# Performance Evaluation of a Compound Power-Split CVT for Hybrid Powertrains
## 4 Numerical Example  
In this section the performance of the type C \"four-port-mechanical-power split device\" (Figure 1c) is evaluated through a numerical example. This kind of architecture is often used as part of a more complex compound transmission and engaged in a specific speed range, as for example shown by [PERSON] et al. [13]. Assuming that we are designing a complex compound power split transmission that changes configuration depending on the speed range, we can decide to engage the topology presented in this paper when the global transmission ratio is between 0.7 and 1.5. The input design parameters are therefore chosen as follows:  
\[\tau_{\text{glm}}=0.7,\qquad\tau_{\text{glM}}=1.5,\qquad\tau_{\text{eCVTm}}=-1. 3,\qquad\tau_{\text{eCVTM}}=1.3 \tag{28}\]  
Assuming inverse proportionality between global and eCVT speed ratios, the corresponding values of the planetary gear ratios are calculated with Equations (12) and (13):  
\[\tau_{\text{W1}}=1.722,\qquad\tau_{\text{W2}}=1.357 \tag{29}\]  
Figure 4 shows how the global transmission ratio varies as a function of the speed ratio of the eCVT (Equation (5)). Although the input parameters obtained so far appear reasonable for conventional eCVT and planetary gear trains, it is important to assess the feasibility of the application by checking the speed of the two electric machines in the eCVT (namely, \(\omega_{5}\) and \(\omega_{6}\)). Assuming a constant input speed of 4000 RPM, Figure 5 shows the speed of the electric machines (Equation (4)), as a function of the global transmission ratio. The highest speed is reached by the electric machine connected to shaft 6, which spins at a rate of 7360 RPM when the global transmission ratio is 0.7. Considering that conventional electric machines can work up to 9000 RPM [22], it is safe to accept these speed ranges. The fraction of power through the eCVT in the ideal case \(\left|\text{P}_{\text{eCVT}}/\text{P}_{\text{i}}\right|_{\text{ideal}}\) is plotted in Figure 6 as a function of the global transmission ratio, as obtained from Equation (19). The efficiency of the whole system is shown in Figure 7 as a function of the global transmission ratio, where the efficiency of the eCVT is assumed equal to 0.9. In the plots of Figures 6 and 7, two different areas can be identified and each one corresponds to a different type of power flow. For values of the global speed ratio between 1.357 and 1.5 power flow (f) is established in the transmission (Figure 3f). In this case, the efficiency is calculated with Equation (24) because the power through the eCVT flows from left to right. For \(\tau_{\text{gl}}\) between 0.7 and 1.357, power flow (b) occurs, and the efficiency is  
Figure 4: Global transmission ratio as a function of eCVT transmission ratio, given the input design parameters of Equations (28) and (29). calculated with Equation (27) because the power through the eCVT has changed direction, flowing from right to left. The use of the type C architecture for this application is justified by the improved efficiency of the whole system, which is much higher than the efficiency of the eCVT for most of the range of the global transmission ratio (\(0.88<\tau_{\rm gl}<1.5\)). This is explained by the fact that in the same region, the power flowing through the eCVT, that is the only source of power losses, is lower than the input power (\(\left|\rm{P_{eCVT}/P_{i}}\right|_{ideal}<1\)). This condition is verified by definition for the case of power flow (f), because the input power splits twice before entering the eCVT, first between shafts 1 and 3, and then between shafts 5 and 2 Figure 3f). This means that power flow (f) is always advantageous. In the case of power flow (b), the power through the eCVT is low when the global transmission ratio is between 0.88 and 1.357, indicating that no recirculation occurs. However, when the global  
Figure 5: Speed of the electric machines as a function of the global transmission ratio.