**Measurement of Power Factor II as cos phi**

Power factor is a purely relational number which is calculated as the quotient of active power / apparent power.

The “cos phi” measured quantity which has traditionally been used instead of power factor (active power / apparent power) is, for the most part, a result of the measuring technology utilized to date: Separate measurement of active power and apparent power with subsequent division (active power / apparent power) as required for determining power factor has not been implemented in the past with traditional cos phi transducers due to the time-consuming, cost intensive measuring technique.

Instead, the technically much easier to implement measurement of phase displacement of current and voltage (phi angle, interval between current and voltage zero-crossings) has been substituted. As a rule, the utilized measuring transducers generate an output signal which is linearly proportional to the phi angle (not to cos phi), for example -20 mA ...0...20 mA.

The desired cosine function was implemented at the scales of the downstream instruments by means of accordingly non-linear scale divisions (scale graduation proportional to the cosine curve, figure 1).

1) Non-linear scale 2) Linear scale

The most important **advantage** of this method has been its simple, cost-effective technical implementation.

The **disadvantages** result from the following two factors:

– Firstly, downstream connection of indicators or analysis modules is problematic if these devices only allow for a linear relationship between the input and the display (for example digital indicators, in which case the desired characteristic cosine curve cannot be calibrated into most device types, thus resulting in erroneous interpretation).

– Secondly, and most importantly, measurement results are only correct for undistorted curves. Incorrect measurement results are obtained in the case of distorted signals (distortion results in additional zero-crossings, which means that the interval between current and voltage zero-crossings is no longer determined by phase displacement).

However, if the basic requirements for this measurement are clearly recognized and adhered to (amongst other factors strictly sinusoidal measured quantities), it can still be used today. But these ideal conditions no longer prevail within today’s electrical systems, and the above described traditional cos phi measurement is thus in need of replacement.

The microprocessor technology utilized in multi-transducers (M1004, M563, DME4...) allows for a transition from differential angle measurement to true power factor measurement. In order to plainly signify the abandonment of the traditional “cos phi” measurement, the terms power factor I and power factor II were introduced in order to establish a differentiation between the two measuring methods.

As opposed to differential angle measurement, the two measured quantities assure a linear relationship between the measured quantity and the measuring transducer’s analog output signal (figure 2). In addition to this, harmonics are taken into consideration by the measuring method (up to the 16th harmonic).

The **power factor I** measuring method results in a physically, mathematically accurate cos phi value calculated as the quotient of active power and apparent power. When this measuring method is used, the preceding plus or minus sign is based upon active power (plus for energy import, minus for energy export; apparent power itself has no preceding plus or minus sign).

Thus power factor I also indicates **export or import**.

PF = P_{w}/S_{s}

**However, in actual practice the most common requirement is to identify the type of load (inductive or capacitive). Power factor II takes this requirement into consideration.**

As opposed to **power factor I**, the preceding plus or minus sign for power factor II does not indicate the direction of energy flow, but rather the **type of load**. In order to assure that indication is only dependent upon load type (and not the direction of energy flow), only the active power value is used in the calculation. The preceding sign itself is determined by means of the measurement of fundamental reactive phase power (by definition, a plus sign indicates an inductive load with import, and a minus sign indicates a capacitive load).

Power factor II is thus calculated as follows.

LF = sgn Qn * |P_{w}| / P_{s} |
(inductive: Q+ for import, Q– for export) |

It must be noted that in generating the active power value, power factor II can only be used as a measured quantity for a single direction of energy flow.

In the event that a **four quadrant power factor measurement** is required, the power factor I method should be used, and indication of load type should be ascertained by means of reactive power limit monitoring (e.g. set the limit value to 0 mA).

alibration based upon the above formula for power factor II would result in a sudden change at some point within the output signal (figure 3). In order to account for this, device calibration for power factor II is calculated instead as follows:

Power factor II = sgn Qn * (1 - |Pactive| / Papparent)

A desired measuring range of, for example, cap. 0.5...1...ind. 0.5 (i.e. -0.5...1...+0.5) corresponding to, for example, -20...0...+20 mA, thus results in internal dimensioning of -0.5...0...+0.5. In this way the characteristic curve crosses zero, and can be represented without a jump (figure 4).

Figure 3 Figure 4