The traditional operational amplifier in the memristor equivalent circuit has problems such as high power consumption and large noise. To solve this problem, two extremely simplified operational amplifiers based on simplified differential pairs was designed. The power consumption and noise analysis of two simplified operational amplifiers and traditional operational amplifiers were carried out through simulation analysis. The results showed that this simplified operational amplifier circuit had the lowest power consumption and the best anti-noise performance when compared with three traditional operational amplifiers. The total power consumption was 15 mW, the equivalent output noise voltage was 11.55 nV·Hz-1/2, and the noise figure was 35.873 dB. Based on the two simplified operational amplifiers, a second-order charge-controlled memristor equivalent circuit was designed. Through theoretical analysis, circuit simulation, and the hardware circuit board-based experiment, the memristive characteristics of the equivalent circuit were analyzed and verified.