DC-DC Converters Feedback and Control

1 DC-DC Converters Feedback and Feedback generalities Conditions for stability Poles and zeros Phase margin and quality coefficient Undershoot and crossover frequency Compensating the converter Compensating with a TL431 Watch the optocoupler! Compensating a DCM flyback Compensating a CCM flyback Simulation and bench results Feedback generalities Conditions for stability Poles and zeros Phase margin and quality coefficient Undershoot and crossover frequency Compensating the converter Compensating with a TL431 Watch the optocoupler! Compensating a DCM flyback Compensating a CCM flyback Simulation and bench results is Feedback ? A target is assigned to one or several state-variables, Vout= 12 V. A circuitry monitors Voutdeviations related to Vin, Iout, T etc. If Voutdeviates from its target, an error is created and fed-backto the power stage for action.

2 The action is a change in the Control variable: duty-cycle (VM), peak current (CM) or the switching voltageVinDC-DCOutput voltageVoutcontrolactionCompensating for the converter shortcomings!VoutRthVthInput Feedback Implementation Voutis permanently compared to a reference voltage Vref. The reference voltage Vrefis precise and stable over temperature. The error, , is amplified and sent to the Control input. The power stage reacts to reduce as much as it = +-+-VinVoutRupperRlowerError amplifier - GVrefModulator - GPWMdPower stage - HVp +-+-VinVoutRupperRlowerError amplifier - GVrefModulator - GPWMdPower stage - HVp Feedback generalities Conditions for stability Poles and zeros Phase margin and quality coefficient Undershoot and crossover frequency Compensating the converter Compensating with a TL431 Watch the optocoupler!

3 Compensating a DCM flyback Compensating a CCM flyback Simulation and bench results or Negative Feedback ?H(s)Vin(s)Vout(s)+ Do we want to build an oscillator? ()()()() ()()0lim1inoutinVsHsVsVsGsH s = + To sustain self-oscillations, as Vin(s)goes to zero, quotient must go infinite= 1 Sign is neg for: = -180 ()()()() ()1outinVsHsVsHsGs=+Open-loop gain T(s)G(s)The Plant Error in for Oscillations when the open-loop gain equals 1 (0 dB) cross over point total rotation is -360 : -180 for H(s)and -180 for G(s) we have self-sustaining oscillating conditionsLoop gain |H(s)|Loop phase argH(s)-180 0 dBGain is 1atfc = -180 Total phase delay atfc:-180 H(s)power stage-180 G(s)opamptotal = -360 Need for Phase Margin we needphasephasemargin whenT(s)= 0 dB we needgaingainmargin when argT(s)= -360 gainmarginPhase margin:The margin before the loopphase rotation argT(s)reaches -360 atT(s)= 0 dBGain margin.

4 The margin before the loopgain T(s)reaches 0 dB at afreq. where argT(s)= -360 CrossoverfrequencyfcT(s)= 0 db(volts) in degreesPlot1210 0 dBargT(s)= -360 T(s) Feedback generalities Conditions for stability Poles and zeros Phase margin and quality coefficient Undershoot and crossover frequency Compensating the converter Compensating with a TL431 Watch the optocoupler! Compensating a DCM flyback Compensating a CCM flyback Simulation and bench results and Zeros A plant (power stage) loop gain is defined by:()()()NsHsDs= solving for N(s)= 0, the roots are called the zeroszeros solving for D(s)= 0, the roots are called the polespoles()()()5301sks kHssk++=+numeratordenominator1215301zzps ksksk= = = ======Two zerosOne and Zeros A polelagsthe phase by -45 at its cutoff frequency0()11() 11outinVssVssRC ==++R21kC110nFV1AC = 1 VinVout01RC =-90 delayfor f = atcutoffCutoff frequency-3 dB-1 slope-20 dB/decade|Vout(s)|argVout(s) and Zeros A zerobooststhe phase by +45 at its cutoff frequency0() 1sGs =+ db(volts)Plot101101001k10k100kfrequency in in db(volts)Plot101101001k10k100kfrequency in in degreesPlot22101001k10k100k1 Meg10 Megfrequency in in db(volts)

5 Plot101101001k10k100k1 Meg10 Megfrequency in degreesplot22101001k10k100k1 Meg10 Megfrequency in in db(volts)plot101101001k10k100k1 Meg10 Megfrequency in degreesplot2201RC =00()() 11outinsVssRCsVssRC ==++The general form of a zero:R21kC110nFV1AC = 1 VinVoutCutoff frequency-3 dB+1 slope+20 dB/decade+45 atcutoff+45 atcutoff+1 slope+20 dB/decadeCutoff frequency-3 dB0 0 90 90 |Vout(s)|argVout(s)|Vout(s)|argVout(s) Right Half-Plane Zero In a CCM boost, Ioutis delivered during the off time: ()1outdLIII D== TswD0 TswId(t)tIL(t)inVLId0 TswD1 TswId(t)tIL(t) dIL1inVLId1IL0 If Dbrutallyincreases, D'reduces and Ioutdrops! What matters is the inductor current slew-rate Occurs in flybacks, buck-boost, Cuk etc.()LdV Right-Half-Plane-Zero With a RHPZ we have a boostin gain but a lagin phase!

6 In db(volts)Plot111101001k10k100k1 Megfrequency in in degreesPlot22|Vout(s)|argVout(s)-90 +1 slope+20 dB/decade0() 1sGs =+LHPZRHPZ0() 1sGs = Feedback generalities Conditions for stability Poles and zeros Phase margin and quality coefficient Undershoot and crossover frequency Compensating the converter Compensating with a TL431 Watch the optocoupler! Compensating a DCM flyback Compensating a CCM flyback Simulation and bench results much Margin? The RLCF ilter let us study an RLClow-pass filter, a 2ndorder system23R1{R}1L1{L}C1{C}Voutparametersf0 =235kL=10uC=1/(4* ^2*f0^2*L)w0=({L}*{C})^ ((({C}/(4*{L}))^ )*2*{Q})Vin()211 TsLCsRCs=++()222211211rrrrTsssssQ ==++ ++1rLC =4 CRL =12Q =zeta rresonant freq. damping factorQ quality RLC Response to an Input Step changingQaffects the transient #6, vout#5, vout#4, vout#3, vout in voltsPlot17891011Q= 1Q= 5 Fast response and no overshoot!

7 Q< over dampingQ= critical dampingQ> under dampingOvershoot = 65% Asymptotically is the Analogy withT(s)? in the vicinity of the crossover point, T(s)combines: one pole at the origin, 0and one high frequency pole, 29 Link the closed-loopresponse to the open-loopphase margin:()0211 Tsss = + in db(volts) in degrees210 0 dB-2-1()()2020111 TsssTs =+++()()22111rrTsssTsQ =+++(OL)(CL)Link open-loop mwith closed-loop QClose theloopT(s) Open-Loop m a Qfactor of (critical response) implies a mof 76 a 45 mcorresponds to a Qof : oscillatory response!0 tan ()2+()14tan () 3602 Q on the Design Criteria compensate the open-loop gain for a phase margin of 70 make sure the open-loop gain margin is better than 15 dB never accept a phase margin lower than 45 in worst in #a, vout2, vout2#b, vout2#d in voltsPlot22315PM = 10 PM = 25 PM = 45 PM = 76 (),,outcoutfC f I () Feedback generalities Conditions for stability Poles and zeros Phase margin and quality coefficient Undershoot and crossover frequency Compensating the converter Compensating with a TL431 Watch the optocoupler!

8 Compensating a DCM flyback Compensating a CCM flyback Simulation and bench results in in db(volts)plot12 A DC-DC conv. combines an inductor and a capacitor As fis swept, different elements dominateZout,OLDC-DC Output ImpedanceZout(dB )f (Hz)RLfLoutCoutResr41 Lout100u2 Rlf10m3 Resr1mCout1000uFI1AC = 1 VoutA buck equivalent circuitf0To avoid stability issues,fc>> f022001lflfRZRZ + Crossover region()1||outoutLfesroutZsLRRsC =++ Open-loop the At the crossover frequencyZout,CL Zout,OL1101001k10k100kfrequency in #b, vdbout, vdberr in db(volts)Plot1235|Zout,CL||Zout,OL||T(s) |fc|Zout,CL| |Zout,OL| the Output Impedance the closed-loop output impedance is dominated by Cout()(),122cos11 121 2out CLcoutcoutmZfCT sfC +20406080012()11cTf+ m ZoutimprovesZoutdegradesOpen-loop phasemargin affects theclosed-loop Example with a Buck Let s assume an output capacitor of 1 mF The spec states a 80 mV undershoot for a 2 A step How to select the crossover frequency?

9 2outoutcoutIVfC 2outcoutoutIfVC 248012cfkHzmm = 1@440241outCZkHzmkm == Select a 1000- F capacitor featuring less than a 40-m the Right Crossover Frequency Compensate the converter for a 4 kHz fc101001k10k100kfrequency in in db(volts) in degreesPlot1434 kHz m= 70 Compensated open-loop gainBuck operated in Load the = = 100uFs = 100kdacPWM switch VMp8R7{R3}C3{C3}13R2{R2}C1{C1}C2{C2}I1G( s)H(s)PWMgain the load variesfrom 100 mA to the Obtained in in voltsPlot1570 mV()22co0s4mmIV = Feedback generalities Conditions for stability Poles and zeros Phase margin and quality coefficient Undershoot and crossover frequency Compensating the converter Compensating with a TL431 Watch the optocoupler! Compensating a DCM flyback Compensating a CCM flyback Simulation and bench results do we Stabilize the converter ?

10 1. Select the crossover frequencyfc(assume 4 kHz)2. Provide a high dc gain for a low static error and good input rejection 3. Shoot for a 70 phase margin atfc4. Evaluate the needed phase boost atfcto meet (3)5. Shape the G(s)path to comply with 1, 2 and 3()()(),,1sc OLsc CLAsAsTs=+|H(s)| @ fcArgH(s)@ fcOpen-loop Bode plot of the power stage, H(s)PhaseGain101001k10k100kfrequency in db(volts) , Provide Mid-Band Gain at Crossover1. AdjustG(s)to boost the gain by +21 dB at crossover Create the so-called mid-band gain|H(s)|= -21 dBArgH(s)= -175 PhaseGain101001k10k100kfrequency in db(volts)Plot1 TailorG(s)toexhibit a gainof +21 dB@ dB@fcPush thegain in hertz-360-1800180360p in in db(volts)Plot1810 Second, Provide High Gain in DC2. An integrator provides a high dc gain but rotates by -270 This is the origin pole12C1100n4R110kE11kV1AC = 1 Vout60 dB-20 dB per decadeslope -1-180 by invertingop amp-90 by poleat the origin-270 #a in in unknownPlot311101001k10k100kfrequency in in unknownPlot21 Third, Evaluate the Phase Boost atfcargH(s)argG(s)argH(s)G(s)Phase boost atfcargH(s) at 4 kHz m()arg270360cmHfBOOST + = ()arg9070175 90 155mcBOOSTH f = = + = -175 +155 -113 m= 70 argH(s)argG(s)+ do We Boost the Phase atfc?