6.2.1. Capacity and Capacitor

Figs. 6.2. to 6.4. show certain phenomena between two charged plates. Two plates provided with connections and separated by a dielectric are called capacitor (Fig. 6.6.). This component is capable of storing a certain charge when a certain voltage is present. This storage capability is called capacity of a capacitor.

Fig. 6.6. Design of a capacitor

1 - Conducting metal plates (electrodes)
2 - dielectric
3 - Connections

C = Q/U

[C] = (A·· s)/V = F

(6.2.)

where:

C	capacity
Q	charge
U	voltage

Since the unit 1 F (farad) is very great, the capacity of the capacitors manufactured only reaches fractions of 1 F. These fractions are designated by the prefixes specified by legal regulation:

1 pF = 1 picofarad = 10^-12 F
1 nF = 1 nanofarad = 10^-9 F
1 µF = 1 microfarad = 10^-6 F

The storage capacity of a capacitor is dependent on the area of the electrodes, the distance between them and the type of dielectric.

C = eA/d

(6.3.)

where:

e 1)	dielectric constant
A	area of the electrodes
d	distance between the electrodes

¹⁾ e Greek letter epsilon

The material constant is usually stated for the dielectric inquestion in the form of the product of the absolute dielectric constant times the relative dielectric constant.

e = e₀ · e_r

(6.4.)

where:

e0	absolute dielectric constant
er	relative dielectric constant

The absolute dielectric constant applies to vacuum and is

e₀ = 8.86 · 10^-12 (A·· s)/(V·· m)

Table 6.2. e_r of Some Insulating Materials

Insulating material	er
air	1
paper	2
transformer oil	2.5
rubber	2.7
porcelain	5
Epsilon (special ceramic compound for the production of capacitors)	up to 10,000

Like resistors, capacitors can be connected in series or in parallel. The total capacity obtained in this way is to be determined. Fig. 6.7. shows the series connection of two capacitors. The two capacitors have the same charge Q. The following holds:

Q_AB = Q₁ = Q₂
U_AB = U₁ + U₂

Fig. 6.7. Capacitors connected in series

When dividing the voltage equation by the charge, we have

U_AB/Q = U₁/Q + U₂/Q

After inversion, we obtain from equation 6.2.

1/C = U/Q

and, for the total capacity of a series connection of capacitors we have

1/C_equ = 1/C₁ +1/C₂

(6.5.)

This equation has the same structure as the equation for the determination of R_equ of a parallel connection of resistors.

The parallel connection of two capacitors is shown in Fig. 6.8. The same voltage is applied to the two capacitors, and each capacitor has tored a charge in accordance with its capacity.

Fig. 6.8. Capacitors connected in parallel

Thus, we have

U = U_C1 = U_C2
Q_AB = Q₁ + Q₂

After division, we obtain

Q_AB/U = Q₁/U +Q₂/U

and with equation 6.2. we have

C_equ = C₁ + C₂

(6.6.)

This equation has the same structure as the equation for the determination of R_equ of a series connection of resistors.

From the equations 6.5. and 6.6., the following general statement can be derived: In a series connection of capacitors, the total capacity is always smaller than the smallest individual capacity, and in a parallel connection of capacitors, the total capacity is always greater than the greatest individual capacity.

Example 6.1.

Two capacitors with a capacity of 470 nF and of 680 nF have to be connected in series and then in parallel. Determine the total capacity of each of the two types of connections!

Given:

C₁ = 470 nF

To be found:

C in series connection and in parallel connection

Solution:

Series connection of C₁ and C₂

1/C_equ = 1/C₁ +1/C₂ = (C₂ + C₁)/(C₁ · C₂)
C_equ = (C₁ · C₂)/(C₁ + C₂)
C_equ = (470 nF · 680 nF)/(470 nF + 680 nF)
C_equ = 277.9 nF

Parallel connection of C1 and C2

C_equ = C₁ + C₂
C_equ = 470 nF + 680 nF
C_equ = 1150 nF
C_equ = 1/15 µF

In series connection, a total capacity of 277.9 nF is obtained while in parallel connection the total capacity is 1.15/µF.