## 4 Harmonic Analysis of the DFIG  
DFIG generates harmonics through the stator or rotor voltage sources: the harmonic frequencies for stator and rotor are \(f_{sh}~{}=~{}hf_{es}\) and \(f_{rh}~{}=~{}hf_{\sigma r}\), respectively. Two fundamental frequencies are available: one in the stator (\(f_{es}\)) and another in the rotor (\(f_{\sigma r}\)). To perform the frequency-domain analysis of the DFIG at harmonic frequencies by analyzing the generator with its independent voltage sources, it is necessary that only the effects of the stator are considered. This is done by feeding the DFIG by the stator with frequency \(f_{sh}\) and keeping the rotor short-circuited (Figure 5). Equation (28) describes the circuit as follows:  
\[ \begin{array}{l}\mathbf{U_{sh}}~{}=~{}-(r_{s}+j\hbar\omega_{es}L_{ls})\mathbf{I_{sh}}+ j\hbar\omega_{es}L_{M}\mathbf{I^{\prime}_{rh}}\\ 0~{}=~{}r^{\prime}r/s_{h}+j\hbar\omega_{es}(\mathbf{I^{\prime}_{lr}}+L_{M})\mathbf{I_{ rh}}-j\hbar\omega_{es}L_{M}\mathbf{I_{sh}}\end{array}  \tag{28}\]  
where the harmonic slip seen from stator is given by Equation (29)  
\[s_{h}~{}=~{}\frac{\pm\hbar\omega_{es}-\omega_{r}}{\pm\hbar\omega_{es}} \tag{29}\]  
In Equation (29), the (+) and (-) represent positive and negative harmonic sequences. By solving the Equation (28), the stator and rotor harmonic current phasors in the time-domain are obtained by Equation (30)  
\[ \begin{array}{l}\mathbf{i_{sh}}=|l_{sh}|cos(\hbar\omega_{es}t+\varphi_{sh})\\ \mathbf{i_{rh}}=|l^{\prime}r_{rh}|cos\Big{(}s_{h}\hbar\omega_{es}t+\varphi^{\prime }n_{\pm}\pm\theta_{eff}\Big{)}\end{array}  \tag{30}\]  
where \(\pm\)\(\theta_{eff}\) is (-) for the positive and (+) for negative sequence. On the contrary, feeding the DFIG by the rotor with frequency \(f_{rh}\) keeps the stator in short-circuit (Figure 5), and the equation (31) represents the circuit,  
\[ \begin{array}{l}0~{}=~{}r^{\prime}s/s_{rh}+j\hbar\omega_{\sigma r}(\mathbf{L^{ \prime}_{ls}}+L_{M})\mathbf{I^{\prime}_{sh}}-j\hbar\omega_{er}L_{M}\mathbf{I_{rh}}\\ \mathbf{V_{rh}}~{}=~{}r_{r}+j\hbar\omega_{er}(L_{lr}+L_{M})\mathbf{I_{rh}}-j\hbar\omega _{es}L_{M}\mathbf{I^{\prime}_{sh}}\end{array}  \tag{31}\]  
where the harmonic slip seen from rotor is given by Equation (32)  
\[ \begin{array}{l}s_{rh}~{}=~{}\frac{\pm\hbar\omega_{er}+\omega_{r}}{\pm\hbar \omega_{er}}\end{array}  \tag{32}\]  
Similarly, the (+) and (-) signs in Equation (32) represent the positive and negative harmonic sequence, respectively. By considering Equation (4), the stator and rotor current phasors in the time-domain are obtained by Equation (33),  
\[ \begin{array}{l}\mathbf{i_{rh}}~{}=~{}|l_{rh}|\cos(\hbar\omega_{er}t+\varphi _{rh})\\ \mathbf{i_{sh}}~{}=~{}|l^{\prime}s_{hl}|\cos\Big{(}s_{rh}\hbar\omega_{er}t+ \varphi^{\prime}_{sh}+\theta_{eff}\Big{)}\end{array}  \tag{33}\]  
Figure 5: Harmonic equivalent circuit of the induction generator in steady-state seen from stator. where \(\pm\)\(\theta_{eff}\) is (-) for the positive and (+) for the negative sequence. Finally, the effects of the harmonic voltage sources in the stator and rotor summed to provide the complete solution for the DFIG are given by Equations (34) and (35):  
\[\mathbf{i_{sh}} = \left|I_{sh}\right|\cos(h\omega_{es}t+\varphi_{sh})+\left|l^{ \prime}{}_{sh}\right|\cos(s_{rh}h\omega_{es}t+\varphi^{\prime}{}_{sh}\pm \theta_{eff}) \tag{34}\] \[\mathbf{i_{rh}} = \left|I_{rh}\right|\cos(h\omega_{er}t+\varphi_{rh})+\left|l^{ \prime}{}_{rh}\right|\cos(s_{h}h\omega_{es}t+\varphi^{\prime}{}_{rh}\pm\theta _{eff}) \tag{35}\]  
Regarding the zero sequence components, in a DFIG only the wye connections with insulated neutral and delta connect the stator phases of the induction generator. The wye connection with the neutral connected to the electrical grid is not used in these machines. This indicates that homopolar currents are absent in the stator of these machines. Therefore, Figure 2 is invalid since the induction generator operates as two decoupled windings for the zero sequence.