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DC-DC Converters Feedback and Control

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DC-DC Converters Feedback and Control DC-DC Converters Feedback and Control Agenda 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 Conclusion www.onsemi.com 2 Agenda 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 Conclusion www.onsemi.com 3 What is Feedback? A target is assigned to one or several state-variables, e.g. Vout = 12 V. A circuitry monitors Vout deviations related to Vin, Iout, T° etc. If Vout deviates from its target, an error is created and fed-back to the power stage for action. The action is a change in the control variable: duty-cycle (VM), peak current (CM) or the switching frequency. Compensating for the converter shortcomings! Input voltage Output voltage DC-DC Rth Vin Vout Input voltage Vth Vout Vin action control www.onsemi.com 4 The Feedback Implementation Vout is permanently compared to a reference voltage Vref. The reference voltage Vref is precise and stable over temperature. The error,ε =−VVrefα out, is amplified and sent to the control input. The power stage reacts to reduce ε as much as it can. Power stage - H Vout Control d variable Error amplifier - G Rupper + - Vin - - α + + Vp Modulator - G V PWM ref Rlower www.onsemi.com 5 Agenda 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 Conclusion www.onsemi.com 6 Positive or Negative Feedback? Do we want to build an oscillator? The « Plant » + ε Vin(s) V (s) Error voltage H(s) out − G(s) Vs() Hs( ) out = Vsin () 1+ HsGs() () Open-loop gain T(s) ⎡ Hs( ) ⎤ Vs= lim Vs To sustain self-oscillations, as Vin(s) out() ⎢ in ()⎥ goes to zero, quotient must go infinite Vsin ()→0 ⎣⎢1+ GsHs() () ⎦⎥ Sign is neg for: ϕ = -180° = 1 www.onsemi.com 7 Conditions 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) ¾ we have self-sustaining oscillating conditions Total phase delay at fc: 180 Loop gain |H(s)| -180° H(s) power stage Gain is 1 -180° G(s) opamp 90.0 at fc total = -360° 0 0 dB Loop phase arg H(s) -90.0 ϕ = -180° -180 2221 -180° 1 10 100 1k 10k 100k frequency in hertz www.onsemi.com 8 The Need for Phase Margin we need phase margin when T(s) = 0 dB we need gain margin when arg T(s) = -360° 180 80.0 T(s) Phase margin: phase phase margin The margin before the loop gain arg T(s) phase rotation arg T(s) 90.0 40.0 = -360° reaches -360° at T(s) = 0 dB 0° 0 0 2 in db(volts) Plot1 0 dB vdbout ph_vout in degrees gain margin Crossover -90.0 -40.0 1 frequency fc T(s) = 0 dB Gain margin: The margin before the loop -180 -80.0 gain T(s) reaches 0 dB at a 10 100 1k 10k 100k freq. where arg T(s) = -360° www.onsemi.com 9 Agenda 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 Conclusion www.onsemi.com 10 Poles and Zeros A plant (power stage) loop gain is defined by: Ns( ) numerator Hs()= D()s denominator solving for N(s) = 0, the roots are called the zeros solving for D(s) = 0, the roots are called the poles 5k Two zeros fHz==796 z1 2π skz = −5 ()s ++530ks( k) 1 30k sk=−30 fkHz==4.77 Hs()= z z2 sk+1 2 2π sk=−1 p1 1k fHz==159 One pole p1 2π www.onsemi.com 11 Poles and Zeros A pole lags the phase by -45°at its cutoff frequency 0 Vin Vout 20.0 Cutoff frequency R2 -3 dB 1k 0 -20.0 V1 -40.0 C1 -1 slope AC = 1 1 10nF -60.0 |Vout(s)| -20 dB/decade 0 -20.0 -45° at Vsout () 11 -40.0 == cutoff Vs() 1+ sRC s -60.0 in 1+ -80.0 ω0 argV (s) out 2 10 100 1k 10k 100k 1Meg 10Meg 1 ω = 0 RC -90° delay for f = ∞ www.onsemi.com 12 Poles and Zeros A zero boosts the phase by +45°at its cutoff frequency 0 0 40.0 1 20.0 +1 slope |V (s)| +1 slope out 0 +20 dB/decade 1 30.0 +20 dB/decade -20.0 20.0 plot1plot1 inin db(volts) db(volts) Plot1Plot1 vdboutvdbout in in db(volts) db(volts) vdb2vdb2 -40.0 10.0 Cutoff frequency -60.0 -3 dB 0 |Vout(s)| 10 100 1k 10k 100k 1Meg 10Meg frequency in hertz 90.0 2 Cutoff frequency 90.0 argVout(s) 70.0 -3 dB 90° 70.0 90° 50.0 inin degrees degrees Plot2Plot2 50.0 +45° at plot2plot2 +45° at ph_v2ph_v2 in in degrees degrees 30.0 ph_voutph_vout cutoff 30.0 cutoff 10.0 0° argV (s) 10.0 0° out 2 10 100 1k 10k 100k 10 100 1k 10k 100k 1Meg 10Meg frequency in hertz frequency in hertz Vin Vout C1 s The general form of a zero: 10nF s Vsout () sRC ω0 Gs()=+ 1 == V1 R2 Vs() 1+ sRC s ω AC = 1 in 1+ 0 1k ω 1 0 ω = 0 RC www.onsemi.com 13 The Right Half-Plane Zero In a CCM boost, Iout is delivered during the off time: Iout==II d L (1 − D) Id(t) Id(t) I IL0 L1 Vin Vin L L IL(t) IL(t) Id0 Id1 dˆ t t D0Tsw D1Tsw Tsw Tsw If D brutally increases, D' reduces and Iout drops! dVL ( t) What matters is the inductor current slew-rate dt Occurs in flybacks, buck-boost, Cuk etc. www.onsemi.com 14 The Right-Half-Plane-Zero With a RHPZ we have a boost in gain but a lag in phase! 1 40.0 |Vout(s)| 20.0 +1 slope +20 dB/decade LHPZ 0 Plot1 vdbout in db(volts) -20.0 s -40.0 Gs()=+ 1 ω0 180 RHPZ argVout(s) 90.0 0 s Plot2 Gs()=− 1 ph_vout in degrees ω -90.0 -90° 2 0 -180 1 10 100 1k 10k 100k 1Meg frequency in hertz www.onsemi.com 15 Agenda 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 Conclusion www.onsemi.com 16 How much Margin? The RLC Filter let us study an RLC low-pass filter, a 2nd order system R1 L1 {R} {L} 1 Vout Ts()= 2 3 2 1 LCs++ RCs 1 C1 Vin 11 {C} Ts()== ssss22 22++21ζ ++ 1 ωωωωrrrrQ 1 parameters ω = r LC f0=235k L=10u C 1 C=1/(4*3.14159^2*f0^2*L) ζ = R Q = w0=({L}*{C})^-0.5 4L 2ζ Q=10 zeta R=1/((({C}/(4*{L}))^0.5)*2*{Q}) ωr resonant freq. ζ damping factor Q quality coeff. www.onsemi.com 17 The RLC Response to an Input Step changing Q affects the transient response Q = 5 Q < 0.5 over damping 1.80 Q = 1 Q = 0.5 critical damping Q > 0.5 under damping Q = 0.707 1.40 in volts Overshoot = 65% vout vout#3, 1.00 1011789 Plot1 vout#4, Asymptotically stable vout#5, 600m vout#6, Q = 0.5 Fast response and no overshoot! 200m Q = 0.1 5.00u 15.0u 25.0u 35.0u 45.0u ti i d www.onsemi.com 18 Where is the Analogy with T(s)? in the vicinity of the crossover point, T(s) combines: one pole at the origin, ω0 and one high frequency pole, ω2 9 Link the closed-loop response to the open-loop phase margin: 1 Ts()= (OL) 180 80.0 T(s) phase ⎛⎞ss⎛⎞ ⎜⎟⎜⎟1+ gain ⎝⎠ω02⎝⎠ω 90.0 40.0 Close the loop Ts( ) 1 0° 0 0 2 = 2 (CL) 0 dB 1+Ts() ss vdbout in db(volts) ph_vout in degrees + +1 ωω02 ω 0 -90.0 -40.0 -1 1 Ts( ) 1 Link open-loop -2 ϕm = 2 -180 -80.0 1+Ts ss with closed-loop Q () + +1 ωω2 Q 10 100 1k 10k 100k rr www.onsemi.com 19 Closed-Loop Q Versus Open-Loop ϕm a Q factor of 0.5 (critical response) implies a ϕm of 76° a 45° ϕm corresponds to a Q of 1.2: oscillatory response! 10 Q 7.5 1 4 2 ϕm ()1+ tan()φ 5 tan()φ 2.5 0.5 0 0 255075100 76° 360 φ⋅ 2⋅π www.onsemi.com 20 Summary 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 case 5.12 PM = 10° PM = 25° PM = 45° 5.06 PM = 76° in volts vout2#d 5.00 2315 Plot2 vout2#b, vout2, vout2#a, fC( out4.94,, f cΔ I out ) 4.88 fPM( ) 300u 900u 1.50m 2.10m 2.70m time in seconds www.onsemi.com 21 Agenda 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 Conclusion www.onsemi.com 22 DC-DC Output Impedance A DC-DC conv.
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