Introduction

In the previous Timing 201 article, Timing 201 #7: The Case of the Dueling PLLs – Part 1, I referred to a Silicon Labs white paper that describes Silicon Labs’ DSPLL nested dual-loop architecture as used in the Si538x wireless jitter attenuators. I first discussed the general motivation for a dual-loop PLL and compared the cascaded (series) dual-loop PLL versus the nested dual-loop PLL architectures.

The practical advantages of the nested dual-loop approach in this example were to reduce the number of tuned oscillators from 2 to 1 and to eliminate the need for a sensitive external voltage control line. The tradeoff for a nested feedback control loop is that the inner loop must be faster than the outer loop. If the loop speeds (or bandwidths) are comparable, then the loops will contend or “duel” with each other.

In this Part 2 follow-up post, I will discuss in more detail how to calculate the phase noise of both these dual-loop PLL approaches.

Some Simplifying Assumptions

To emphasize the basic ideas without getting bogged down in too much detail, I will make the following simplifying assumptions:

1. Jitter generation, i.e., intrinsic PLL noise, is negligible. I only care about how the PLLs filter their respective phase noise sources, and how they contribute to the output clock phase noise.
2. Phase noise scaling is not needed. Assume that the phase noise data I use is taken at the same carrier frequency for both PLLs. For example, it’s as if you had a 54 MHz XO or VCXO, and the input and output clock frequencies were all 54 MHz, and the VCO phase noise had been scaled down to 54 MHz. (Otherwise you would need to scale the phase noise data.)
3. Identical phase noise sources will be used to compare the two topologies. In particular, the XO and VCXO phase noise will be assumed to be identical. This is generally not the case as VCXOs are usually noisier than XOs for the same nominal frequency. Why this assumption is being taken will be made clear later in this article.
4. The PLL acts as a low pass filter (LPF) or high pass filter (HPF) to the input and VCO phase noise respectively. The filter is modeled with the corner frequency at the bandwidth (BW) frequency. The filter skirt slope or steepness magnitude is 20 dB/dec x the PLL order. So, a 2nd order PLL will be modeled with 40 dB/dec filter slopes.    

Series Dual-Loop Phase Noise Calculations

The figure below is from the cited white paper and previous post. The two PLLs are in series with each other.  

You will recall that the first PLL, PLL1, is narrowband (NB) and the second PLL, PLL2, is wideband. To calculate the phase noise, we will go left to right through the following steps.

1. Low pass filter the input clock phase noise based on PLL1’s bandwidth (BW).
2. High pass filter the VCXO phase noise based on PLL1’s BW.
3. Add their contributions together in a power sense to yield the total PLL1 phase noise input to PLL2.
4. Low pass filter the input to PLL2 based on PLL2’s BW.
5. High pass filter the VCO phase noise based on PLL2’s BW.
6. Finally, add these last two contributions together in a power sense to yield the total output phase noise

These calculations have been done in the attached spreadsheet Timing_201_7_The_Case_of_the_Dueling_PLLs - Part 2.xlsx. See the “Series Dual-Loop” worksheet. The PLLs are assumed to be 2nd order with the NB PLL BW = 100 Hz and the WB PLL BW = 1 MHz. These parameters can be changed in the spreadsheet, but practically speaking, NB PLLs will be on the order of mHz to kHz. WB PLLs are typically 500 kHz to 2 MHz. 

The resulting plot is as follows.

Nested Dual-Loop Phase Noise Calculations

The figure below is also repeated from the cited white paper and previous post. In this case, the two PLLs are nested with respect to each other. 

Now the inner loop (IL) PLL is WB and the outer loop (OL) PLL is NB. To calculate the phase noise, we will proceed from the “inside out” through the following steps.

1. Low pass filter the XO phase noise based on the IL PLL’s BW.
2. High pass filter the VCO phase noise based on the IL PLL’s BW.
3. Add their contributions together in a power sense to yield the total IL phase noise. Recall that the IL PLL acts as the “VCO” of the OL PLL.
4. Low pass filter the input phase noise based on the OL PLL’s BW.
5. High pass filter the total IL phase noise based on the OL PLL’s BW.
6. Add these last two contributions together in a power sense to yield the total output phase noise. 

These calculations have also been done in the attached spreadsheet Timing_201_7_The_Case_of_the_Dueling_PLLs - Part 2.xlsx. As before, to keep things “apples to apples”, the PLLs are assumed to be 2nd order with the NB PLL BW = 100 Hz and the WB PLL BW = 1 MHz.  See the “Nested Dual-Loop” worksheet. The resulting plot is as follows.

You will note how similar these plots are to each other. Let’s overlay the output phase noise plots together for comparison.
 

Series versus Nested Dual-Loop Phase Noise Plots

The output phase noise plots are overlaid on top of each other for direct comparison. As shown, they look identical. Further, if you experiment with the bandwidths of each PLL, for each topology, the best output phase noise is generally obtained by making PLL1 (or OL PLL) and PLL2 (or IL PLL) narrowband and wideband respectively. Even apart from stability considerations, both topologies benefit from wide separation between the bandwidths.  

Why the Topological Equivalence?

This result is not necessarily to be expected so let’s walk through the steps again and compare the approaches.

1. The total series dual-loop phase noise can be written out as:

LPF_WB (LPF_NB (Input) + HPF_NB (VCXO)) + HPF_WB (VCO)
LPF_WB * LPF_NB (Input) + LPF_WB * HPF_NB (VCXO) + HPF_WB (VCO)

The use of WB and NB notation versus PLL1, PLL2, IL PLL, and OL PLL is to help us make direct comparison between the two sets of calculations. LPF_WB means apply a wideband low pass filter to the phase noise in parenthesis. Two filter terms multiplied indicates applying both filters. 

2. Similarly, the total nested dual-loop phase noise can be written out as:

LPF_NB (Input) + HPF_NB (LPF_WB (XO) + HPF_WB (VCO))
LPF_NB (Input) + HPF_NB * LPF_WB (XO) + HPF_NB * HPF_WB (VCO)

3. If the bandwidths involved are the same, and the VCXO phase noise equals the XO phase noise, then the center terms above are equivalent. These can be thought of as the “bandpass” terms.
    
4. Further, assuming sufficient separation between bandwidths, we can make the following approximations:

LPF_WB * LPF_NB (Input) ≈ LPF_NB (Input)
HPF_NB * HPF_WB (VCO) ≈ HPF_WB (VCO)

In other words, the NB LPF and the WB HPF dominate the calculations for these LPF and HPF terms. This is why these topologies produce equivalent total phase noise when all else is equal.
 

Conclusion

I hope you have enjoyed this Timing 201 article. In this Part 2 follow-up post, I have discussed in more detail how to calculate the phase noise of both the series and nested dual-loop approaches. Finally, by using a simplified example with widely separated bandwidths, we can see that the approaches are essentially equivalent. There are particular advantages to the nested dual-loop approach which arise from an alternate practical implementation that yields phase noise equivalent to the series dual-loop approach.  

As always, if you have topic suggestions or questions appropriate for this blog, please send them to kevin.smith@silabs.com with the words Timing 201 in the subject line. I will give them consideration and see if I can fit them in. Thanks for reading. Keep calm and clock on.

Cheers,
Kevin