# Design Optimization of the Magnet-Free Synchronous Homopolar Motor of a Subway Train
## 4 Average Loss Calculation  
Usually, the subway stations are located near of each other. Therefore, the duration of the cruising is not too much. Moreover, the required power in the cruising mode is much smaller than the one needed in other operation modes excluding stops. Therefore, the cruising mode and coasting were excluded from consideration.  
Average value of some variable \(A\) during operation cycle depicted in Figure 4 is defined by equation:  
\[<A>=\frac{\int\limits_{0}^{t_{cycle}}{dt}}{t_{cycle}}=\frac{\int\limits_{cycle} {Adt}}{\int\limits_{cycle}{dt}} \tag{1}\]  
where \(t_{cycle}\) is the duration of the cycle; \(t\) is time.  
The mechanical subsystem of the train can be represented by:  
\[J\frac{\mathrm{d}\Omega}{\mathrm{d}t}=\frac{2\pi}{60}J\frac{\mathrm{d}n}{ \mathrm{d}t}=T \tag{2}\]  
where \(\Omega\) is the motor mechanical angular frequency in radians per second; \(n=60\cdot\Omega/(2\cdot\pi)\) is the motor rotational speed in revolutions per minute; \(J\) is the total equivalent inertia of the train at the motor shaft; \(T\) is the equivalent torque at the motor shaft. Neglecting the friction and the slopes, the equivalent torque \(T\) is equal to the torque produced by the motor.  
By expressing \(\mathrm{d}t\) from (2) and substituting it into (1) the average value can be expressed as:  
\[<A>\ =\ \frac{\int\limits_{cycle}{A}{dn}}{\int\limits_{cycle}{\frac{dn}{T}}} \tag{3}\]  
Let's estimate the value of integral in the numerator of (3) with a help of quadrature formulas. The operation mode of the motor changes in operation points 1-5 (see Figure 4), and there is no derivative of the torque with respect to speed. Therefore, it is reasonable to use the equation of first algebraic order (i.e., trapezoidal rule). Summing up the products of arithmetic means of the integrands \(A/T\) and the speed differences at the separate segments from Figure 4 (the segments from the point 2 to point 3, from 3 to 4, from 4 to 5, and from 5 to 1) results in the following:  
\[\begin{split}\int\limits_{cycle}{\frac{A}{T}}{dn}& \approx\left(\frac{A_{2}}{T_{2}}+\frac{A_{3}}{T_{0}}\right)\frac{n_{ \max}-n_{g}}{2}+\left(\frac{A_{3}}{T_{0}}+\frac{A_{4}}{T_{0}}\right)\frac{n_{ g}}{2}+\ldots\\ &\ldots+\left(\frac{A_{4}}{T_{0}}+\frac{A_{5}}{T_{0}}\right) \frac{n_{m}}{2}+\left(\frac{A_{5}}{T_{0}}+\frac{A_{1}}{T_{1}}\right)\frac{n_{ \max}-n_{m}}{2},\end{split} \tag{4}\]  
where \(A_{1}\)-\(A_{5}\) are the values of the calculated function \(A\) at operating points 1-5 (see Table 1), respectively.  
Approximation of the denominator of (3) can be expressed from (4) by substituting \(A\)\(\equiv\) 1. Substituting (4) into (3) as well as \(T_{2}=T_{0}\cdot\eta_{g}/n_{\text{max}}\), \(T_{1}=T_{0}\cdot n_{m}/n_{max}\), it is possible to find similarly summand in each product with \(A_{i}\). Then the average loss estimation during operation cycle can be obtained as a weighted-average losses in points 1-5 of the operation cycle as:  
\[<A>\approx\sum\limits_{i=1,2,3,4,5}w_{i}A_{i} \tag{5}\]  
In (5), the weight coefficients \(w_{i}\) normalized on the basis unity sum of all coefficients can be evaluated by:  
\[w_{i}=\frac{W_{i}}{\sum\limits_{1}^{5}W_{i}} \tag{6}\]  
In (6), the coefficients \(Wi\) are calculated as according to (7):  
\[W_{1}=\frac{n_{\text{max}}(n_{\text{max}}-n_{m})}{n_{m}};\ W_{2}=\frac{n_{\text{ max}}(n_{\text{max}}-n_{g})}{n_{g}};\ W_{3}=n_{\text{max}};\ W_{4}=(n_{g}+n_{m});\ W_{5}=n_{\text{max}}. \tag{7}\]  
Average value estimate (5) is relevant for any function \(A(t)\) in the considered cycle from Figure 3. Here, this Equation (5) is used to evaluate average electric losses in the electric machine <\(U_{loss\,d}\)>. The losses during coasting are not taken into account due to the fact that they are small. Only the acceleration and deceleration phases are considered.