The streaming potential effect has a wide range of applications in geophysics. The core streaming potential experiment requires that there is no external circuit at both ends of the core, but a measurement circuit must be introduced to measure the voltage between both ends of the core which will cause an external circuit. In order to analyze the effect of measurement circuits on the streaming potential experiment, this paper proposes a core current source model, i.e., the core in the streaming potential experiment is regarded as a circuit composed of a current source whose output current is equal to the seepage current and the core resistance. By changing the resistance value of the external circuit, it is found that the seepage current is not affected by the external resistance but by the excitation pressure. Experiments on the streaming potential of 20 sandstone cores under distilled water, 0.01 mol/L, 0.02 mol/L, 0.05 mol/L, 0.1 mol/L, 0.2 mol/L, 0.4 mol/L, and 0.6 mol/L sodium chloride solutions revealed that the effect of the external circuit on the streaming potential signal increased with decreasing mineralization. For distilled water-saturated sandstone cores, the effect of the external circuit was about 2%.