MC12181 View Datasheet(PDF) - Motorola => Freescale

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Description

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MC12181

125 – 1000 MHz Frequency Synthesizer

Motorola => Freescale 

MC12181

Figure 8. Closed Loop Frequency Response for ζ = 1

Natural Frequency

10

3dB Bandwidth

0

–10

Technically, Kv and Kp should be expressed in Radian

units [Kv (RAD/V), Kp (A/RAD)]. Since the component

design equation contains the Kv × Kp term. the 2π

cancels and the values can be epressed as above.

Figure 9. Design Equations for the 2nd Order System

–20

–30

–40

–50

–60

0.1

1.0

10

100

1.0 k

Hz

To simplify analysis further a damping factor of 1 will be

selected. The normalized closed loop response is illustrated

in Figure 8 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

15 kHz/2.5 or 6.0 kHz (37.7 krads) with a damping coefficient,

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

ǒ Ǔ ǒ Ǔ ǒ Ǔ ǒ Ǔ + ) ) ) + ) ) ) T(s)

RoCos 1

NCo s2

KpKv

RoCos

1

2z

wo

s

1

1

wo2

s2

2z

wo

s

1

ǒ Ǔ ǒ Ǔ Ǹ ǒ Ǔ + ³ + ³ + NCo

KpKv

1

wo2

wo

KpKv

NCo

Co

KpKv

Nwo2

ǒ Ǔ ǒ Ǔ ǒ Ǔ + ³ + ³ + RoCo

2z

wo

z

woRoCo

2

2z

Ro woCo

where Nt = Total PLL Divide Ratio — 8×N where (N = 25...40)

Kv = VCO Gain — Hz/V

Kp = Phase Detector/Charge Pump Gain — A

= ( |IOH| + |IOL| ) / 2

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 8 × N) 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 7.

Step 5: Correlate this loop bandwidth to the loop natural

frequency 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 in Figure 9, a

math tool or spread sheet is useful to select the

values for Ro and Co.

Appendix: Derivation of Loop Filter Transfer Function

The purpose of the loop filter is to convert the current from

the phase detector to a tuning voltage for the VCO. The total

transfer function is derived in two steps. Step 1 is to find the

voltage generated by the impedance of the loop filter. Step 2

is to find the transfer function from the input of the loop filter to

its output. The “voltage” times the “transfer function” is the

overall transfer function of the loop filter. To use these

equations in determining the overall transfer function of a PLL

multiply the filter’s impedance by the gain constant of the

phase detector then multiply that by the filter’s transfer

function (Figure 10 contains the transfer function equations

for 2nd, 3rd and 4th order PLL filters.)

MOTOROLA RF/IF DEVICE DATA

7

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