# Capacitors and Inductors in DC Circuits !

Capacitors are components, in which static electrical charges are stored. In its most basic structure a capacitor consists of two metal plates, which constitute the electrodes of the capacitor. Due to the charge separation an electrical potential difference (voltage) *U* is formed between the electrodes. The subsequent graphic image shows an example of a plate capacitor with plate surface area *A* and plate separation *d*, which carries the charge *Q*. Due to the charge separation an electric field forms between the plates (not shown here).

Normally there is insulating material, a so-called *dielectric*, positioned between the plates (not shown above). There is a linear relationship between the charge and voltage as given by the following expression:

The variable *C* is termed the *capacitance* of the capacitor, which is measured in units called *Farads* (unit symbol: F). The greater the capacitance of a capacitor, the more charge is required to generate a certain voltage between its electrodes. You can take a swimming pool as an analogy for this, whereby the capacitance corresponds to the floor space of the pool, the charge is the amount of water in the pool and the voltage is the level of water. The greater the pool's floor space (capacitance), the more water (charge) you need to put in to achieve a certain water level (voltage).

The capacitance of a capacitor can be considered to be constant. It depends solely on the geometric design and the dielectric material used. For a plate capacitor the following expression is true:

Where e_{0} is the electrical field constant (permittivity of free space) and has a value of 8.8542·10^{-12} AS/Vm, e_{r} is the dielectric constant or relative permittivity (which has no unit), *A* is the surface area of the plate and *d* is the distance between the plates. If a capacitor is connected to a via a charging resistor *R* to a DC voltage *U*_{0}, it charges up to the said voltage, whereby the capacitor voltage increases in accordance with an *e*xponential function from 0 V to its end value *U*_{0} (100%) (charging curve of a capacitor, see graph below, left). If you then disconnect the capacitor from the voltage source and short it out, discharging occurs which is the inverse to the charging process (see graph below, right).

**Inductance of an inductor:**

In addition to an electrical field, such as that which appears between the plates of a charged capacitor, there exists a second type of field in electrical engineering, namely a *magnetic* field. Whereas an electrical field arises in the proximity of static charges, an magnetic field is associated with moving charge carriers, i.e. an electrical current.

An inductor (or coil) is equivalent to multiple conductor loops in series, which are then permeated by magnetic field lines when a current flows through it. The strength of the magnetic field is characterised by its magnetic flux. If the magnetic flux of the inductor is varied (e.g. by changing the current intensity), then a so-called *self-induction voltage* is generated, the magnitude of which depends on the current's rate of change and on the coil's size and design. The inductance *L* of the coil is then a measure for the capacity of the inductor to generate self-induction voltage. For an oblong coil the following relationship is true:

Here *µ*_{0} is the magnetic field constant, *µ*_{r} the relative permeability of the coil core, *N* the number of windings, *l* the length of the coil and *A* its cross-section (see the following graphic).

The unit of inductance is the *Henry* (unit symbol H, 1 H = 1 Vs/A). A coil has an inductance of 1 H if the self-induction voltage of 1 V is induced for a constant variation in the coil current of 1 A per second.

**Switching a coil on and off:**

If a coil is located in a DC circuit, then the current flowing through the coil is constant - if the initial switch-on process is neglected - so that no self-induction voltage is induced. In this case the coil acts like an ohmic resistor with a (normally very low) resistance value that is the product of specific resistance of the coil material as well as length of wire in the coil and its cross-section.

When the coil is switched on, its magnetic field first has to start building up; and a self-induction voltage is generated by the flux variation. This self-induced voltage counteracts the applied voltage. As a result the current in the circuit does not rise abruptly (as would be the case for a resistive load), but rather the current increases only gradually to a certain end value. When the coil is switched off, the reverse happens: Here a self-induction voltage is produced by the collapse of the magnetic field. This self-induced voltage has the same direction as the voltage previously applied and can assume extreme values in coils with strong magnetic fields. The self-induction voltage initially maintains the current flow through the coil so that the current does not collapse abruptly, but instead drops to zero in a gradual fashion.

The subsequent graphic shows what happens during switch off. In the switched-on state (diagram on the left), the current *I* flows through the coil *L*. If the circuit is now broken (diagram on the right), the following process takes place. The coil initially maintains the current due to the energy built up in the magnetic field. Since this can no longer flow via the voltage source, it flows as shown across the resistor *R*_{L}connected in parallel to the coil. The energy of the magnetic field is then converted into thermal energy, so that the current dissipates rapidly. This dissipation process occurs exponentially (exponential curve) as in the case of the capacitor, whereby the time constant in this case is given by the quotient comprising the inductance and ohmic resistance.

If there is no resistor *R*_{L} in the circuit, then a very extreme voltage spike occurs which might easily cause damage to any sensitive components (e.g. ICs) found in a circuit. For that reason connected inductors frequently have so-called free-wheeling diodes connected to them in parallel, which can short-circuit these voltage spikes across the coil and thus ensure that the energy generated in the coil can be converted into thermal energy in the coil itself.