Datasheet

s
2Sf
Z1
+1
s
2Sf
Z2
+1
s
s
2Sf
P1
+1
s
2Sf
P2
+1
G
EA
= K
m
100 1k 10k 100k 1M 10M
-80
-60
-40
-20
0
20
40
60
-360
-320
-280
-240
-200
-160
-120
-80
-40
0
GAIN (dB)
FREQUENCY (HZ)
PHASE (°)
GAIN
PHASE
1
f
ESR
=
2SC
OUT
R
ES
R
O
+ R
DCR
f
LC
=
1
2S
L
OUT
C
OUT
(R
O
+ R
ESR
)
G
PWM
=
V
in
ÂV
ramp
LM21212-2
SNVS715A –MARCH 2011–REVISED MARCH 2013 www.ti.com
The power train consists of the output inductor (L) with DCR (DC resistance R
DCR
), output capacitor (C
0
) with
ESR (effective series resistance R
ESR
), and load resistance (R
o
). The error amplifier (EA) constantly forces FB to
0.6V. The passive compensation components around the error amplifier help maintain system stability. The
modulator creates the duty cycle by comparing the error amplifier signal with an internally generated ramp set at
the switching frequency.
There are three transfer functions that must be taken into consideration when obtaining the total open loop
transfer function; COMP to SW (Modulator) , SW to V
OUT
(Power Train), and V
OUT
to COMP (Error Amplifier).
The COMP to SW transfer function is simply the gain of the PWM modulator.
(10)
where ΔV
RAMP
is the oscillator peak-to-peak ramp voltage (nominally 0.8 V). The SW to COMP transfer function
includes the output inductor, output capacitor, and output load resistance. The inductor and capacitor create two
complex poles at a frequency described by:
(11)
In addition to two complex poles, a left half plane zero is created by the output capacitor ESR located at a
frequency described by:
(12)
A Bode plot showing the power train response can be seen below.
Figure 31. Power Train Bode Plot
The complex poles created by the output inductor and capacitor cause a 180° phase shift at the resonant
frequency as seen in Figure 31. The phase is boosted back up to -90° because of the output capacitor ESR zero.
The 180° phase shift must be compensated out and phase boosted through the error amplifier to stabilize the
closed loop response. The compensation network shown around the error amplifier in Figure 30 creates two
poles, two zeros and a pole at the origin. Placing these poles and zeros at the correct frequencies will stabilize
the closed loop response. The Compensated Error Amplifier transfer function is:
(13)
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