# Brushless Field Excitation Scheme for Wound Field Synchronous Machines
## 2 Proposed Topology and Its Operating Principle
### Operating Principle  
Two customary inverters, i.e., the three-phase inverter and single-phase inverter, are used to supply the armature current (\(I_{abc}\)) to the machine. The three-phase inverter supplies 10 A (peak) current at the frequency of 60-Hz, i.e., the fundamental current component, to the armature winding; however, the single-phase inverter which is connected to the star-connected armature winding's common point supplies 10 A (peak) current at the frequency of 180-Hz, i.e., the third harmonic current component. Figure 3a illustrates the output current of the three-phase inverter (\(I_{INV\text{-}1}\)). The output current of the single-phase inverter (\(I_{INV\text{-}2}\)) is shown in Figure 3b. As the two inverters are operating at different output frequencies, the composite output of both inverters contains fundamental as well as third harmonic current components. The third harmonic current component generates current in the 8-pole rotor harmonic winding which will later be employed for rotor field excitation after the rectification. Figure 3c shows the composite shape of the three-phase armature current (\(I_{abc}\)). To illustrate the harmonic contents of the armature current, an FFT (Fast Fourier Transform) plot is demonstrated in Figure 4 which indicates the two components, i.e., fundamental and third harmonic components, of the armature current of phase A. This figure shows that the magnitude of the third harmonic component is around 2.5 A, whereas the magnitude of the fundamental of the armature currents is 10 A.  
Figure 3: \(Cont.\)  
Figure 2: Armature and rotor windings configuration.  
The composite output currents of three-phase and single-phase inverters as given by (1):  
\[\begin{split} i_{a}=I(\sin\omega t+n\sin 3\omega t)\\ i_{b}=I\left[\sin(\omega t-\frac{2\pi}{3})+n\sin 3(\omega t- \frac{2\pi}{3})\right]\\ i_{c}=I\left[\sin(\omega t+\frac{2\pi}{3})+n\sin 3(\omega t+ \frac{2\pi}{3})\right]\end{split} \tag{1}\]  
Here \(n\) is the weightage per third harmonic component. The above input armature current results in an MMF which can be calculated using the following equation:  
\[\begin{split} F_{a}=i_{a}N_{\phi}[\sin\theta_{e}]\\ F_{b}=i_{b}N_{\phi}\left[\sin(\theta_{e}-\frac{2\pi}{3})\right]\\ F_{c}=i_{c}N_{\phi}\left[\sin(\theta_{e}+\frac{2\pi}{3})\right] \end{split} \tag{2}\]  
Here \(N_{\phi}\) is the number of turns per phase, \(\theta_{e}=\omega t+\theta_{o}\), \(\theta_{e}\) is the electrical angle, and \(\theta_{o}\) is the initial position of the rotor.  
The net MMF generated due to the three-phase input armature currents is given by Equation (3):  
\[F_{abc}(\theta_{e},i_{abc})=\frac{3}{2}I_{1}N_{\psi 1}\cos(\omega t-\theta_{e}) +I_{3}N_{\psi 1}\cos 3(\omega t+\theta_{e}) \tag{3}\]  
Figure 4: FFT plot for phase A of the armature currents.  
Figure 3: (**a**) Three-phase inverter, (**b**) single-phase inverter, and (**c**) armature currents.  
The pulsating third harmonic MMF component shown in Equation (3) is generated in the harmonic winding of the rotor as:  
\[\lambda_{h}=\frac{N_{h}N_{\psi 1}(\frac{3}{2}I_{1}\cos(\omega t-\theta_{e})+nl_{3} \cos(3\omega t+3\theta_{e}))}{R_{g}} \tag{4}\]  
Here \(N_{h}\) is the number of turns of the rotor harmonic, and \(R_{g}\) is the reluctance of the airgap.  
Equation (5) is used to calculate the magnitude of the generated voltages in the rotor harmonic winding.  
\[emf_{h}=6\frac{d\lambda_{h}}{dt}=\frac{18N_{h}N_{\psi 1}I_{3}\omega\sin 3(2 \omega t+\theta_{0})}{R_{g}} \tag{5}\]  
The generated harmonic current is rectified and supplied to the rotor main field winding to achieve the proposed brushless operation for WFSM.  
The described operating principle of the proposed brushless technique for WFSM is illustrated in Figure 5.