we have

(6)  

From the last equation, we have

(7)  

This is the expression for the impedance of the inductor.

Impedance of a capacitor component

We start from the differential equation describing the capacitor component

(8)  

where is the capacitance. By applying the Laplace transform to this equation under the zero initial conditions, we obtain

(9)  

From the last equation, we have

(10)  

This is the expression for the impedance of the capacitor.

Impedance of a resistor component

We start from the Ohm’s law

(11)  

where is resistance. By applying the Laplace transform to this equation, we obtain

(12)  

This is the expression for the impedance of the resistor.