# Thermal and Acoustic Design of a Shelter for High-Voltage Electrical Equipment
## 2 Theoretical Background
### Acoustics  
To understand how the noise emitted by a switchgear influences the surrounding environment, it is important to characterise its sound power. The main issue is that the noise generated by the switchgear is impulsive by nature. A standard procedure to find the sound  
Figure 1: Picture of a high-voltage switchgear [16]. power level from sound pressure level measurements is described in the ISO 3744:2020 standard [18]. For impulsive noise sources, the standard gives a specific approach, requiring capturing the acoustic energy contained in each operation of the device under test. In any case, a reference measurement surface must be set around the noise source. For transient noise signals, it is necessary, along with measuring the sound pressure levels, to also measure the time of the event during which the measurements are made. This is important as it helps to find the single-event time-integrated sound pressure level (\(L_{E}\)). From the measured sound pressure levels in one-third octave bands, the \(L_{E}\) level of each band for that microphone is calculated, according to ISO 3744, as  
\[L_{E}=10\text{log}_{10}\left[\frac{1}{t_{0}}\int_{t_{1}}^{t_{2}}\frac{p_{A}^{ 2}(t)}{p_{0}^{2}}dt\right] \tag{4}\]  
where \(t_{0}\) (=1 s) is the reference time, \(t_{1}\) and \(t_{2}\) are the beginning and end time instances of the event, \(p_{A}(t)\) is the instantaneous sound pressure and \(p_{0}\) (=20 uPa) is the reference sound pressure. However, usually, measurement instruments return equivalent sound pressure levels. Thus, to find \(L_{E}\) starting from the equivalent sound pressure level \(L_{\text{eq}}\) and the duration of the event \(\Delta t\), the following equation can be used:  
\[L_{E}=L_{\text{eq}}+10\text{log}_{10}(\Delta t) \tag{5}\]  
The next step is to take an average of all the \(L_{E}\) levels thus obtained over the microphones located on the measurement surface for each one-third octave band of interest. This is performed by a logarithmic average given by:  
\[\overline{L_{E}}=10\text{log}_{10}\left(\frac{\Sigma_{i=1}^{N}10^{0.1L_{E}}}{ N}\right) \tag{6}\]  
where \(\overline{L_{E}}\) is the position-averaged single-event time-integrated sound pressure level (\(L_{E}\)) of all the microphones for each one-third octave band, \(L_{E,i}\) is the \(L_{E}\) level measured for each microphone and \(N\) is the number of microphones. It is then possible to find a frequency-averaged value for the entire measurement surface. This is performed by summing the \(L_{E}\) levels of each one-third octave band. The following expressions are used to obtain this. \[\overline{L_{E,\text{overall}}}=10\text{log}_{10}\left(\Sigma_{fc,i}10^{0.1 \overline{L_{E}(f_{c,i})}}\right) \tag{7}\]  
where \(\overline{L_{E,\text{overall}}}\) is the frequency and position-averaged \(L_{E}\) and \(\overline{L_{E}(f_{c,i})}\) is the position-averaged \(L_{E}\) for each one-third octave band with the centre frequency \(f_{c,i}\). The next step is to find the sound energy level (\(L_{j}\)) that can be analogous to sound power levels (\(L_{\varpi}\)) compressed into one second. To find \(L_{j}\) levels from \(L_{E}\), the following expression is used. \[L_{J}=\overline{L_{E,\text{overall}}}+10\text{log}_{10}\left(\frac{S}{S_{0}} \right)+K_{1}+K_{2} \tag{8}\]  
where \(S\) is the surface area of the measurement surface, \(S_{0}\) (=1 m\({}^{2}\)) is the reference measurement surface, \(K_{1}\) is the background noise correction and \(K_{2}\) is the reverberation time correction. When the difference of the \(L_{E}\) between the noise emitted by the source and the background noise is higher than 15 dB and the measurements are made outdoors, as in the case at hand, \(K_{1}\) and \(K_{2}\) can be neglected. For assessing the annoyance of impulsive noise, a time interval over which the events occur must be considered. BS 4142:2014 [19] is a suitable standard applicable to check the level of annoyance in residential areas.