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Glossary · Energy industry & energy transition

cos φ

cos phi power factor active power Reactive power apparent power alternating current phase angle

cos φ, pronounced "cos phi", is in a sinusoidal AC system the cosine of the phase angle φ between voltage and current. The value shows what share of the electrical apparent power is converted into active power. Under ideal sinusoidal conditions, cos φ = P / S applies, where P is the active power and S is the apparent power. The value has no unit and lies between 0 and 1 in absolute terms. In German regulations such as the Low-Voltage Connection Ordinance (Niederspannungsanschlussverordnung), cos φ is explicitly called the displacement factor (Verschiebungsfaktor).

At cos φ = 1, current and voltage are in step with each other over time. All of the apparent power is then used as active power, for example for a motor, for heating or for electronic devices. If cos φ is less than 1, part of the apparent power is contained in reactive power. Calculated over a whole period, reactive power performs no usable work. The energy merely oscillates back and forth between the grid and the electric or magnetic fields of the equipment. Even so, current flows for it. It loads lines, transformers, switchgear and inverters.

The three types of power have their own units: active power in watts (W, kW), reactive power in var (var, kvar) and apparent power in volt-amperes (VA, kVA). For ideal sinusoidal quantities they form the power triangle with S² = P² + Q², where Q is the reactive power. A useful rule of thumb follows from this: the ratio Q/P corresponds to tan φ. At cos φ = 0.95, the reactive power is about 33 percent of the active power, at cos φ = 0.90 about 48 percent and at cos φ = 0.80 exactly 75 percent.

Phase angle, inductive and capacitive

A phase angle arises when current and voltage do not reach their peak values at the same time (see phase position and phase angle). With predominantly inductive equipment such as electric motors, transformers or chokes, the current lags behind the voltage. With predominantly capacitive equipment such as capacitor banks or long cable runs, the current leads the voltage. In both cases the magnitude of cos φ falls below 1, although the technical cause is different.

This distinction matters for grid operation because inductive and capacitive reactive power influence the voltage in opposite ways. Depending on grid level, load and line type, reactive power can support the voltage or aggravate voltage problems. The statement "cos φ = 0.95" alone is therefore not sufficient. It must always be accompanied by information on whether operation is inductive or capacitive and which sign convention applies. The NAV therefore explicitly specifies a range between 0.9 capacitive and 0.9 inductive.

In practice there are several pairs of terms for this: underexcited and overexcited, inductive and capacitive, lagging and leading. They are not used the same way in every document. Generation plants are usually described from the generator's point of view, consumers usually from the consumer's point of view. An inverter operated underexcited absorbs inductive reactive power from the grid's perspective and thus behaves similarly to a motor. Grid connection rules, metering concepts and manufacturer specifications must therefore be read in their respective context.

Distinction from the general power factor

cos φ is often equated with the power factor. This is only correct if current and voltage are purely sinusoidal. Real grids contain many nonlinear loads and power electronic devices. Their current is often not sinusoidal, even if the fundamental of the current is almost in phase with the voltage.

The general power factor λ is the ratio of active power to apparent power, even with distorted current and voltage waveforms. It captures the phase shift of the fundamental and, in addition, the distortion caused by harmonics. With sinusoidal voltage, the following approximately applies: λ = (I₁ / I) · cos φ₁. Here I₁ is the RMS value of the fundamental of the current and I is the total RMS value. Strictly speaking, cos φ describes only the displacement factor of the fundamental. A device can therefore have a cos φ close to 1 and still a significantly poorer power factor if its current is heavily distorted.

This is particularly important for electronic power supplies, frequency converters, chargers and other power electronic installations. Modern devices often have active power factor correction, which improves phase position and current waveform at the same time. In technical documents, one should nevertheless check carefully whether "power factor", "displacement factor" or explicitly cos φ is meant.

Why cos φ matters in the power system

The lower cos φ is, the more current flows for the same active power. A worked example: A 400 V three-phase installation is to draw 100 kW of active power. At cos φ = 1, about 144 A flow per phase conductor. At cos φ = 0.8 it is about 180 A, the apparent power is 125 kVA and the reactive power 75 kvar. The resistive heat losses in lines grow with the square of the current. In the example they therefore rise by about 56 percent, although not a single additional kilowatt-hour of useful energy arrives at the consumer. Cables and transformers must be designed for this current and thermal load, not just for the active power.

For grid operators, cos φ therefore influences utilisation, voltage control and connection capacity. An installation with unfavourable reactive power behaviour can load the grid more heavily than its kilowatt rating suggests. Conversely, installations with controllable reactive power can contribute to voltage control. Today this concerns not only large power plants, but also wind farms, photovoltaic systems, battery storage, industrial plants and charging infrastructure.

Legal and regulatory framework in Germany

For consumers on the low-voltage grid, § 16 paragraph 2 of the Low-Voltage Connection Ordinance (NAV) stipulates: electricity must be used with a displacement factor between cos φ = 0.9 capacitive and 0.9 inductive. If this range is not met, the grid operator can require the installation of sufficient compensation equipment. The NAV applies to grid connections at low voltage. For connections at medium and high voltage, the respective grid connection contracts and technical connection rules apply.

For generation plants, the technical connection rules require that they be able to provide reactive power. VDE-AR-N 4105 applies in the low-voltage grid, VDE-AR-N 4110 in the medium-voltage grid. The required minimum capability depends on plant size; typical values are cos φ = 0.95 or 0.90. The threshold values have changed between editions of VDE-AR-N 4105. Older accounts give 13.8 kVA, newer secondary sources 4.6 kVA. Which setting actually applies is determined by the grid operator: a fixed cos φ, an active-power-dependent characteristic curve cos φ(P) or a voltage-dependent reactive power characteristic curve Q(U). With cos φ(P), the required reactive power contribution depends on the active power currently being fed in. For large plants, the reactive power ranges are usually specified as PQ diagrams.

Economic consequences: reactive energy and compensation

Reactive power is not only a matter of electrical engineering calculations. It helps determine how much grid capacity remains for active power. Many German distribution system operators charge a fee for reactive energy at offtake points with quarter-hour demand metering. Typically it becomes due when the inductive reactive energy drawn in a period amounts to more than 50 percent of the active energy. This corresponds to an average cos φ of about 0.9 inductive. Billing is then in ct/kvarh. Limit values, time windows and prices differ depending on the grid operator and voltage level. Some price sheets also charge for drawn capacitive reactive energy or require a narrower range outside low voltage.

For industrial and commercial customers with large motors, welding equipment, converters or compressors, cos φ can therefore cost money. Reactive power compensation with capacitor banks, chokes or power electronic compensators reduces the current, avoids reactive energy charges and creates reserves at the connection. It must, however, be designed appropriately. An overcompensated installation feeds capacitive reactive power into the grid. This can cause new voltage problems or resonances with harmonics. Capacitor banks in networks rich in harmonics are therefore usually built with detuning reactors.

For generation plants and storage, the relationship is different. An inverter can usually control reactive power very quickly. This is valuable for voltage control, but costs current reserve. A current-limited inverter at cos φ = 0.9 can, in calculation terms, deliver only 90 percent of its rated apparent power as active power, unless it was dimensioned larger. The ability to control reactive power is therefore linked to connection rules, remuneration, grid planning and equipment costs.

Typical misunderstandings

  • cos φ is not an efficiency. A motor with cos φ = 0.85 can have a high or a low efficiency. cos φ describes how the apparent power divides into active and reactive components. Efficiency describes what part of the active power drawn is delivered as useful output, for a motor as mechanical power.

  • Reactive power is not "wasted energy" in the sense of permanent consumption. It oscillates back and forth periodically. It still has an effect, because the necessary current loads equipment and causes real active losses. More precisely: reactive power supplies no usable energy, but it does stress the electrical infrastructure.

  • cos φ = 1 is not the best target everywhere. At an individual consumer connection, a high cos φ usually makes sense. In the distribution and transmission grid, by contrast, reactive power is often needed deliberately to keep voltages within permissible limits. The goal of grid operation is permissible voltages, sufficient reserves and manageable reactive power flows, not zero reactive power at every point.

  • cos φ does not directly change the measured active energy in kilowatt-hours. A household appliance with a poorer power factor can draw the same number of kilowatt-hours as another appliance, but causes higher currents. Households without demand metering are usually not charged separately for reactive energy. For commercial and industrial customers with demand metering, however, it can be billed.

Classification in the power system

cos φ links a local property of individual equipment with questions of overall grid operation. A single motor, inverter or compensator changes the phase position at its connection point. Many such installations together shape voltage profiles, losses and the utilisation of entire grid sections. With the electrification of heat, transport and industry, more and more installations connected via power electronics are being added, such as heat pumps, charging points, battery storage and electrolysers. Whether they are well compatible with the grid depends not only on installed capacity. Current waveform, phase position, control and connection conditions also matter.

cos φ therefore does not describe a general state of the power system, but a specific electrical relationship: the phase shift between voltage and current of the fundamental. Its strength is the clear link between active, reactive and apparent power. Its limit is reached when distorted current waveforms or complex control schemes occur. The general power factor then describes the situation better. Anyone who uses cos φ correctly distinguishes between usable power, electrical loading and the role of reactive power in the grid.

Sources

Related glossary entries

Edited by eCMS on 11.11.2026 18:07

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