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JAJSUV0 June 2024 LM5171

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7.1.1.1 Current Loop Small Signal Model

Figure 7-1 shows the current loop block diagram of each phase in buck mode. V_HV is the input while V_LV is the output.

Figure 7-1 Buck Loop Block Diagram

The inner current loop should be designed first. The average current-mode control loop of buck mode can be modeled asFigure 7-2

Figure 7-2 Current Loop Block Diagram

The buck mode duty cycle (d) to channel inductor current (i_Lm) transfer function is determined by the following:

Equation 23.

G_{i d_B K} (s) = \frac{{\hat{i}}_{L m}}{\hat{d}} = \frac{V_{H V}}{R_{O U T_B K}} \times \frac{1 + \frac{s}{ω_{Z_i l_B K}}}{1 + \frac{s}{ω_{0_B K} \times Q_{B K}} + \frac{s^{2}}{ω_{0_B K}^{2}}}

where

Equation 24.

R_{O U T_B K} = \frac{V_{L V}}{n_{p} {\times I}_{L m a x}}

Equation 25.

ω_{Z_i l_B K} = \frac{1}{R_{O U T_B K} \times C_{O U T_B K}}

Equation 26.

ω_{0_B K} = \frac{1}{\sqrt{L_{m} \times C_{O U T_B K}}}

Equation 27.

Q_{B K} = \frac{1}{ω_{0_B K}} \times \frac{1}{\frac{L_{m}}{R_{O U T_B K}} + (R_{E S R_B K} + R_{C S} + R_{S}) \times C_{O U T_B K}}

L_m is the power inductor,
R_CS is the current sense resistor,
R_S is the equivalent total resistance along the current path excluding R_CS,
C_{OUT_BK} is the total output capacitance in buck mode.
R_{ESR_BK} is the total output capacitor equivalent series resistance (ESR).

Figure 7-3 shows the current loop block diagram in boost mode. V_LV is the input while V_HV is the output.

Figure 7-3 Boost Loop Block Diagram

The average current-mode control loop of boost mode is the same as buck as shown in Figure 7-2. But the transfer function of the boost power stage G_id(s) and G_vd(s) is different from that of buck power stage.

The boost mode duty cycle (d) to channel inductor current (i_Lm) transfer function is determined by the following:

Equation 28.

G_{i d_B S T} (s) = \frac{{\hat{i}}_{L m}}{\hat{d}} = \frac{2 \times V_{L V}}{{D'}^{3} \times R_{O U T_B S T}} \times \frac{1 + \frac{s}{ω_{Z_i l_B S T}}}{1 + \frac{s}{ω_{0_B S T} \times Q_{B S T}} + \frac{s^{2}}{ω_{0_B S T}^{2}}}

where

Equation 29.

D' = \frac{V_{L V}}{V_{H V}}

Equation 30.

R_{O U T_B S T} = \frac{{V_{H V}}^{2}}{V_{L V} \times I_{L m a x}}

Equation 31.

ω_{Z_i l_B S T} = \frac{2}{R_{O U T_B S T} \times C_{O U T_B S T}}

Equation 32.

ω_{0_B S T} = \frac{D'}{\sqrt{L_{m} \times C_{O U T_B S T}}}

Equation 33.

Q_{B S T} = \frac{D'}{ω_{0_B S T}} \times \frac{1}{\frac{L_{m}}{D' \times R_{O U T_B S T}} + \frac{(R_{C S} + R_{S}) \times C_{O U T_B S T}}{D'} + R_{E S R_B S T} \times C_{O U T_B S T}}

C_{OUT_BST} is the total output capacitance for each phase in boost mode.
R_{ESR_BST} is the total output capacitor equivalent series resistance (ESR) for each phase in boost mode.

When we select the current loop cross over frequency at 1/6 of switching frequency, G_{id_BK}(s) can be simplified. For the numerator, s×R_{OUT_BK}×C_{OUT_BK} dominates. And for the denominator, s²/ω_{0_BK}² dominates.Equation 23 can be simplified as:

Equation 34.

G_{i d_B K} (s) = \frac{V_{H V}}{R_{O U T_B K}} \times \frac{1 + \frac{s}{ω_{Z_i l_B K}}}{\frac{s^{2}}{{ω_{0_B K}}^{2}}} = \frac{V_{H V}}{s \times L_{m}}

Similarly, Equation 28 can be simplified as:

Equation 35.

G_{i d_B S T} (s) = \frac{2 \times V_{L V}}{{D'}^{3} \times R_{O U T_B S T}} \times \frac{\frac{s}{ω_{Z_i l_B S T}}}{\frac{s^{2}}{{ω_{0_B S T}}^{2}}} = \frac{V_{H V}}{s {\times L}_{m}}

It can be observed that the same duty cycle (d) to channel inductor current (i_Lm) transfer function is shared by both buck and boost mode:

Equation 36.

G_{i d} (s) = \frac{V_{H V}}{s {\times L}_{m}}

So compensator for buck current loop and boost current loop can also be shared.

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7.1.1.1 Current Loop Small Signal Model