Harmonic distortion analysis of switched-capacitor based second-order sigma-delta modulator induced by operational amplifier's limited slew rate

A detailed discussion on genesis, determinant and algorithmic method of the harmonic distortion of Switched-Capacitor integrator and Switched-Capacitor based second-order Sigma-Delta modulator is presented. In many instances, people make harmonic distortion analysis of signal through expanding periodic signal into Fourier series or making Fourier transformation for non-periodic signal respectively. Actually, due to the complexity of transient response of Switched-Capacitor integrator, the output expression of second-order Sigma-Delta modulator loop filter is also not trivial, so we will be in great trouble if traditional method is employed when its harmonics are analyzed and calculated in such a way. A fact that is noteworthy is no matter how complicated the output response of the second integrator of the modulator is, its harmonic expression has the same functional form whether in the whole time domain or at the discrete time moments. From this we turn the harmonic analysis of the continuous-time transient response into another thing, namely computing the harmonic coefficients of the transient response at discrete time moments. The least square method is adopted in process of calculation. When developing the output expression of second-order Sigma-Delta modulator, the following point is clarified, that the harmonic distortion is dominated by the first Switched-Capacitor integrator, owing to the suppression resulted from the closed feedback of harmonic distortion of the second stage is much greater than the first stage. In addition, it is avoided that the harmonic distortion is once again brought into the feedback signal in calculation, through the way that first the additive harmonic error model is utilized to get the transfer function, second replaced by gain model in the form of power series. And by doing so, both the feasibility and simplicity of the analysis and mathematical derivation are enhanced effectively.