# Explaining capacitance formula of electrolytic capacitor

As stated in this article at page 5, capacitance of a capacitor is generally expressed with the following formula:

My question is about surface area S, where it is described as "On aluminum electrolytic capacitor, āSā is effective surface area of anode foil enlarged to 60 to 150 times of the projected area through etching process.". What is the etching process here? And how to understand "anode foil enlarged to 60 to 150 times of the projected area"?

Second question about constant 8.855. Is it widely accepted constant value, or changing at some conditions?

• Etching introduces bumps and pits. More bumps and pits => more surface. – PlasmaHH Apr 28 '17 at 15:04

To understand 'Projected area' imagine the plate unrolled so it was flat, and parallel to the ground with the sun directly overhead. The projected area is the area of the shadow.

The surface area of an etched plate is increased by making it rougher. The capacitance of the finished capacitor increases proportionally. There are limits because the oxide must be relatively thick on a high voltage capacitor so you cannot have etched features that are too fine.

Here is a microphotograph from this TDK site:

8.8541878E-12 is the permittivity of free space in SI units. They are using cm so there is a power of 10 multiplier. The constant $\epsilon_0$ is exactly $\epsilon_0 = \frac{1}{c^24 \pi \cdot 10^{-7}}$ where c is the speed of light, defined as exactly 299,792,458 m/s.

The key to the high density of capacitance in aluminum electrolytic capacitors is the fact that the etching causes a large increase in surface area. On top of that, the thin oxide layer that is built up can be very thin, also aiding in getting these high values of capacitance:

Take a look at this picture, from wikipedia (By user Elcap, own work):

• So it means that the electrolytic capacitor made using the foil without etching would have 60 - 150 times less capacitance? – Ernestas Gruodis Apr 28 '17 at 15:53
• Correct. Of course, the 60-150 number depends on the exact process of etching and such. – Joren Vaes Apr 28 '17 at 16:55