# RMS phase jitter, not sure the right formula

In derivation of IPN power we normally approximate the area under the noise curve by dividing the area into trapezoids, where each trapezoid represents a specific region. The following plot is one example. Here are some questions:

• Why don't we include the area below 0-10kHz as well? And also why does integration stop at 2fo?
• There's a formula that relates the bandwidth to the IPN power, given as $$\ P_{noise}=S_1+10log(4BW_1) \$$, where $$\S_1\$$, which has the unit dBc/Hz, and $$\BW_1\$$ are specified in the following figure, The document given to me only cited this equation from a conference journal held at an ISSCC public meeting somewhere in 2009. Couldn't find its source unfortunately. Anyway I can relate the RMS phase jitter to the noise power given by the document as such:

$$\sqrt{4S_1BW_1}$$

$$\sqrt{10^{\frac{S_1}{10}}.10^{log(4BW_1)}}$$

$$\sqrt{10^{\frac{S_1+10log(4BW_1)}{10}}}$$

$$\sqrt{10^{\frac{P_{noise}}{10}}}$$

• How does this equation reconcile with the equation given in the first image, in which RMS phase error is reported on as $$\\sqrt{2\times 10^{\frac{A}{10}}}\$$. There seems to be a 2 missing in the document's calculated noise power. So does it really miss a 2?

• Since $$\S_1\$$ is measured at f-3dB it looks to me that some area will be missed from integration. Because $$\S_1\$$ is slightly less than the dc gain of the noise profile. Does that add to the inaccuracy of the second formula for RMS phase jitter?

Brief Background for Phase Noise
Ideal free-running oscillator will have a completely periodic output and the Fourier Transform of the output waveform will have spikes at the fundamental and its harmonics. Because of the noise sources present inside the constituent elements (transistor, resistor etc.) the oscillator output will have some noise as well. The noise on the amplitude gets decayed so we are left with just the phase noise. The Fourier transform of the waveform will now look like this. The phase noise at low frequency offset according to the various Phase Noise models varies as $$\\frac{1}{f^3}\$$. Clearly, these theories are not valid at very low offsets since they predict infinite power at low frequencies. But, usually, we don't care about the phase noise at very low offsets since there effect would be visible only after very long time.
When a PLL loop is added around the oscillator, its phase noise gets high pass filtered and the output phase noise has the following plot.

Why don't we include the area below 0-10kHz as well? And also why does integration stop at 2fo?

The phase noise models are not accurate enough at low frequency offset. The integration is usually done till $$\\frac{f_o}{2}\$$ because we usually care about the noise around the oscillator center frequency.

How does this equation reconcile with the equation given in the first image, in which RMS phase error is reported on as 2×10A10−−−−−−−√. There seems to be a 2 missing in the document's calculated noise power. So does it really miss a 2?

Firstly, I want to clarify that the two images are showing phase noise of different systems. The first image is for the free-running oscillator and the second one the phase noise at the output of a PLL.
I am not sure what you are trying to do with your calculations (your second step doesn't make sense and seems wrong!), but both the noise expressions are calculating the area under the respective curve. Nothing seems to be missing.

Since S1 is measured at f-3dB it looks to me that some area will be missed from integration. Because S1 is slightly less than the dc gain of the noise profile. Does that add to the inaccuracy of the second formula for RMS phase jitter?

No area is missed, the DC component of phase noise at PLL output is $$\S_1\$$. Once again, the first plot is only for a free running oscillator.

Below 10KHz, the PLL noise floor is controlling the phasenoise, which will be assumed to be some ratio of the XTAL reference.

And if you view the phasenoise as the PhaseModulation component (the quadrature energy) of random noise summed with a pure sinusoid, then phasenoise bandwidth will either be sidebands of modulation, or will be additive random noise in some bandlimited positive-feedback system. If there is a 2nd harmonic, then that can sustain sidebands as well.

Hope this helps. Its a good question.