NCP1650
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28
in the introduction to this analysis, this is not analyzed
separately.
The equation for the gain is good for frequencies below
the pole. There is a single pole due to the output filter. Since
the NCP1650 is a current mode converter, the inductor is not
part of the output pole as can be seen in that equation.
Calculating the Loop Gain
At this point in the design process, all of the parameters
involved in this calculation have been determined with the
exception of the pole−zero pair on the output of the voltage
error amplifier.
All equations give gains in absolute numbers. It is
necessary to convert these to the decibel format using the
following formula:
A(dB) + 20 Log
10
(A)
For example, the voltage divider would be:
A +
5.6 k
560 k ) 5.6 k
+ .0099
A(dB) + 20 Log
10
.0099 +*40 dB
The gain of the loop will vary as the input voltage changes.
It is recommended that the compensation for the voltage
error amplifier be calculated under high line, full load
conditions. This should be the greatest bandwidth that the
unit will see.
By necessity, the unity gain (OdB) loop bandwidth for a
PFC unit, must be less than the line frequency. If the
bandwidth approaches or exceeds the line frequency, the
voltage error amplifier signal will have frequency
components in its output that are greater than the line
frequency. These components will cause distortion in the
output of the reference amplifier, which is used to shape the
current waveform. This in turn will cause distortion in the
current and reduce the power factor.
Typically the maximum bandwidth for a 60 Hz PFC
converter is 10 Hz, and slightly less for a 50 Hz system. This
can be adjusted to meet the particular requirements of a
system. The unity gain bandwidth is determined by the
frequency at which the loop gain passes through the 0 dB
level.
For stability purposes, the gain should pass through 0 dB
with a slope of –20 dB/decade for approximately one decade
on either side of the unity gain frequency. This assures a
phase margin of greater than 45°.
The gain can be calculated graphically using the equations
of Figure 43 as follows:
Divider:
Calculate V’/Vo in dB, this value is constant so
it will not change with frequency.
Reference Signal:
Calculate V
ref
/V
e/a
using the peak level
of the AC input signal at high line that will be seen on pin 5.
Convert this to dB. This is also a constant value.
Modulator and Output Stage:
Calculate the gain in dB for
DVo/DVref. Calculate the pole frequency. The gain will be
constant for all frequencies less than f
p
. Starting at the pole
frequency, this gain will drop off at a rate of 20 dB/decade.
Plot the sum of these three values. Figure 43 shows a gain
of 35.5 dB until the pole of the output filter is reached at
0.3 Hz. After that, the gain is reduced at a rate of
20 dB/decade.
Figure 43. Open Loop Gain Less Error Amp
FREQUENCY (Hz)
0.01 0.1 1 10 100 100
30
20
10
0
−10
−20
−30
−40
LOOP GAIN
WITHOUT
ERROR AMP
A typical error amplifier bode plot is shown in Figure 44.
The zero is used to offset the pole of the output filter. The
output filter pole will typically be lower than the unity gain
loop bandwidth, so the zero will be necessary.
This plot shows a forward gain of 7.0 dB at 10 Hz. To
compensate for this the error amplifier should have a gain of
–7.0 dB (0.45) at 10 Hz, and a zero at 0.4 Hz. The gain at
10 Hz is determined by the resistor since it is well past the
zero. The resistor can be calculated by the equation:
R
7
+ A
v
ńG
m
+ .45ń.0001 + 4.5 kW
4.7 kW is the closest standard value. Using this, the
capacitor can be calculated based on the zero frequency of
0.4 Hz. This would give a value for C
7
of:
C +
1
2 @ p @ 4.7 k @ 0.4 Hz
+ 85 mF
Using these values (4.7 kW and 86 mF), the open loop gain
plot would be:
Figure 44. Open Loop Gain of Voltage Loop
FREQUENCY (Hz)
0.01 0.1 1 10 100 100
40
20
0
−20
−40
VOLTAGE
LOOP BODE
PLOT
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