Next-generation ultrasound probes will require in-probe digitization. In previous work, the transducer was used as the loop filter of a sigma-delta modulator to obtain a compact element-matched ADC. However, the performance of the previous design can be improved potentially. The aim of this project is to upgrade the element-matched first-order sigma-delta system into a second-order system with minimum changes of the previous structure in order to obtain an SQNR improvement. Five main parts, the parasitic capacitance compensation, first stage, second stage, tracking quantizer and DAC, are discussed and compared with the first-order system. With some modifications of the coefficients and the structure, the system is upgraded with second-order behaviour which has 6 dB higher SQNR than the first-order system. In order to reduce the change of the whole structure, with the SQNR compromise, the second-order system can be modified in some ways, such as shifting the second-order noise shaping to the first-order at some level and optimizing the transimpedance of the first stage. This thesis will give a background about the first-order system and represent the ways to modify and compensate the second-order system.