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Real Integrator (no Delay)

Models a real discrete integrator without delay considering op-amp saturation, finite gain, finite bandwidth and slew rate.

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Description

The Real Integrator (no Delay) block models a real discrete integrator without delay considering op-amp saturation, finite gain, finite bandwidth and slew rate.

The transfer function in the z-domain of an ideal integrator without delay is

Analog circuit implementations of the integrator deviate from this ideal behavior due to several non-ideal effects. One of the major causes of performance degradation in Switched-Capacitor (SC) Sigma-Delta modulators is the incomplete transfer of charge in the SC integrators. This non-ideal effect is a consequence of the operational amplifier non-idealities, namely finite gain and bandwidth, slew rate and saturation voltages.

DC Gain

The dc gain of the ideal integrator is infinite. In practice, however, the actual gain is limited by circuit constraints and in particular by the operational amplifier open-loop gain A0 (considering also the intergator feedback factor). The consequence of this integrator "leakage" is that only a fraction a of the previous output of the integrator is added to each new input sample. The limited dc gain of the integrator increases the in-band noise. The transfer function of the integrator with leakage becomes

The dc gain of the integrator H0, therefore, becomes

and hence the parameter alpha is given by

The effect of the finite open-loop dc gain on the integrator coefficient and hence on the modulator coefficients, is considered together with the operational amplifier finite bandwidth and slew-rate (because of A0 the actual integrator coefficient is multiplied by alpha).

Bandwidth and Slew Rate

The finite bandwidth and the slew-rate of the operational amplifier are modeled with a building block placed in front of the integrator, which implements a MATLAB function. The effect of the finite bandwidth and the slew-rate are related to each other and may be interpreted as a non-linear gain. In fact, finite bandwidth and slew-rate in SC circuits lead to a non-ideal transient response within each clock cycle, thus producing an incomplete or inaccurate charge transfer to the output at the end of the integration period. The evolution of the output node during the nth integration period is given by

where Vs = Vin(nTs - Ts/2), alpha is the integrator leakage (which accounts for the operational amplifier finite gain A0) and tau = 1/(2PiGBW) is the time constant of the integrator (GBW is the unity gain frequency of the integrator loop-gain during the considered clock phase). The slope of this curve reaches its maximum value when t = 0, resulting in

We must consider now two separate cases:

1. The value of the slope is lower than the operational amplifier slew-rate SR (taking into account all of the capacitors connected to the operational amplifier output during the considered clock phase). In this case no slew-rate limitation appears and the evolution of v0 is exponential during the whole clock period (until t = Ts/2).
2. The value of the slope is larger than SR. In this case, the operational amplifier is in slewing and, therefore, the first part of the transient of v0 (t < t0) is linear with slope SR. The following equations hold (assuming t0 < Ts/2):

Imposing the condition for the continuity of the derivatives in t0, we obtain

If t0 > Ts/2 the transient is linear for the whole clock period.

A MATLAB function implements the above equations to calculate the value reached by v0(t) at time Ts, which will be different from Vs due to the gain, bandwidth and slew-rate limitations of the operational amplifier. The slew-rate and bandwidth limitations produce harmonic distortion reducing the total signal-to-noise and distortion ratio (SNDR) of the Sigma-Delta modulator.

Saturation

The swing of signals in a Sigma-Delta modulator is a major concern. It is therefore important to take into account the saturation levels of the operational amplifier used. This can simply be done using a saturation block inside the feedback loop of the integrator.

Parameters

- Sample Time: Period of the sampling signal in s (Ts = 1/fs, where fs is the sampling frequency)
- Finite Gain: Gain of the op-amp used in the integrator, taking into account the integrator feedback factor
- Saturation: Saturation voltage of the op-amp used in the integrator in V
- Slew-Rate: Slew-rate of the op-amp used in the integrator, taking into account any load
- Gain-Bandwidth: Gain-bandwith product of the op-amp used in the integrator, taking into account also the integrator feedback factor and any load

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