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JAJSUV0 June 2024 LM5171

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7.1.1.3 Voltage Loop Small Signal Model

When the current loop compensator is designed, the outer voltage loop can then be analyzed.

A system with n_p phases is shown in Figure 7-4.

Figure 7-4 n_p phases system

The equavilent inductance and resistance are determined by

Equation 50.

L_{m n p} = \frac{L_{m}}{n_{p}}

Equation 51.

R_{S n p} = \frac{R_{S}}{n_{p}}

Equation 52.

R_{C S n p} = \frac{R_{C S}}{n_{p}}

Equation 53.

R_{f n p} = \frac{R_{f}}{n_{p}}

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

Equation 54.

G_{i d n p_B K} (s) = \frac{{n_{p} \times \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 n p_B K} \times Q_{n p B K}} + \frac{s^{2}}{ω_{0 n p_B K}^{2}}}

where

Equation 55.

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

Equation 56.

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

Equation 57.

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

Equation 58.

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

For n_p phase, the equivalent open loop gain T_inp(s) can be obtained as

Equation 59.

T_{i n p} (s) = G_{c i} (s) \times \frac{1}{V_{M}} \times G_{i d} (s) \times R_{f n p}

where

Figure 7-5 shows the outer voltage control loop and inner current loop.

LM5171 Voltage Loop and Current Loop
Block Diagram

Figure 7-5 Voltage Loop and Current Loop Block Diagram

We can get ISET to output voltage (v_O) close loop transfer function as:

Equation 60.

G_{v s} (s) = \frac{{\hat{v}}_{L V}}{{\hat{v}}_{I S E T}} = \frac{G_{c i} (s) \times \frac{1}{V_{M}} \times G_{v d} (s)}{1 + T_{i n p} (s)}

When selecting the crossover frequency of the buck voltage loop lower than the current loop crossover frequency, G_vs(s) can be simplified. For the denominator, T_inp(s) dominates, Equation 60 can be written as:

Equation 61.

G_{v s} (s) = \frac{{\hat{v}}_{L V}}{{\hat{v}}_{I S E T}} = \frac{G_{c i} (s) \times \frac{1}{V_{M}} \times G_{v d} (s)}{T_{i n p} (s)} = \frac{G_{v d} (s)}{G_{i d} (s) \times R_{f n p}}

The buck power plant duty cycle (d) to output voltage (v_LV) transfer function is determined by :

Equation 62.

G_{v d_B K} (s) = \frac{{\hat{v}}_{L V}}{\hat{d}} = V_{H V} \times \frac{1 + \frac{s}{ω_{Z_v l_B K}}}{1 + \frac{s}{ω_{0 n p_B K} \times Q_{n p B K}} + \frac{s^{2}}{ω_{0 n p_B K}^{2}}}

where

Equation 63.

ω_{Z_v l_B K} = \frac{1}{R_{E S R_B K} \times C_{O U T_B K}}

Substituting Equation 62 into Equation 61, a simplified ISET to output voltage (V_LV) transfer function is determined by the following:

Equation 64.

G_{v s_B K} (s) = \frac{{\hat{v}}_{L V}}{{\hat{v}}_{I S E T}} = K_{d c_B K} \times \frac{1 + \frac{s}{ω_{Z_v l_B K}}}{1 + \frac{s}{ω_{Z_i l_B K}}}

where

Equation 65.

K_{d c_B K} = \frac{R_{O U T_B K}}{R_{f n p}}

Similarly, the boost power plant duty cycle (d) to output voltage (v_HV) transfer function is determined by :

Equation 66.

G_{v d_B S T} (s) = \frac{{\hat{v}}_{H V}}{\hat{d}} = \frac{V_{L V}}{{D'}^{2}} \times \frac{(1 + \frac{s}{ω_{Z_v l_B S T}}) (1 - \frac{s}{ω_{R H P Z}})}{1 + \frac{s}{ω_{0 n p_B S T} \times Q_{n p B S T}} + \frac{s^{2}}{ω_{0 n p_B S T}^{2}}}

where

Equation 67.

ω_{Z_v l_B S T} = \frac{1}{R_{E S R_B S T} \times C_{O U T_B S T}}

Equation 68.

ω_{R H P Z} = \frac{R_{O U T_B S T} \times {D'}^{2}}{L_{m n p}}

Substituting Equation 66 into Equation 61, a simplified ISET to output voltage (V_HV) transfer function is determined by the following:

Equation 69.

G_{v s_B S T} (s) = \frac{{\hat{v}}_{H V}}{{\hat{i}}_{s e t}} = K_{d c_B S T} \times \frac{(1 + \frac{s}{ω_{Z_v l_B S T}}) (1 - \frac{s}{ω_{R H P Z}})}{1 + \frac{s}{ω_{Z_i l_B S T}}}

where

Equation 70.

K_{d c_B S T} = \frac{R_{O U T_B S T} \times D'}{2 \times R_{f n p}}

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7.1.1.3 Voltage Loop Small Signal Model