# Torque Characteristic Analysis of a Transverse Flux Motor Using a Combined-Type Stator Core
## 3 Theoretical Analysis of Torque Characteristic of TFM
### Electromagnetic Torque  
The torque of CS-TFM can be seen as the interaction between the rotor core and the air-gap magnetic field, which is the sum of the permanent magnet field and the armature current magnetic field. It can also be explained in the magnetic field modulation method as shown in Figure 3: the distribution of magnetic field excited by armature current magnetomotive force (MMF) is modulated by the rotor teeth, and the MMF of permanent magnet can be replaced by the magnetizing imaginary current \(I_{\rm m}\) at the boundaries of the permanent magnet. Then, the torque of TFM is generated by the interaction of a modulated armature magnetic field and imaginary current. Thus, the electromagnetic torque can be calculated conveniently by the classical electromagnetic equation of \"BIL\". From Figure 3, the air-gap flux density of armature MMF is a periodic function about position, the period is two times that of permanent magnet pole-pitch \(\tau_{\rm m}\) When the center-line of rotor teeth is aligned to the center-line of permanent magnet (as the position shown in Figure 2a), the position is signed as \"origin\", and the expression of air-gap flux density can be written as Fourier series:  
\[B=B_{0}+B_{n}\text{sin}\left(\frac{n\tau\tau x}{\tau_{\rm m}}\right), \tag{1}\]  
Figure 2: The permanent magnet flux path and induction principle of no-load BMF for CS-TFM: (**a**) permanent magnet flux path at position a—positive maximum flux linkage; (**b**) permanent magnet flux path at position c—negative maximum flux linkage; and (**c**) principle of no-load BMF generation. \[B_{0}=\frac{1}{\tau_{\rm m}}\left(\int_{0}^{0.5I_{\rm tr}}B_{\rm teeth}dx+\int_{0.5I_{\rm tr}}^{\tau_{\rm m}}B_{\rm slot}dx\right), \tag{2}\]  
\[B_{\rm n}=\frac{1}{\tau_{\rm m}}\left(\int_{0}^{0.5I_{\rm tr}}B_{\rm teeth}\sin \left(\frac{n\pi x}{\tau_{\rm m}}\right)dx+\int_{0.5I_{\rm tr}}^{\tau_{\rm m}}B _{\rm slot}\sin\left(\frac{n\pi x}{\tau_{\rm m}}\right)dx\right), \tag{3}\]  
where \(B_{0}\) is the constant component, \(B_{\rm n}\) is the harmonic component, \(B_{\rm teeth}\) is the flux density in the teeth area, \(B_{\rm slot}\) is the flux density in the slot area, \(\tau_{\rm m}\) is the permanent magnet pole-pitch, \(I_{\rm tr}\) is the circumferential rotor teeth length, and \(x\) is the distance between position and origin. The air-gap flux density is the product of air-gap permeance and armature winding MMF [19]; however, the permeance calculation is generally difficult in transverse flux motors as the flux distributions are three-dimensional. To avoid that complexity, the space distribution of the permeance can be obtained by using quasi-flux tubes with boundaries determined by straight line and semicircular segments drawn to approximately maximize the permeance of each path [20], as shown in Figure 4.  
Then, the flux density in teeth area and slot area can be expressed as:  
\[B_{\rm teeth}=\frac{\
u_{0}Ni_{a}}{h_{\rm m}+8}, \tag{4}\]  
Figure 4: Equivalent quasi-flux tubes with boundaries determined by straight line and semicircular segments for permeance calculation. Figure 3: Magnetic field distribution by the armature magnetomotive force (MMF) and imaginary current distribution equivalent to the permanent magnet MMF. \[B_{\rm slot}=\frac{\mu_{0}Ni_{a}}{h_{\rm m}+\delta+\pi\alpha/2}, \tag{5}\]  
\[B=B_{0}+B_{n}\text{sin}\left(\frac{n\tau\tau x}{\tau_{\rm m}}\right), \tag{1}\]0  
where \(h_{\rm m}\) is the thickness of permanent magnet, \(\delta\) is the air-gap length, \(\mu_{0}\) is the permeability of vacuum, \(N\) is the armature winding turns, \(i_{\rm a}\) is the instantaneous armature current, \(I\) is the Root-Mean-Square (RMS) of armature current, \(\omega=2\pi f\) is the electrical angular velocity and \(\varphi\) is the initial phase angle of armature current.