SLUAAJ0 February   2024 TPS51397A , TPS54308 , TPS54320 , TPS54350 , TPS54620 , TPS54622 , TPS54821 , TPS54824 , TPS563300 , TPS566231 , TPS566235 , TPS566238 , TPS568230 , TPS56C215 , TPS62933 , TPS62933F , TPS62933O

 

  1.   1
  2.   Abstract
  3.   Trademarks
  4. 1Introduction
  5. 2Comparison of Feedback Sense Methods with Second Stage Filter
  6. 3Transfer Function Derivation of PCM Converter with Second Stage Filter and Hybrid Sense
  7. 4Overall Loop Model
  8. 5Zero and Pole Analysis
  9. 6Stability Design Method
  10. 7Design Example and Experimental Validation with TPS62933F
  11. 8Summary
  12. 9References
  13.   A Appendix

Transfer Function Derivation of PCM Converter with Second Stage Filter and Hybrid Sense

Figure 3-1 shows the PCM buck converter including second stage filter and hybrid sense.

GUID-20231017-SS0I-TQ3D-LDGP-HFJL339CD7CT-low.svgFigure 3-1 Simplified Schematic for PCM Buck Converter with Second Stage Filter and Hybrid Sense

Figure 3-2 is the overall control block diagram of PCM buck converter with second stage filter and hybrid sense. The variable notation refers to application note[5].

GUID-20231017-SS0I-S0RK-XTV5-GLP68QJGSCGS-low.svgFigure 3-2 Control Implementation of PCM Buck Converter with Second Stage Filter and Hybrid Sense

Among all the variables, the meanings of below ones are same as normal PCM buck converter without second stage filter:

  • Gdi is the duty cycle to iL transfer function.
  • GEA is the transfer function of the error amplifier with certain compensation.
  • Fm is the gain of PCM PWM comparator.
  • Ri is the current sensing resistor.
  • He is the transfer function model of inductor current sampling-hold effect.

The following transfer functions are new added or the meanings are changed from normal buck converter without second stage filter:

  • ZO is the transfer function of output impedance (combining 1st stage and second stage filter).
  • G2 is the transfer function of second stage filter.
  • GFF is the transfer function of the feed-forward path for vo1 in feedback network.
  • GFB is the transfer function of the feedback path for vo2 in feedback network.

This application note does not introduce the derivation method of those transfer function expressions which have same meanings as normal PCM buck converter without second stage filter. The expressions are listed as Equation 1 to Equation 4.

Equation 1. Gdi(s)=i^L(s)d^(s)VINsL
Equation 2. GEA(s)=v^COMP(s)-v^FB(s)GmCCOMP1+sRCOMPCCOMPs1+sRCOMPCO_EA
Equation 3. Fm=fSWSn+Se
Equation 4. He(s)=sfSWesfSW-1≈1-s2fSW+s2πfSW2

where, Sn is the on-time slope of the sensed-current waveform and Se is the external ramp slope.

Equation 5. Sn=RiVIN-VOL
Equation 6. Se=VSe×fSW

After adding second stage filter, the output impedance structure of buck converter changes from Figure 3-3 to Figure 3-4.

GUID-20231017-SS0I-3HCH-0K5J-RRF99H1TTS0C-low.svgFigure 3-3 Output Impedance Structure with One Stage Filter
GUID-20231017-SS0I-G2GT-D6DG-NSTZSKVMD9K2-low.svgFigure 3-4 Output Impedance Structure with Two Stage Filter

Since the output capacitors used in the design of a low-ripple design is generally low-ESR MLCC, to simplify the derivation of the loop model, the capacitor ESR effect is ignored here. From Figure 3-4, we can get the expression of the output impedance ZO of two stage filter as:

Equation 7. ZO(s)=C2L2RLs2+L2s+RLC2COL2RLs3+COL2s2+C2+CORLs+1

G2 reflects the low pass filter effects of second stage filter L2 and C2. The expression is:

Equation 8. G2(s)=RLC2L2RLs2+L2s+RL

The feedback signal VFB includes VO1 information coupled through feed-forward path and VO2 information through feedback voltage divider. According to the superposition theorem, we can get:

Equation 9. GFF(s)=v^FB(s)v^O1(s)|v^O2(s)=0=CffR1R2sCffR1R2s+R1+R2
Equation 10. GFB(s)=v^FB(s)v^O2(s)|v^O1(s)=0=R2CffR1R2s+R1+R2

Based on the relation of Figure 3-2, it can be derived:

Equation 11. GFB-TOTAL(s)=v^FB(s)v^O1(s)=GFF(s)+GFB(s)G2(s)