Small-Signal $z$ -Domain Analysis of Digitally Controlled Converters

As the performance of microcontrollers has increased rapidly during the last decade, there is a growing interest to replace the analog controllers in low power switching converters by more complicated and flexible digital control algorithms. Compared to high power converters, the control loop bandwidths for converters in the lower power range are generally much higher. Because of this, the dynamic properties of the uniformly-sampled pulse-width modulators (PWMs) used in low power applications become an important restriction to the maximum achievable bandwidth of the control loop. Though frequency- and Laplace-domain models for uniformly-sampled PWMs are very valuable as they improve the general perception of the dynamic behavior of these modulators, the direct discrete design of the digital compensator requires a $z$ -domain model for the combination modulator and converter. For this purpose a new exact small-signal $z$ -domain model is derived. In accordance with the zero-order-hold equivalent commonly used for “regular” digital control systems, this $z$ -domain model gives rise to the development of a uniformly-sampled PWM equivalent of the converter. This $z$ -domain model is characterized by its capability to quantify the different dynamics of the converter for different modulators, its ease of use and its ability to predict the values of the control variables at the true sampling instants of the real system.