CPLL Tutorial Cont'd.
Given a charge-pump PLL with α = 4 and loop bandwidth of 1 kHz, the transient response for a step in frequency of 1 kHz is shown in Fig. 5-2. The solid line is exact and corresponds to (5.11), the dashed line is approximate and corresponds to (5.12).
Fig. 5-2 Frequency Step: Δw = 2pi103
The effect of loop bandwidth on acquisition time is demonstrated in the following example.
Given a charge-pump PLL with a α = 4 and loop bandwidths of 2 kHz and 1 kHz, the transient response for a step in frequency of 1 kHz is shown in Fig. 5-3. Notice that the smaller bandwidth requires longer acquisition time.
Fig 5-3 Frequency step: ƒBW=2 kHz, 1 kHz.
5.2. Noise Bandwidth
The input to the PLL will contain some phase noise. The noise is a random signal and is assumed white, with a power spectral density given by 4, 7
The output noise power spectral density of a random process passed through a linear filter is given by8
Given that the PLL closed-loop response is h(s), then the mean square output noise power is
The noise bandwidth Bn is defined7 such that
Substituting (4.28) into (5.15) and solving for Bn(w) yields
From which the noise bandwidth is
Equation (5.18) indicates the output noise can be minimized by decreasing the loop bandwidth.
6. Reference Suppression
6.1. Jitter Due to Leakage
When the loop is locked, both U and D outputs from the PD would ideally be low, causing no charge-pump current to flow. However, in non- ideal circuits, there will be some leakage current Ik, due mostly to the shunt loading of the VCO3. The leakage current must be nulled out by the loop, since the average current into the filter during steady-state must be zero. The loop therefore, adjusts the phase of the VCO to produce small pulses on either the U or D outputs, depending on the polarity of the leakage. The width of the pulses is such that the average pump current iavg equals the leakage current Ik.
Negative leakage flows into the pump and out of the loop-filter. This decreases the VCO control voltage vC which decreases the frequency. In order to compensate for this negative leakage, the loop must provide pulses on the U input (Fig. 6-1).
Positive leakage flows out of the pump and into the loop-filter. This increases the voltage on vC which increases the VCO frequency. In order to compensate for this positive leakage, the loop provides pulses on the D input (Fig. 6-2).
The average pump current will equal the leakage current.
Therefore, the pulse width of the pump current due to leakage is
For the 2nd-order loop, the pulses required to cancel leakage cause the pump current to drive the loop-filter impedance
Note that the reference frequency is typically many orders of magnitude larger than the loop- bandwidth k.
The reference frequency is typical orders of magnitude larger than the loop bandwidth k so the impedance of the capacitor is negligible. Thus, the reference pulses manifest as instantaneous voltage jumps
The frequency of the VCO follows the voltage steps Δv. The frequency excursion for each pulse is
Where the following constants have been used
The phase excursion (radians) during each cycle is called ripple or jitter denoted by Øj (radians).
Substituting (3.17) into (6.8) yields
6.2. Reference suppression Filter
Many applications may be able to tolerate the reference jitter given by (6.8). However, certain applications (e.g., frequency synthesis) may require a reduction in reference jitter to maintain spectral purity of the clock signal 4. This can be done by adding a reference suppression filter, which is simply a capacitor C2 in parallel with the 1st –order loop-filter as shown in Fig. 6-3 4,7
The addition of C2 adds an additional pole at w2 making the PLL a 3rd-order system. The open and closed-loop gains (straight line approx.) for the 3rd-order PLL are shown in Fig. 6-4 and Fig. 6-5 respectively.
It should be observed that if w2 >>k, then C2 will not appreciably affect the loop transfer function for frequencies near or below the loop bandwidth. This implies that the loop can, for practical purposes, be treated like a 2nd-order loop as described by (4.23) with a rate of closure of -20dB per decade as desired.3,7
As a rule of thumb, to ensure stability with moderate peaking, the following constraints are recommended7.
About the Author
Jeffrey Pattavina has worked for 30 years in the data and voice communications industry, specializing in: broadband access, high-reliability IP streaming, and TDM carrier-class communication systems. Mr. Pattavina holds a Master of Science degree in electrical engineering from Northeastern University. He has authored four patents and seven technical publications in electronics, reliability, and communication systems.
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