# Performance Evaluation of a Compound Power-Split CVT for Hybrid Powertrains
## 2 Kinematic Model of a Compound Power-Split eCVT  
The compound power-split eCVT analyzed in this paper is shown in Figure 2 and it is one of the possible arrangements of the \"four-port-mechanical-power split device\". The shafts are numbered from 1 to 6. In this set-up, the shafts that drive the central gears of the  
Figure 1: Compound eCVT based on the four-port-mechanical power-split device: (**a**) type A arrangement; (**b**) type B arrangement; (**c**) type C arrangement; (**d**) type D arrangement. PGs (shafts 1, 2, 3, and 4) are connected to the input and output shafts, while the planet carriers (shafts 5 and 6) are connected to the eCVT. At this stage, it is not specified which one of the two central gears (i.e., sun and ring) of PG1 is connected to shaft 1 or shaft 2, and the same applies to PG2. It is a choice of the designer that depends on the value of the transmission ratio required by the specific application. The transmission ratios for the PGs and the eCVT are given by (using Willis equation for PG1 and PG2 [21]):  
\[\tau_{\text{W1}}=\frac{\omega_{2}-\omega_{5}}{\omega_{1}-\omega_{5}};\ \ \ \ \ \tau_{\text{W2}}=\frac{\omega_{4}-\omega_{6}}{\omega_{3}-\omega_{6}};\ \ \ \ \ \ \tau_{\text{eCVT}}=\frac{\omega_{6}}{\omega_{5}} \tag{1}\]  
with:  
\[\omega_{1}=\omega_{3}=\omega_{\text{i}};\ \ \ \ \ \omega_{2}=\omega_{4}= \omega_{\text{o}} \tag{2}\]  
where \(\omega_{\text{j}}\) represents the angular speed of shaft j, \(\omega_{\text{i}}\) is the input speed, and \(\omega_{\text{o}}\) is the output speed. It is important to specify that the planetary gear ratios \(\tau_{\text{W1}}\) and \(\tau_{\text{W2}}\) are design parameters and therefore constants defined by the geometry of the gears. On the other hand, the global and eCVT speed ratios are kinematic variables, and the design choice regards their spread rather than their instantaneous value that depends on the input and output speeds. The following speed ratios can be derived from Equations (1) and (2):  
\[\frac{\omega_{5}}{\omega_{\text{i}}}=\frac{\tau_{\text{W2}}-\tau_{\text{W1}}} {\tau_{\text{eCVT}}(\tau_{\text{W2}}-1)-(\tau_{\text{W1}}-1)};\ \ \ \ \frac{\omega_{6}}{\omega_{\text{o}}}=\frac{\tau_{\text{eCVT}}(\tau_{\text{W2}} -\tau_{\text{W1}})}{\tau_{\text{eCVT}}(\tau_{\text{W2}}-1)-(\tau_{\text{W1}} -1)} \tag{3}\]  
\[\frac{\omega_{5}}{\omega_{\text{o}}}=\frac{\tau_{\text{W2}}-\tau_{\text{W1}}} {\tau_{\text{eCVT}}\tau_{\text{W1}}(\tau_{\text{W2}}-1)-\tau_{\text{W2}}( \tau_{\text{W1}}-1)};\ \ \ \ \frac{\omega_{6}}{\omega_{\text{o}}}=\frac{\tau_{\text{eCVT}}(\tau_{\text{W2}} -\tau_{\text{W1}})}{\tau_{\text{eCVT}}\tau_{\text{W1}}(\tau_{\text{W2}}-1)- \tau_{\text{W2}}(\tau_{\text{W1}}-1)} \tag{4}\]  
Rearranging and combining Equations (1) and (2), the global speed ratio can be derived as a function of the transmission ratios of PGs and eCVT:  
\[\tau_{\text{gl}}=\frac{\omega_{\text{O}}}{\omega_{\text{i}}}=\frac{\omega_{2} }{\omega_{1}}=\frac{\tau_{\text{W1}}\tau_{\text{eCVT}}(\tau_{\text{W2}}-1)- \tau_{\text{W2}}(\tau_{\text{W1}}-1)}{\tau_{\text{eCVT}}(\tau_{\text{W2}}-1)- (\tau_{\text{W1}}-1)} \tag{5}\]  
Figure 2: Shafts indexing with positive direction of power on all shafts. The analytical expression of the partial derivative of the global transmission ratio with respect to the speed ratio of the eCVT is derived from Equation (5):  
\[\frac{\partial\tau_{\text{gl}}}{\partial\tau_{\text{eCVT}}}=\frac{(\tau_{\text{W 2}}-1)(\tau_{\text{W1}}-\tau_{\text{W2}})(1-\tau_{\text{W1}})}{\left(\tau_{\text {eCVT}}(\tau_{\text{W2}}-1)-(\tau_{\text{W1}}-1)\right)^{2}} \tag{6}\]  
The sign of Equation (6) varies according to the numerator. Therefore, the function \(\tau_{\text{gl}}(\tau_{\text{eCVT}})\) is monotonic for a given set of planetary gear trains. The type of proportionality (namely direct or indirect) between \(\tau_{\text{gl}}\) and \(\tau_{\text{eCVT}}\) is a design choice.