To obtain high-pressure gas, air-driven boosters are widely used. In this paper, a new pneumatic pressure booster (named expansion energy used booster, short for EEU booster), which makes use of the expansion power of compressed air in driving chambers is proposed. To set a foundation for the study on optimization of the booster, the basic mathematical model of working processes is set up. By selecting the appropriate reference values, the basic mathematical model is transformed to a dimensionless expression for modeling simulation. In this way, the dimensionless output flow characteristics of the booster can also be found. Through analysis, it can be seen that, first, the dimensionless output flow of the booster is mainly determined by the dimensionless Piston Stroke-set (the piston stroke, when the driving chambers stopped to charge air, is defined to be Piston Stroke-set), the dimensionless output pressure of the booster and the dimensionless area of the piston in the driving chambers, the study on optimization of the booster can be done based on the analysis of the influence of the three dimensionless parameters on the dimensionless average output flow and the efficiency. Lastly, the mathematical model is verified experimentally. This research can be referred to in the design of EEU boosters and the study on optimization of the EEU booster.