Acquisition State

The acquisition state is the initial state in our PLL design, whose purpose is to find the possible region of reference frequency from 1kHz to 1MHz quickly. Because there is no prior information about this frequency, we have to get useful information from the PFD and handle the information adequately.

The traditional three-state PFD only provides the Up and Down outputs as shown in Section 2.3.1. It is not enough for the requirement of wide capture range. An example is given as Figure 3.1, where we only get the information that the difference between two frequencies is very large from the exaggerated length of Up/Down, but can’t distinguish the difference between these two cases. A PFD which can make an accurate judgment when there is a huge difference between the reference frequency and DCO frequency is necessary.

After some observation, we find it is possible to count the trigger times of positive edge continuously when the Up/Down is determined already. The concept is equivalent to use the reference frequency as the sampling frequency to sample the DCO signal. If one DCO period

freq_1 = 660kHz freq_2 = 15kHz

Up Down

freq_1 = 500kHz freq_2 = 15kHz

Up Down

Figure 3.1 Two cases with large frequency difference

is sampled k times, the trigger times should be k-1, as shown in Figure 3.2. The trigger times can also represent the cycles which the leader exceeds. Although we can’t identify the precise reference frequency according to the trigger times, we know the ratio of the reference frequency with respect to DCO frequency approximately.

On the other hand, we divide the 1kHz-1MHz into multiple bands (Figure 3.3). Each band has its own center frequency, and the difference between adjacent center frequencies is double as frequency increases. Then, we use a Divider that can control the DCO to shift its center frequency from one band to the others. Combining above ideas, we can spend only several DCO signal periods to count the trigger times and then shift the DCO signal to the appropriate center frequency closest to the reference frequency.

Figure 3.2 The trigger positions

1kHz 1MHz

Figure 3.4: (a) The boundary in unit of period (b) The boundary in unit of frequency

The decision rule of trigger times is closely related to the approach of dividing bands.

For simplify, we use the Divider to divide bands, which helps us to use a simple decision rule for selecting the right band according to the trigger times. As shown in Figure 3.4a, the most intuitive approach in dividing bands is to select the middle of two adjacent center frequencies as the boundary. But our design is based on digital circuits, therefore we replace the frequency with the period as time unit, given as

where Tn and Tn+1 are any two adjacent center periods, and Tn+1 is twice of Tn. Then

3 1

4 3

2

= +

= _{n n}

boundary

f f

f

As shown in Figure 3.4b, when the reference frequency is over 4/3 times or under 2/3 times than the current DCO center frequency, it jumps to other band.

For the decision rule of trigger times, we have to know the relationship between trigger times and reference frequency. Suppose the initial phase difference between the DCO frequency and the reference frequency is zero for simplifying the problem, and let the reference frequency be 1.6 times of the DCO frequency. This example is shown in Figure 3.5, where (a) shows the phase difference detected by PFD, and (b) shows them on the unit circle.

We could find that the trigger happens when the phase difference is beyond a cycle. Therefore, we can use a simple equation to calculate the number of trigger times.

When the reference frequency is m times of the DCO frequency, the phase difference ψ will be

Figure 3.5 The relation between triggers and phase differences

0

Thus the total number of trigger times Ntri_total is

 ^{( 1 )} 

But in our plan, the trigger times have to recount once again per cycle, and we turn the Ntri as following:

Finally, we build a table with different m for observing the trend about trigger times, as shown in Table 3.1.

According to Table 3.1, we find that it is almost 3 rounds per cycle about the boundary trigger times because of our way of dividing bands. For example, at m=4/3 if we observe the sum of any three adjacent Ntri that is over 1 (0+0+1), the reference frequency is over 4/3 times of the DCO frequency at least and then double the DCO frequency. Moreover, our purpose is to judge whether the reference frequency is over the boundary or not, therefore we just have to observe two rounds. We take the maximum sum of any two adjacent k points as a threshold and show it in Table 3.2. If the sum of two trigger times is over the threshold, the DCO frequency band will be changed immediately.

Table 3.1 The trigger times for several m

multiples trigger times k

m=16/3 4 4 5 4 4 5 4 4 5 4 … m=4 3 3 3 3 3 3 3 3 3 3 … m=8/3 1 2 2 1 2 2 1 2 2 1 … m=2.5 1 2 1 2 1 2 1 2 1 2 … m=2 1 1 1 1 1 1 1 1 1 1 … m=1.6 0 1 0 1 1 0 1 0 1 1 … m=4/3 0 0 1 0 0 1 0 0 1 0 …

Table 3.2 The decision rule of band change

multiples thresholds change level

m=32/3 20 *16

m=16/3 9 *8

m=8/3 4 *4

m=4/3 1 *2

We spend two DCO periods to determine where the reference frequency is located if it leads the DCO frequency. Otherwise, if the reference frequency lags t he DCO frequency, it needs two reference periods, too.

In addition to determining the band which the reference frequency is located, we hope the DCO frequency could be closer to the reference frequency in the acquisition state. Recall in Section 2.3.2, we use a high frequency signal to count the length of Up/Down output of PFD. But which frequency should we choose? According to Figure 3.1 or Figure 3.2, we find the maximum length of Up/Down is limited to a DCO signal period. It inspires us to use the harmonic frequency of DCO signal instead of a constant high frequency signal to count.

Figure 3.6 (a) Use quadruple DCO frequency to count the phase difference (b) Phase difference level

With the harmonic DCO frequency, we can transform the length into the phase ratio easily.

When we select the quadruple DCO frequency to count the Up/Down, both positive and negative edges can be used. In this way, all the lengths are classified into eight levels and these levels are filled within 2π exactly as shown in Figure 3.6. The DCO will change its frequency according to different levels. A larger level makes a greater change. Therefore, the ratio between the Up/Down and the DCO period is more significant than the actual length itself.

Next, we observe the variation of levels. The levels should be converged when the DCO frequency is close to the reference frequency. Here we set a condition that if the same levels appear three times in a row, the acquisition state will be ended. As shown in Figure 3. 7, when the specified condition occurs, the phase difference ψn should be

2

φ

−

φ

φ

Figure 3.7 The phase difference converges to the same level

and

There is about 6.25% difference between fref and fdco after the acquisition state. Naturally, the 6.25% is an approximation because the DCO frequency is varied each time; the actual frequency difference will be much less than 6.25%.

In summary, in the acquisition state we respectively use the reference frequency and the multiple DCO frequency to count the phase difference and then choose the appropriate band.

Next, the system will get into the tracking state.