7.7. Discrete State Equations of Boost and Buck Converters

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7.7. Discrete State Equations of Boost and Buck Converters

Fig. 7.26 presents the block diagram of the digital controller of a dc–dc converter. Moreover, Fig. 2.27 presents the control signal u(t) in continuous control, average control, and control from D/A.

Suppose that an operation cycle of a boost converter lasts from the time instant kT_s to (k + 1)T_s as shown in Fig. 7.28.

Assuming that the switching period T_s is too small, the following approximate discrete time model holds:

$\frac{\vec{x} (k + 1) - \vec{x} (k^{'})}{t_{off}} = A_{1} \vec{x} (k^{'}) + {BV}_{in}$ $\frac{\vec{x} (k + 1) - \vec{x} (k^{'})}{t_{off}} = A_{1} \vec{x} (k^{'}) + {BV}_{in}$

(7.125)

$\vec{x} (k + 1) = (I + t_{off} A_{1}) \vec{x} (k^{'}) + B t_{off} V_{in}$ $\vec{x} (k + 1) = (I + t_{off} A_{1}) \vec{x} (k^{'}) + B t_{off} V_{in}$

(7.126)

where I = 2 × 2 unit matrix.

Figure 7.26 Block diagram of the digital controller of a dc–dc converter.

Figure 7.27 Different types of control signals.

Figure 7.28 One operational discrete cycle of the boost converter.

These equations are valid from the instant time kT_s to k^′T_s. Since the switching period T_s is too small, the following discrete time model is applied:

$\frac{\vec{x} (k^{'}) - \vec{x} (k)}{t_{on}} = A_{2} \vec{x} (k) + B V_{in}$ $\frac{\vec{x} (k^{'}) - \vec{x} (k)}{t_{on}} = A_{2} \vec{x} (k) + B V_{in}$

(7.127)

$\vec{x} (k^{'}) = (I + t_{on} A_{2}) \vec{x} (k) + B t_{on} V_{in}$ $\vec{x} (k^{'}) = (I + t_{on} A_{2}) \vec{x} (k) + B t_{on} V_{in}$

(7.128)

Substituting Eq. (7.128) into Eq. (7.126) yields:

$\vec{x} (k + 1) = (I + t_{off} A_{1}) (I + t_{on} A_{2}) \vec{x} (k) + (I + t_{off} A_{1}) B t_{on} V_{in} + B t_{off} V_{in}$ $\vec{x} (k + 1) = (I + t_{off} A_{1}) (I + t_{on} A_{2}) \vec{x} (k) + (I + t_{off} A_{1}) B t_{on} V_{in} + B t_{off} V_{in}$

(7.129)

Substituting the boost converter state matrices A₁, A₂, and B into Eq. (7.129), the following discrete state equation is obtained:

$[\begin{matrix} x_{1} (k + 1) \\ x_{2} (k + 1) \end{matrix}] = [\begin{matrix} (1 - \frac{t_{off}}{RC}) (1 - \frac{t_{on}}{RC}) & \frac{t_{off}}{C} \\ - \frac{t_{off}}{L} (1 - \frac{t_{on}}{RC}) & 1 \end{matrix}] [\begin{matrix} x_{1} (k) \\ x_{2} (k) \end{matrix}] + [\begin{matrix} \frac{t_{on} t_{off}}{LC} \\ \frac{(t_{on} {+ t}_{off})}{L} \end{matrix}] V_{in}$ $[\begin{matrix} x_{1} (k + 1) \\ x_{2} (k + 1) \end{matrix}] = [\begin{matrix} (1 - \frac{t_{off}}{RC}) (1 - \frac{t_{on}}{RC}) & \frac{t_{off}}{C} \\ - \frac{t_{off}}{L} (1 - \frac{t_{on}}{RC}) & 1 \end{matrix}] [\begin{matrix} x_{1} (k) \\ x_{2} (k) \end{matrix}] + [\begin{matrix} \frac{t_{on} t_{off}}{LC} \\ \frac{(t_{on} {+ t}_{off})}{L} \end{matrix}] V_{in}$

(7.130)

Furthermore, assuming t_ont_off << (RC)², t_ont_off < LC, and knowing that t_on + t_off = T_s, Eq. (7.130) becomes:

$[\begin{matrix} x_{1} (k + 1) \\ x_{2} (k + 1) \end{matrix}] = [\begin{matrix} 1 - \frac{T_{s}}{RC} & \frac{t_{off}}{C} \\ - \frac{t_{off}}{L} & 1 \end{matrix}] [\begin{matrix} x_{1} (k) \\ x_{2} (k) \end{matrix}] + [\begin{matrix} 0 \\ \frac{T_{s}}{L} \end{matrix}] V_{in}$ $[\begin{matrix} x_{1} (k + 1) \\ x_{2} (k + 1) \end{matrix}] = [\begin{matrix} 1 - \frac{T_{s}}{RC} & \frac{t_{off}}{C} \\ - \frac{t_{off}}{L} & 1 \end{matrix}] [\begin{matrix} x_{1} (k) \\ x_{2} (k) \end{matrix}] + [\begin{matrix} 0 \\ \frac{T_{s}}{L} \end{matrix}] V_{in}$

(7.131)

Using the same procedure, the following equations are obtained for the buck converter:

$\frac{\vec{x} (k^{'}) - \vec{x} (k)}{t_{on}} = A_{1} \vec{x} (k) + B_{1} V_{in}$ $\frac{\vec{x} (k^{'}) - \vec{x} (k)}{t_{on}} = A_{1} \vec{x} (k) + B_{1} V_{in}$

(7.132)

$\vec{x} (k^{'}) = (I + t_{on} A_{1}) \vec{x} (k) + B_{1} V_{in} t_{on}$ $\vec{x} (k^{'}) = (I + t_{on} A_{1}) \vec{x} (k) + B_{1} V_{in} t_{on}$

(7.133)

$\frac{\vec{x} (k + 1) - \vec{x} (k^{'})}{t_{off}} = A_{2} \vec{x} (k^{'}) + B_{2} V_{in}$ $\frac{\vec{x} (k + 1) - \vec{x} (k^{'})}{t_{off}} = A_{2} \vec{x} (k^{'}) + B_{2} V_{in}$

(7.134)

$\begin{matrix} \vec{x} (k + 1) = (I + t_{off} A_{2}) \vec{x} (k^{'}) \\ B_{2} = 0 \end{matrix}$ $\begin{matrix} \vec{x} (k + 1) = (I + t_{off} A_{2}) \vec{x} (k^{'}) \\ B_{2} = 0 \end{matrix}$

(7.135)

$\vec{x} (k + 1) = (I + t_{off} A_{2}) [(I + t_{on} A_{1}) \vec{x} (k) + B_{1} V_{in} t_{on}]$ $\vec{x} (k + 1) = (I + t_{off} A_{2}) [(I + t_{on} A_{1}) \vec{x} (k) + B_{1} V_{in} t_{on}]$

$\vec{x} (k + 1) = (I + t_{off} A_{2}) (I + t_{on} A_{1}) \vec{x} (k) + ({I + t}_{off} A_{2}) B_{1} V_{in} t_{on}$ $\vec{x} (k + 1) = (I + t_{off} A_{2}) (I + t_{on} A_{1}) \vec{x} (k) + ({I + t}_{off} A_{2}) B_{1} V_{in} t_{on}$

(7.136)

$[\begin{matrix} x_{1} (k + 1) \\ x_{2} (k + 1) \end{matrix}] = [\begin{matrix} 1 - \frac{t_{off}}{RC} & \frac{t_{off}}{C} \\ - \frac{t_{off}}{L} & 1 \end{matrix}] [\begin{matrix} 1 - \frac{t_{on}}{RC} & \frac{t_{on}}{C} \\ - \frac{t_{on}}{L} & 1 \end{matrix}] [\begin{matrix} x_{1} (k) \\ x_{2} (k) \end{matrix}] + [\begin{matrix} 1 - \frac{t_{off}}{RC} & \frac{t_{off}}{C} \\ - \frac{t_{off}}{L} & 1 \end{matrix}] [\begin{matrix} 0 \\ \frac{1}{L} \end{matrix}] V_{in} t_{on}$ $[\begin{matrix} x_{1} (k + 1) \\ x_{2} (k + 1) \end{matrix}] = [\begin{matrix} 1 - \frac{t_{off}}{RC} & \frac{t_{off}}{C} \\ - \frac{t_{off}}{L} & 1 \end{matrix}] [\begin{matrix} 1 - \frac{t_{on}}{RC} & \frac{t_{on}}{C} \\ - \frac{t_{on}}{L} & 1 \end{matrix}] [\begin{matrix} x_{1} (k) \\ x_{2} (k) \end{matrix}] + [\begin{matrix} 1 - \frac{t_{off}}{RC} & \frac{t_{off}}{C} \\ - \frac{t_{off}}{L} & 1 \end{matrix}] [\begin{matrix} 0 \\ \frac{1}{L} \end{matrix}] V_{in} t_{on}$

$[\begin{matrix} x_{1} (k + 1) \\ x_{2} (k + 1) \end{matrix}] = [\begin{matrix} (1 - \frac{t_{off}}{RC}) (1 - \frac{t_{on}}{RC}) - \frac{t_{off} t_{on}}{LC} & (1 - \frac{t_{off}}{RC}) \frac{t_{on}}{C} + \frac{t_{off}}{C} \\ - (1 - \frac{t_{on}}{RC}) \frac{t_{off}}{L} - \frac{t_{on}}{L} & \frac{t_{off} t_{on}}{LC} + 1 \end{matrix}] [\begin{matrix} x_{1} (k) \\ x_{2} (k) \end{matrix}] + [\begin{matrix} \frac{t_{off} t_{on}}{LC} \\ \frac{t_{on}}{L} \end{matrix}] V_{in}$ $[\begin{matrix} x_{1} (k + 1) \\ x_{2} (k + 1) \end{matrix}] = [\begin{matrix} (1 - \frac{t_{off}}{RC}) (1 - \frac{t_{on}}{RC}) - \frac{t_{off} t_{on}}{LC} & (1 - \frac{t_{off}}{RC}) \frac{t_{on}}{C} + \frac{t_{off}}{C} \\ - (1 - \frac{t_{on}}{RC}) \frac{t_{off}}{L} - \frac{t_{on}}{L} & \frac{t_{off} t_{on}}{LC} + 1 \end{matrix}] [\begin{matrix} x_{1} (k) \\ x_{2} (k) \end{matrix}] + [\begin{matrix} \frac{t_{off} t_{on}}{LC} \\ \frac{t_{on}}{L} \end{matrix}] V_{in}$

(7.137)

Assuming t_ont_off << (RC)² and t_ont_off < LC, Eq. (7.137) becomes:

$[\begin{matrix} x_{1} (k + 1) \\ x_{2} (k + 1) \end{matrix}] = [\begin{matrix} 1 - \frac{T_{s}}{RC} & \frac{T_{s}}{C} \\ - \frac{T_{s}}{L} & 1 \end{matrix}] [\begin{matrix} x_{1} (k) \\ x_{2} (k) \end{matrix}] + [\begin{matrix} 0 \\ \frac{t_{on}}{L} \end{matrix}] V_{in}$ $[\begin{matrix} x_{1} (k + 1) \\ x_{2} (k + 1) \end{matrix}] = [\begin{matrix} 1 - \frac{T_{s}}{RC} & \frac{T_{s}}{C} \\ - \frac{T_{s}}{L} & 1 \end{matrix}] [\begin{matrix} x_{1} (k) \\ x_{2} (k) \end{matrix}] + [\begin{matrix} 0 \\ \frac{t_{on}}{L} \end{matrix}] V_{in}$

(7.138)

Eq. (7.138) represents the buck converter discrete state equation.

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Table of Contents for 7.7. Discrete State Equations of Boost and Buck Converters

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7.7. Discrete State Equations of Boost and Buck Converters

Table of Contents for
7.7. Discrete State Equations of Boost and Buck Converters