Abstract: Abstract: For overcoming the limitations of trapped-oil pressure test on a higher rotate speed or greater outlet pressure and the defects of trapped-oil pressure simulation operation with accumulated error and so on, the analytic formula for seeking the maximum trapped-oil pressure value was established in an external gear pump. In the compression stage of a complete trapped-oil process with one state of smaller backlash value and another state of biggish backlash value; Firstly, the backlash flow made by pressure difference was calculated by the slender-hole flow formula substituted for a conventional short-hole flow formula; secondly, from the simplified paths of axial trapped-oil leakage, axial trapped-oil leak flow was derived; according to the above calculated trapped-oil leak flows; thirdly, the special transient position was found that the rate of trapped-oil volume on time was equal to total trapped-oil leak flow, then the maximum trapped-oil pressure value was calculated and was verified by test results of the current literature. All results calculated by analytic formula showed that, under the parameters of that modulus was 4.75 mm, the number of teeth was 10, the center distance was 58 mm, the gear width was 20 mm and pressure angle was 20°50', when the relief groove was just closed and backlash value was varied from 30 to 200 μm, the deviation rate of the maximum trapped-oil pressure position and the closed position of the relief groove was down from 18.2% to 3.5%. The bigger the backlash value was, the closer the maximum trapped-oil pressure position was to the closed relief groove position; the magnitude order of shear flow made by speed was ?6 and the flow made by pressure difference was ?4, in comparison to differential pressure flow, shear flow could be neglected; the maximum trapped-oil pressure was occurred in between the theoretical position and the actual position of the relief groove, and the maximum trapped-oil pressure value was linearly influenced by the outlet pressure and rotate speed; The slender-hole flow formula that flow rate had linear relationship with the pressure difference was more suitable for calculating backlash flow than conventional short-hole flow formula that flow rate had linear relationship with the square root of pressure difference, etc. Finally, the it is concluded that the built analytic formula can quickly calculate the maximum trapped-oil pressure and its position, because of omitting a lot of process operation and reducing its accumulated error, the result is more accurate, and a new reference is provided to estimate the maximum trapped-oil pressure and the layout of relief groove.