# Performance Evaluation of a Compound Power-Split CVT for Hybrid Powertrains
## 3 Power Flows  
The compound power split devices are subject to several types of internal power flows because of the presence of two planetary gear sets. The objective of this section is to derive the specific conditions at which each power flow occurs in the type C \"four-port-mechanical-power split device\". The conditions for each power flow in terms of planetary gears speed ratios are studied in two cases, depending on the sign of the global transmission ratio. The schematics of the twelve possible types of power flows are shown in Figure 3, while the conditions for each power flow type in terms of power ratio signs are described in Table 1, where Pj is the power across shaft j. In this paper, Pj is assumed positive when entering the planetary gears. The positive direction of power for all shafts is shown in Figure 2.  
To obtain analytical expressions for the power ratios across the various shafts, the torque ratios must be computed first. Assuming no losses in the planetary gear trains, the following equations can be derived:  
\[\frac{\mathrm{T}_{1}}{\mathrm{T}_{2}}=-\tau_{\mathrm{W1}};\ \ \ \ \ \ \frac{\mathrm{T}_{3}}{\mathrm{T}_{4}}=-\tau_{ \mathrm{W2}};\ \ \ \ \ \ \frac{\mathrm{T}_{1}}{\mathrm{T}_{5}}=\frac{\tau_{ \mathrm{W1}}}{1-\tau_{\mathrm{W1}}};\ \ \ \ \ \ \frac{\mathrm{T}_{3}}{\mathrm{T}_{6}}=\frac{\tau_{ \mathrm{W2}}}{1-\tau_{\mathrm{W2}}} \tag{14}\]  
From Equations (3), (4), and (14), the following expressions for the power ratios can be obtained:  
\[\frac{\mathrm{P}_{1}}{\mathrm{P}_{2}}=\frac{\mathrm{T}_{1}\omega_{\mathrm{i}} }{\mathrm{T}_{2}\omega_{\mathrm{o}}}=-\frac{\tau_{\mathrm{W1}}}{\tau_{\mathrm{ gl}}};\ \ \ \ \ \frac{\mathrm{P}_{3}}{\mathrm{P}_{4}}=\frac{\mathrm{T}_{3}\omega_{\mathrm{i}} }{\mathrm{T}_{4}\omega_{\mathrm{o}}}=-\frac{\tau_{\mathrm{W2}}}{\tau_{\mathrm{ gl}}};\ \ \ \ \ \frac{\mathrm{P}_{1}}{\mathrm{P}_{5}}=\frac{\mathrm{T}_{1}\omega_{\mathrm{i}} }{\mathrm{T}_{5}\omega_{5}}=\frac{\tau_{\mathrm{W1}}}{\tau_{\mathrm{gl}}- \tau_{\mathrm{W1}}};\ \ \ \ \ \frac{\mathrm{P}_{3}}{\mathrm{P}_{6}}=\frac{\mathrm{T}_{3}\omega_{ \mathrm{i}}}{\mathrm{T}_{6}\omega_{6}}=\frac{\tau_{\mathrm{W2}}}{\tau_{\mathrm{ gl}}-\tau_{\mathrm{W2}}} \tag{15}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Power Flow** & \(\frac{\mathrm{P}_{1}}{\mathrm{P}_{2}}\) & \(\frac{\mathrm{P}_{3}}{\mathrm{P}_{4}}\) & \(\frac{\mathrm{P}_{4}}{\mathrm{P}_{5}}\) & \(\frac{\mathrm{P}_{4}}{\mathrm{P}_{5}}\) & \(\frac{\mathrm{P}_{4}}{\mathrm{P}_{5}}\) \\ \hline
**(a)** & \(<0\) & \(<0\) & \(>0\) & \(>0\) \\
**(b)** & \(<0\) & \(<0\) & \(<0\) & \(<0\) \\
**(c)** & \(>0\) & \(>0\) & \(<0\) & \(<0\) \\
**(d)** & \(>0\) & \(<0\) & \(<0\) & \(<0\) \\
**(e)** & \(>0\) & \(<0\) & \(<0\) & \(>0\) \\
**(f)** & \(<0\) & \(<0\) & \(<0\) & \(>0\) \\
**(g)** & \(<0\) & \(<0\) & \(>0\) & \(<0\) \\
**(h)** & \(<0\) & \(>0\) & \(>0\) & \(<0\) \\
**(i)** & \(>0\) & \(>0\) & \(<0\) & \(<0\) \\
**(l)** & \(<0\) & \(>0\) & \(<0\) & \(<0\) \\
**(m)** & \(<0\) & \(<0\) & \(<0\) & \(<0\) \\
**(n)** & \(<0\) & \(<0\) & \(>0\) & \(>0\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Power ratio signs required to obtain each power flow type.  
Figure 3: Power flow types within a type C compound transmission: twelve possible flows with different power circulations.