MC12179

6

MOTOROLA RF/IF DEVICE DATA

Figure 6. Graphical Analysis of Optimum Bandwidth

–150

–140

–130

–120

–110

–100

–90

–80

–70

–60

Crystal Reference

15dB NF of the Noise

Contribution from Loop

20*log(256)

VCO

Optimum Bandwidth

10

100

1k

10k

100k

1M

Hz

Figure 7. Closed Loop Frequency Response for

ζ = 1

–60

–50

–40

–30

–20

–10

0

10

0.1

1k

Hz

1

10

100

Natural Frequency

3dB Bandwidth

To simplify analysis further a damping factor of 1 will be

selected. The normalized closed loop response is illustrated

in Figure 7 where the loop bandwidth is 2.5 times the loop

natural frequency (the loop natural frequency is the

frequency at which the loop would oscillate if it were

unstable). Therefore the optimum loop bandwidth is

15kHz/2.5 or 6kHz (37.7krads) with a damping coefficient,

ζ ≈ 1. T(s) is the transfer function of the loop filter.

T(s)

+

RoCos ) 1

NCo

KpKv

s2

) RoCos ) 1

+

2

z

1

wo2

s2

)

2

z

NCo

KpKv

+

1

wo2

³ wo +

KpKv

NCo

³ Co [

KpKv

N

wo2

RoCo +

2

z

wo ³ z +

woRoCo

2

³ Ro +

2

z

woCo

Figure 8. Design Equations for the 2nd Order System

In summary, follow the steps given below:

Step 1: Plot the phase noise of crystal reference and the

VCO on the same graph.

Step 2: Increase the phase noise of the crystal reference by

the noise contribution of the loop.

Step 3: Convert the divide–by–N to dB (20log 256 – 48 dB)

and increase the phase noise of the crystal

reference by that amount.

Step 4: The point at which the VCO phase noise crosses the

amplified phase noise of the Crystal Reference is the

point of the optimum loop bandwidth. This is

approximately 15 kHz in Figure 6.

Step 5: Correlate this loop bandwidth to the loop natural

frequency and select components per Figure 8. In

this case the 3.0 dB bandwidth for a damping

coefficient of 1 is 2.5 times the loop’s natural

frequency. The relationship between the 3.0 dB loop

bandwidth and the loop’s “natural” frequency will

vary for different values of

ζ. Making use of the

equations defined above in a math tool or spread

sheet is useful. To aid in the use of such a tool the

equations are summarized in Figures 9 through 11.

Let:

NCo

KpKv

+ 1

wo2

,RoCo +

2

z

wo

Figure 9. Loop Parameter Relations

Let: Ca + aCo ,Cx + bCo ,A + 1 ) a , and B + 1 ) a ) b

Let: RoCo +

1

w3

,RxCx +

1

w4

,Ro(Ca ) Cx) +

1

w5

Let: K3w3 + wo ,K4w4 + wo ,K5w5 + wo

Freescale Semiconductor, Inc.

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