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Phase Margin Test for Power-Stage of DC-DC Buck Converter

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Phase Margin Test for Power-Stage of DC-DC Buck Converter Proceedings of International Conference on Technology and Social Science 2020 (ICTSS 2020) Dec. 3, 2020 Phase Margin Test for Power-Stage of DC-DC Buck Converter MinhTri Tran*, Yasunori Kobori, Anna Kuwana, and Haruo Kobayashi Gunma University, Japan Kobayashi Laboratory Gunma Outline 1. Research Background • Motivation, objectives and achievements • Characteristics of an adaptive feedback network 2. Analysis of Power-Stage of DC-DC Converter • Operating regions of 2nd-order systems • Phase margin of power-stage of DC-DC converter 3. Ripple Reduction for DC-DC Converters • LC Harmonic Notch filter • Nichols chart of power stage of buck converter 4. Conclusions 1. Research Background Noise in Electronic Systems Performance of a system Performance of a device Signal power Output SNR Signal to SNR Figure of F Noise Ratio: Noise power Merit: Input SNR Common types of noise: Device noise: • Electronic noise • Flicker noise, • Thermal noise, • Thermal noise, • Intermodulation noise, • White noise. • Cross-talk, DC-DC converters • Impulse noise, • Overshoot, • Shot noise, and • Ringing • Transit-time noise. • Ripple 2 1. Research Background Effects of Ripple and Ringing on Electronic Systems • Ringing is overshoot/undershoot voltage or current when it’s seen on time domain. • Ripple is wasted power, and has many undesirable effects on analog circuits. Ringing does the following Ripple does the following things: things: • Causes EMI noise, • Increases current flow, • Heats components, • Increases noise, • Consumes the power, • Creates the distortion, • Decreases the performance, • Causes digital circuits • Damages the devices. to operate improperly3 . 1. Research Background Objectives of Study • Investigation of behaviors of power-stage of DC-DC converters • Ripple reduction for DC-DC buck converter using LC Harmonic Notch filter • Measurement of self-loop function in power- stage of DC-DC buck converter • Observation of phase margin at unity gain on the Nichols chart 4 1. Research Background Achievements of Study Alternating current conservation Implemented circuit for deriving self-loop function LC V L() inc Stability test for notch power-stage filter Vtrans Derivation of self-loop function Phase margin of power stage Incident Transmitted current current Phase margin 49 degrees 131o Phase margin 46 degrees 134o 1. Research Background Approaching Methods Simplified power-stage Balun transformer Measurement set up Design of DC-DC buck converter input PWM CONTROL BLOCK output 6 1. Research Background Characteristics of Adaptive Feedback Network Block diagram of a typical adaptive feedback system DC voltage Reference voltage DC voltage DC voltage DC voltage + Ripple voltage Adaptive feedback is used to control the output source along with the decision source (DC-DC Buck converter). Transfer function of an adaptive feedback network is significantly different from transfer function of a linear negative feedback network. Loop gain is independent of frequency variable (referent voltage, feedback voltage, and error voltage are DC voltages). 7 1. Research Background Loop Gain in Feedback Control Systems Adaptive feedback systems Inverting amplifier Input Output Vin + A Vout +- G - F Transfer function G Transfer function A 1 H 1 H GF : loop gain 1 GF Aβ : loop gain 1 A Nichols plot of loop gain Gain reduction Loop Gain BW =100 Hz GBW =10 MHz 1. Research Background Self-loop Function in A Transfer Function Linear system Model of a linear system Input Output n b0( j ) ... bn 1 ( j ) b n H () H () n V () a0( j) ... an 1 ( j ) a n Vin () out Transfer function A() : Numerator function V () A() H() : Transfer function H () out Vin () 1 L() L() : Self-loop function Variable: angular frequency (ω) oPolar chart Nyquist chart oMagnitude-frequency plot Bode plots oAngular-frequency plot oMagnitude-angular diagram Nichols diagram 9 1. Research Background Alternating Current Conservation Transfer function Self-loop function Vout () 1 V V V Z H () inc trans L() inc in V () Z in in 1 ZZinZ o u tVtrans out Z out Z L() in ; 10 mH inductance Z out Incident Transmitted current current Simplified linear system Derivation of self-loop function 10 1. Research Background Limitations of Conventional Methods o Middlebrook’s measurement of loop gain Applying only in feedback systems (DC-DC converters). o Replica measurement of loop gain Using two identical networks (not real measurement). o Nyquist’s stability condition Theoretical analysis for feedback systems (Lab tool). o Nichols chart of loop gain Only used in feedback control theory (Lab tool). 11 Outline 1. Research Background • Motivation, objectives and achievements • Characteristics of an adaptive feedback network 2. Analysis of Power-Stage of DC-DC Converter • Operating regions of 2nd-order systems • Phase margin of power-stage of DC-DC converter 3. Ripple Reduction for DC-DC Converters • LC Harmonic Notch filter • Nichols chart of power stage of buck converter 4. Conclusions 2. Analysis of Power-Stage of DC-DC Converter Behaviors of Second-order Transfer Function 1 Second-order transfer function: H () 2 1a0 ( j ) a 1 j Case Over-damping Critically damping Under-damping 2 2 2 Delta 1 a1 2 1 a1 2 1 a1 2 a1 4 a 0 0 a1 4 a 0 0 a1 4 a 0 0 () a02 a 0 a02 a 0 a02 a 0 1 1 1 a0 a0 Module a0 1 2 2 6dB 2 2 2 2 2 2 2 2 2 1a a 1 a a 2a1 a 11 2 a 1 a 1 1 2 1 1 1 1 2 a1 H () 2a 2 a a 2 a 2 a a a2 a 2 a a 2 a 2 a 0 0 0 0 0 0 0 0 0 0 0 0 2a 0 2 2 1a1 1 a 1 Angular a2 a a 2 a 0 0 0 0 arctan arctan 2a0 arctan arctan 2 2 a1 a 1 a a 1 a a 1 2 arctan 1 1 1 1 2a 2 a 2a 2 a a 2 a 2 a a 0 0 () 0 0 0 0 0 0 a1 a1 2a 2a0 0 2a0 cut H () H ()cut () cut () cut cut H()cut () cut 2a a1 0 a1 2 2 a1 2 13 2. Analysis of Power-Stage of DC-DC Converter Behaviors of Second-order Self-loop Function Second-order self-loop function: L() j a0 j a 1 Case Over-damping Critical damping Under-damping 2 2 a2 4 a 0 Delta () a1 4 a 0 0 a1 4 a 0 0 1 0 2 2 2 2 2 2 L() a0 a 1 a0 a 1 a0 a 1 a a 0 0 a0 () arctan arctan arctan 2 a1 2 a1 2 a1 a o 1 5 2 L( ) 1 ( ) 76.3 o L( ) 1 ( ) 76.3o 1 1 1 L(1 ) 1 ( 1 ) 76.3 1 1 2a0 a 1 ( ) 63.4o ( ) 63.4o ( ) 63.4o 2 2 L( ) 5 2 L(2 ) 5 2 L(2 ) 5 2 2a0 a o 1 o o ( ) 45 ( ) 45 L(3 ) 4 2 3 3 L(3 ) 4 2 ( 3 ) 45 L(3 ) 4 2 3 a0 14 2. Analysis of Power-Stage of DC-DC Converter Operating Regions of 2nd-Order System •Under-damping: 1 Transient response H 1 () 2 ; 2 j j 1 L1() j j; 1 •Critical damping: H 2 () ; j2 2 j 1 L () j 2 2 j; 2 1 H 3 () ; •Over-damping: j 2 3 j 1 2 L3 () j 3 j; Bode plot of transfer function Nichols plot of self-loop function 1dB -6dB PM PM 82o PM 52o -10dB 76.3o 98o 103.7o 128o 15 2. Analysis of Power-Stage of DC-DC Converter Implemented Circuit of Åkerberg-Mossberg LPF Schematic of Åkerberg-Mossberg LPF Measurement set up Transfer function & self-loop function b0 H 2 ; a0 j a 1 j 1 2 L a0 j a 1 j ; R6 where, b0 ; R1 RRRR a3 R R C C;; a 3 5 6 C 0 5 6 1 2 1 2 16 RRR4 4 2 2. Analysis of Power-Stage of DC-DC Converter Measurement Results of Åkerberg-Mossberg LPF Bode plot of transfer function Nichols plot of self-loop function 20 dB Phase margin 12dB 103o 40 degrees o 0 dB 85 140o Transient response Over-damping: Phase margin is 95 degrees. Critical damping: Phase margin is 77 degrees. Under-damping: Phase margin is 40 degrees. 17 2. Analysis of Power-Stage of DC-DC Converter Behaviors of power-stage of DC-DC Converter Parallel RLC low-pass filter Transfer function & self-loop function: V 1 H (); out V 2 in 1a0 j a 1 j 2 L(); a0 j a 1 j L Where: a0 LC;; a 1 R 0 1LC ; ZLZCLC 0; 1 0 ; Operating regions 2 1 R ZZRLC / 2 •Over-damping: LC2 L ZZRLC 2 2 1 R ZZRLC / 2 Balanced charging- •Critical damping: LC2 L 2 discharging time condition 1 R ZZRLC / 2 •Under-damping: LC2 L 18 2. Analysis of Power-Stage of DC-DC Converter Phase Margin of Power-Stage of DC-DC Converter Simplified power-stage Measured results Under-damping Bode region plot of transfer L1 = 220 uH, C1 = 100 uF, R1 = 5 Ω, function Implemented circuit Phase margin Nichols 46 degrees plot of self-loop function 134o 19 Outline 1. Research Background • Motivation, objectives and achievements • Characteristics of an adaptive feedback network 2. Analysis of Power-Stage of DC-DC Converter • Operating regions of 2nd-order systems • Phase margin of power-stage of DC-DC converter 3. Ripple Reduction for DC-DC Converters • LC Harmonic Notch filter • Nichols chart of power stage of buck converter 4. Conclusions 3. Ripple Reduction for DC-DC Converters Ripple Reduction using LC Harmonic Notch Filter Schematic diagram of DC-DC buck converter Transfer function & self-loop function 2 b0 j 1 Simplified model of power-stage H 4 3 2 a0 j a 1 j a 2 j a 3 j 1 4 3 2 L a0 j a 1 j a 2 j a 3 j Where, b0 L 2 C 2;; a 0 L 1 C 1 L 2 C 2 LLCL1 2 2 1 a1;;; a 2 L 1 C 1 L 2 C 2 L 1 C 2 a 3 RRLL 21 3.
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