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10
SP8861
Fig. 9 Standard form of second order loop filter
Fig. 10 Modified form of second order loop filter
LOOP CALCULATIONS
Many frequency synthesiser designs use a second order
loop with a loop filter of the form shown in Fig. 9.
In practice, an additional RC time constant (shown dashed
in Fig. 9) is often added to reduce noise from the amplifier. In
addition, any feedthrough capacitor or local decoupling at the
VCO will be added to the value of C
. These additional
components in fact form a third order loop and, if the values
are chosen correctly, the additional filtering provided can
considerably reduce the level of reference frequency sidebands
and noise without adversely affecting the loop settling time.
The calculations of values for both types of loop are shown
below.
Second Order Loop
For this filter, two equations are required to determine the
time constants
t
1
(= C
1
R
1
) and
t
2
(= C
1
R
2
); the equations are:
…(1)
…(2)
K
u
K
0
v
n2
N
t
1
=
2
z
v
n
t
2
=
where
K
u
is the phase detector gain factor in V/radian
K
0
is the VCO gain factor = 2
p
3
10MHz/V
N
is the division ratio from VCO to reference frequency
v
n
is the natural loop frequency = 500Hz
z
is the damping factor = 0·7071
The SP8861 phase detector is a current source rather than
a conventional voltage source and has a gain factor specified
in
μ
A/radian. Since the equations deal with a filter where R
1
is feeding the virtual earth point of an operational amplifier
from a voltage source, R
sets the input current to the filter –
similar to the circuit shown in Fig. 10 – where a current source
phase detector is connected directly to the virtual earth point
of the operational amplifier.
The equivalent voltage gain of the phase detector can be
calculated by assuming a value for R
and calculating a gain
in V/radian which would produce the set current.
The digital phase detector used in the SP8861 is linear
over a range of 2
p
radians and therefore the phase detector
gain is given by:
Phase detector current setting
2
p
For R
1
= 1k
and assuming a value of phase detector current
of 50
μ
A, the phase detector gain is therefore:
50
μ
A
2
p
=
0·00796V/radian
μ
A/radian
+
R1
C1
R2
R3
C2
FROM PHASE
DETECTOR
TO VCO
+
C1
R2
PHASE
DETECTOR
K
u
=
K
u
=
3
10
3
This value can now be inserted in equation 1 to obtain a value
for C
1
and equation 2 used to determine a value for R
2
.
Example
Calculate values for a second order loop with the following
parameters:
Frequency to be synthesised
Reference frequency
=100kHz
= 800MHz
=
8000
0·0796
3
2
p3
10
6
(2
p3
500)
2
3
8
3
10
3
∴
t
1
=
6·334
μ
s
Division ration N
=
From equation (1),
t
1
=
800MHz
100kHz
From equation (2),
t
2
=
Now, since
t
1
=
C
1
R
1
,
C
1
=
2
3
0·7071
2
p3
500
∴
t
2
=
450
μ
s
6·334
3
10
2
6
∴
C
1
=
6·33nF
10
3
and, since
t
2
=
C
1
R
2
,
R
2
=
∴
R
2
=
71k
4·5
3
10
2
4
6·33
3
10
2
9
Third Order Loop
The third order loop is normally as shown in Fig. 11. Fig. 12
shows the circuit redrawn to use an RC time constant after the
amplifier, allowing any feedthrough capacitance on the VCO
line to be included in the loop calculations. Where the modified
form in Fig. 12 is used, it is advantageous to connect a small
capacitor C
X
of typically 100pF (shown dashed) across R
2
to
reduce sidebands caused by the amplifier being forced into
non-linear operation by the phase comparator pulses
Three equations are required to determine the time
constants
t
1
,
t
2
, and
t
3
, where
for Fig. 11
and for Fig. 12
The equations are:
t
1
=
C
1
R
1
t
2
=
R
2
(C
1
1
C
2
)
t
3
=
C
2
R
2
t
1
=
C
1
R
1
t
2
=
C
1
R
2
t
3
=
C
2
R
3
t
2
=
1
v
n2
t
3
2
…(4)
2
tan
F
0
1
t
3
=
…(5)
v
n
…(3)
t
1
=
K
u
K
0
v
n2
N
1
1
v
n
1
1
v
n
2
t
2
2
t
3
2
2
1
2
1
cos
F
0