# On the Variation of Cup Anemometer Performance Due to Changes in the Air Density
## 1 Introduction
This change in air density is an effect that alters the accuracy of an anemometer's calibration curve (which relates the frequency of its output signal, \(f\), or the frequency of rotor rotation, \(f\), to the incident air velocity over the instrument, \(V\)):  
\[V=Af+B. \tag{3}\]  
In the above equation, which is the transfer function of the anemometer, the coefficients \(A\) and \(B\) are called the calibration coefficients of the instrument. Based on the authors' search of the available literature, three scientific papers dealing with this problem can be highlighted. The first is the article by [PERSON] and [PERSON] in 1937 [27], the second is the technical report by Deutsche WindGuard GmbH (Varel, Germany) in 2018 [28], and the third is the publication of these latest results in 2023 [29]. This paper analyzes the effect of changes in air density on the performance of a cup anemometer (i.e., on the measured wind speed). This change in air density is produced by the location of the instrument, often in places with a high altitude in relation to the place where it has been calibrated and where the air density is significantly lower (e.g., air density values around \(\rho=0.35\) kg\(\cdot\)m\({}^{-3}\) have been observed in the Tibetan Plateau [30]). In addition, the possibility of using the cup anemometer in rare environments should not be overlooked. Examples include the cup anemometer designed to be carried on a probe designed at NASA's Jet Propulsion Laboratory and intended to land on Mars [31], and the cup anemometer carried on the Venera 9 mission (1975) launched in 1975 for the planet Venus [32]. Furthermore, it should also be mentioned that quite recently, a cup anemometer was included in a stratospheric balloon mission (TASEC-Lab) as a wind speed sensor [33]. The work conducted in the above-mentioned publications is reviewed in Section 2 of this paper, where the data extracted from two of them [28; 29] are analyzed. With this information, the impact of density variations on three different wind turbines' Annual Energy Production (AEP) due to wind speed measurement errors is analyzed in Section 3. Finally, the conclusions of this work are included in Section 4. Effect of Air Density Variations on Cup Anemometers Performance--Research Carried Out in 1937 and 2018  
The research by [PERSON] and [PERSON] [27] is particularly interesting and, unfortunately, appears to have been ignored by the scientific community since it does not appear in the Web of Science database and has only 10 citations in Google Scholar at the time of writing the present paper. These two authors studied the performance of a flow meter for fluids of different densities, air, and water whose morphology resembles that of a cup anemometer (see Figure 2). In the work of these authors, it is shown that there is a fundamental parameter in the dynamics of the instrument, which is the relationship between friction and pressure forces. However, the effect of lowering the fluid density is greater than the effect of increasing it. Therefore, at low air densities, the cup anemometer should change its performance in relation to the one expected due to a different ratio between the aerodynamic and the friction forces (that is, the general assumption of neglecting the friction term in Equation (1) is no longer valid). [PERSON] and [PERSON] proposed the root of the dynamic pressure as the fundamental variable to represent the performances of the studied instrument:  
\[V\sqrt{\rho}=V_{0}\sqrt{\rho_{0}}. \tag{4}\]  
Thus, Figure 2 represents graphically the number of revolutions per unit of distance traveled by the wind. The points represented in the graph of this figure represent the one-to-one relationship between the rotation frequency, \(f\), and the wind speed, \(V\), represented by the calibration curve of the instrument, i.e., Equation (3).