What is the formula for the impedance of a capacitor in the s-domain?

The impedance of a capacitor in the s-domain, which is used in Laplace transforms and control systems, is represented by the formula 1/sC. In this context, "s" is a complex frequency variable and "C" is the capacitance of the capacitor.

When analyzing circuits in the frequency domain, capacitors are characterized by their ability to store energy in an electric field. The impedance formula derives from the relationship between current and voltage in a capacitor. The current through the capacitor is proportional to the derivative of the voltage with respect to time (dV/dt), which in the s-domain can be represented as sV. By rearranging the relationship and expressing impedance (Z) as the ratio of the voltage across the capacitor (V) to the current through it (I), we get Z = V/I.

For a capacitor, the relationship is defined as I = C(dV/dt). In Laplace terms, this becomes I = sCV. Therefore, rearranging the equation gives us V/I = 1/(sC), confirming that the impedance of the capacitor in the s-domain is indeed 1/sC. This understanding is vital for analyzing circuits that involve capacitive elements, particularly when using Laplace transforms to