Power flow in the air gap of linear electrical machines by utilization of the Poynting vector: Part 1 ‐ Analytical expressions

Abstract

Analytical solutions and estimations for the power flow in the air gap of linear electrical machines of different geometries are derived from Poynting's theorem. The different geometries considered are flat one‐sided, multi‐sided, and tubular linear electrical machines. The radial power flow for all considered geometries is dependent on the area of the air gap, the electric field, the magnetic field, and the load angle. The tangential power flow for both flat one‐sided and tubular linear electrical machines is dependent of the area of the air gap, number of poles, the electric field, the magnetic field, and the load angle. The number of poles could be increased to decrease the tangential power flow in flat linear electrical machines. The expression for the tangential flow in tubular linear electrical machines is so complicated that it is difficult to draw conclusions from it.

Power flow in the air gap of linear electrical machines by utilisation of the Poynting vector: Part 2 ‐ simulations

“…Six periods can be seen in the figures, and the simulation is done over three electrical periods. Thereby the frequency is twice the electrical frequency, which is expected from the results in Part 1 [8], and also confirmed by the frequency analysis.…”

“…Two flat and two tubular linear generators were simulated and compared to analytical expressions from Part 1 [8]. The flat and tubular designs had different power flows, both normal and tangential.…”

“…To compare with the results in Part 1 [8], both the amplitudes and the frequencies of the power flows are also of interest and analysed both for the normal and for the tangential power flow.…”

“…In Part 1 [8], it is suggested from the equations that increasing the number of poles can decrease the power flow in the longitudinal direction with no or little effect on the power flow in the normal direction. To see how well this holds, all four generators are also simulated with eight poles, on top of the six poles used in the base case.…”

“…In Part 1 [8], the normal power flow in a flat LEM is found to be: $$\begin{equation} P_{\perp ,\text{flat}} = \frac{1}{2}bL\hat{E}_{y\text{R}}\hat{H}_{z\text{S}}\sin \delta . \end{equation}$$where b is the stack length of the LEM, L is the length of the LEM, $$\hat{E}_{y\text{R}}$$ is the top‐value of the electric field in the $$\hat{y}$$‐direction due to $$\hat{H}_{z\text{R}}$$, $$\hat{H}_{z\text{R}}$$ is the top‐value of the magnetic field in the $$\hat{z}$$‐direction by the translator, $$\hat{H}_{z\text{S}}$$ is the top‐value of the magnetic field in the $$\hat{z}$$‐direction by the stator, and δ is the load angle.…”