## 2 Materials and Methods
### Short-Circuit Resistance and Load Losses of Each Harmonic  
The circulation of a harmonic current of order \(h_{z}=f_{hz}/f_{1}\) and RMS value \(I_{hz}\) through the phase (\(z=A,B,C\)) of the secondary transformer causes the following losses:  
\[P_{cc}(h_{z})=R_{cc}(h_{z})\cdot I_{hz}^{2} \tag{18}\]  
where \(R_{cc}(h_{z})\) is the short-circuit resistance referred to as the secondary characteristic of the harmonic current of order \(h_{z}\). The load losses in each phase (\(z=A,B,C\)) of the transformer must be equal to those caused by all the current harmonics of that phase, as follows:  
\[P_{cc,z}=\sum_{h_{z}=1}^{h_{z,max}}P_{cc}(h_{z})=\sum_{h_{z}=1}^{h_{z,max}}R_{ cc}(h_{z})\cdot I_{hz}^{2} \tag{19}\]  
Comparing the last expression with (8), the short-circuit resistance \(R_{cc}(h_{z})\) is expressed as follows:  
\[R_{cc}(h_{z})=R_{DCN}+R_{ECN}\cdot I_{z}^{2}+R_{OSN}\cdot I_{z}^{0.8} \tag{20}\]  
From Expression (20), it follows that:  
1. The short-circuit resistance of each phase harmonic current, referred to as the secondary current, \(R_{cc}(h_{z})\), only depends on the order \(h_{z}=f_{hz}/f_{1}\) (or the frequency \(f_{hz}\)) of the harmonic currents of each phase (\(z=A,B,C\)), a difference in the resistances \(R_{cc,z}\), which also depend on the RMS values of the currents, as deduced from Equations (12) and (13);
2. As it is usual that, in practice, the currents have harmonics of the same frequencies in the three phases (\(h_{A}=h_{B}=h_{C}=h\)), the resistances of each harmonic, \(R_{cc}(h_{z})\), are usually the same in the three phases of the transformer and, therefore, these resistances can be termed as \(R_{cc,h}\) and expressed, in general, as follows: \[R_{cc,h}=R_{cc}(h)=R_{DCN}+R_{ECN}\cdot I^{2}+R_{OSLN}\cdot I^{0.8}\] (21)  
These properties give the short-circuit resistances of each harmonic (\(R_{cc,h}\)) the status of characteristic parameters of three-phase transformers. For each transformer, the short-circuit resistances (\(R_{cc,h}\)) of its order \(h\) harmonic currents always have the same values in the three phases, regardless of the load powered by the transformer. Short-Circuit Resistance and Load Losses for the Fundamental Frequency and Combined of All Non-Fundamental Frequency Harmonics  
Among all the short-circuit resistances of each harmonic (\(R_{cc,h}\)), the one corresponding to the fundamental frequency (\(h=1\)) stands out, because these harmonic currents transfer the useful active and reactive powers in power systems. The expression of the fundamental short-circuit resistance, based on Equation (21), is as follows:  
\[R_{cc,1}=R_{DCN}+R_{ECN}+R_{OSLN}=R_{ccN} \tag{22}\]  
This matches the nominal short-circuit resistance (\(R_{ccN}\)), expressed by (15). The short-circuit resistances \(R_{cc,1}\) characterize the phase load losses (\(P_{cc,1z}\)) as follows:  
\[P_{cc,1z}=R_{cc,1}\cdot I_{1z}^{2} \tag{23}\]  
and characterize the total load losses (\(P_{cc,1}\)) of the transformer caused by the fundamental frequency currents (\(I_{1z}\)), as follows:  
\[P_{cc,1}=R_{cc,1}\cdot\sum_{z=A,B,C}I_{1z}^{2} \tag{24}\]  
The short-circuit resistances of non-fundamental frequency harmonics (\(R_{cc,h},\ h\
eq 1\)) increase considerably with the order (\(h\)), or with the frequency (\(f_{h}\)), of each harmonic, as deduced from Equation (21). However, the effective short-circuit resistance of each phase (\(R_{cc,z}\)) is usually only slightly higher than \(R_{cc,1}\) in practice. This shows that \(R_{cc,z}\) cannot be obtained as the sum of the short-circuit resistances of each harmonic (\(R_{cc,z}\
eq\sum R_{cc,h}\)). However, the resistances \(R_{cc,z}\) are related and can be obtained based on the short-circuit resistances of the harmonics.