AN042 EN

Application Note
Kunhung Chen
AN042 – December 2015
The Calculation of Output DC Offset for ACOT™ Control Buck
Converter with Feed-forward Compensator
Abstract
With the increasing focus on reliability and size consideration of converter design, the ceramic capacitors become
more and more popular in NB applications. Hence, the Advanced Constant On-Time (ACOT™) control topology has
been developed to provide stable operation for ceramic output capacitors without complicated external
compensating networks. Generally, the stability is always the top concern for the designer. In many cases, in order
to enlarge the noise margin and transient speed of feedback loop, the traditional voltage divider is replaced with
feed-forward compensator. However, an additional dc offset will be generated on the output voltage which comes
from output voltage ripple and feed-forward compensator due to valley control of output. Especially, the different
pole and zero placement of feed-forward compensator will make the distortion and phase movement on feedback
signal. This may influence the regulating accuracy and maximum value of output voltage. In this application note,
detailed analysis and derivation about dc offset will be presented and discussed.
Table of Contents
1. The Introduction of ACOT™ Control and Feed-forward Compensator ................................................................ 2
2. The Calculation of Output DC Offset in DEM ..................................................................................................... 4
3. The Calculation of Output DC Offset in CCM ..................................................................................................... 7
4. Verification via Simulation and Experiment ........................................................................................................ 8
5. Conclusions ..................................................................................................................................................... 11
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© 2015 Richtek Technology Corporation
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The Calculation of Output DC Offset for ACOT™ Control
Buck Converter with Feed-forward Compensator
1. The Introduction to ACOT™ Control and Feed-forward Compensator
Before starting to calculate the quantity of dc offset, it deserves to reserve some time to introduce and understand
the mechanism of ACOT™ control topology. Moreover, the feed-forward compensator characteristics and effects on
feedback signal will be discussed in the later.
Figure 1. A Buck Converter with ACOT™ Control Method
Figure 1 illustrates a standard ACOT™ control Buck converter. Unlike traditional constant on-time control method
which requires large output capacitor ESR to generate the current ramp signal on feedback voltage for stable
operation, an internal pulse-shaping-regulator (PSR) is applied to generate the equivalent inductor current ramp
voltage. By comparing the composed signal of PSR and feedback voltage with reference voltage, an on-time
one-shot circuit will be triggered as the composed signal below reference voltage. Besides, for constant frequency
operation in CCM, a frequency locked loop is applied to adjust on-time period dynamically. Contrarily, fixed on-time
enables the reduction of switching frequency during the light load operation, and the smaller switching loss will
improve the light load efficiency in DEM. Moreover, faster transient response and no need for additional slope
compensator make ACOT™ control method become more popular.
The operating behavior of ACOT™ control loop in CCM and DEM are demonstrating in Figure 2 and Figure 3
separately. In CCM operation, the PSR circuit generates the ramp signal by subtracting VB with VA to acquire the
signal of VC, where VA and VB are the internal signals via detecting the switching node signal. The composed signal
of VC and reference voltage will then be used to compare with feedback voltage in closed loop control. Meanwhile,
the dc offset of ramp ripple can be properly eliminated by an internal sample and hold circuit of PSR. However, due
to the valley control of output voltage, the output ac ripple will create another dc offset on output voltage. This may
influence the accuracy of control and restrict the design margin of output voltage. In the other hand, while in DEM
operation, the three different conditions of switch conduction are considered and keep ramp voltage flat as inductor
current reduce to zero. This is indispensable for stable loop control in DEM operation. Likewise, the output ac ripple
will contribute additional dc offset on output voltage.
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The Calculation of Output DC Offset for ACOT™ Control
Buck Converter with Feed-forward Compensator
VLX
VLX
iL
iL
I0
I0
VA
VA
SW1/2 open
VB
VB
VFB
VFB
V0
ΔVFB
V0
ΔVFB
VX
VX
VC
VREF
VREF
VC
Figure 2. ACOT™ Control in CCM
Figure 3. ACOT™ Control in DEM
As mentioned in previously, the feed-forward compensator is usually added to improve the noise margin and
transient performance. For a typical feed-forward compensator as Figure 4, one pole and zero are generated to
perform as a high-pass filter which means the gain magnitude will change according to different frequency locations
of output signal. Also there is an additional phase lead on feedback that affects the dc offset on output voltage. The
transfer function from VOUT to VFB can be derived as equation (1), and an example is given to make more easily
realize in Figure 5. The dc gain of A(s) is equal to 1 / (1 + R1 / R2), and staring to increase at the frequency of zero
and then decrease at the frequency of pole. The pair of pole and zero provides a phase leading on feedback signal,
and the maximum phase leading is 51.8˚ for this example. In the example bode plot, the gain and phase are
-13.4dB and 46.7˚ respectively at switching frequency.
s
1
VFB  s 
1
R1  Cff
A s  


R1
s
VOUT  s 
1
1
1
R2
 R1 / /R2   Cff
1
(1)
Feed-forward compensator
VOUT
R1
Buck
Converter
Cff
VFB
R2
GND
Figure 4. Feed-forward compensator in feedback loop
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The Calculation of Output DC Offset for ACOT™ Control
Buck Converter with Feed-forward Compensator
Feedforward Compensator
fs
Gain
Phase
fS = 500kHz, Cff = 22pF,
dB
R1 = 22kΩ, R2 = 3kΩ
60
 10
40
 15
 20
3
110
degree
5
80
20
4
110
5
110
6
110
7
110
0
8
110
Frequency
Figure 5. Bode Plot of A(s)
2. The Calculation of Output DC Offset in DEM
For a Buck converter with constant on-time control and operates in light load, the inductor current may reaches to
zero when off-time is large enough to discharge the inductor current. If the current reaches to zero, the low-side
switch will then turn off and high-side switch remains off. At this moment, there is no current flowing through the
inductor. However, the high-side switch will remain off until output voltage decreases lower than the reference
voltage. During the both off state, the residual charge in the output capacitors is discharged by load current.
Therefore, the switching frequency will vary with the load current in DEM. The relation of switching frequency and
load current can be derived as :
Fs  Io  
2LIo

2  Vin
VinTon
 1

 Vout

(2)
, where the Fs is the switching frequency of converter, VIN is the input voltage of converter, Vout is the output voltage
of converter, Io is the load current and Ton is the on-time period of high-side switch. It can be observed that switching
frequency is proportional to the output load.
Due to the piecewise linear characteristic of inductor current in DEM, the output voltage ripple consists of a lot of
components with different frequencies. It makes it infeasible to look output voltage ripple as a single frequency
signal. Fortunately, all of the periodical signal can be decomposed to a composition of variety of sine and cosine
functions. In the light of linearity and time shift properties of Fourier transform, it makes a sense that multiplying the
gain and phase at specific frequency for each decomposed function when considering about the feed-forward
compensator. Therefore, the output voltage ripple can be expanded as a Fourier series. Before define the steady
state equation of output voltage ripple, the inductor current should be realized. The following equations state the
variation rate of the inductor current at different switching status for a single switching cycle.
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The Calculation of Output DC Offset for ACOT™ Control
Buck Converter with Feed-forward Compensator
For the period during high-side switch is turned on, 0  t  TON :
diL  t,Io 
dt
V  VOUT
 IN
L
(3)
For the period during low-side switch is turned on, TON  t  TON  Toff :
diL  t,Io 
dt
V
  OUT
L
(4)
For the period during all switches are turned off, TON  Toff  t  Ts Io  :
diL  t ,Io 
dt
0
(5)
, where Ts(Io) is the switching period at Io and T off is the turning on duration of low-side switch.
Then the output voltage ripple can be derived from above equations, as stated in below :
VOUT ,ripple  t,Io  
1 t
  iL  t,Io   Io  dt
Co 0 
(6)
, where Co is the output capacitor and it should be noticed that both of iL and Vout,ripple are the functions of time and
Io, the switching period will change with load current. It should be noticed that only AC ripple is considered in
equation (6), the DC value is not important here.
Next step, the Fourier series is adopted for representing the function of output voltage ripple. Following are the
equations of output voltage ripple in Fourier series with N order expression :
VOUT ,ripple _ N  t,Io  
ao Io 
2
N 
 2 nt 
 2 nt  
  an Io   cos 
  bn Io   sin 
 

n 1 
 Ts Io  
 Ts Io   
a0 Io  
Ts Io 
1
  VOUT ,ripple _ N  t ,Io  dt
Ts  Io  Ts Io 
an Io  
Ts Io 
 2 nt 
1
  VOUT ,ripple _ N  t ,Io   cos 
dt
 Ts Io  
Ts Io  Ts Io 


(9)
bn Io  
Ts Io 
 2 nt 
1
  VOUT ,ripple _ N  t,Io   sin 
dt
 Ts Io  
Ts Io  Ts Io 


(10)
(7)
(8)
The function of output voltage ripple can be represented as a series of coefficients time sine and cosine functions of
different frequencies. The coefficients can be derived by multiply to sine or cosine functions of different frequencies
and apply the integral to acquire the average value. Where the a0 (I o) is the coefficient of dc component when output
load current is equal to Io, the an (Io) are the coefficients of cosine functions of different frequencies for different load
current and the bn (Io) are the coefficients of sine functions of different frequencies for different load current.
As the Fourier expression of output voltage ripple is defined, the next step is to use these functions as the input for
feed-forward compensator, A(s). In order to process the multiplication of two functions in time-domain more
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The Calculation of Output DC Offset for ACOT™ Control
Buck Converter with Feed-forward Compensator
efficiently, then apply the convolution property of Fourier transform. The convolution property makes it possible for
two functions in time domain to multiply each other in frequency domain. Therefore, the description of feedback
signal in time domain can be derived as following :
VFB,ripple _ N  t ,Io  
a0  Io 
2
 2n t
  n   
N 
  n 

 A 0    A 
 an Io   cos 
 arg  A 

   





Ts  Io 
n 1  Ts  Io  
  Ts  Io     


 2n t
  n   
N 
  n 

   A
 bn  Io   sin 
 arg  A 

   





T
I
T
I
T
I
 s  o 
n1  s  o  
  s  o     


(11)
, where the A(0) is the dc gain of feed-forward compensator, | A (n / Ts (Io) ) | is the gain at specific frequency of (n /
Ts (Io) ) and arg ( A (n / Ts (Io) ) ) is the phase shift at the frequency of (n / Ts (Io) ).
Because of the feed-forward compensator and output voltage ripple, an additional dc offset will be generated. In
order to describe the value of dc offset, it is a good way to discuss with a picture. As shown in Figure 6, there are
three signals used to describe the origin of this offset , for the first one, the red waveform with solid line, is named
“Original VFB without phase shift”, it displays the feedback signal only with magnitude change but no phase shift. In
this case, there is only partial dc offset produced by the ripple voltage of VFB. For the second one, the red waveform
with dotted line named “Original VFB with phase shift”, the phase shift is also considered, and it can be noticed that
the VFB is lower than the Vc when the gate signal of high-side MOSFET (UG) is triggered. Due to the valley control
mechanism, the final VFB in loop control will then be the blue waveform with solid line, is named “VFB with
Feed-forward in loop”. It can be observed that an additional dc offset, y1, has been generated. The formula of y1
and the average of feedback voltage ripple can be derived as :
y1  VFB,ripple _ N  0,Io 
VFB,ripple _ N _ av Io  
(12)
1 Ts Io 
 VFB,ripple _ N  t,Io  dt  y1
Ts Io  0
VFB with Feedforward in
loop
(13)
Original VFB with
phase shift
Original VFB without
phase shift
VREF
Δy1
VC
Gate
signal
UG
t
Figure 6. The relation of feedback signal and internal ramp signal in DEM
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The Calculation of Output DC Offset for ACOT™ Control
Buck Converter with Feed-forward Compensator
3. The Calculation of Output DC Offset in CCM
When converter operates in CCM, the frequency is well regulated as a constant. Unlike in DEM as discussed before,
inductor current is always above zero. At stable operation, the duty cycle can be decided by the input and output
voltage.
V
D  OUT
VIN
(14)
For the period during high-side switch is turned on, 0  t  DTs :
diL  t 
dt
V  VOUT
 IN
L
(15)
For the period during low-side switch is turned on, DTs  t  Ts :
diL  t 
dt

VOUT
L
(16)
, where Ts(Io) is the switching period at Io and T off is the turning on duration of low-side switch.
Then the output voltage ripple can be derived from above equations, as stated in below :
VOUT ,ripple  t,Io  
1 t
   iL  t   Io  dt
Co 0
(17)
, where Co is the output capacitor, the dc value of Io will not change the switching frequency in CCM, therefore, only
ac ripple will be considered in following derivation.
Same as the analysis in DEM condition, the Fourier series of CCM output voltage ripple is also derived as equation
(18)~(21).
N 
 2 nt 
 2 nt  
a
VOUT ,ripple _ N  t   o   an  cos 
  bn  sin 

2 n 1 
 Ts 
 Ts  
a0 
1 Ts
  VOUT ,ripple _ N  t  dt
Ts Ts
an 
 2 nt 
1 Ts
  VOUT ,ripple _ N  t   cos 
 dt
Ts Ts
 Ts 
(20)
bn 
 2 nt 
1 Ts
  VOUT ,ripple _ N  t   sin 
 dt
Ts Ts
 Ts 
(21)
(18)
(19)
Apply the convolution property to process the multiplication of two functions in frequency domain then transfer back
to time-domain as equation (22).
VFB,ripple _ N  t  
N 
 2n t
  n   
a0
  n 

 A  0     A    an  cos 
 arg  A     
2
T
T
T
n 1
  s    
  s

 s
N 
 2n t
  n
  n 
   A    bn  sin 
 arg  A 
T
T
n 1
s

s


  Ts


AN042
  

   
   

(22)
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The Calculation of Output DC Offset for ACOT™ Control
Buck Converter with Feed-forward Compensator
As depicted in Figure 7, three waveforms are used to describe the origin of additional dc offset which comes from
phase shift. Similar story as describe in DEM section, an additional dc offset value of y2 has been generated and
make final output voltage to locate under or above the preset value.
y 2  VFB,ripple _ N  0 
VFB,ripple _ N _ av 
(23)
1 Ts
 VFB,ripple _ N  t  dt  VFB,ripple _ N  0 
Ts 0
(24)
Original VFB with
phase shift
Δy2
Original VFB without
phase shift
VREF
VC
VFB with Feedforward in
loop
UG
Figure 7. The relation of feedback signal and internal ramp signal in CCM
4. Verification via Simulation and Experiment
The detailed derivation and description of the origin of output dc offset has been given in the previous sections. The
major purpose here is to verify the accuracy of the derived formula. A practical example of RT6220 ACOT™ control
converter is selected to verify the calculation result. The detailed setup for simulation and experiment are listed in
the Table 1. And the results are depicted in Figure 8 and Figure 9. The comparison results between them will be
discussed later.
Table 1. Simulation and Experiment Parameters Setup
Operating Condition
VIN
Vref
Voltage Divider
Cff
DEM (Io = 0.15~1.6A)
7.4V/12V/19V
0.6V
R1 = 162k, R2 = 36k
5pF/22pF
CCM (Io = 3A)
7.4V/12V/19V
0.6V
R1 = 162k, R2 = 36k
1pF~47pF
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The Calculation of Output DC Offset for ACOT™ Control
Buck Converter with Feed-forward Compensator
VIN=12V, VO=3.3V, R1=162k, R2=36k, Cff=5pF,
Io=0.1A to 1.6A
VIN=12V, VO=3.3V, R1=162k, R2=36k, Cff=22pF,
Io=0.1A to 1.6A
3.330
3.360
3.355
Vout_average (V) 1
Vout_average (V)
3.325
3.350
3.345
3.340
3.335
3.330
3.325
3.320
3.315
3.310
3.305
3.320
3.315
3.300
0
0.5
1
1.5
2
0
0.5
1
Output Load (A)
Simulation
Mathematic
1.5
2
Output Load (A)
Experiment
Simulation
Mathematic
(a)
Experiment
(b)
VIN=19V, VO=3.3V, R1=162k, R2=36k, Cff=22pF,
Io=0.1A to 1.6A
VIN=7.4V, VO=3.3V, R1=162k, R2=36k, Cff=22pF,
Io=0.1A to 1.2A
3.340
3.370
3.365
3.335
Vout_average (V)
Vout_average (V)
3.360
3.355
3.350
3.345
3.340
3.335
3.330
3.330
3.325
3.320
3.315
3.325
3.310
3.320
0
0.5
1
1.5
2
0
0.5
Output Load (A)
Simulation
Mathematic
1
1.5
Output Load (A)
Experiment
Simulation
(c)
Mathematic
Experiment
(d)
Figure 8. The comparison of simulation, mathematic and experiment results under DEM condition
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The Calculation of Output DC Offset for ACOT™ Control
Buck Converter with Feed-forward Compensator
VIN=12V, VO=3.3V, R1=162k, R2=36k,
Cff=1pF~44pF
VIN=19V, VO=3.3V, R1=162k, R2=36k,
Cff=1pF~44pF
3.330
3.345
3.340
Vout_average (V)
Vout_average (V)
3.325
3.320
3.315
3.310
3.335
3.330
3.325
3.320
3.315
3.310
3.305
3.305
0
10
20
30
40
50
0
10
Feedforward Capacitor (pF)
Simulation
Mathematic
20
30
40
50
Feedforward Capacitor (pF)
Simulation
Experiment
(a)
Mathematic
Experiment
(b)
VIN=7.4V, VO=3.3V, R1=162k, R2=36k,
Cff=1pF~44pF
3.314
Vout_average (V)
3.312
3.310
3.308
3.306
3.304
3.302
0
10
20
30
40
50
Feedforward Capacitor (pF)
Simulation
Mathematic
Experiment
(c)
Figure 9. The comparison of simulation, mathematic and experiment results under CCM condition
In Figure 8, the average output voltage with regard to different output load under DEM is depicted. From Figure 8(a)
to Figure 8(c), the converter setup with the same feed-forward capacitor of 22pF, and with different input voltage
changes from 7.4V to 19V separately. The result appears that the mathematic results almost fit the simulation result
with deviance smaller than 0.2%. Especially as the duty approaches to 0.5, take Figure 8(a) as an example, due to
the output voltage ripple is very similar to sinusoidal waveform, the mathematic result can well estimate the
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The Calculation of Output DC Offset for ACOT™ Control
Buck Converter with Feed-forward Compensator
simulation result with deviance smaller than 0.07%. However, the experiment result seems much different with
simulation and mathematic results at some conditions. Possible reasons may be the PCB layout, noise disturbance,
the regulation of internal LDO and the parasitic components…etc. Nevertheless, the factors that affect the average
output voltage in experiment are hard to be implemented in simulation and calculation. In other hands, it can be
observed that the feed-forward capacitor will affect the average output voltage. Compare Figure 8(c) with Figure 8(d),
the Cff value of prior one is 22pF, the maximum output voltage occurs as load current is 1A instead of no load, and
the Cff of latter one is 5pF, the maximum output voltage occurs as no load. It represents that the feed-forward
compensator plays an important role in design.
In Figure 9, the relationships of average output voltage and feed-forward capacitor in CCM are presented. As the
input voltage changed, the trend of output voltage will also be different. That is, it can’t take one result as the
reference for every condition. Unlike the comparison results in DEM, the experiment result is much similar to
simulation and mathematic results in CCM for each condition. The maximum deviance between simulation and
experiment result is smaller than 0.15%. Moreover, the impact of thermal issue on voltage regulation is not
considered in this CCM guess model, one may misunderstand it as the same with the reason of feed-forward
compensator.
5. Conclusions
In the application note, the derivation and description of dc offset in both CCM and DEM are well presented. An
example has also given to verify the accuracy of mathematic result, the deviance of simulation and mathematic is
always smaller than 0.2% whether in CCM or DEM. The well prediction of the mathematic result can reduce the
effort and time on simulation setup during design. However, there are still many challenges on the estimation for
practical hardware implement. Like the PCB layout, noise disturbance, the regulation of internal LDO and the
parasitic components…etc. Many of them are not easy to be forecast and modeled. After all, the additional dc offset
which is generated from feed-forward compensator can’t be ignored in practical application, and the optimum design
for converter can be completed through accurate mathematic analysis.
Next Steps
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responsibility for use of any circuitry other than circuitry entirely embodied in a Richtek product. Information furnished by Richtek is believed to be accurate and
reliable. However, no responsibility is assumed by Richtek or its subsidiaries for its use; nor for any infringements of patents or other rights of third parties which may
result from its use. No license is granted by implication or otherwise under any patent or patent rights of Richtek or its subsidiaries.
AN042
© 2015 Richtek Technology Corporation
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