Note that (4.23) represents a second-order system, due to the s2 in the denominator of h(s). We will now establish a useful approximation for h(s) by consider the following polynomial.
Therefore, if k >> w1 we can substitute the right hand side of (4.27) for the denominator in (4.23) yielding
Equation (4.28) is a very convenient result because it reduces the 2nd-order loop given by (4.23) into a simple single-pole, lst-order system.
4.4. Loop Bandwidth
The loop-bandwidth determines which input phase jitter will be tracked and which will be rejected or minimized. It is desirable to track low frequency phase jitter on the input clock (i.e., wander) in order to ensure bit timing is maintained in elastic-stores and receive buffers.
From (4.28) the -3dB bandwidth is readily seen to be approximately equal to
The associated phase-error transfer function is
4.5. Rate of Closure (Stability)
The rate of closure is defined as the slope of the open-loop gain as it crosses the 0dB axis (Fig. 4-5). A rate of closure of -40dB per decade results in a marginally stable system2, because it consists of two uncompensated poles, each one contributing 90º of phase shift. Recall that for loop stability, the open-loop phase shift must be less than 180º at unity gain (i.e., 0dB). So to ensure stability, a rate of closure of -20dB or less is desirable.
Now consider the open-loop gain given by (4.21) for very low frequencies such that w << K and w << w1. For this case g(s) is proportional to 1/s2.
The double pole at D.C., causes the open-loop gain to start out with a -40dB per decade slope (Fig. 4-5). Equation (4.28) shows that the 0dB axis is crossed at k radians. Thus to ensure stability, the zero in the loop-filter w1 must be less than k.
The ratio α = k/w1 is defined as the damping ratio, which plays a similar role as the damping constant. To ensure stability a damping ratio of at least 4 is typical7
4.6 Peaking and Jitter Accumulation
To ensure stability of the loop, we require k > w1. However, even if the loop is stable the proximity of w1 to k will affect peaking in the closed-loop transfer function and it may be desirable to make w1 much less than k.
In most cases, peaking in the transfer function should be kept to a minimum by controlling the damping ratio α. For example, when cascading a number of PLL's, such as in a chain of repeaters, peaking in the transfer function can cause jitter to accumulate.1, 4, 7
The magnitudes of the closed-loop gain 20Log|h(s)|, for a PLL with k = 2pi 103 and various damping ratios are shown in Fig. 4-6. As predicted by (4.29) the loop bandwidth is approximately k/2pi.
Consider a single PLL with transfer function h(s). Jitter is contributed at each stage in the chain. The jitter introduced in the kth PLL will propagate through k stages. Assuming the jitter source ¢s(s) is the same in each PLL, then the composite transfer function is 4,7
The magnitudes of the closed-loop gain 20Log|h(s)| for 50 cascaded PLLs is shown in Fig. 4-7.The total mean-squared jitter is proportional to the area under the curves. Notice that decreasing the loop bandwidth k, and/or increasing the damping ratio, α will reduce the accumulated R.M.S. jitter.
5. Transient Response
The following sections derive equations for the transient response of the CP-PLL; specifically when a step-change in phase and a step-change in frequency are applied to the input.
5.1. Response to a Step in Phase
The response to a step change in phase ΔØ u(t) is found by multiplying (4.9) and (4.31) yielding
From (4.12) we already know the steady-state phase-error for this case is zero.
The transient response is determined by summing the residues of Øe(s) 9
From which the transient response for a step change in phase of magnitude ΔØe is
If we impose the condition that k >> w1 then the phase-error is closely approximated by
Given a charge-pump PLL with α = 4 and loop bandwidth of 1 kHz, the transient phase response for a step change in phase of 10 radians is shown in Fig. 5-1. The solid line is exact and corresponds to (5.5), the dashed line is approximate and corresponds to (5.6). The figure confirms the steady-state error is zero.
The response to a step change in frequency of magnitude Δw u(t) is found by multiplying (4.15) and (4.31) which gives
The steady-state phase error is
The steady-state phase-error is zero, as expected, since the loop-filter has a pole at D.C.
The transient response is determined by summing the residues of Øe(s)
From which the transient response is
If we impose that k >> w1 then the phase-error is approximated by