# CAPACITOR IN ALTERNATING CURRENT AC CIRCUITS !

Capacitance and capacitors Capacitors are components which store static electric charge. A capacitor essentially consists of two metal plates serving as electrodes. Charge separation leads to an electric potential difference (voltage) U between the electrodes. The diagram below shows an example of a plate capacitor with a plate area A and plate spacing d carrying a charge Q. The charge separation produces an electric field (not shown here) between the plates.

Between the plates, there is usually an insulator or dielectric (not shown here). The following linear relationship exists between the charge and voltage:

The value C is termed the capacitance of the capacitor which is expressed in units called Farad (F). The larger the capacitance, the greater the quantity of stored charge needed to produce a defined potential between the capacitor's electrodes. As an analogy, take a swimming pool where the floor area corresponds to capacitance, water volume to charge quantity and water level to potential. The larger the floor area (capacitance), the more water (charge) is needed to achieve a particular depth (voltage). A capacitor's value can be considered constant, dependent only the geometric design and the dielectric material employed. The following relationship applies to plate capacitors (and also approximately to closely spaced concentric cylinders such as in the case of paper capacitors):

e0 is the electric field constant, also called the permittivity of free space, which has a value of 8.8542·10-12 AS/Vm, er is the (unitless) relative permittivity, A is the plate area and d the plate spacing.

**Charging and discharging of capacitors:**

The processes of charging and discharging a capacitor are already familiar from the treatment of *DC circuits*: Connecting a capacitor via a charging resistor *R* to a direct voltage *U*_{0} causes the capacitor to be charged to this voltage. The potential rises exponentially from 0 V to the final value of *U*_{0}. The charging current drops, also according to an exponential function, from an initial maximum value to zero (the capacitor's charging curve is shown on the left in the following diagram). If the capacitor is then disconnected from the voltage source and short-circuited, a discharging process takes place that is the converse of charging (shown on the right in the following diagram).

Charging and discharging currents flow in opposite directions. The rate at which the capacitor is charged or discharged depends on its capacitance and the resistance *R *of the circuit branch in which it is located (here represented by the equivalent resistor). It is characterised by a *time constant **T* = *R*·*C*. After this amount of time, the capacitor attains exactly 63% of its final voltage during charging or loses 63% of its initial voltage during discharging. Once the capacitor is fully charged, current ceases to flow, i.e. the capacitor *blocks direct current*.

If, after charging, the capacitor is disconnected from the voltage source without being short-circuited, it retains its charge and potential for a theoretically indefinite period of time. Under real circumstances, however, the capacitor always undergoes a certain amount of intrinsic self-discharging.

**Series and parallel connection of capacitors:**

**Parallel connection of capacitors:**

The diagram below shows an example of capacitors connected in parallel. In this case, the same voltage *U* is present across all capacitors.

The capacitors carry the following charges:

The total charge *Q* that is generated by a direct current source is equal to the sum of the individual charges:

Summing the values in parentheses gives the total capacitance *C *for a parallel circuit:

In other words:

The total capacitance of capacitors connected in parallel is equal to the sum of the individual capacitances. |

Connecting capacitors in parallel is equivalent to increasing the plate area.

**Series connection of capacitors:**

Connecting several capacitors in series to a voltage source as shown in the diagram below causes a charge *Q* to be transported through the entire circuit.

The effect of this is that each of the capacitors must have this same charge Q induced across it. The voltage *U*_{i} across each of the components is then dependent on the corresponding capacitances:

The total potential is equal to the sum of the individual potentials:

Consequently, the total capacitance of a series-connection is given by the following:

In other words:

For capacitors connected in series, the inverse of the total capacitance is equal to the sum of the inverses of the individual capacitances. |

In this case, the total capacitance is less than the smallest individual capacitance. Connecting capacitors in series is equivalent to increasing the plate spacing.

What is the total capacitance C of the following circuit comprising four capacitors?

**Reactance of a capacitor:**

The instantaneous value *p*(*t*) of the power consumed by a capacitor is the product of the instantaneous current and voltage. Because these two variables are separated in phase by 90° in a capacitor, however, the power consumption characteristic has a frequency that is double that of the voltage and current themselves, as shown by the green curve in the diagram below. This characteristic comprises regions in which the voltage and current are in the same direction, causing the capacitor to act as a load, and regions of equally long duration where the voltage and current oppose each other so that the capacitor acts as a source of power (like a battery).

The diagram represents consumption of energy as a positive value and regions where energy is being supplied as a negative value. In other words, the electrical energy alternates between capacitor-like and battery-like modes. With a resistive load, the power consumed is termed *active power* (in which case electrical energy is simply converted into heat energy), whereas for a capacitor the consumption is described in terms of *reactive power*. Instead of an *ohmic resistance*, the capacitor is said to have a *reactance* *X*_{C} defined as the quotient of the rms voltage *U* and rms current *I*:

The unit of reactance is the same as for an ohmic resistance, the *ohm*.

As indicated earlier, a capacitor's ability to "conduct" alternating current increases as the frequency rises and the capacitance increases. The formula for capacitive reactance is:

In qualitative terms:

Capacitive reactance decreases as the frequency and capacitance increase. |

**Example:** at a mains frequency of *f* = 50 Hz, a 1 µF capacitor has a reactance of

**Sinusoidal alternating voltage for a capacitor:**

If a capacitor is subjected to a *sinusoidal* alternating voltage of the form:

the resulting capacitor current *i* is also sinusoidal and leads the voltage by the following angle:

The current is therefore determined by the equation:

The voltage and current characteristics are displayed in the diagram below.

The capacitor voltage attains its maximum value at the instant when the current is zero, and vice versa.

In a capacitor circuit, the current leads the applied voltage by j = 90°. |

**Phase shift in a RC element:**

If a sinusoidal voltage is applied to an ohmic resistance *R*, the resulting current is in phase with the voltage, i.e. the corresponding vectors *I* and *U* of the rms values point in the same direction (illustrations on the left below). A capacitor acts as a pure reactance *X*_{C}, where the voltage trails the current by a phase angle of j = 90° so that the two corresponding vectors form a right angle (illustrations in the centre below). If an ohmic resistance and capacitor are connected in series to form an *RC element*, the resulting overall resistance includes both active and reactive components. In this case, the voltage lags behind the current by a phase angle j that lies somewhere between 0° and 90° depending on the frequency, resistance and capacitance (illustrations on the right below). The voltage vector *U* here is formed through geometric addition of the partial voltages *U*_{R} (in phase with the current) and *U*_{C} (lagging behind the current by 90°).

As in the case of DC circuits, Ohm's law also applies here to the rms values of the current and voltage.

The impedance *Z* is the term for the *apparent resistance* of an AC circuit. It is composed of both the active resistance and the reactance. These values need to be added *geometrically* in the same way as the various voltages. This is demonstrated in the following illustration which shows a *triangle of resistances* for an RC element.

The impedance *Z* can be easily calculated from the triangle of resistances, which being right-angled, results in the following relationship:

Taking the square root of both sides of the equation and replacing *X*_{C} with the relationship ascertained earlier leads to the following:

The phase angle j can also be determined from the triangle of resistances as follows:

Given known values of w or *f*, *R* and j, this equation can be used to determine the capacitance by resolving in terms of *C*. The following relationship is then obtained:

Alternatively (as described in the next experiment), the phase shift between the capacitor voltage *U*_{C} and supply voltage *U* can be measured. The capacitance is then derived from the following formula:

**Example**: a resistor *R* = 220 W connected in series with a capacitor *C* = 10 µF are subjected to an alternating voltage *U* = 220 V with a frequency *f* = 50 Hz. Calculate the capacitor's reactance, impedance, current *I*, both partial voltages *U*_{R} and *U*_{C} as well as the phase angle j between *U* and *I*.

Result:

**Rectangular alternating voltage for a capacitor:**

To investigate the behaviour of a capacitor in an AC circuit, let us first work through a theoretical scenario. As shown in the left-hand part of the diagram below, a capacitor is connectable via a switch S and a resistor *R* to one of two direct voltage sources (e.g. batteries) of different polarities. Varying the switch position then supplies the RC network with a periodically changing rectangular alternating voltage. As a result, alternating current flows "through" the capacitor (right-hand side of diagram). The capacitor is first charged, during which process the charging current falls and the potential across the capacitor rises. On reversal of the voltage polarity, a charging current starts to flow in the opposite direction; the capacitor discharges, its potential drops to zero and then rises again in the opposite direction. The dashed curve shows how the capacitor potential (voltage) characteristic would have continued had the switch stayed in position. When it switches, however, the capacitor charges up again and so on. In other words, the capacitor appears to "conduct" alternating current by way of repeated re-charging. Its AC resistance is finite in contrast to its "infinite" DC resistance.

The following animation illustrates these relationships.

The diagram shows that current flows through the capacitor while the voltage is still building up across the plates and is at its highest before the voltage has risen very much. Similarly, discharge current flows strongly before the capacitor voltage has dropped perceptibly. One speaks here of a *phase shift* between the current and voltage. In capacitors, the current leads the voltage, in contrast to resistive loads, for instance, where the current and voltage are in phase.

The curve also reveals that the current is at its highest at the beginning, after which it drops sharply. The higher the switching frequency, the higher the average current, since the charge/discharge current characteristic is within a range where the current is high for a greater proportion of the time. As a result, the effective resistance of a capacitor can be expected to drop as the frequency of the alternating current increases. Also to be expected is a rise in the alternating current as the capacitance increases since the charging phase then takes longer and the current remains high for a greater proportion of the time. In other words, AC resistance can be expected to be smaller as the capacitance increases.