# Alternating current AC

Three-phase machines work with a three-phase AC. Before three-phase machines can be explained in detail, a short introduction into AC calculations is given here.

An alternating voltage dependent on time is defined as:

with amplitude v, zero phase angle frequency m given by:

cpv of the voltage and the angular

The associated current i with the zero phase angle y and the amplitude t is:

The phase angle ç between current and voltage is:

If the phase angle is positive, the voltage is ahead of the current (leading). If the phase angle is negative, the voltage is behind (lagging). Figure 5.16 shows the curve of current and voltage as a function of time. With zero phase angles yv = 0 and y = -n/4, the phase angle of this example is y = +n/4, i.e. the voltage is ahead of the current.

Another common description is the vector diagram of the amplitudes. Here, the amplitude of the voltage is usually the vertical reference vector. The other amplitudes are added relatively with their phase angle. The amplitudes of the current and voltage of the example above are shown in the vector diagram of the amplitudes in Figure 5.16.

Figure 5.16 Current and Voltage as a Function of Time and Vector Diagram of the Amplitudes i and v (y = n/4)

For a description of alternating quantities, the definition of an average value is interesting. Since the positive and negative parts of a sinusoidal oscillation cancel each other out when calculating the arithmetic average value, the root mean square value (rms) is used in electrical engineering. The definition of the root mean square value of a function v(t) with the periodic time T = 1/f is as follows:

For sinusoidal currents and voltages the rms values are:

The vector diagram of the rms values is similar to that of the amplitudes shown in Figure 5.16, except that the lengths of the vectors are shorter. The vector diagram is also used in mathematics for complex numbers. Therefore, the vectors of current and voltage can be interpreted as complex quantities. However, the real axis (Re) of the vector diagram for the currents and voltages is drawn vertically, whereas in mathematics it is usually drawn horizontally. Complex quantities are underlined in the following expressions.

With the complex voltage V = V • ejq,v = V • ej0 = V of the example above, the complex value of the rms value of the current with the phase angle cpl and the imaginary unit j (j2 = -1) becomes:

Electronic components such as capacitors or inductors cause a phase displacement between current and voltage. For the description in the complex number system, an imaginary resistance, the so-called reactance X, is introduced.

Figure 5.17 shows the series connection of a resistance and inductance and the associated vectors of the currents and voltages. The voltage V1 is chosen as a reference value and is drawn onto the real axis (çv = 0). In this example the current I is turned by the zero phase angle ç = 3n/4. Hence, the phase angle between current and voltage is ç = -3n/4. This value in the example is chosen arbitrarily.

With the current I = I • ejçi, the voltage across the resistance R becomes:

Figure 5.17 Series Connection of Resistance and Inductance with Vector Diagram

The vector of the voltage VR has the same direction as the vector of the current I. The voltage across an inductance L with reactance X = a>L and j = ej2, becomes:

The voltage V2 can now be calculated as:

The translation of the voltage vectors VR and VL provides the vector V2 that closes the loop in the vector diagram.

The instantaneous power p(t) is calculated similarly to the DC power as:

The active power P is the difference between the positive and negative areas defined by the curve p(t) and the horizontal time axis. Using

the active power P of a harmonic voltage and current curve with the phase angle y becomes:

If the phase displacement between current and voltage is ±n/2 (±90°), the positive areas are equal to the negative areas. Thus, the active power becomes zero. However, this does not mean that there is really no power. The remaining power oscillates between consumer and generator. A lower level of oscillating power is also available at other phase angles. This power is called reactive power Q and is estimated by:

The reactive power Q is positive for positive phase angles. It is then called lagging reactive power. If the phase angle and the reactive power are negative, it is called leading reactive power. Sometimes a different definition for the relation between currents and voltages is used where the phase angle changes by 180° and therefore all the signs change as well. This can cause some confusion. The so-called apparent power is defined as:

The unit of the active power P is W (Watts). The unit of the reactive and apparent power is also the product of the units of voltage and current. To avoid confusion and to distinguish between the three power types, other units are defined for the reactive and apparent power. The reactive power Q is expressed by the unit var (Volt Ampere reactive) and the apparent power S with the unit VA (Volt Ampere). The power factor cos (5.63)

describes the ratio of the active power P to the apparent power S. Since the cosine of negative and positive phase angles provides the same value, the power factor is annotated with the supplemental labels 'inductive' or 'capacitive' in many cases. This indicates clearly if the phase angle is positive or negative.