SLUAAJ0 February 2024 TPS51397A , TPS54308 , TPS54320 , TPS54350 , TPS54538 , TPS54620 , TPS54622 , TPS54821 , TPS54824 , TPS563300 , TPS566231 , TPS566235 , TPS566238 , TPS568230 , TPS56C215 , TPS62933 , TPS62933F , TPS62933O

4 Overall Loop Model

After adding second stage filter, the transfer functions of the current inner loop are not changed from normal PCM buck converter. The expression of transfer function from control to inductor current G_ci is given in the Loop Response Considerations in Peak Current Mode Buck Converter Design application note:

Equation 12.

G_{ci} (s)= \frac{{\hat{i}}_{L} (s)}{{\hat{v}}_{COMP} (s)} = \frac{1}{R_{i}} \frac{1}{1+s × [\frac{V_{Se} f_{SW} L+ (0.5 V_{IN} - V_{O}) R_{i}}{V_{IN} R_{i} f_{SW}}]}

To prevent sub-harmonic oscillation, it is necessary to avoid the existence of right-half-plane poles in the inner-loop closed-loop transfer function, so the inductance constraint range obtained according to Equation 12 is:

Equation 13.

L> \frac{R_{i} (V_{O} -0.5 V_{IN})}{V_{Se} f_{SW}}

Based on Figure 3-2, the open loop response of PCM buck converter with second stage filter and hybrid sense is:

Equation 14.

G_{open} (s) = G_{EA} (s) × G_{ci} (s) × Z_{O} (s) × G_{FB-TOTAL} (s)

Substituting Equation 11 to Equation 14, we can get the open loop transfer function as:

Equation 15.

G_{open} (s) = G_{EA} (s) × G_{ci} (s) × Z_{O} (s) × [G_{FF} (s)+ G_{FB} (s) G_{2} (s)]

There is some simplification in the derivation of this small signal model, but it can mostly reflect the actual loop response, especially the response in middle frequency range near bandwidth affecting stability.

The small signal modeling accuracy is verified with TPS62933F under the operating condition: V_in=24V, V_out=1.2V, I_out=3A, F_sw=500kHz, L=2.2uH, C_o=90uF, L₂=20nH, C₂=47uF, R₁=5kohm, R₂=10kohm, C_ff=680pF.

The comparison of loop response between small signal model and simplis simulation for the converter with second stage filter is shown as Figure 4-1.

Figure 4-1 Comparison of Loop Response Between Small Signal Model and Simplis Simulation