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For a given clock-cycle n, the instantaneous exponentially-decaying current I, (t) resulting from the charge transfer is given by equation (4). Recall that the feedback value is sampled on Cp,c during the first clock half-cycle (when @, is high) and then the sampled voltage is transferred to loop filter during the second clock half-cycle (when @, is high). For a total integrated charge of Kpac * y(n) «Ts to be delivered by the SCR DAC during @, of clock-cycle n, A commonly used solution to alleviate DAC sensitivity to PWJ is the switched-capacitor- resistor (SCR) DAC with exponentially-decaying waveform, shown in Figure 16. The exponentially-decaying waveform (Figure 8) of the SCR DAC makes the amount of charge transferred to the loop per clock-cycle less dependent on the exact timing of the DAC clock- edges [4, 9]. — Figure 16 For a given clock-cycle n, the instantaneous exponentially-decaying current I, (t) resulting from the charge transfer is given by equation (4). Recall that the feedback value is sampled on Cp,c during the first clock half-cycle (when @, is high) and then the sampled voltage is transferred to loop filter during the second clock half-cycle (when @, is high). For a total integrated charge of Kpac * y(n) «Ts to be delivered by the SCR DAC during @, of clock-cycle n, A commonly used solution to alleviate DAC sensitivity to PWJ is the switched-capacitor- resistor (SCR) DAC with exponentially-decaying waveform, shown in Figure 16. The exponentially-decaying waveform (Figure 8) of the SCR DAC makes the amount of charge transferred to the loop per clock-cycle less dependent on the exact timing of the DAC clock- edges [4, 9].

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Figure 2. AX ADC (AX pulse density modulation + digital decimation and filtering). models for additive errors induced by clock-jitter in AZ modulators are given so that to help designers characterize the sensitivities of various types of AX architectures to clock-jitter. The material in this chapter is organized as follows. Section 2 gives a general background about the types of errors caused by clock-jitter in different classes of switched circuits and signal waveforms. The critical sources of jitter induced errors in a AX loop are identified for DT and CT AX modulators and a comparison between the two types, in terms of sensitivity to clock-jitter, is done in Section 3. Section 4 provides detailed sensitivity analysis for CT AZ modulators to clock-jitter in the feedback DAC sampling-clock. In Section 5, Simulink models, based on the analysis of Section 4, for the additive errors generated by clock-jitter in CT AX modulators are shown and the robustness of these models is verified by CT simulations in Matlab/Simulink. Simulations results show good agreement with the theoretical expectations. Finally, conclusions are drawn in Section 6. theoretical expectations. Finally, conclusions are drawn in Section 6.

Figure 3. Typical Phase-Noise profile in a VCO. 396 MATLAB -A Fundamental Tool for Scientific Computing and Engineering Applications — Volume 1

Figure 4. T/H Circuit. (a) Schematic view and clock waveform. (b) Effect of aperture jitter on sampled

Figure 5. SNR variation with the input frequency due to aperture jitter for different rms jitter values. 398 MATLAB-—A Fundamental Tool for Scientific Computing and Engineering Applications — Volume 1 2.2. Charge transfer jitter

Figure 6. (a) Non-inverting switched-capacitor discrete-time integrator. (b) Time-domain waveforms far clack nhaceac innit cional and charoa flaw fram (.. to 6. 400 MATLAB -A Fundamental Tool for Scientific Computing and Engineering Applications — Volume

Figure 7. SNR variation with the input frequency due to charge transfer jitter for different rms jitter values. (a) T = 0.05 T,. (b) T= 0.1T,. (c) T= 0.2 Ts. The plots in Figure 7 show the limitation on the achievable SNR vs. the signal frequency due to charge transfer jitter for different values of the rms jitter oj. Typical values of a = 0.5 and B =1 have been considered. The results are provided for t = 0.05 T,, 0.1 T;, and 0.2 T;. As can be seen from the plots in Figure 7, for a given clock frequency, the SNR limitation due to charge transfer jitter is much more relaxed compared to the aperture jitter error (Figure 5). This result was expected because from equation (11), the effect of the jitter induced noise is reduced by an exponential factor indicating that charge transfer error in SC circuits should be less critical. This also can also be explained intuitively by noting that for the exponentially-decaying waveform in Figure 8, the amplitude of the pulse is rather low at the end of the clock-cycle and hence the amount of charge that varies over one clock period due to jitter is significantly reduced. However, for a given rms jitter and sampling frequency, the SNR limitation due to charge transfer jitter degrades as the discharging time-constant T increases. This is because the value of the charge transfer current at the end of the clock- cycle (discharge phase) is varying exponentially with t, thus for a given timing error At, the error in the amount of charge transferred is higher.

Figure 8. Jitter-tolerant exponentially-decaying waveform.

Figure 9. Pulse-width jitter in switched-current circuits.

Figure 10. SNR variation with the input frequency due to pulse-width jitter for different rms jitter walhiec Analysis and Modeling of Clock-Jitter Effects in Delta-Sigma Modulators 405

Figure 11. SNR variation with the sampling frequency due to different types of jitter induced errors for arms jitter of 10 ps.

Figure 13. Modulation of in-band desired signal and shaped quantization noise by phase-noise in the DAC sampling clock. 408 MATLAB -A Fundamental Tool for Scientific Computing and Engineering Applications — Volume ‘1

In DT AX modulators, the sampling takes place at the modulator input. The SC integrator in Figure 14 is commonly used as an input stage for DT loop filters in AZ modulators. The sampling aperture jitter errors due to the sampling switch (Sz) will be added to the signal at the input and hence will directly appear at the modulator output without any suppression. As mentioned earlier, the feedback signal (Vpac) doesn’t experience aperture jitter because the feedback signal is DT and also it has discrete amplitude levels. Thus, timing errors cannot result in a sampled value that is different from the original feedback one. Timing errors at switch S2 cause charge transfer jitter errors being added at the input stage. However, the charge transfer jitter errors at S2 are very small owing to the high robustness

° This is provided that the solution adopted to improve the DAC tolerance to clock jitter errors is not adding high power overhead and thus not increasing the total power consumption.

The expressions for the IBJN due to input signal and shaped quantization noise are given by extra requirements on the SR or GBW of the integrating amplifier. which for At(n) « t can be approximated by

Figure 17. Simulink Modeling for DACs and jitter induced additive errors in the feedback of a CT AZ modulator. (a) RZ DAC. (b) NRZ DAC. (c) SCR DAC. Analysis and Modeling of Clock-Jitter Effects in Delta-Sigma Modulators 415

Table 2. Modulator specifications and performance summary Analysis and Modeling of Clock-Jitter Effects in Delta-Sigma Modulators 417

Figure 19. Dynamic-range of the adopted AX modulator.

Figure 22. IBJN plots for the AZ modulator in Figure 18 using different DAC types. 6. Conclusion