SBAA661 February   2025 LMX1205

 

  1.   1
  2.   Abstract
  3.   Trademarks
  4. 1Flicker Noise, Noise Floor, and Total Noise
    1. 1.1 Flicker Noise
    2. 1.2 Noise Floor
    3. 1.3 Total Noise
  5. 2Finding the Slew Rate
    1. 2.1 Finding the Slew Rate With an Oscilloscope
    2. 2.2 Calculating the Slew Rate From Power and Frequency
  6. 3Impact of Slew Rate on Phase Noise
    1. 3.1 Modeling of Input Clock Slew Rate, Inherent Device Noise, and Output Jitter
    2. 3.2 Slew Rate Impact on Flicker Noise and Noise Floor
  7. 4Application of Slew Rate Rules to PLL Synthesizers
    1. 4.1 PLL Flicker Noise
    2. 4.2 PLL Figure of Merit
    3. 4.3 Other Areas in PLLs Where Slew Rate has an Impact on Performance
    4. 4.4 Improving PLL Slew Rate for Better Performance
  8. 5Application of Slew Rate Rules to Data Converters
  9. 6Summary
  10. 7References
  11.   Appendix A: Relating Slew Rate, Power, and Frequency
  12.   Appendix B: Relating Slew Rate, Frequency, Jitter, and Phase Noise
  13.   Appendix C: Equations for Data Converters
    1. 8.1 Relating Sampled Signal Slew Rate to SNR
    2. 8.2 Justification That SNR Decreases 1dB per 1dB With Input Power for Slew Rate Limited Case
  14.   Appendix D: Calculations for Data Converter Example

Appendix B: Relating Slew Rate, Frequency, Jitter, and Phase Noise

Consider the case of a perfectly clean input clock with slew rate SR into a buffer. For the buffer, there is some internal voltage noise which we consider to be constant. In symbolic terms, this can be stated that with slew rate (SR) and rms jitter ( and voltage jitter (which is:

Equation 28. σ v =   σ t × S R

In this case, the unknown is the jitter that results from the voltage noise.

Equation 29. σ t =   σ v S R

Now equation Equation 29 is just for one rising edge of the signal. In actuality, at every rising edge there is a different jitter. In this case, the number of cycles depends on the measurement time (t).

Equation 30. n =   t f

Now variance relates to noise power. So consider measuring the power over the timer interval t. If the jitter in equation Equation 29 is correlated from one period to the next, then it appears as 1/f noise, also known as flicker noise. As jitter is the standard deviation of the time error, if one takes the average of n cycles, this average can be calculated.

Equation 31. σ f l i c k e r =   n ×   σ t t = σ t × f

If the jitter in equation Equation 29 is not correlated from one period to the next, then it appears as noise floor. As jitter is the standard deviation of the time error, if one takes the average of n cycles, this average can be calculated.

Equation 32. σ f l o o r =     n × σ t n = σ t × f

Recall that jitter and phase noise are related.

Equation 33. σ =   2 × ∫ a b L ( w ) × d w 2 π × f

For the integral, we are only interested in the phase noise as it changes with frequency and slew rate, so we are not concerned with the integration limits and we roll this up into a constant.

Equation 34. σ = 2 × 10 P N / 10 × ∫ a b L ( w ) × d w 2 π × f = c o n s t a n t × 10 P N / 10 f

Equation Equation 34 can be arranged as follows.

Equation 35. P N =   20 × log ⁡ σ + c o n s t a n t

In the case of flicker noise, combine Equation 31 and Equation 35 to get:

Equation 36. P N f l i c k e r =   20 × log ⁡ σ t × f + c o n s t a n t

This can be rearranged to get the rule for 1/f noise.

Equation 37. P N f l i c k e r ∝   20 × log ⁡ σ t + 20 × l o g ⁡ ( f )

This result can be combined with Equation 29 to obtain:

Equation 38. P N f l i c k e r ∝   20 × log ⁡ σ v S R + 20 × l o g ⁡ ( f )

As the voltage noise, is a constant, Equation 38 can be rearranged to get the fundamental relationship between flicker noise, slew rate, and frequency.

Equation 39. P N f l i c k e r ∝   20 × log ⁡ f - 20 × l o g ⁡ ( S R )

In the case of noise floor, we combine with Equation 32 with Equation 35 to get:

Equation 40. P N f l o o r =   20 × log ⁡ σ t + 20 × l o g ⁡ ( σ t × f ) + c o n s t a n t

This can be rearranged to get the rule for noise floor.

Equation 41. P N f l o o r ∝   20 × log ⁡ σ t + 10 × l o g ⁡ ( f )

Now combine this result with Equation 29 to get:

Equation 42. P N f l o o r ∝   20 × log ⁡ σ v S R + 10 × l o g ⁡ ( f )

As the voltage noise, is a constant, Equation 41 can be rearranged to get the final result.

Equation 43. P N f l o o r ∝   10 × log ⁡ f - 20 × l o g ⁡ ( S R )