Question 3.5: System Phase Noise Calculation Estimate the phase noise for ...
System Phase Noise Calculation
Estimate the phase noise for the synthesizer designed in Example 3.4. The VCO has a phase noise of −120 dBc/Hz at a 1-MHz offset (it bottoms out at −130 dBc/Hz), and the charge pump puts out a noise current of 10 pA/√Hz. Ignoring PFD, divider, and reference noise sources, plot the phase noise. In addition, what would the phase noise of an equivalent integer-N design be?
From Example 3.4, we know the charge pump current, and we know that the K_{phase} will be 100 μA/rad. Now, in the case of the integer-N synthesizer, the reference must be 1 MHz in order to get a step size of 1 MHz. Therefore, the division ratio will be 4,000. Knowing that we want a loop bandwidth of 150 kHz means that we need a natural frequency of 75 kHz (assuming a damping constant of 0.707), and this means that, for the integer-N design, C_{1} and R are 141.5 pF and 21.2 kΩ respectively. Now, we will assume that the VCO follows the 20 dB/dec rule just outlined. Therefore, we can come up with a linear expression for the phase noise of the VCO based on (3.72):
\varphi ^{2}_{VCO} \left(\Delta \omega \right) =\frac{C} {\Delta \omega ^{2} } +D (3.72)
C=\log ^{-1} \left(\frac{PN_{VCO} } {10} \right) \cdot \Delta \omega ^{2} =\log ^{-1} \left(\frac{-120} {10} \right) \cdot \left(2\pi \cdot 1MHz\right)^{2} =39.5\frac{rad^{4} }{Hz^{2} }
Since the VCO bottoms out at −130 dBc/Hz, we can determine the D term of
the VCO phase noise equation (3.72):
\varphi ^{2}_{VCO} \left(\Delta \omega \right) =\frac{C} {\Delta \omega ^{2} } +D (3.72)
D=\log ^{-1} \left(\frac{PN_{VCO} } {10} \right) =\log ^{-1} \left(\frac{- 130}{10} \right) =10^{-13} \frac{rad^{2} }{Hz}
This can be turned into an equation that has units of voltage instead of units
of voltage squared:
The output noise current from the charge pump can be input- referred by dividing by K_{phase}:
Noise_{CP} =\frac{i_{n} }{K_{phase} } =\frac{10\frac{\rho A}{\sqrt{Hz} } }{100\frac{\mu A}{rad} } =100n \cdot \frac{rad}{\sqrt{Hz} }The noise from the loop filter must also be moved back to the input:
Noise_{LPF} \left(\omega \right) =\frac{1}{K_{phase} } \left|\frac{\sqrt{\frac{4kT} {R} } j\omega }{j\omega + \frac{1}{C_{2} R } } \right|Clearly, noise from the LPF is dependent on filter-component values as well as the phase detector gain. Now, input-referred noise from the loop filter and the charge pump can both be substituted into (3.78),
\frac{\varphi _{noise out} \left(s\right) }{\varphi _{noise I} \left(s\right) } =\frac{\frac{IK_{VCO} } {2\pi \cdot C_{1} } \left(1+ RC_{1} s\right) } {s^{2} + \frac{IK_{VCO} } {2\pi \cdot N} Rs + \frac{IK_{VCO} } {2\pi \cdot NC_{1} } } (3.78)
while noise from the VCO can be substituted into (3.80)
\frac{\varphi _{noise out} \left(s\right) }{\varphi _{noise II} \left(s\right) } =\frac{s^{2} } {{s^{2} + \frac{IK_{VCO} } {2\pi \cdot N} Rs + \frac{IK_{VCO} } {2\pi \cdot NC_{1} } } } (3.80)
to determine the contribution to the phase noise at the output. Once each component value at the output is calculated, the overall noise can be computed (noting that noise adds as power). So, for instance, in the case of the noise due to the charge pump, the output phase noise for the fractional-N case is [from (3.78)]
\frac{\varphi _{noise out} \left(s\right) }{\varphi _{noise I} \left(s\right) } =\frac{\frac{IK_{VCO} } {2\pi \cdot C_{1} } \left(1+ RC_{1} s\right) } {s^{2} + \frac{IK_{VCO} } {2\pi \cdot N} Rs + \frac{IK_{VCO} } {2\pi \cdot NC_{1} } } (3.78)
\varphi _{noise out –CP } \left(s\right) =\frac{2.22\cdot 10^{13} \left(1+ 3\cdot 10^{- 6} s\right) } {s^{2}+ 6.66 \cdot 10^{5} s + 2.22\cdot 10^{11} } 100n\left(\frac{rad} {\sqrt{Hz} } \right)
Therefore, to plot phase noise in dBc/Hz, we take
PN_{CP} \left(\Delta \omega \right) 20 \log \left[\left|\frac{2.22\cdot 10^{13} \left(1+ 3\cdot 10^{- 6} j\Delta \omega \right) } {\left(j\Delta \omega\right) ^{2}+ 6.66 \cdot 10^{5} j\Delta \omega + 2.22\cdot 10^{11} } \right| 100n\right] \left(\frac{rad} {\sqrt{Hz} } \right)The results of this calculation and similar ones for the other noise sources are shown in Figure 3.31. The total phase noise is computed by
\varphi _{total} =\sqrt{\varphi ^{2}_{noise out –CP } + \varphi ^{2}_{noise out –VCO } + \varphi ^{2}_{noise out –LPF } }and is shown in the figure.
If we assume that the numbers given so far have been for SSB phase noise, then we can also compute the integrated phase noise of this design as
IntPN_{rms}=\frac{180\sqrt{2} }{\pi } \sqrt{\int\limits_{f=10KHz} ^{f=10MHz} {\varphi ^{2}_{total} df } } =0.41^{\circ }Note that, in this example, the loop filter noise is quite low and could have been ignored safely. Also note that, due to the frequency response of the filter even in-band, noise from the loop filter is attenuated at lower frequencies. Inside of the loop bandwidth, the total noise is dominated by the charge pump. Note that, out of band, the noise is slightly higher than the VCO noise. This is because the charge pump is still contributing. This can be corrected by making the loop bandwidth slightly smaller and, thus, attenuating the out-of-band contribution of the charge pump by a few more decibels.
With the integer-N numbers, the phase noise is shown in Figure 3.32. Note that with integer N, the noise is completely dominated by the charge pump, both inside and outside of the loop bandwidth. In order to reduce the effect of charge pump noise, the loop bandwidth in this case should be reduced by at least two orders of magnitude. Note also the dramatic change in the in-band phase noise performance between the two designs. While the fractional design has −100 dBc/Hz of in-band noise, this design has a performance of only −67 dBc/Hz. Note that the two numbers are different by 20 log(40), which is the ratio of the two divider values, as would be expected.
