AD9548
CALCULATING DIGITAL FILTER COEFFICIENTS
It can also be shown that the adjusted open-loop bandwidth leads
to T2 (the secondary time constant of the second order loop filter)
expressed as
The digital loop filter coefficients (α, β, γ, and δ (see Figure 40))
relate to the time constants (T1, T2, and T3) associated with the
equivalent analog circuit for a third order loop filter (Figure 66).
FROM
CHARGE
PUMP
1
TO
VCO
R3
T2 =
2
ωC
(
T1 + T3
)
C3
C1
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.
C2
Figure 66. Third Order Analog Loop Filter
30,517,578,125
K =
fS
233
The design process begins by deciding on two design parameters
related to the second order loop filter shown in Figure 67: the
desired open-loop bandwidth (fP) and phase margin (θ).
U
D = S + +1
V
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
FROM
TO
CHARGE
VCO
PUMP
C1
C2
2
2
ωC 2T2 D
T1 K
1+
(
ωCT1
)
)(1+
(
)
ωCT3
)
)
α =
Figure 67. Second Order Analog Loop Filter
2
1+
(
ωCT2
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
⎛
⎞
− 32
1
1
⎜
⎜
⎟
⎟
β =
γ =
δ =
+
fS T1 T2
⎝
⎠
1− sin(θ)
T1 =
− 32
fST1
ωP cos(θ)
where ωP = 2πfP .
32
fST3
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 introduced
by R3 and C3 at some specified frequency offset (fOFFSET) from the
PLL output frequency.
Calculation of the coefficient register values requires the
application of some special functions described as follows:
The if() function
y = if(test_statement, true_value, false_value)
ATTEN
10
10
−1
where test_statement is a conditional expression (for example, x <
3), true_value is what y equals if the conditional expression is true,
and false_value is what y equals if the conditional expression is
false.
T3 =
ωOFFSET
where
.
ωOFFSET = 2πfOFFSET
Note that ATTEN is the desired excess attenuation in decibels.
Furthermore, ATTEN and ωOFFSET should be chosen so that
The round() function
y = round(x)
1
T3 ≤
5 fP
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
2
⎡
⎢
⎤
(
T1 + T3
)
tan(θ)
T1T3 +
(
T1 + T3
)
⎥
−1
2
ωC =
1+
2
T1T3 +
(
T1 + T3
)
[
(
T1 + T3
)
tan(θ)
]
⎢
⎣
⎥
⎦
Rev. 0 | Page 107 of 112