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3 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.

Figure 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].

Figure 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:

G_di is the duty cycle to i_L transfer function.
G_EA is the transfer function of the error amplifier with certain compensation.
F_m is the gain of PCM PWM comparator.
R_i is the current sensing resistor.
H_e 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:

Z_O is the transfer function of output impedance (combining 1^st stage and second stage filter).
G₂ is the transfer function of second stage filter.
G_FF is the transfer function of the feed-forward path for v_o1 in feedback network.
G_FB is the transfer function of the feedback path for v_o2 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.

G_{di} (s)= \frac{{\hat{i}}_{L} (s)}{\hat{d} (s)} ≈ \frac{V_{IN}}{sL}

Equation 2.

G_{EA} (s)= \frac{{\hat{v}}_{COMP} (s)}{- {\hat{v}}_{FB} (s)} ≈ \frac{G_{m}}{C_{COMP}} \frac{1+s R_{COMP} C_{COMP}}{s (1+s R_{COMP} C_{O_EA})}

Equation 3.

F_{m} = \frac{f_{SW}}{S_{n} + S_{e}}

Equation 4.

H_{e} (s)= \frac{\frac{s}{f_{SW}}}{e^{\frac{s}{f_{SW}}} -1} ≈1- \frac{s}{2 f_{SW}} + \frac{s^{2}}{{(π f_{SW})}^{2}}

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

Equation 5.

S_{n} = R_{i} \frac{V_{IN} - V_{O}}{L}

Equation 6.

S_{e} = V_{Se} × f_{SW}

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

Figure 3-3 Output Impedance Structure with One Stage Filter

Figure 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 Z_O of two stage filter as:

Equation 7.

Z_{O} (s)= \frac{C_{2} L_{2} R_{L} s^{2} + L_{2} s+ R_{L}}{C_{2} C_{O} L_{2} R_{L} s^{3} + C_{O} L_{2} s^{2} + (C_{2} + C_{O}) R_{L} s+1}

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

Equation 8.

G_{2} (s)= \frac{R_{L}}{C_{2} L_{2} R_{L} s^{2} + L_{2} s+ R_{L}}

The feedback signal V_FB includes V_O1 information coupled through feed-forward path and V_O2 information through feedback voltage divider. According to the superposition theorem, we can get:

Equation 9.

G_{FF} (s)= \frac{{\hat{v}}_{FB} (s)}{{\hat{v}}_{O1} (s)} |_{{\hat{v}}_{O2} (s)=0} = \frac{C_{ff} R_{1} R_{2} s}{C_{ff} R_{1} R_{2} s+ R_{1} + R_{2}}

Equation 10.

G_{FB} (s)= \frac{{\hat{v}}_{FB} (s)}{{\hat{v}}_{O2} (s)} |_{{\hat{v}}_{O1} (s)=0} = \frac{R_{2}}{C_{ff} R_{1} R_{2} s+ R_{1} + R_{2}}

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

Equation 11.

G_{FB-TOTAL} (s)= \frac{{\hat{v}}_{FB} (s)}{{\hat{v}}_{O1} (s)} = G_{FF} (s)+ G_{FB} (s) G_{2} (s)