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45

Converter Modeling Theory

Field: Power Electronics

Sequence of Expressions

Let x⃗(t)\vec{x}(t) be the state vector and u⃗(t)\vec{u}(t) be the input vector. The averaged model is derived by approximating the time-varying dynamics dx⃗dt=f(x⃗,u⃗,s(t))\frac{d\vec{x}}{dt} = f(\vec{x}, \vec{u}, s(t)) using the average operator ⟨⋅⟩\langle \cdot \rangle: dx⃗dt≈A⃗avgx⃗+B⃗avgu⃗\frac{d\vec{x}}{dt} \approx \vec{A}_{avg}\vec{x} + \vec{B}_{avg}\vec{u} where A⃗avg=⟨∂f∂x⃗⟩\vec{A}_{avg} = \langle \frac{\partial f}{\partial \vec{x}} \rangle and B⃗avg=⟨∂f∂u⃗⟩\vec{B}_{avg} = \langle \frac{\partial f}{\partial \vec{u}} \rangle, representing the time average over the switching period TsT_s.
Define the state vector x⃗(t)∈Rn\vec{x}(t) \in \mathbb{R}^n and the input vector u⃗(t)∈Rm\vec{u}(t) \in \mathbb{R}^m. The fundamental continuous-time behavior is modeled by the linear time-invariant (LTI) system: x⃗˙(t)=A⃗x⃗(t)+B⃗u⃗(t)\dot{\vec{x}}(t) = \vec{A}\vec{x}(t) + \vec{B}\vec{u}(t) where A⃗∈Rn×n\vec{A} \in \mathbb{R}^{n \times n} and B⃗∈Rn×m\vec{B} \in \mathbb{R}^{n \times m} are the system matrices.
The duty cycle DD is defined as the ratio of the ON time TonT_{on} to the switching period TsT_s: D=TonTsD = \frac{T_{on}}{T_s} For a DC-DC converter, the average output voltage VoutV_{out} is related to the input voltage VinV_{in} and the duty cycle DD by the voltage conversion ratio: VoutVin≈D1−D\frac{V_{out}}{V_{in}} \approx \frac{D}{1-D} (assuming ideal components and steady state).
Let s(t)∈{0,1}s(t) \in \{0, 1\} be the switching state. The instantaneous voltage vsw(t)v_{sw}(t) across the inductor is defined piecewise: vsw(t)=s(t)Vin−(1−s(t))Voutv_{sw}(t) = s(t) V_{in} - (1-s(t)) V_{out} The system dynamics are governed by the state equation dx⃗dt=f(x⃗,u⃗,s(t))\frac{d\vec{x}}{dt} = f(\vec{x}, \vec{u}, s(t)), where ff incorporates the switching function s(t)s(t) and its associated voltage/current relationships.
Linearization around the steady-state operating point (x⃗0,u⃗0)(\vec{x}_0, \vec{u}_0) yields the small-signal model (SSM) for the deviation variables Δx⃗=x⃗−x⃗0\Delta\vec{x} = \vec{x} - \vec{x}_0 and Δu⃗=u⃗−u⃗0\Delta\vec{u} = \vec{u} - \vec{u}_0: d(Δx⃗)dt=A⃗ssΔx⃗+B⃗ssΔu⃗\frac{d(\Delta\vec{x})}{dt} = \vec{A}_{ss}\Delta\vec{x} + \vec{B}_{ss}\Delta\vec{u} where A⃗ss=∂f∂x⃗∣(x⃗0,u⃗0)\vec{A}_{ss} = \left. \frac{\partial f}{\partial \vec{x}} \right|_{(\vec{x}_0, \vec{u}_0)} and B⃗ss=∂f∂u⃗∣(x⃗0,u⃗0)\vec{B}_{ss} = \left. \frac{\partial f}{\partial \vec{u}} \right|_{(\vec{x}_0, \vec{u}_0)} are the Jacobian matrices.
The system dynamics are modeled by applying Kirchhoff's laws to lumped elements. For a general circuit, the state vector x⃗\vec{x} comprises capacitor voltages v⃗c\vec{v}_c and inductor currents i⃗L\vec{i}_L. The governing equations take the form: C⃗dv⃗cdt=I⃗source−G⃗v⃗c−i⃗L\vec{C} \frac{d\vec{v}_c}{dt} = \vec{I}_{source} - \vec{G}\vec{v}_c - \vec{i}_L and di⃗Ldt=1L(Vin−Ri⃗L−switching terms)\frac{d\vec{i}_L}{dt} = \frac{1}{L} (V_{in} - R \vec{i}_L - \text{switching terms}) where C⃗\vec{C} and L⃗\vec{L} are diagonal matrices of capacitance and inductance.
At the switching boundaries tkt_k and tk+1t_{k+1}, the physical continuity constraints must be satisfied. Specifically, the inductor current iL(t)i_L(t) and capacitor voltages vc(t)v_c(t) must be continuous: iL(tk+1)=iL(tk)andvc(tk+1)=vc(tk)i_L(t_{k+1}) = i_L(t_k) \quad \text{and} \quad v_c(t_{k+1}) = v_c(t_k) These conditions are used to define the initial state for the next switching interval.
The inductor current iL(t)i_L(t) is approximated by its average value iˉL\bar{i}_L. The state dynamics are simplified by assuming diˉLdt≈0\frac{d\bar{i}_L}{dt} \approx 0 over the switching period TsT_s. The resulting averaged model relates the average current iˉL\bar{i}_L to the average voltage drop ⟨vL⟩\langle v_L \rangle: diˉLdt=1L⟨vL⟩≈1L(Vin−Vout)DTs\frac{d\bar{i}_L}{dt} = \frac{1}{L} \langle v_L \rangle \approx \frac{1}{L} (V_{in} - V_{out}) \frac{D}{T_s}
Given the state-space representation x⃗˙(t)=A⃗x⃗(t)+B⃗u⃗(t)\dot{\vec{x}}(t) = \vec{A}\vec{x}(t) + \vec{B}\vec{u}(t) with initial condition x⃗(0)=x⃗0\vec{x}(0) = \vec{x}_0, the Laplace transform yields the solution X⃗(s)\vec{X}(s): \vec{X}(s) = (s\vec{I} - \vec{A})^{-1} \left( \vec{x}_0 + \vec{B}\ring{\vec{U}}(s) \right) The transfer function G(s)G(s) relating the output Y⃗(s)\vec{Y}(s) to the input U⃗(s)\vec{U}(s) is then defined as: G(s)=Y⃗(s)/U⃗(s)=C⃗(sI⃗−A⃗)−1B⃗G(s) = \vec{Y}(s) / \vec{U}(s) = \vec{C}(s\vec{I} - \vec{A})^{-1} \vec{B}