LMX2332LSLB datasheet(18/21 Pages) NSC

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Application Information

A block diagram of the basic phase locked loop is shown in

Figure 1.

LOOP GAIN EQUATIONS

A linear control system model of the phase feedback for a

PLL in the locked state is shown in

Figure 2. The open loop

gain is the product of the phase comparator gain (K

φ), the

VCO gain (K

VCO/s), and the loop filter gain Z(s) divided by

the gain of the feedback counter modulus (N). The passive

loop filter configuration used is displayed in

Figure 3, while

the complex impedance of the filter is given in

Equation (1).

(1)

The time constants which determine the pole and zero fre-

quencies of the filter transfer function can be defined as

(2)

and

T2 = R2

• C2

(3)

The 3rd order PLL Open Loop Gain can be calculated in

terms of frequency,

ω, the filter time constants T1 and T2,

and the design constants Kφ,KVCO, and N.

(4)

From

Equations (2), (3) we can see that the phase term will

be dependent on the single pole and zero such that the

phase margin is determined in

Equation (5).

φ(ω) = tan −1 (ω • T2) − tan−1 (ω • T1) + 180˚

(5)

A plot of the magnitude and phase of G(s)H(s) for a stable

loop, is shown in

Figure 4 with a solid trace. The parameter

p shows the amount of phase margin that exists at the point

the gain drops below zero (the cutoff frequency wp of the

loop). In a critically damped system, the amount of phase

margin would be approximately 45 degrees.

If we were now to redefine the cut off frequency, wp’, as

double the frequency which gave us our original loop band-

width, wp, the loop response time would be approximately

halved. Because the filter attenuation at the comparison fre-

quency also diminishes, the spurs would have increased by

approximately 6 dB. In the proposed Fastlock scheme, the

higher spur levels and wider loop filter conditions would exist

only during the initial lock-on phase — just long enough to

reap the benefits of locking faster. The objective would be to

open up the loop bandwidth but not introduce any additional

complications or compromises related to our original design

criteria. We would ideally like to momentarily shift the curve

Figure 4 over to a different cutoff frequency, illustrated by

the dotted line, without affecting the relative open loop gain

and phase relationships. To maintain the same gain/phase

relationship at twice the original cutoff frequency, other terms

in the gain and phase

Equation (4) and Equation (5) will have

to compensate by the corresponding “1/w” or “1/w

2” factor.

Examination of equations

Equations (2), (3) and Equation (5)

indicates the damping resistor variable R2 could be chosen

to compensate the “w”’ terms for the phase margin. This im-

plies that another resistor of equal value to R2 will need to be

switched in parallel with R2 during the initial lock period. We

must also insure that the magnitude of the open loop gain,

H(s)G(s) is equal to zero at wp’ = 2wp. K

vco,Kφ, N, or the net

product of these terms can be changed by a factor of 4, to

counteract the w

2 term present in the denominator of Equa-

tion (2) and Equation (3). The K

φ term was chosen to com-

plete the transformation because it can readily be switched

DS012806-14

FIGURE 1. Basic Charge Pump Phase Locked Loop

DS012806-15

FIGURE 2. PLL Linear Model

DS012806-16

FIGURE 3. Passive Loop Filter

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