SAMPLED-DATA SIMULATION OF OVER-SAMPLED SIGMA-DELTA MODULATORS - Computer Methods for Analysis of Mixed-Mode Switching Circuits - page 194

Once is available, simulation proceeds in a sampled-data
manner. As compared with the analysis of sigma-delta modulators with
clocked quantizes, the following additional computation in the analysis
of sigma-delta modulators with unclocked quantizers is required :

i) Evaluation of the switching variable of the quantizer in each integra-

tion step.

ii) If a change in the sign of the switching variable is detected, the exact

switching time is allocated by solving

iii) Computation of and subsequently using the

sampled-data simulation for periodically switched linear circuits.

iv) Computation of the consistent initial condition at time instants im-

mediately after switching using the two-step algorithms presented in
Chapter 5.

4. Modeling of Digital-to-Analog Data Converters
The function of a digital-to-analog converter (DAC) in a single-bit
sigma-delta modulator is to select two distinct reference voltages (or
currents), based on the output of the quantizers. Fig.9.4 shows the
typical configurations of DACs in single-bit sigma-delta modulators. The
single-bit DAC is a switched linear circuit with the controlling signal
be the output of the quantizer of the sigma-delta modulator and two
voltage inputs with constant voltages for voltage-mode DACs or two
current inputs with constant currents for current-mode DACs.

5. Modeling of Other Blocks
When the input to the integrators is small, the integrators can be
treated as a linear element, and can be modeled at the circuit level us-
ing linear elements such as resistors, inductors, capacitors, dependent
and independent sources, and ideal switches. The non-idealities of inte-
grators, such as finite bandwidth, finite input impedance, and non-zero
output impedance of the operational amplifier of the integrators, can
also be modeled conveniently using circuit-level models.