Conclusiones generales

In the previous two sections the transfer function of the integrator was derived consid-ering finite opamp gain and finite bandwidth separately. If both these are taken into account simultaneously the derivation get slightly more complicated. Using a single pole model of the integrator the transfer function is given by [14]

where is the relative settling error on clock phase According to Sec. 6.6.2 is given by

We conclude that when the finite opamp gain is included the finite bandwidth will introduce both gain errors and integrator leakage.

6.9.4 Noise

In Sec. 6.7 the noise of S/H amplifiers were derived. The noise in the integrator can be calculated in a similar way. The z-domain output noise from the opamp noise source ( see Fig. 6-20) can be derived based on charge conservation. We assume in the following the opamp gain is infinite and we neglect the influence of finite bandwidth in the opamp on the transfer function. On clock phase

we have the total charge

6.9 SC Integrator 173

where denotes a voltage at time In the following clock phase the total charge is

The total charge in the two phases is conserved and the output signal at can be calculated as

In the following clock phase the output is held by capacitor C, but we also have a noise contribution from the noise source. The total charge on C and pC is conserved, i.e.

We can now get an expression for the output signal at as

The z-domain transfer function is

where is the noise contribution on and is the contribution on The noise can be referred to the input by dividing the output noise by the trans-fer function of the integrator, (6-93), i.e.

If the signal frequency is much lower than the sampling frequency, which is the case in an oversampled sigma-delta modulator, the second term in (6-119) will dominate since the first term is high-pass filtered. Assuming a single-stage opamp the input

174 Chapter 6. Analog Functional Blocks

referred noise is (see Sec. 6.7)

where is a factor taking the noise contributions from the current sources in the opamp into account. In the above analysis only the opamp noise was considered. The kT/C noise from the passive sampling should also be considered. The total noise would then be

Again it is interesting to note that the noise only depends on the capacitor sizes and not the transconductance of the input device. If the noise contribution factor, is small and the load capacitance at the output is large, the kT/C -noise from the switch resistance will dominate.

6.10 SUMMARY

In this chapter we have discussed two important analog functional blocks in data con-verters, the MDAC and the integrator. Both SC and SI circuits were considered. We showed how to calculate the noise and maximum speed of the circuits based on first order models. The maximum speed of the circuits is determined by the bandwidth and the required number of clock phases. The maximum speed normally decreases as the gain of the circuit increases. The thermal noise of the circuits can be referred to the input as a noise voltage. We showed that the noise voltage power for all the circuits is only determined by the capacitors in the circuit.

The results in this chapter will be used in chapter 9 and chapter 11 to investigate the effect of circuit imperfections on pipelined and oversampled sigma-delta converters respectively.

REFERENCES

[l]

[2]

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6.10 Summary 175

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7 BASIC ANALOG CIRCUIT