MC12181
Figure 8. Closed Loop Frequency Response for
味
= 1
Natural Frequency
10
3dB Bandwidth
0
鈥?0
鈥?0
dB
鈥?0
鈥?0
鈥?0
鈥?0
0.1
1.0
10
Hz
100
1.0 k
Technically, Kv and Kp should be expressed in Radian
units [Kv (RAD/V), Kp (A/RAD)]. Since the component
design equation contains the Kv
脳
Kp term. the 2蟺
cancels and the values can be epressed as above.
Figure 9. Design Equations for the 2nd Order System
In summary, follow the steps given below:
Step 1: Plot the phase noise of crystal reference and the
VCO on the same graph.
Step 2: Increase the phase noise of the crystal reference by
the noise contribution of the loop.
Step 3: Convert the divide鈥揵y鈥揘 to dB (20log 8
脳
N) and
increase the phase noise of the crystal reference by
that amount.
Step 4: The point at which the VCO phase noise crosses the
amplified phase noise of the Crystal Reference is the
point of the optimum loop bandwidth. This is
approximately 15 kHz in Figure 7.
Step 5: Correlate this loop bandwidth to the loop natural
frequency per Figure 8. In this case the 3.0 dB
bandwidth for a damping coefficient of 1 is 2.5 times
the loop鈥檚 natural frequency. The relationship
between the 3.0 dB loop bandwidth and the loop鈥檚
鈥渘atural鈥?frequency will vary for different values of
味.
Making use of the equations defined in Figure 9, a
math tool or spread sheet is useful to select the
values for Ro and Co.
Appendix: Derivation of Loop Filter Transfer Function
The purpose of the loop filter is to convert the current from
the phase detector to a tuning voltage for the VCO. The total
transfer function is derived in two steps. Step 1 is to find the
voltage generated by the impedance of the loop filter. Step 2
is to find the transfer function from the input of the loop filter to
its output. The 鈥渧oltage鈥?times the 鈥渢ransfer function鈥?is the
overall transfer function of the loop filter. To use these
equations in determining the overall transfer function of a PLL
multiply the filter鈥檚 impedance by the gain constant of the
phase detector then multiply that by the filter鈥檚 transfer
function (Figure 10 contains the transfer function equations
for 2nd, 3rd and 4th order PLL filters.)
To simplify analysis further a damping factor of 1 will be
selected. The normalized closed loop response is illustrated
in Figure 8 where the loop bandwidth is 2.5 times the loop
natural frequency (the loop natural frequency is the
frequency at which the loop would oscillate if it were
unstable). Therefore the optimum loop bandwidth is
15 kHz/2.5 or 6.0 kHz (37.7 krads) with a damping coefficient,
味 鈮?/div>
1. T(s) is the transfer function of the loop filter.
T(s)
+
+
+
RoCos
NCo
K pK v
s2
)
RoCos
)
1
)
1
+
2
z
w
o
s
2
w
o2
s
1
)
)
1
2
z
w
o
s
)
1
NCo
KpKv
w
o2
2
z
w
o
1
魯
w
o
+
魯
z
+
Kp Kv
NCo
魯
魯
Co
+
+
N
w
o2
2
z
KpKv
RoCo
w
oRoCo
2
Ro
w
oCo
where Nt = Total PLL Divide Ratio 鈥?8脳N where (N = 25...40)
Kv = VCO Gain 鈥?Hz/V
Kp = Phase Detector/Charge Pump Gain 鈥?A
= ( |IOH| + |IOL| ) / 2
MOTOROLA RF/IF DEVICE DATA
7
1
2
3
4
5
6
7
8
9
