SP8853A/B
C1
C1
R2
R2
R1
FROM PHASE
DETECTOR
PHASE
DETECTOR
−
−
R3
TO VCO
+
+
C2
Fig. 9 Standard form of second order loop filter
Fig. 10 Modified form of second order loop filter
Example
Calculate values for a second order loop with the following
parameters:
LOOP CALCULATIONS
Many frequency synthesiser designs use a second order
loop with a loop filter of the form shown in Fig. 9.
Frequency to be synthesised
Reference frequency
= 800MHz
=100kHz
800MHz
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 C2. These additional
components in fact form a third order loop and, if the values
are chosen correctly, the additional filtering provided can
considerablyreducethelevelofreferencefrequencysidebands
and noise without adversely affecting the loop settling time.
The calculations of values for both types of loop are shown
below.
Division ration N
=
100kHz
= 8000
0·079632p3106
From equation (1),
From equation (2),
t1 =
(2p3500)2383103
t1 = 6·334µs
230·7071
2p3500
t2 =
Second Order Loop
For this filter, two equations are required to determine the
time constants t1 (=C1R1) and t2 (= C1R2); the equations are:
t2 = 450µs
6·33431026
Now, since t1 = C1R1 , C1
=
103
KuK0
vn2N
…(1)
t1 =
t2 =
C1 = 6·33nF
4·531024
6·3331029
2z
vn
…(2)
and, since t2 = C1R2 , R2
=
where
R2 = 71kΩ
Ku is the phase detector gain factor in V/radian
K0 is the VCO gain factor = 2p310MHz/V
Third Order Loop
N
is the division ratio from VCO to reference frequency
vn is the natural loop frequency = 500Hz
is the damping factor = 0·7071
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
linetobeincludedintheloopcalculations. Wherethemodified
form in Fig. 12 is used, it is advantageous to connect a small
capacitor CX of typically 100pF (shown dashed) across R2 to
reduce sidebands caused by the amplifier being forced into
non-linear operation by the phase comparator pulses
z
The SP8853 phase detector is a current source rather than
a conventional voltage source and has a gain factor specified
in µA/radian. Since the equations deal with a filter where R1
is feeding the virtual earth point of an operational amplifier
from a voltage source, R1 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 R1 and calculating a gain
in V/radian which would produce the set current.
The digital phase detector used in the SP8853 is linear
over a range of 2p radians and therefore the phase detector
gain is given by:
Three equations are required to determine the time
t , t , and t , where
constants
1
2
3
for Fig. 11
t1 = C1R1
t2 = R2 (C11C2)
t3 = C2R2
t1 = C1R1
t2 = C1R2
t3 = C2R3
and for Fig. 12
Phase detector current setting
Ku =
µA/radian
The equations are:
2p
1
2
11vn2 t22
11vn2 t32
KuK0
vn2N
For R1 = 1kΩ and assuming a value of phase detector current
of 50µA, the phase detector gain is therefore:
…(3)
…(4)
…(5)
t1 =
t2 =
50µA
3103
Ku =
1
2p
vn2t32
= 0·00796V/radian
1
2tan F0 1
This value can now be inserted in equation 1 to obtain a value
for C1 and equation 2 used to determine a value for R2.
cos F0
t3 =
vn
10
MITEL [ MITEL NETWORKS CORPORATION ]
