AD9547SLASHPCBZ datasheet(99/104 Pages) AD

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AD9547

Rev. 0 | Page 99 of 104

CALCULATING THE DIGITAL FILTER COEFFICIENTS

The digital loop filter coefficients (α, β, γ, and δ, as shown in

Figure 38) relate to the time constants (T1, T2, and T3) that are

associated with the equivalent analog circuit for a third-order loop

filter (see Figure 64).

FROM

CHARGE

PUMP

VCO

Figure 64. Third-Order Analog Loop Filter

The design process begins by deciding on two design parameters

related to the second-order loop filter shown in Figure 65: the

desired open-loop bandwidth (fP) and the phase margin (θ).

FROM

CHARGE

PUMP

VCO

Figure 65. Second-Order Analog Loop Filter

An analysis of the second-order loop filter leads to its primary

time constant, T1. It can be shown that T1 is expressible in terms

of fP and θ as

)

cos(

)

sin(

−

where ωp = 2πfP

An analysis of the third-order loop filter leads to the definition

of another time constant, T3. It can be shown that T3 is expressible

in terms of the desired amount of additional attenuation intro-

duced by R3 and C3 at some specified frequency offset (fOFFSET)

from the PLL output frequency.

OFFSET

ATTEN

10 10

−

where ωOFFSET = 2πfOFFSET

Note that ATTEN is the desired excess attenuation in decibels (dB).

Furthermore, ATTEN and ωOFFSET should be chosen so that

≤

With an expression for T1 and T3, it is possible to define an adjusted

open-loop bandwidth (fC) that is slightly less than fP. It can be

shown that ωC (fC expressed as a radian frequency) is expressible

in terms of T1, T3, and θ (phase margin) as follows:

()

[]

⎥

⎦

⎤

⎢

⎣

⎡

−

)

tan(

)

tan(

It can also be shown that the adjusted open-loop bandwidth

leads to T2 (the secondary time constant of the second-order

loop filter), which is expressed as

()

Calculation of the digital loop filter coefficients requires a

scaling constant, K (related to the system clock frequency, fS),

and the PLL feedback divide ratio, D.

125

578

517

where S, U, and V are the integer and fractional feedback divider

values that reside in the profile registers.

Keep in mind that the desired integer feedback divide ratio is one

more than the stored value of S (hence, the +1 term in the

equation for D in this equation). This leads to the digital filter

coefficients given by

(

)

(

) ()

()

()2

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

Calculation of the coefficient register values requires the appli-

cation of some special functions, which are described as follows:

The if() function

y = if (test_statement, true_value, false_value)

where:

test_statement is a conditional expression (for example, x < 3).

true_value is what y equals if the conditional expression is true.

false_value is what y equals if the conditional expression is false.

The round() function

y = round(x)

If x is an integer, then y = x. Otherwise, y is the nearest integer to x.

For example, round(2.1) = 2, round(2.5) = 3, and round(−3.1) = −3.

The ceil() function

y = ceil(x)

If x is an integer, then y = x. Otherwise, y is the next integer to

the right on the number line. For example, ceil(2.8) = 3,

whereas ceil(−2.8) = −2.

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