## 2 Principle of Hydrostatic Power Split CVT
### Basic Transmission Equations  
Consider that the rotation speed of each planetary gear shaft meets the conditions [20]:  
\[n_{s}+kn_{r}-(1+k)n_{c}=0 \tag{1}\]  
where \(n_{s}\), \(n_{r}\), and \(n_{c}\) are the rotation speed of sun gear, ring gear, and carrier of the planetary gear set, respectively, r/min; \(k\) is the standing ratio of planetary gear set, which is the ratio of sun gear and ring gear speed with a fixed planet carrier.  
The speed relationship between pump shaft and motor shaft is as follows:  
\[n_{m}=n_{p}\varepsilon \tag{2}\]  
where \(n_{m}\) and \(n_{p}\) are the rotation speeds of the fixed displacement motor (MF) and the variable displacement pump (PV), r/min, respectively; \(\varepsilon\) is the displacement ratio of the swash plate axial piston units.  
It is easy to deduce that the transmission ratio of the CVT in each range is:  
\[n_{R1}(\varepsilon)=\frac{k_{2}i_{4}i_{5}-i_{2}i_{3}\varepsilon}{(1+k_{2})i_{ 2}i_{3}i_{4}i_{5}i_{12}}n_{e} \tag{3}\]  
Figure 2: Schematic diagram of clutch hydraulic control system.  
\[n_{R2}(\varepsilon) =\frac{(1+k_{1})i_{4}i_{5}+i_{2}i_{3}\varepsilon}{k_{1}i_{2}i_{3}i_{4} i_{5}i_{7}i_{11}}n_{e} \tag{4}\] \[n_{R3}(\varepsilon) =\frac{k_{2}i_{4}i_{5}-i_{2}i_{3}\varepsilon}{(1+k_{2})i_{2}i_{3}i_ {4}i_{5}i_{9}i_{12}}n_{e}\] (5) \[n_{R4}(\varepsilon) =\frac{(1+k_{1})i_{4}i_{5}+i_{2}i_{3}\varepsilon}{k_{1}i_{2}i_{3} i_{4}i_{5}i_{6}i_{11}}n_{e} \tag{6}\]  
where \(n_{Rx}\) is the output speed of the transmission in range \(R_{x}\) (x = 1, 2, 3, 4), which is a function of the displacement ratio \(\varepsilon\), r/min; \(i_{2}\)-\(i_{12}\) are the transmission ratios of gear pairs \(g_{2}\)-\(g_{12}\), respectively; \(k_{1}\) and \(k_{2}\) are the standing ratios of the planetary gear set \(p_{1}\) and \(p_{2}\), respectively; \(n_{e}\) is the engine speed, r/min.  
In order to achieve excellent acceleration performance, the speed regulation range of two adjacent ranges must meet the equal ratio condition:  
\[\frac{n_{Rx}(\varepsilon^{*})}{n_{Rx}(-\varepsilon^{*})}=\varphi \tag{7}\]  
where \(\varphi\) is the common ratio of each range; \(\varepsilon^{*}\) is the displacement ratio corresponding to the maximum tractor speed in range \(R_{x}\). \(\varepsilon^{*}=-1\) if \(x=1\) or \(x=3\), and \(\varepsilon^{*}=1\) if \(x=2\) or \(x=4\).  
According to Equations (3)-(7), the following relationship can be obtained:  
\[\frac{i_{4}i_{5}}{i_{2}i_{3}}=\frac{\varphi+1}{k_{2}(\varphi-1)} \tag{8}\]  
\[\frac{i_{4}i_{5}}{i_{2}i_{3}}=\frac{\varphi+1}{(1+k_{1})(\varphi-1)} \tag{9}\]  
Comparing Equations (8) and (9), it is easy to get:  
\[1+k_{1}=k_{2} \tag{10}\]  
In addition, in order to make each range connect at the maximum pump displacement, the connection conditions shall also be met:  
\[n_{R1}(-1)=n_{R2}(-1) \tag{11}\]  
\[n_{R2}(1)=n_{R3}(1) \tag{12}\]  
\[n_{R3}(-1)=n_{R4}(-1) \tag{13}\]  
By substituting Equations (3)-(6) into Equations (11)-(13), the following relationship can be obtained:  
\[\frac{i_{8}i_{12}}{i_{7}i_{11}}=\frac{\varphi(k_{2}-1)}{1+k_{2}} \tag{14}\]  
\[\frac{i_{7}i_{11}}{i_{9}i_{12}}=\frac{\varphi(k_{2}+1)}{k_{2}-1} \tag{15}\]  
\[\frac{i_{9}i_{12}}{i_{6}i_{11}}=\frac{\varphi(k_{2}-1)}{1+k_{2}} \tag{16}\]  
In conclusion, the output speed of the CVT calculated by Equations (3)-(6) was reasonable only when Equation (10) and Equations (14)-(16) were satisfied. According to Equations (3)-(6), the relationship between the output speed of the transmission in each range and the displacement ratio of the swash plate axial piston units could be calculated. The actual output speed of the transmission could be obtained by testing at different engine speeds, ranges, and displacement ratios. The calculation results of the transmission output speed were consistent with the test results, as shown in Figure 3.  
It can be seen from the figure that the transmission had the same displacement ratio and the same output speed before and after the shift, so the shift process under ideal conditions should be relatively smooth.