V2 Control

8/12/2019 V2 Control

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Modeling of V2Current-Mode Control

Jian Li and Fred C. Lee

Center for Power Electronics SystemsThe Bradley Department of Electrical and Computer Engineering

Virginia Polytechnic Institute and State University

Blacksburg, VA 24061 USA

Abstract-Recently, the V2type of constant on-time control has

been widely used to improve light-load efficiency. In V2

implementation, the nonlinear PWM modulator is much morecomplicated than usual, since not only is the inductor currentinformation fed back to the modulator, but the capacitor voltageripple information is also fed back to the modulator. Generallyspeaking, there is no sub-harmonic oscillation in constant on-time control. However, the delay due to the capacitor rippleresults in sub-harmonic oscillation in V2constant on-time control.So far, there has been no accurate model to predict instabilityissue due to the capacitor ripple. This paper presents a newmodeling approach for V2 constant on-time control. The powerstage, the switches and the PWM modulator are treated as a

single entity and modeled based the describing function method.The model for the V2 constant on-time control achieved by thenew approach can accurately predict sub-harmonic oscillation.Two solutions are discussed to solve the instability issue. Theextension of the model to other types of V2current-mode controlis also shown in the paper. Simulation and experimental resultsare used to verify the proposed model.1

I. INTRODUCTIONCurrent-mode control architecture has been widely used for

years [1]-[6]. For practical implementation, the equivalent

series resistance (ESR) of the output capacitor can be used as

the sensing resistor [7], which means that the output voltage

ripple, which includes the current information, can be directly

used as the ramp for the duty cycle modulation, as shown inFig. 1. Later, the control structure can be implemented with

an additional compensation loop, which is called the V2

current-mode control [8], [9], as shown in Fig. 2.

Based on different modulation schemes, these two

architectures consist of constant-frequency peak voltage

control, constant-frequency valley voltage control, constant

on-time control, constant off-time control and hysteretic

control. Among all of these control structures, the constant

on-time control is the most widely used to improve light-load

1This work was support by Analog Devices, C&D Technologies,CRANE, Delta Electronics, HIPRO Electronics, Infineon, Intel,International Rectifier, Intersil, FSP-Group, Linear Technology,LiteOn Tech, Primarion, NXP, Renesas, National Semiconductor,

Richtek, Texas Instruments.This work also made use of Engineer Research Center Shared

Facilities supported by the National Science Foundation under NSFAward Number EEC-9731677.

Any opinions, findings, and conclusions or recommendationsexpressed in this material are those of the authors and do notnecessarily reflect those of the National Science Foundation.

efficiency, since the switching frequency can be lowered to

reduce the switching-related loss, as shown in Fig. 3.

Figure 1. Ripple regulator

Figure 2. V2control architecture

Figure 3. V2constant on-time control structure

In the V2 implementation, the nonlinear PWM modulator

becomes much more complicated, because not only is the

inductor current information fed back to the modulator, but the

capacitor voltage ripple information is fed back to the

modulator as well. Generally speaking, there is no sub-

harmonic oscillation in the constant on-time control. However,

the delay due to the capacitor ripple results in sub-harmonic

oscillation in the V2constant on-time control, as shown in Fig.

4. The system is stable when used with the OSCON capacitor,

because the inductor current information dominates the total

output voltage ripple; meanwhile sub-harmonic oscillation

occurs when the ceramic capacitor is used with the constant

on-time control, since the capacitor voltage ripple is too large.

This phenomenon also occurs in peak voltage control [8].

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(a)

(b)Figure 4. Sub-harmonic oscillation in V2constant on-time control:

(a) OSCON capacitor (560F/6m), and (b) ceramic capacitor(100F/1.4m)

Modeling of the V2control becomes a challenge due to the

complexity of the PWM modulator. Previously, R. Ridleys

model [10] was widely used to design the converter with

current-mode control. However, its not reasonable to extendR. Ridleys model to the V2implementation, since his model

is based on discrete-time analysis, which does not consider the

influence of the capacitor ripple. This is why the models used

in [11]-[13] cant accurately predict the influence from the

capacitor ripple in the peak voltage control; all of them are

extension of R. Ridleys model. In [14], the Krylov-

Bogoliubov-Mitropolsky (KBM) algorithm is used to improve

the accuracy of the model, but numerical computation is

required. So far, there is no accurate model for V2current-

mode control.

Recently, a new modeling approach has been proposed to

model current-mode control [15]. In this modeling approach,

the describing function (DF) method [16]-[18] is used tomodel the non-linear current-mode modulator to obtain the

transfer function from the control signal vc to the output

voltage vo. A similar methodology can be applied here as well.

In Section II, constant on-time control is used as an example

to illustrate the modeling process and the prediction of the

sub-harmonic oscillation is shown in the proposed model.

Then, Section III describes two solutions to solve the

instability issue. In Section IV, the extension of this model to

other V2 types of current-mode control is presented.

Simulation and experiment are used to verify the model in

Section V. Finally, a summary is given in Section VI.

II. PROPOSED MODELING APPROACH FOR V2CONSTANT ON-TIME CONTROL

The V2 constant on-time control is used to exemplify the

proposed modeling approach. As shown in Fig. 5, the non-

linear constant on-time modulator consists of switches, the

output voltage, the comparator and the on-time generator. Its

reasonable to treat these components as a single entity to

model instead of breaking them into parts. In the proposed

modeling approach, the describing function (DF) method is

used to model the non-linear current-mode modulator to

obtain the transfer function from the control signal vc to the

output voltage vo.

Figure 5. Modeling Methodology

As shown in Fig. 5, a sinusoidal perturbation with a small

magnitude at the frequency fmis injected through the control

signal vc; then based on the perturbed output voltage

waveform, the describing function from the control signal vc

to the output voltage vo can be found by mathematical

derivation. Before applying the DF method, it is necessary to

make several assumptions: (i) the magnitude of the inductor

current slopes during the on-period and the off-period stay

constant separately; (ii) the magnitude of the perturbation

signal is very small; and (iii) the perturbation frequency fmand

the switching frequency fsare commensurable, which meansthat Nfs = Mfm, where N and M are positive integers.

Following the modulation law of V2constant on-time control,

the duty cycle and the output voltage waveforms are shown in

Fig. 6.

Because the on-time Ton is fixed, the off-time Toff ismodulated by the perturbation signal vc(t):

)2sin()( 0 ++= tfrrtv mc , where r0 is the steady-state DC

value of the control signal,0r is the magnitude of the

perturbation, and is the initial angle. Based on the

modulation law, it is found that:

Figure 6. Perturbed waveform

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)()(

)1(1

)(

/]/)()([)()(

)1(1

iofffioffic

o

Tt

Tt LoLonnioffic

TsTtv

CdtRtvtiTsTtv ioffi

ioffi

++=

++ +

+

(1)

where, Toff(i) is the ith cycle off-time, soinCon LVVRs /)( = ,

soCof LVRs /= , Lsis the inductance of the inductor, RCois the

ESR of the output capacitors, Co is the capacitance of the

output capacitors, RL is the load resistor, iL(t) is the inductor

current, and vo(t) is the output voltage. Assuming

)()( ioffoffioff TTT += , where Toffis the steady state off-time, and

Toff(i)is the ithcycle off-time perturbation, tican be calculated

as:

=++=

1

1 )())(1(

i

k koffoffoni TTTit . Based on (1), it is found

that:

oLm

m

ioffic

m

ioffic

oLm

m

ioffic

m

ioffic

iofficioffic

iofficioffic

i

Coo

offon

i

Coo

off

f

CRf

fTtv

fTtv

CRf

fTtv

fTtv

TtvTtv

TtvTtv

TRC

TTT

RC

Ts

++

++

+

++

++

++

++=

+

+

2

)22

()22

(

2

)22

()22

(

)]()([

)]()([

])2

21()

21[(

)1(1)2(2

)()1(1

)1(1)2(2

)()1(1

1

(2)

The perturbed duty cycle d(t) and the perturbed inductor

current iL(t) can be expressed by (3) and (4):

])()([)(1

)()(=

=M

i

onioffiioffi TTttuTttutd

(3)

00 ])([)( Lt

s

o

s

in

L idtL

V

tdL

V

ti += (4)where, u(t)=1 when t>0, and iL0 is the initial value of the

inductor current. Then, the Fourier analysis can be performed

on the inductor current:

++ =

onMoffMm

TTttfj

LmiLm dtetiNfjc)(

0

2

)( )(/2 (5)

where, cm(iL) is the Fourier coefficient at the perturbation

frequency fm for the inductor current. Based on the result in

[15], the coefficient can be calculated as (6):

ms

j

in

Tfj

Coo

offon

Coo

off

oLm

j

TfjTfj

f

s

iLm fjL

eV

eRC

TT

RC

T

CRf

eee

s

f

cswm

swmonm

2)

2

21()

21(

)2

1)(1)(1(

2

222

)(

++

=

(6)

where, Tswis the steady-state switching period. Next, Fourier

coefficient cm(vo) of the output voltage vo can be calculated

based on (7):

12)(

)12()()(

++

+=

moCoL

moCoLiLmvom

fjCRR

fjCRRcc

(7)

The Fourier coefficient at the perturbation frequency fmfor

the control signal vc(t) isj

er , so the describing function

from the control signal to the output voltage in the s-domain

can be calculated as (8):

1)()1(

)2

21()

21(

)]/(11)[1)(1(

)(

)(

+++

+

+

+=

sCRRsCRR

sLV

eRC

TT

RC

T

sCRee

s

f

sv

sv

oCoL

oCoL

s

in

sT

Coo

offon

Coo

off

oL

sTsT

f

s

c

o

sw

swon

(8)

This transfer function is effective at frequencies even

beyond half of the switching frequency if there is no outer

loop compensation. A similar method in [15] is used to

simplify the transfer function as (9):

)1(

)1(

)1(

1

)(

)(

2

2

2

22

2

1

2

11

s

Q

s

sCR

s

Q

ssv

sv oCo

c

o

++

+

++

(9)

where, 1= /Ton, Q1= 2/, 2= Tsw/[( RCoCo- Ton/2) ],

and RL>>RCo. The simplification is valid for up to half of the

switching frequency. When the duty cycle is relatively small,

the transfer function can be further simplified as (10):

)1/()1()(

)(2

2

2

22

s

Q

ssCR

sv

svoCo

c

o +++ (10)

From the transfer function, it is clear that the double poles at

half of the switching frequency may move to the right half-

plane according to different capacitors parameters. The

critical condition for stability is RCoCo>Ton/2, which clearly

shows the influence of the capacitance ripple.

The control-to-output voltage transfer function comparison

between the peak current-mode control and the V2constant-on

time is shown in Fig. 7. In the peak current-mode control, the

Q factor of the double poles at half of the switching frequency

is determined by the duty cycle: if there no external ramp,when the duty cycle is larger than 0.5, two poles will move to

the right half-plane and the system becomes unstable. In V2

constant on-time control, the Q factor is not only related to the

on-time Ton, but also related to the capacitor parameters. The

critical condition RCoCo>Ton/2 reflects the interaction between

the ESR and the capacitance of the output capacitor, which

means those two parameters must be considered at the same

time. Different types of capacitors result in different system

performance. When fs= 300 KHz and D 0.1, the parameters

of the OSCON capacitors (560F/6m) meet the critical

condition, so the system is stable. However, the parameters

of the ceramic capacitors (100F/1.4m) cannot satisfy the

critical condition, so sub-harmonic oscillation occurs, as

shown in Fig. 4.

The output impedance can be also derived based on similar

methodology. As shown in Fig. 8, a sinusoidal perturbation

with a small magnitude at the frequency fmis injected through

the output current io, then based on the perturbed output

voltage waveform, the describing function from the output

current io to the output voltage vo can be found out by

mathematical derivation.

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In the s-domain, the output impedance is derived as (11):

)1

(]1

)2

21()

21(

)1

)(1)(1(

[)(sC

RsL

V

eRC

TT

RC

T

sCRee

s

fsZ

o

Co

s

in

sT

Coo

offon

Coo

off

o

Co

sTsT

f

s

o

sw

swon

++

+

+

=

(11)

The simplified output impedance is expressed as (12):

)1

(]1

)1(

)1(

)1(

1[)(

2

2

2

22

2

1

2

11

sCR

s

Q

s

sCR

s

Q

ssZ

o

CooCo

o +

++

+

++

(12)

When the duty cycle is relatively small, the outputimpedance can be further simplified as (13):

)1(

)1(]

)1()

2(

2[)(

2

2

2

22

2

2

2

s

Q

s

sCRs

C

DR

C

TTsZ oCo

o

Co

o

onono

++

++ (13)

where, D is the steady-state duty cycle.

The Bode plot of the output impedance is shown in Fig. 9.It is found that the output impedance is very low throughout a

wide frequency range. This is why this control can deal with

the transient response even without the outer loop

compensation.

III. POSSIBLE SOLUTIONS FOR ELIMINATING THE SUB-HARMONIC OSCILLATION

In order to eliminate sub-harmonic oscillation due to the

capacitor ripple using the V2 constant on-time control, two

possible solutions are proposed: the first is adding the

inductor current ramp; the second solution is adding an

external ramp. Detailed analysis of these two approaches is

presented below.

A.

Solution I: Adding the inductor current rampAs discussed above, the capacitor voltage ripple is

detrimental to the system stability, so an additional current

loop can be introduced to enforce the current feedback

information and reduce the influence of the capacitor voltage

ripple, as shown in Fig. 10.

Figure 10. Solution I: adding the inductor current ramp

Following the same modeling methodology, the control-to-

output transfer function can be derived as (14):

)1(

)1(

)1(

1

)(

)(

2

2

2

23

2

1

2

11

s

Q

s

sCR

s

Q

ssv

svoCo

c

o

++

+

++

(14)

where, Ri is the sensing gain of the additional current loop,

soiCof LVRRs /)( += , and }]2/)/{[(3 onoiCosw TCRRTQ += .

By comparing (10) and (14), we see that adding the inductor

current ramp equivalently increases the ESR of the output

capacitors. The sensing gain of the inductor current Rican be

used as a design parameter to eliminate sub-harmonicoscillation for various output capacitors.

The output impedance can also be derived as (15):

)1

(]1

)1(

)1(

)1(

1[)(

2

2

2

23

2

1

2

11

sCR

s

Q

s

sCR

s

Q

ssZ

o

CooCo

o +

++

+

++

(15)

Figure 8. Model methodology for output impedance

(a) (b)Figure 7. Control-to-output transfer function comparison:

(a) peak current-mode control, and (b) V2constant on-time controlFigure 9. Output impedance

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Moreover, when the duty cycle is relatively small, (15) can

be further simplified as (16):

)1(

)1(}1)]2

(2

)1({[

)(

2

2

2

23

2

2

2

s

Q

s

sCRsRRC

T

R

T

CR

D

RsZ

oCoCoi

o

on

i

on

oiio

++

+++

(16)

An example using the ceramic capacitors is shown in Fig.11 and Fig. 12. The parameters are: Vin= 12V, Vo= 1.2V, Ls

= 300nH, Ton = 0.33s, and the output capacitor consists of

eight ceramic capacitors (100F/1.4m). The control-to-

output transfer function and the output impedance are plotted

to show the influence of the inductor current ramp.

Comparing Fig. 9 with Fig. 12, it is found that the output

impedance stays a constant value Ri at low frequency when

there is an additional current loop. In some applications, such

as voltage regulators for microprocessors where constant

output impedance is required, this control structure can be

used to meet the design target. For other applications without

such a requirement, the outer loop compensation can be used

to lower the output impedance in the frequency range withinthe control bandwidth.

B. Solution II: Adding the external rampAs we know, the external ramp is used to eliminate sub-

harmonic oscillation in the peak current-mode control. A

similar concept can be used in V2constant on-time control, as

shown in Fig. 13. The external digital ramp starts to build up

at the end of the on-time period and resets at the beginning of

the on-time period in every switching cycle.

Figure 13. Solution II: adding the external ramp

The control-to-output transfer function can be derived as

(17):

2

2

2

2

21

2

2

2

22

2

2

2

21

2

1

2

11

)1)(1(

)1)(1(

)1(

1

)(

)(

sTCRs

ss

Q

ss

Q

s

sCRs

Q

s

s

Q

s

sv

sv

swoCo

f

e

oCo

c

o

+++++

+++

++

(17)

where, sethe magnitude of the external ramp.

When the duty cycle and the external ramp are small, (17)

can be further simplified as (18):

)1(

)1(

)(

)(

2

2

2

24

s

Q

s

sCR

sv

svoCo

c

o

++

+ (18)

where, }]2/)1/2/{[(4 onoCofesw TCRssTQ += . The Q factor

is damped due to the external ramp.

The output impedance can be derived as (19):

)1

(]1

)1)(1(

)1)(1(

)1(

1

[)(

22

2

2

21

2

2

2

22

2

2

2

21

2

1

2

11

sC

R

sTCRsss

Qss

Qs

sCRs

Q

s

s

Q

s

sZ

o

Co

swoCo

f

e

oCo

o +

+++++

+++

++

(19)

Based on the parameters used in the previous section, the

control-to-output transfer function and the output impedance

are plotted in Fig. 14 and Fig. 15 to show the influence of the

external ramp.

IV. EXTENSION TO OTHER TYPES OF V2CURRENT-MODECONTROL

The proposed model strategy can be extend to other types of

V2 current-mode control structures, including constant off-

time, constant-frequency peak voltage control, and constant-

frequency valley voltage control structures. When usingconstant frequency modulation, an external ramp is added to

help stabilize the system. Peak voltage control is used as

example for further illustration. The parameters are: input

voltage Vin = 12V, switching frequency fs = 300KHz, and

external ramp se= 0. The transfer functions for peak voltage

control are derived as (20) and (21):

Figure 11. The control-to-outputtransfer function with different Ri

Figure 12. The output impedancewith different Ri

Figure 14. The control-to-outputtransfer function with different

external ramps

Figure 15. The output impedancewith different external ramps

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2

2

2

2

212

2

2

25

2/)()1)(1(

)1(

)(

)(

sTss

TsCRsss

Q

ss

Q

s

sCR

sv

sv

sw

fn

offfoCoef

oCo

c

o

+

++++

+

(20)

]1

2)(

)1)(1(

)1([

)1

()(

2

2

2

2

21

2

2

2

25

+

++++

+

+

sTss

TsCRsss

Q

ss

Q

s

sCR

sCRsZ

sw

fn

offfoCoef

oCo

o

Coo

(21)

where, ]))/(2

1/[(15

onfnnsw

swoCo

TsssT

TCRQ

++

+= .

The control-to-output transfer function is plotted based on

different duty cycles and different capacitor parameters, as

shown in Fig. 16 and Fig. 17. One common characteristic of

the peak voltage control and the peak current-mode control is

that sub-harmonic oscillation occurs when the duty cycle is

larger than a critical value. The difference between these two

control structures is that this critical duty cycle value is 0.5 for

the peak current-mode control, while this value for the peak

voltage control is less than 0.5 which is related to the capacitor

parameters. As shown in Fig. 17, different capacitor

parameters may result in sub-harmonic oscillation in the peak

voltage control even when D = 0.1. It is clear that sub-

harmonic oscillation is more likely occurs when using peak

voltage control due to the influence of the capacitor ripple.

Based on the duality principle, the properties of the constant

off-time control and the valley voltage control can be easily

found based on the previous analysis on the constant on-time

control and the peak voltage control.

V.

SIMULATION AND EXPERIMENTAL VERIFICATIONThe SIMPLIS simulation tool is used to verify the proposed

model for V2constant on-time control. The parameters of the

buck converter are as follows: Vin = 12V, Vo = 1.2V, Ton =

0.33s, fs300KHz, and Ls= 300nH. The control-to-output

transfer function and the output impedance are plotted using

the simulation results. As shown in Fig. 18 and Fig. 19, the

proposed model can accurately predict the system response.

The two approaches to eliminate the sub-harmonic

oscillation in V2 constant on-time control are also verified

through the SIMPLIS simulation. For the case of adding an

inductor current ramp, the control-to-output transfer function

and the output impedance are shown in Fig. 20 and Fig. 21.

For the case of adding an external ramp, the results are shown

in Fig. 22 and Fig. 23. All the results show the validity of the

model achieved by the proposed modeling approach.

Red curve: proposed model; Blue curve: Simplis simulationFigure 16. The control-to-output

transfer function with different dutycycles: capacitor (560F/6m)

Figure 17. The control-to-output transferfunction with different capacitor

parameters: D = 0.1

Figure 18. Control-to-outputtransfer function comparison:output capacitor (56F/6m)

Figure 19. Output impedancecomparison: output capacitor

(56F/6m)

Red curve: proposed model; Blue curve: Simplis simulation Red curve: proposed model; Blue curve: Simplis simulation

Figure 20. Control-to-output transferfunction for Solution I (8 output

ca acitors: 100 F/1.4m): Ri= 1m

Figure 21. Output Impedance for SolutionI (8 output capacitors: 100F/1.4m): Ri

= 1m

Figure 22. Control-to-outputtransfer function for Solution II (8out ut ca acitors: 100 F/1.4m)

Figure 23. Output Impedance forSolution II (8 output capacitors:

100 F/1.4m)

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The experimental verification is done based on the

LM34919 evaluation board from National Semiconductor.

The experiment setup is shown in Fig. 24, and the parameters

are: Vin= 12V, Vo= 5V, Fs= 800kHz, Ls= 15F, Rx= Ry=

2.49K, and RL = 10. Experimental results with different

values for Coare shown in Fig. 25. The proposed model can

accurately predict the double poles at half of the switching

frequency. Experiment verification for adding the inductor

current ramp is done based on the same evaluation board with

some modifications as shown in Fig. 26. The inductor

information is sensed using sensing resistor Ri, and the

feedback information is the sum of the inductor current and

the output voltage. Experiment results with different valuesfor Riare shown in Fig. 27. The experiment results agree with

the model results very well.

VI. CONCLUSIONSThis paper presents a new modeling approach for V2

constant on-time control. The power stage, the switches and

the PWM modulator are treated as an entity and modeled

based the describing function method. The model achieved by

the proposed modeling approach can accurately predict sub-

harmonic oscillation due to the capacitor ripple using V2

constant on-time control. Two possible solutions are modeled

and analyzed to solve the instability issue. The extension to

other types of V2current-mode control is also demonstrated.

Simulation and experimental results are used to verify the

proposed model.

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(a) (b)Figure 24. Experiment setup (1)

Figure 25. Control-to-output transfer function comparison: (a) RCo= 0.39, Co = 210F, and (b) RCo= 0.39,Co = 20.47F (Dashed line: proposed model; Solid line: measurement)

(a) (b)Figure 26. Experiment setup (2)

Figure 27. Control-to-output transfer function comparison: (a) Ri= 0.15, Co = 210F, and (b) Ri= 0.39, Co= 210F (Dashed line: proposed model; Solid line: measurement)

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V2 Control