study of subharmonic oscillation mechanism and effect of circuit propagation delay for buck...

788 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 60, NO. 3, MARCH 2013

Study of Subharmonic Oscillation Mechanismand Effect of Circuit Propagation Delay for BuckConverters With Constant On-Time Control

Ting Qian

Abstract—This paper explores the subharmonic oscillationmechanism for constant on-time control, which has been widelyused for power supply controllers in industry. As well as ana-lyzing the stability criterion, this research reveals that the circuitpropagation delay can significantly increase the demand of

value for stable operation in practical applications. Athorough study is provided based on the inductor current infor-mation and the charge variations of the output capacitor. Thus,critical condition for subharmonic oscillation is derived, and theeffect of circuit propagation delay is also quantified. Based ondetailed analysis, a prototype with constant on-time control andadded inductor current ramp is built. Experimental results verifythe theoretical analysis.

Index Terms—Circuit propagation delay, constant on-time con-trol, subharmonic oscillation.

I. INTRODUCTION

I NTHE FIELDOF dc-dc power converter design, there is anincreasing demand for fast load transient response. For ex-

ample, in computing power systems, it is often desirable to havethe output voltage overshoot and undershoot during load tran-sient to be less than 5% of the averaged output voltage. Conven-tional control methods, such as voltage mode or current modecontrol [1]–[7], usually have difficulty to achieve very fast loadtransient response due to loop speed limitation.Output ripple voltage based control methods [8]–[25] seem

to be efficient solutions to improve the load transient response.Among them, constant on-time control has been widely usedfor power supply controllers in industry [18], [25]. The outputripple voltage is directly used as the ramp for duty ratio mod-ulation. When step load change occurs, feedback of the outputripple voltage makes it possible to adjust the equivalent dutyratio promptly, and thus, minimize the output voltage overshootor undershoot. However, there are also some restrictions for thebasic constant on-time control due to the demand of certain

value for stable operation. When low capac-itors are used, the delay due to the capacitance related ripplecan lead to subharmonic oscillation, whose mechanism is dif-ferent from the subharmonic oscillation or instability of con-

Manuscript received November 14, 2011; revised February 17, 2012; ac-cepted February 27, 2012. Date of publication October 19, 2012; date of currentversion February 21, 2013. This paper was recommended by Associate EditorE. Alarcon.The author is with Texas Instruments, Warwick, RI 02886 USA (e-mail:

[email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TCSI.2012.2215748

ventional control methods [26]–[30]. More recently, constanton-time control with added inductor current ramp [18]–[20],[22], [31] are beginning to see applications in power supply con-trollers. The sensed inductor current signal is combined with theoutput ripple voltage for off-time regulation. The attraction isthat lower is allowed, and thereby, the load transient per-formance is further improved.Previously, several modeling approaches [9]–[18] have

been proposed to estimate the loop stability of ripple voltagebased control. The small signal models in [9]–[13] are derivedbased on the PWM model in [5], which is developed for peakcurrent control. The approach in [14] utilizes Krylov-Bogoli-ubov-Mitropolsky (KBM) algorithm to improve the accuracyof the model. Also, the influence of capacitor ripple is con-sidered to predict the subharmonic instability when lowcapacitors are utilized. More recently, the modeling approachesin [16]–[18] are proposed to accurately predict the subharmonicoscillation for constant on-time control. The approach in [16]utilizes both time- and s-domain analyses to obtain the critical

value for subharmonic oscillation. In [17], [18], the timedomain relation between a small signal sinusoidal perturbationand the resulted output voltage perturbation is calculated andtranslated to frequency domain. According to the analysis,

is considered to be the critical condition( is the on-time with a constant value) to avoid subharmonicoscillation. For constant on-time control, subharmonic oscilla-tion is defined as the phenomenon that the off-time is unable toconverge to a stable value [14]–[18], [32].Of course, the critical condition of is

favorable for understanding the subharmonic oscillation mech-anism. However, this conclusion is derived based on the as-sumption that all the conditions are ideal. In practical imple-mentations, subharmonic oscillation often occurs even when thevalue of is noticeably higher than . This is pri-marily due to the fact that the circuit propagation delay, whichis mainly determined by the comparator propagation delay anddriver delay, influences the stability of constant on-time control.Although [32] has some consideration for the effect of controlcircuit delay, more detailed analysis is needed to understand thedesign trade-offs for power supply controllers (or power con-verters) using constant on-time control.The purpose of this paper is to explore the subharmonic os-

cillation mechanism for constant on-time control. A thoroughstudy is provided based on the inductor current informationand the charge variations of the output capacitor. Thus, crit-ical condition for subharmonic oscillation is derived, and the

1549-8328/$31.00 © 2012 IEEE

QIAN: STUDY OF SUBHARMONIC OSCILLATION MECHANISM AND EFFECT OF CIRCUIT PROPAGATION DELAY 789

Fig. 1. Synchronous buck converter with constant on-time control and addedinductor current ramp. Traditional voltage mode constant on-time control canbe derived when in the above diagram is zero. (Outputripple voltage feedback) is compared with for off-time regulation.

effect of circuit propagation delay is also quantified. Also, con-stant on-time control with added inductor current ramp is uti-lized for analysis in order to further expand the applicability.According to the detailed analysis, subharmonic oscillation canbe avoided when the demand of

is satisfied, where represents the total cir-cuit propagation delay, is the equivalent current sensing re-sistance and is the gain of the output ripple voltage feed-back. is the on-time, which has a constant value. rep-resents the equivalent series resistance of the output capacitor,and is the output capacitance. When there is no direct in-ductor current feedback, the critical condition can be simplifiedas .

II. OPERATION PRINCIPLE OF CONSTANT ON-TIME CONTROL

This section briefly explains the basic operation principle ofconstant on-time control [10], [18], [19]. Fig. 1 shows the struc-ture of a buck converter with constant on-time control and addedinductor current ramp, and Fig. 2 illustrates the circuit’s re-lated waveforms. Obviously, traditional voltage mode constanton-time control can be derived when there is no direct inductorcurrent feedback . In Fig. 1, is the input voltage.The output ripple voltage feedback is achieved by deducting theoutput voltage from the reference voltage , and thengoes through an amplifier (The gain is .). The output of theamplifier is compared with the sensed inductor current signal

to adjust the off-time. When a switching cycle begins,switch is turned on during the on-time, which has a constantvalue, . After that, the off-time begins by turning on switch

and turning off switch . According to the ideal timingshown in Fig. 2(a), the value of is below thevalue of during the off-time. The off-time finishes when

reaches . Then the switching cycle isreset, and the on-time is triggered again.For the convenience of illustration in Section III, the timing of

constant on-time control with circuit propagation delay is shown

Fig. 2. Waveforms for constant on-time control: (a) waveforms in ideal condi-tion; (b) waveforms with circuit propagation delay .

in Fig. 2(b). The circuit propagation delay, , is defined as thetime interval from the time that reaches tothe end of the off-time. In general, the circuit propagation delayprimarily includes the comparator propagation delay, the driverdelay and the delay due to the remaining logic circuits. Amongthem, the delay due to the remaining logic circuits is relativelynegligible. The MOSFET turn-on delay is included in thedriver delay.It should be noted that constant on-time control can use P con-

troller for output regulation mainly due to two reasons: 1) Con-stant on-time control directly utilizes the output ripple voltagefeedback and the sensed inductor current signal, which have lowmagnitudes, as the ramp for PWMmodulation. Therefore, evenby using P controller, high dc accuracy is still achievable forthe output voltage since negligible output voltage error is suffi-cient to adjust the duty cycle. On the other hand, conventionalvoltage mode or current mode control cannot achieve high dcaccuracy with P controller because much higher ramp is usedfor PWM modulation. 2) Constant on-time control utilizes theoutput ripple voltage feedback to promptly reflect the load cur-rent change. Thus, load transient performance is not limited bycrossover frequency of the loop.

III. SUBHARMONIC OSCILLATION ANALYSIS FOR CONSTANTON-TIME CONTROL WITH CIRCUIT PROPAGATION DELAY

As has been introduced in Section I, the critical condition ofto avoid subharmonic oscillation (for


Fig. 3. Inductor current waveform for subharmonic oscillation analysis.

constant on-time control) in previous literatures is still basedon the assumption that the control circuits are ideal [17], [18].However, due to the effect of control circuit propagation delay,subharmonic oscillation often occurs even when the value of

is noticeably higher than .To deal with this problem, this paper provides more thorough

subharmonic oscillation analysis for constant on-time control byquantifyingtheeffectofcircuitpropagationdelay.Fig.3(a)showsthe transientwaveformof a buck converterwith constant on-timecontrol and added inductor current ramp when the inductor cur-rent is perturbed. The inductor current and PWMduty cyclefluc-tuate over and below the steady state value in twoconsecutive cy-cles. Also, Fig. 3(b) shows a simplified control circuit diagram.The inductor current increases and decreaseswith slopes of

and , respectively.Since the ripplesofand are negligible in comparisonwith their dcvalue, andare approximated to be constant values. is the inductance valueof the output filtering inductor. is the peak-to-peak inductorcurrent ripple at steady state. The shadowed current-time areasrepresent the amounts of charge variations on the output capac-itor defined as , respectively.Assume the inductor current is perturbed, and the deviations

of the inductor current (at the end of each cycle) from the steadystate value are , and in three consecutive cycles.The perturbation can be attenuated with time to avoid subhar-monic oscillation when . At the same time,the time points that reaches the value ofare defined as , and for the three consecutive cycles.Based on this definition, it can be concluded that the changesof and are equal during the time in-terval of or . Thus, the relations between ,

and are obtained. Since is re-quired for perturbation attenuation, the conclusion is derivedthat subharmonic oscillation can be avoided when the demandof is satisfied. If theabove demand is not satisfied, the deviation will keep increasinguntil a minimum off-time occurs in every two consecutive cy-cles. Therefore, the eventual subharmonic oscillation waveformshows as double pulses.

In this case, the analysis is based on the assumption that theeffect of ESL is negligible. In fact, the existence of ESL re-duces the delay caused by the capacitor ripple, and is beneficialfor subharmonic oscillation elimination. Therefore, this analysisonly considers the worst case condition, which ignores the pos-itive role of ESL.It should be noted that the proposed approach is also appli-

cable to hysteretic structures [33] derived by the topology inFig. 1. In fact, the structure in [33] has no subharmonic oscil-lation since it utilizes direct information of the inductor cur-rent ramping without the effect of capacitor ripple delay. Ofcourse, because there is no feedback of the ripple voltageto promptly reflect the load current change, its load transient re-sponse is not as fast as conventional hysteretic control.According to Fig. 3(a), the first switching cycle starts at timeand finishes at time . reaches the value of

at , but the inductor current starts to increase at dueto circuit propagation delay. The length of is equal to. The deviation of the inductor current from the steady state

value is equal to at .The second switching cycle starts at time and fin-

ishes at time . At the end of the on-time, the devia-tion of the inductor current from the steady state valueis still equal to . The length of is . Fromto , the changes of and

areand , respectively. Sinceis equal to at and , their changes

(and ) are also equal during this interval. There-fore, the relation between charge variations and inductor currentdeviations can be derived as follows:

(1)

where the charge variation divided by the capacitance value isequal to the capacitor voltage variation.


Similarly, the third switching cycle starts at time and endsat time . At the beginning and the end of the on-time, thedifference between the actual inductor current and the steadystate value is still equal to . The length of is .From to , the changes of and are

and , respectively. Similar as the secondcycle, the relation between charge variations and inductor cur-rent deviations is derived as

(2)

Also, according to Fig. 3(a), the amounts of charge variationscan be derived as

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

By combining (1), (3)–(6), the formula related with andis obtained as (11) at the bottom of the page.

According to (11), the relation between and is de-rived as (12), at the bottom of the page.Thus, to satisfy the demand of for perturbation

attenuation, (13) is derived.

(13)

Similarly, by combining (2), (7)–(10), (14) at the bottom ofthe page, is derived.According to (14), the relation between and is de-

rived as (15), at the bottom of the page. Thus, (16) is derived tosatisfy the demand of .

(16)

It can be clearly seen that (13) matches (16). As a result, inorder to realize , the demands of (13) and(16) need to be satisfied. Alternatively, this implies that subhar-monic oscillation can be avoided when

.

IV. EXPERIMENTAL RESULTS

To verify the proposed subharmonic oscillation analysis forconstant on-time control, a prototype is built by using discretecomponents for the control circuit. The specification is:• input voltage: 12 V;• output voltage: 1.2 V;• output current: 3 A;• switching frequency: 200 kHz.The circuit consists of a buck converter and the control circuit

that utilizes constant on-time control with added inductorcurrent ramp. is set to 6. An inductor with a value of 1

is utilized. The inductor DCR is 5.6 . The equivalent

(11)

(12)

(14)

(15)


TABLE ISTABILITY OF CONSTANT ON-TIME CONTROL (WITH CIRCUIT PROPAGATION DELAY) AT VARIOUS CONDITIONS

Fig. 4. Waveforms to show the total circuit propagation delay: (a) 65ns circuit propagation delay (TLV3502 used as a comparator for off-timeregulation); (b) 420 ns circuit propagation delay (LM393 used as acomparator for off-time regulation). CH1, input pulse for the controlcircuit (1 V/div); CH2, output pulse for the control circuit (5 V/div).

is set to various values to verify the above analysis.The output capacitors have a total capacitance value of 88

. is adjusted (also dominated) by connecting varioushigh precision resistors (Milli-ohm level resistance) in serieswith the capacitors. The purpose of using external resistorsis to prevent the effect of capacitor variation, whichis due to temperature and ripple frequency variation, fromcausing unnecessary confusion.To demonstrate the subharmonic oscillation analysis in

various conditions, two different comparators with differentpropagation delay times (TLV3502 with 4.5 ns delay andLM393 with 300 ns delay) are utilized. The total circuitpropagation delays are shown as 65 ns and 420 ns in Fig. 4.ISL6612 is utilized as the driver. The detailed experimentalresults are listed in Table I. Notice that all the results matchthe conclusion in (13) and (16).

A. Comparator With Negligible Propagation Delay Time (4.5ns) for Off-Time Regulation

Fig. 5 shows the waveforms when using TLV3502 (4.5 nspropagation delay) as a comparator to regulate the off-time. Thetotal circuit propagation delay is measured to be 65 ns, whichis much lower than the constant on-time (approximately 500 ns).Fig. 5(a)–(c) show the waveforms with subharmonic oscilla-

tion. In Fig. 4(a), the is 1 , and has a value of 1. The calculated value of is approx-

imately 176 ns, which is lower than the value of(approximately 315 ns). In Fig. 4(b), the is 1.5 , and

is 1 . The value of (approxi-mately 220 ns) is also lower than the value of . InFig. 5(c), the is 1 , and is equal to 1.5 . Thevalue of is approximately 220 ns, whichis still lower than . According to (13) and (16), theperturbation is unable to attenuate with time, and subharmonicoscillation occurs.Fig. 5(d) shows stable waveformswithout subharmonic oscil-

lation. In this case, the is 3 , and has a value of1 . is approximately equal to 352 ns,which is higher than (315 ns). Accord to (13) and(16), the perturbation can be attenuated with time. By examiningthe captured signals, it can be clearly seen that the converter hasstable operation, and there is no subharmonic oscillation.


Fig. 5. Waveforms to explain subharmonic oscillation mechanism (Inputvoltage is 12 V, output voltage is 1.2 V, and output current is 3 A. The circuitpropagation delay is approximately 65 ns.): (a) , ,

; (b) , , ; (c) ,, ; (d) , , . CH1, output

voltage ripple (200 mV/div); CH2, switch node (5 V/div); CH3, inductorcurrent (5 A/div). (5 ).

Fig. 6. Waveforms to explain subharmonic oscillation mechanism (inputvoltage is 12 V, output voltage is 1.2 V, and output current is 3 A. The circuitpropagation delay is approximately 420 ns.): (a) , ,

; (b) , , ; (c) ,, ; (d) , , . CH1,

output voltage ripple (200 mV/div); CH2, switch node (5 V/div); CH3, inductorcurrent (5 A/div). (5 ).


B. Comparator With 300 ns Propagation Delay Time forOff-Time Regulation

Fig. 6 shows the waveformswhen using LM393 (300 ns prop-agation delay) as a comparator for off-time regulation. ismeasured to be 420 ns. Therefore, the effect of delay time be-comes remarkable.In Fig. 6(a), the is 1 , and is 1 . The value

of (approximately 176 ns) is lower thanthe value of (approximately 670 ns). Therefore,subharmonic oscillation occurs.Fig. 6(b) and (c) also show the waveforms with subharmonic

oscillation. In Fig. 6(b), the is 3 , and is equalto 1 . The value of is approximately352 ns, which is lower than (approximately 670ns). In Fig. 6(c), the is 3 , and is 3 . Thevalue of (approximately 528 ns) is alsolower than . Therefore, due to the effect of circuitpropagation delay, subharmonic oscillation still occurs although

is already noticeably higher than(approximately 250 ns).Fig. 6(d) shows the waveforms when there is no subharmonic

oscillation. In this case, the is 5 , and is 3 .The value of is approximately 704 ns,which is higher than (670 ns). As a result, theconverter has stable operation without subharmonic oscillation,which experimentally demonstrates the analysis in Section III.According to the above description, the experimental results

are in accordance with the detailed analysis in Section III andclearly demonstrate the stability criterion and the effect of cir-cuit propagation delay.

V. CONCLUSION

This paper mainly focuses on the subharmonic oscillationanalysis of constant on-time control. Generally speaking, thesubharmonic oscillation can be avoided when the output ripplevoltage is dominated by the voltage across the equivalent seriesresistance of the output capacitor. But when lowcapacitors are used, the delay due to capacitance related ripplecan lead to subharmonic oscillation. As illustrated in [17], [18],

is considered to be the critical condi-tion. However, this conclusion is derived based on the assump-tion that the control circuits (including the comparators and thedrivers) are ideal. In fact, the circuit propagation delay can influ-ence the stability of constant on-time control and significantlyincrease the demand of value to avoid subharmonicoscillation.This research explores new subharmonic oscillation mecha-

nism by analyzing the effect of circuit propagation delay in de-tail. The study is based on the inductor current information andthe charge variations of the output capacitor. According to thedetailed analysis in Section III, it is concluded that the demandof should be satis-fied in order to avoid subharmonic oscillation. (When there isno inductor current feedback, the critical condition can be sim-plified as ( .) Also, a prototype is builtby using constant on-time control with added inductor current

ramp for a buck converter. Experimental results verify the the-oretical analysis.

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Ting Qian received the B.S. and M.S. degrees fromZhejiang University, Hangzhou, Zhejiang, China, in1999 and 2002, respectively, and the Ph.D. degreefrom Northeastern University, Boston, MA, in 2007,all in electrical engineering.He is currently with Texas Instruments, Warwick,

RI. His research interests include power convertertopologies, on-chip integration and control schemesof power converters, active power filters, and digitalcontrol on power electronics.

study of subharmonic oscillation mechanism and effect of circuit propagation delay for buck...

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