Introduction to Electrical Engineering  Basic vocational knowledge (Institut für Berufliche Entwicklung, 213 p.) 
6. Electrical Field 
6.2. Capacity 

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
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
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 = e_{0} · e_{r}
where:
e_{0} 
absolute dielectric constant 
e_{r} 
relative dielectric constant 
The absolute dielectric constant applies to vacuum and is
e_{0} = 8.86 · 10^{12} (A·· s)/(V·· m)
Table 6.2. e_{r} of Some Insulating Materials
Insulating material 
e_{r} 
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} = Q_{1} = Q_{2}
U_{AB} = U_{1} + U_{2}
Fig. 6.7. Capacitors connected in
series
When dividing the voltage equation by the charge, we have
U_{AB}/Q = U_{1}/Q + U_{2}/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/C_{1} +1/C_{2}
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} = Q_{1} + Q_{2}
After division, we obtain
Q_{AB}/U = Q_{1}/U +Q_{2}/U
and with equation 6.2. we have
C_{equ} = C_{1} + C_{2}
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:
C_{1} = 470 nF
To be found:
C in series connection and in parallel connection
Solution:
Series connection of C_{1} and C_{2}
1/C_{equ} = 1/C_{1} +1/C_{2} = (C_{2} + C_{1})/(C_{1} · C_{2})
C_{equ} = (C_{1} · C_{2})/(C_{1} + C_{2})
C_{equ} = (470 nF · 680 nF)/(470 nF + 680 nF)
C_{equ} = 277.9 nF
Parallel connection of C1 and C2
C_{equ} = C_{1} + C_{2}
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.
An uncharged capacitor is connected to a direct voltage source according to Fig. 6.9. (switch position 1).
Fig. 6.9. Circuit for charging and
discharging a capacitor
The terminal voltage of the uncharged capacitor is 0. In order that, between the plates of the capacitor, the voltage can be applied to which it is connected, the capacitor must be charged. This means that a charging current must flow at the instant of switching on. The intensity of the current at the instant of switching on is determined by the large difference between charging voltage and terminal voltage of the capacitor and the resistance. With increasing charge of the capacitor, the voltage of it increases and will reach the value of the charging voltage when the process of charging is finished. With increasing charge, the voltage difference between charging voltage and terminal voltage of the capacitor also drops and, consequently, the charging current also drops. At the end of charging, the voltage difference and the charging current are 0. A direct current is no longer allowed to flow now. This is also due to the design of the capacitor because a dielectric (insulating material) is between its two connections.
When the capacitor is now discharged via a resistor according to Fig. 6.9. (switch position 2), at the first instant of discharge, a discharge current must flow which is only limited by the resistance. Since, due to the discharge, the terminal voltage of the capacitor drops, the discharge current must also drop in the course of time. After complete discharge, the terminal voltage and the discharge current have dropped to 0. The course taken by current and voltage during charging and discharging is shown in Fig. 6.10. Charging commences at time t_{1} and discharging at time t_{2}. It is evident that charging and discharging currents suddenly reach their maximum value at the beginning of the charging or discharging process and then they reach the value of 0 after some time. Both in charging and in discharging, the voltage changes its value only slowly. There are no sudden voltage changes in capacitors.
Fig. 6.10. Behaviour or current and
voltage during the charging and discharging of a capacitor t_{ein} =
t_{on} · t_{aus} =
t_{off}
Since the peak value of the discharge current is limited only by the discharge resistance, a capacitor should not be discharged via s short circuit. If this would occur, however, at the first instant of discharge, an extremely high current would flow for a short time which might cause the destruction of the capacitor or a fusing of the shorting bridge. As a capacitor retains its charge for some time after charging without external discharge, particular caution is necessary when working at installations containing capacitors. After disconnection from the mains, the fact that the capacitors are completely discharged must be checked or discharge via a resistor must be effected. In many installations, a discharge resistance is incorporated in order to avoid dangers to man.
During discharging, a capacitor acts as an electrical energy source (during a certain time, its terminal voltage drives a current). The energy stored in a capacitor is written as
W = C/2 · U^{2}
where:
W 
energy 
C 
capacity 
U 
voltage 
The energy that can be stored in a capacitor is relatively small. It is of advantage however, that it is available as a shorttime energy release. Advantage of this effect is taken in a photoflash device and in some spotwelding equipment.
Example 6.2.
Which energy is stored in a capacitor of 47 µF charged up to 100 V?
Given:
U = 100 V
C = 47 µF
To be found:
W
Solution:
W = C/2 · U^{2}
W = 23.5 · 10^{6} (A·· s)/V · 10^{4} V^{2}
W = 0.235 Ws
W = 235 mWs
The energy stored in the capacitor is 235 mWs.
For the different fields of application, a great variety of designs of capacitors is available. In heavy current engineering, primarily paper capacitors in a metal cup are used (Fig. 6.11.). Two metal foils and two paper strips are placed one upon the other in the way shown in the illustration and then properly rolled up. The two metal foils are attached to connections and the roll is mounted in a metal cup. Such a paper capacitor is also know as rolltype capacitor. The MPcapacitor (metalpaper capacitor) is designed in a similar manner; in this case, the foil is replace by a coat of metal which is produced by vapour deposition. These capacitors are smaller than paper capacitors of the same capacity.
Fig. 6.11. Encased capacitor
Fig. 6.11.a Design of the roll
1  Connections of the foils
2  Metal foil
3  Paper strips
Fig. 6.11.b External view of the
capacitor
1  Metal enclosure
2  Connections
Another advantage of MPcapacitors is the fact that, after a breakdown or puncture of the dielectric, the extremely thin metal coat in the close vicinity of the puncture evaporates and, thus, removing the shortcircuit  that is why MPcapacitors are called “selfhealing” capacitors.
For the practical use, the value of the capacity printed on the device and the rated voltage up to which the capacitor may be used have to be observed.
An arrangement consisting of two plates with a dielectric between them is called capacitor. The capacity of a capacitor is a measure of the charge which the capacitor is capable of storing at a certain voltage, and it is also dependent on the design. The total capacity in series connection and in parallel connection of capacitors is expressed by the equations 6.5. and 6.6.When a capacitor is connected to a direct voltage, a current will only flow during charging and discharging. There are not sudden voltage changes in a capacitor. A charged capacitor can retain is charge for a longer period of time (danger!) and it should never be discharged via a short circuit. The capacitor may be used as an energy store.
In heavy current engineering, the paper capacitor arranged in a metal cup is sued. For use, pay particular attention to the value of the capacity and the rated voltage.
Questions and problems:
1. Describe the basic design of a capacitor and, in particular, the design of a paper capacitor!2. Which property of the capacitor is described by the capacity?
3. Explain the course taken by current and voltage during charging and discharging of the capacitor!
4. Why should capacitors not be discharged via a short circuit?
5. What should be strictly observed when working at installations incorporating capacitors?