INTRODUCTION: Modern trends in the development of mobile machinery, particularly heavy tracked vehicles, tractors and military vehicles, focus on improving the energy efficiency, controllability and reliability of transmission systems. In this context, hydrostatic-mechanical transmissions (HMCVTs) with a closed energy transmission loop demonstrate significant potential as an alternative to traditional mechanical and hydrodynamic systems by combining smooth speed control with increased efficiency.A key feature of closed HMCVTs is circulation power transmission, which arises from the interaction of the hydraulic and mechanical circuits. Using planetary mechanisms on both the input and output links of the transmission creates internal closed power flows, or circulations, which affect torsional rigidity, energy losses and dynamic stability.Currently, the scientific literature pays insufficient attention to identifying circulation effects in transmissions with such a layout. This complicates the design and optimisation of HMCVTs that take internal circulation loads into account, as this can lead to the overloading of individual transmission elements, increasing the temperature in the circuit or reducing efficiency.Therefore, this study is relevant because it provides an in-depth analysis of circulation effects in closed HMCVTs with planetary mechanisms at the input and output of the transmission system. The results of such an analysis could inform the development of methods for identifying and compensating for harmful circulation flows. This would contribute to extending the service life, improving the energy efficiency and enhancing the control accuracy of HMCVTs.OBJECTIVES: This work aims to develop and analyse the mathematical basis for describing circulation effects in the closed loop of the HMCVT, with the planetary mechanism located on the input and output links. This goal can be achieved by identifying patterns and forming a mathematical model of the transmission's operation in a closed loop with the planetary mechanism located on the input and output links. The presented mathematical equations describe changes in the kinematic and power indicators of the transmission. These equations are used to structure the internal gear ratio equation of the closed loop of the HMCVT with the planetary mechanism located on the input and output links of the transmission.METHODS: The study used methods of differential and integral calculus to create a mathematical model of the operation of the hydrostatic-mechanical transmission. This model is based on fundamental equations that describe the kinematic, power and energy indicators of continuously variable transmissions with planetary mechanisms. Due to the means of digitization, namely three-dimensional (3D) technologies, it is possible to determine the patterns of distribution of the internal gear ratio by the hydraulic pump control parameter and the planetary mechanism design parameter. This increases the accuracy of the results obtained.RESULTS: The main results of this study are the development of a mathematical model for analyzing changes in the internal gear ratio of the closed loop HMCVT, based on the control parameter of the hydraulic machine and the design parameter of the planetary mechanism. In addition, the analysis of the obtained results using modern digitalization tools, namely spatial drawings, demonstrated the effectiveness of 3D technologies in determining the optimal combinations of transmission schemes that avoid circulation effects.CONCLUSION: This article concludes that, by using spatial illustration, dependencies of the internal gear ratio on the hydraulic pump control parameter and the design parameter of the planetary mechanism have been constructed. This provides the possibility of a broader analysis of patterns. Based on the analysis of circulation effects in the closed loop of the HMCVT, the range of changes to the planetary mechanism's design parameter has been obtained, which alters the internal gear ratio of the closed loop over a wide range.