Abstract general approach optimizing dynamic response buck conver

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Abstract general approach optimizing dynamic response buck conver

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AND8143/D General Approach Optimizing Dynamic Response Buck Converter
Abstract
general approach optimizing dynamic response buck converters presented. basic theory stabilize buck converter with different types compensation networks introduced detail. Using averaging model computer program, three types compensation networks buck converter examined analyzed. K-factor approach determine compensation network components explained. Finally, practical experiment with popular buck controller computing applications introduced. using presented approach, fast transient response system using type-III compensation network evaluated results high performance system with sufficient stability margin. INTRODUCTION subject stability, which pertains closed-loop frequency response switching regulators, received much attention many papers have been published around subject. them have their implementation method considerations. most practicing engineers, seems cloud mystery shrouds feedback control loop stability. This paper seeks remove that shroud, blending theory, simulation tools practical experiment illustrating general approach stabilize buck converter with least effort. ANALYSIS OPEN LOOP BUCK CONVERTER Figure shows feedback system buck converter. First, transfer functions Gp(s) Pulse Width Modulation (PWM) stage power stage identified. These blocks commonly grouped modulator. Gc(s) compensation network transfer function. will discussed next section. modeling low-frequency behavior power switches square-wave power converters explained [1]. circuit buck
modulator shown Figure model power switches shown Figure That represents frequency equivalent circuit buck modulator. transfer function output, Vout with respect duty ratio,
dVout(s) dD(s) Resr Resr
(eq.
When Resr equation simplified
dVout(s) dD(s) LCs2 ResrCs
Resr)
(eq.
where double poles located
(eq.
zero located
2pResrC
Modulator dVout(s) Gc(s) dVc(s) Compensation Network dD(s) Gp(s) (eq.
Figure Feedback Control Loop Buck Converter
Semiconductor Components Industries, LLC, 2004
April, 2004 Rev.
Publication Order Number: AND8143/D
AND8143/D
Resr
output capacitor introduces zero. this point gain curve slope changes dB/decade slope) phase curve turns back towards -90°. shown Figure stage transfer function
dD(s) dVc(s)
(eq.
Figure Buck Modulator Switches Model
where amplitude ramp stage. Therefore, gain this stage simply input voltage divided From (5), open loop transfer function output Vout with respect compensation network control voltage
dVout(s) dVc(s) LCs2
ResrCs
D*IL D*Vin
Resr
Resr)
(eq.
COMPENSATION NETWORK TYPE
Type-I
Figure Low-Frequency Equivalent Circuit Buck Modulator, (1-D)
simplest form compensation network with single-pole roll shown Figure This called Type-I compensation network. transfer function compensation network Figure
Vout R1C1s
(eq.
GAIN (dB) GAIN Slope) PHASE Slope)
PHASE (DEG)
with crossover frequency,
2pR1C1
Type-I compensation network provides single pole origin gain rolls dB/decade slope) forever, crossing unity gain frequency where reactance equal magnitude resistance Type-I compensation network -270° (-180° phase shift with inverting compensation network included) phase shift throughout slope region. Type-I compensation network used systems where phase shift modulator minimal.
-180
RBIAS VREF Vout
Figure Frequency Response Buck Modulator
Figure shows frequency response typical buck modulator. Note that effect complex conjugate poles L-C, will make gain curve rolls dB/decade slope) phase curve towards -180°. roll-off continues until frequency reaches region around where (Equivalent Series Resistance)
where RBIAS
VREFR1 Vin-VREF
Figure Type-I Compensation Network Schematic Diagram
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GAIN (dB) PHASE (DEG) GAIN (dB) GAIN -180 -180 PHASE -270 PHASE -270 GAIN PHASE (DEG)
Figure Frequency Response Type-I Compensation Network
Figure Frequency Response Type-II Compensation Network
Type-II
(eq.
compensation network Figure offers improved buck converter transient response when converter subject output load changes, opposed slow response Type-I compensation network. Figure shows frequency response Type-II compensation network. zero-pole pair been introduced give region frequency where gain flat phase shift introduced. region with constant gain occurs between break frequencies This region must used loop gain crossover. gain break frequencies presented below.
RBIAS VREF Vout
2pR2C1 2pR2C1C2 2pR2C2
(eq.
(eq.
where
Type-III
compensation network depicted Figure give superior transient response. this circuit, network provides pole origin with zero-pole pairs. shown Figure shows frequency gain decreases dB/decade slope) pole origin. gain becomes constant between zero frequencies, After effects second zero cause gain increase dB/decade slope) until approaching flat again after After magnitude response decreases rate dB/decade slope). closed loop compensation crossover should occur between best results. gains pole-zero frequencies calculated from following equations:
(eq.
where RBIAS
VREFR1 Vin-VREF
Figure Type-II Compensation Network Schematic Diagram
R2(R1
(eq.
2pR2C1 2p(R1 R3)C3
(eq. (eq.
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AND8143/D
2pR2C1C2 2pR3C3
RBIAS VREF Vout (eq. (eq.
networks, shown Figures 11c, zero frequency placed factor below loop crossover frequency pole frequency factor above. Since geometric mean zero pole locations, peak phase boost will occur crossover frequency. widely known that phase boost zero-pole pair inverse tangent ratio measurement frequency zero pole frequency. total phase shift then individual zero pole phase contributions. type-II compensation network, phase boost qboost frequency given equation:
qboost -1(K)
(eq.
From this equation shown that:
where RBIAS Vin-VREF Figure Type-III Compensation Network Schematic Diagram
GAIN GAIN (dB) GAIN GAIN -270 FREQ TYPE PHASE (DEG) FREQ
qboost
(eq.
-180 PHASE
Figure Frequency Response Type-III Compensation Network
GAIN
TYPE
K-Factor
K-factor simple mathematical tool defining shape characteristics transfer function, regardless type compensation network used. K-factor measure reduction gain frequencies increase gain high frequencies, arrived controlling location poles zeros feedback compensation networks Bode plot relation loop crossover frequency Figure shows that, Type-I compensation network always This total lack phase boost corresponding increase decrease gain. Type-II Type-III compensation
FREQ
Figure Bode plot characteristics Type-I compensation network, Type-II compensation network, Type-III compensation network, relation factor.
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Type-III compensation network, phase boost qboost frequency given equation:
qboost
(eq.
loss. gain expressed then compensation network gain simply negative modulator gain, that
(eq.
subsequently,
qboost
(eq.
Equations shown Table provide convenient calculate component values each compensation network type discussed previous section. corresponding schematic, please refer Figure gain required compensation network gain crossover frequency must equal modulator loss.
Table Components Type-I, Type-II Type-III Compensation Networks
Type-I Used Used 2pfcGR1 Type-II User-Selected K2-1 Used K2-1 2pfcGR1 2pfcGR1 Used 2pfcGR1 2pfcGR1 2pfcR1 Type-III
Step Choose desired phase margin (using K-factor approach). This margin amount phase desired unity gain. phase margin should large enough provide well-damped transient response accommodate unforeseen excess phase shift possible variations. Phase margin have range 90°, with being good compromise. Step Calculate required phase boost determine value (using K-factor approach). amount phase boost required from zero-pole pair compensation network given formula:
qboost -90°
(eq.
where: desired phase margin (degrees) modulator phase shift (degrees) Step Choose compensation network type determine value (using K-factor approach). Choose compensation network Type-I when phase boost required, compensation network Type-II when required boost less than more practical requirement less than 70°), compensation network Type-III when required phase boost greater than less than 180°. value calculated from Equation (18) (20) Type-II Type-III respectively. Type-I, always equal Step Calculate component values. Based Equations (7)-(16), calculate values compensation network. Otherwise, derive values using K-factor approach Table described previous sections. SELECTION COMPENSATION NETWORK TYPE Type-I compensation network uses minimum number components achieve necessary phase margin. phase margin adjusted choosing unity gain crossover frequency. This type compensation network used converter topologies that exhibit minimal phase shift prior anticipated unity gain crossover frequency. Topologies include forward-mode regulators, such buck, push-pull, half-bridge full-bridge using either voltage current mode control techniques. These converters exhibit relatively phase shift below pole contributed output filter, phase boost required from compensation network stage. Type-I compensation network relatively poor transient load response time unity gain crossover frequency normally occurs frequency. load regulation outstanding very high gain. This type compensation network commonly used systems that require rapid transient load respond.
VREFR1 NOTE: RBIAS VREF
Synthesis Compensation Networks
basic steps synthesize compensation network stabilize feedback loop recommended follows: Step Choose crossover frequency determine phase shift gain. crossover frequency point where want overall loop gain unity. Remember that higher crossover frequency, better transient response power converter. rule thumb, crossover frequency should high enough provide good dynamic regulation enough avoid sub-harmonic instability noise amplification. However, practical limitations restrict range crossover frequency. theoretical limit half switching frequency, practical considerations have proven that crossover frequency figure less than one-fifth switching frequency good choice. Determine phase shift, modulator gain, crossover frequency, Step Determine required compensation network gain. gain, required compensation network gain crossover frequency must equal modulator
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AND8143/D
Type-II compensation network used converters that exhibit single filter pole frequency maximum phase shift 90°. These converters boost, buck-boost fly-back topologies operating discontinuous mode (DCM) operation. Forward-mode converters with current-mode control also included. pole caused output filter capacitor load resistance occurs extremely frequency. order improve transient response characteristic, loop bandwidth needs extended. adding additional zero before first pole, loop bandwidth greatly extended with phase boost hence overall transient response time greatly improved. Type-III compensation network intended converters that exhibit dB/decade roll-off above poles output filter -180° phase lag. These include forward-mode converters such buck, push-pull, half-bridge full-bridge topologies using voltage mode control techniques. Like Type-II compensation network method, Type-III compensation network introduces zeros into error amplifier reduce steep gain slope above double pole caused filter associated -180° phase shift. This extends loop bandwidth. Type-III compensation network achieve very fast transient response provide more than phase boost. They commonly used systems requiring very fast transient respond. CLOSED FEEDBACK LOOP SYSTEM
Close Loop System with Different Types Compensation Network
Type
buck converter then compensated with Type-II compensation network shown Figure with 165.8 3.96 placing zero around resonant frequency buck modulator pole around switching frequency. Again simulation, frequency response shown Figure closed loop response phase margin unity gain bandwidth 19.78 kHz. transient response much better then last trial when type-II compensation network used, however, phase margin good enough.
180d
Type close loop
open loop SEL>> -200d vp(err) open loop vp(err)-vp(out) vP(out) Phase
real life buck converter shown Figure with Resr 0.25 considered. First all, converter with different compensation networks evaluated. With averaged model proposed Ben-Yaakov [3], simulate open loop transfer function whole closed feedback loop system response. schematic buck converter compensated with Type-I compensation network shown Figure With steps suggested previous sections equations listed Table Equation (7), component values calculated. With RBIAS 9.23 break frequency shown Figure ramp size stage reference error amplifier, VREF From simulation results shown Figure closed loop system phase margin, unity gain bandwidth only 1.1415 kHz. high phase margin results very stable system, bandwidth will result very slow transient response. Another compensation network type should considered.
Gain -100 close loop
vdb(err)
vdb(err)-vdb(out) Frequency
vdb(out)
Figure Closed Loop System Buck Converter with Type-I Compensation Network
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180d 180d
Type close loop
Type close loop
open loop Phase -190d vp(err) vp(out) vp(err)-vp(out) -190d vp(err) vp(out) Phase
open loop
vp(err)-vp(out)
Type close loop open loop Type close loop open loop vdb(out) Frequency vdb(err)-vdb(out)
SEL>>
Gain vdb(err)
SEL>> vdb(err)-vdb(out)
Gain vdb(err)
vdb(out) Frequency
Figure Closed Loop System Buck Converter with Type-II Compensation Network
Figure Closed Loop System Buck Converter with Type-III Compensation Network
Type-III compensation network shown Figure applying equations Table choosing crossover frequency less than switching frequency, gain loss modulator obtained from open loop Bode plot shown Figure Then compensation network gains break frequencies calculated. double zero placed around resonant frequency modulator. first pole placed around zero modulator second pole placed around switching frequency. Component values calculated RBIAS frequency response Type-III compensated buck converter shown Figure closed loop system provides 63.66° phase margin unity gain bandwidth 23.48 kHz. moderate phase margin significantly higher bandwidth provide excellent trade-off between stability fast transient response.
Although compensation network type selected, based phase boost requirement, most cases, converter actually designed with three types compensation networks. major difference transient response closed loop system. Type-III compensation network give fastest transient response among three types compensation network. Choosing value depends modulator output current. When output current large, value arbitrary. output current small, should small order avoid loading effects Vout.
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Testing Closed Loop System with Numerical Tool
Semiconductor developed software simulating buck converter behavior with three types compensation network reviewed this paper. example section, Selection Compensation Network Type, next evaluated using this software. Figure shows open loop Bode plot converter modulator itself. Figures illustrate closed loop converter response with Type-I, Type-II Type-III compensation networks, respectively. Results these simulations agreed well with results from PSpice simulation with averaging model above-mentioned section.
Figure Closed Loop Bode Plot Buck Converter with Type-II Compensation Network
Figure Open Loop Bode Plot Buck Converter
Figure Closed Loop Bode Plot Buck Converter with Type-III Compensation Network
Figure Closed Loop Bode Plot Buck Converter with Type-I Compensation Network
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Feedback Loop System with Practical Controller
Semiconductor provides series synchronous buck controller. NCP5210 computing applications require very fast transient response. well known that type-III compensation network give very fast transient response have good phase margin same time. systems requiring fast response, device designer obviously uses Type-III compensation network rather than other types compensation network.
NCP5210 design includes high bandwidth amplifier. This high bandwidth error amplifier provide fast transient response, but, fastest transient response, should compensated with Type-III compensation network. using component values Type-III compensation network derived previous sections, actual circuit experimental transient response this converter with Type-III compensation network captured Figure
Channel Current sourced into buck converter, A/div Channel Buck converter output voltage, mV/div
Figure Transient Response Buck Converter with Type-III Compensation Network
CONCLUSION closed loop system implemented with different types compensation network. Type-I compensation network give good phase margin, bandwidth usually fast transient systems. Type-II compensation network improve transient response phase boost limited less than 90°. Type-III compensation network provides fast transient response sufficient phase margin ensure system stability, cost circuit complexity. Selection compensation type requires detailed understanding target system. this paper, theory compensation types compensation networks explained detail. K-factor approach feedback loop design introduced, and, through examples simulations, benefits tool highlighted.
REFERENCES Yim-Shu Lee. "Computer-Aided Analysis Design Switch-Mode Power Supplies." Marcel Dekker, Inc. Hong Kong. 1993. Venable, Dean. "The Factor: Mathematical Tool Stability Analysis Synthesis." Proc. Powercon 1983. Diego, H1-1 H1-12. Ben-Yaakov, "Average Simulation Converters Direct Implementation Behavioral Relationships." IEEE Applied Power Electronics Conference (APEC, 1993). 510-516. Moyer, Ole. (April, 2004) "Compensation Calculator Software Tool Voltage Mode Compensation Circuits." Semiconductor.
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Abstract general approach optimizing dynamic response buck conver

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