A handbook for quickly querying the average model formulas.

Matrix

# Should we consider the ioi_o as an indepandent variable or not? Refer to the lecture note on page SupII-12.

The most simple condition

[i˙Lu˙C]=A[iLuC]+BVg\begin{bmatrix}\dot{i}_L\\\dot{u}_C\end{bmatrix}=A\begin{bmatrix}i_L\\u_C\end{bmatrix}+BV_g

[igvo]=C[iLuC]\begin{bmatrix}i_g\\v_o\end{bmatrix}=C\begin{bmatrix}i_L\\u_C\end{bmatrix}

Converter A1A_1 A2A_2 B1B_1 B2B_2 C1C_1 C2C_2
Buck [0−1L1C−1RC]\begin{bmatrix}0&-\frac{1}{L}\\\frac{1}{C}&-\frac{1}{RC}\end{bmatrix} [0−1L1C−1RC]\begin{bmatrix}0&-\frac{1}{L}\\\frac{1}{C}&-\frac{1}{RC}\end{bmatrix} [1L0]\begin{bmatrix}\frac{1}{L}\\0\end{bmatrix} [00]\begin{bmatrix}0\\0\end{bmatrix} [1001]\begin{bmatrix}1&0\\0&1\end{bmatrix} [0001]\begin{bmatrix}0&0\\0&1\end{bmatrix}
Boost [000−1RC]\begin{bmatrix}0&0\\0&-\frac{1}{RC}\end{bmatrix} [0−1L1C−1RC]\begin{bmatrix}0&-\frac{1}{L}\\\frac{1}{C}&-\frac{1}{RC}\end{bmatrix} [1L0]\begin{bmatrix}\frac{1}{L}\\0\end{bmatrix} [1L0]\begin{bmatrix}\frac{1}{L}\\0\end{bmatrix} [1001]\begin{bmatrix}1&0\\0&1\end{bmatrix} [1001]\begin{bmatrix}1&0\\0&1\end{bmatrix}
Buck-Boost [000−1RC]\begin{bmatrix}0&0\\0&-\frac{1}{RC}\end{bmatrix} [01L−1C−1RC]\begin{bmatrix}0&\frac{1}{L}\\-\frac{1}{C}&-\frac{1}{RC}\end{bmatrix} [1L0]\begin{bmatrix}\frac{1}{L}\\0\end{bmatrix} [00]\begin{bmatrix}0\\0\end{bmatrix} [100−1]\begin{bmatrix}1&0\\0&-1\end{bmatrix} [000−1]\begin{bmatrix}0&0\\0&-1\end{bmatrix}

Practical Parameters of MOSFETs and diodes

Considering the on-state resistor of power MOSFET is RonR_{on}, and on-state voltage drop of power diode is VDV_D. The direction of the voltage drop is from anode to cathode.

[i˙Lu˙C]=A[iLuC]+B[VgVD]\begin{bmatrix}\dot{i}_L\\\dot{u}_C\end{bmatrix}=A\begin{bmatrix}i_L\\u_C\end{bmatrix}+B\begin{bmatrix}V_g\\V_D\end{bmatrix}

[igvo]=C[iLuC]\begin{bmatrix}i_g\\v_o\end{bmatrix}=C\begin{bmatrix}i_L\\u_C\end{bmatrix}

Converter A1A_1 A2A_2 B1B_1 B2B_2 C1C_1 C2C_2
Buck [−RonL−1L1C−1RC]\begin{bmatrix}-\frac{R_{on}}{L}&-\frac{1}{L}\\\frac{1}{C}&-\frac{1}{RC}\end{bmatrix} [0−1L1C−1RC]\begin{bmatrix}0&-\frac{1}{L}\\\frac{1}{C}&-\frac{1}{RC}\end{bmatrix} [1L000]\begin{bmatrix}\frac{1}{L}&0\\0&0\end{bmatrix} [0−1L00]\begin{bmatrix}0&-\frac{1}{L}\\0&0\end{bmatrix} [1001]\begin{bmatrix}1&0\\0&1\end{bmatrix} [0001]\begin{bmatrix}0&0\\0&1\end{bmatrix}
Boost [−RonL00−1RC]\begin{bmatrix}-\frac{R_{on}}{L}&0\\0&-\frac{1}{RC}\end{bmatrix} [0−1L1C−1RC]\begin{bmatrix}0&-\frac{1}{L}\\\frac{1}{C}&-\frac{1}{RC}\end{bmatrix} [1L000]\begin{bmatrix}\frac{1}{L}&0\\0&0\end{bmatrix} [1L−1L00]\begin{bmatrix}\frac{1}{L}&-\frac{1}{L}\\0&0\end{bmatrix} [1001]\begin{bmatrix}1&0\\0&1\end{bmatrix} [1001]\begin{bmatrix}1&0\\0&1\end{bmatrix}
Buck-Boost [−RonL00−1RC]\begin{bmatrix}-\frac{R_{on}}{L}&0\\0&-\frac{1}{RC}\end{bmatrix} [01L−1C−1RC]\begin{bmatrix}0&\frac{1}{L}\\-\frac{1}{C}&-\frac{1}{RC}\end{bmatrix} [1L000]\begin{bmatrix}\frac{1}{L}&0\\0&0\end{bmatrix} [0−1L00]\begin{bmatrix}0&-\frac{1}{L}\\0&0\end{bmatrix} [100−1]\begin{bmatrix}1&0\\0&-1\end{bmatrix} [000−1]\begin{bmatrix}0&0\\0&-1\end{bmatrix}

ESR of capacitor

[i˙Lu˙C]=A[iLuC]+BVg\begin{bmatrix}\dot{i}_L\\\dot{u}_C\end{bmatrix}=A\begin{bmatrix}i_L\\u_C\end{bmatrix}+BV_g

[igvo]=C[iLuC]\begin{bmatrix}i_g\\v_o\end{bmatrix}=C\begin{bmatrix}i_L\\u_C\end{bmatrix}

Converter A1A_1 A2A_2 B1B_1 B2B_2 C1C_1 C2C_2
Buck [−RcR(R+Rc)L−R(R+Rc)LR(R+Rc)C−1(R+Rc)C]\begin{bmatrix}-\frac{R_cR}{(R+R_c)L}&-\frac{R}{(R+R_c)L}\\\frac{R}{(R+R_c)C}&-\frac{1}{(R+R_c)C}\end{bmatrix} [−RcR(R+Rc)L−R(R+Rc)LR(R+Rc)C−1(R+Rc)C]\begin{bmatrix}-\frac{R_cR}{(R+R_c)L}&-\frac{R}{(R+R_c)L}\\\frac{R}{(R+R_c)C}&-\frac{1}{(R+R_c)C}\end{bmatrix} [1L0]\begin{bmatrix}\frac{1}{L}\\0\end{bmatrix} [00]\begin{bmatrix}0\\0\end{bmatrix} [10RcRR+RcRR+Rc]\begin{bmatrix}1&0\\\frac{R_cR}{R+R_c}&\frac{R}{R+R_c}\end{bmatrix} [00RcRR+RcRR+Rc]\begin{bmatrix}0&0\\\frac{R_cR}{R+R_c}&\frac{R}{R+R_c}\end{bmatrix}
Boost [000−1(R+Rc)C]\begin{bmatrix}0&0\\0&-\frac{1}{(R+R_c)C}\end{bmatrix} [−RcR(R+Rc)L−R(R+Rc)LR(R+Rc)C−1(R+Rc)C]\begin{bmatrix}-\frac{R_cR}{(R+R_c)L}&-\frac{R}{(R+R_c)L}\\\frac{R}{(R+R_c)C}&-\frac{1}{(R+R_c)C}\end{bmatrix} [1L0]\begin{bmatrix}\frac{1}{L}\\0\end{bmatrix} [1L0]\begin{bmatrix}\frac{1}{L}\\0\end{bmatrix} [100RR+Rc]\begin{bmatrix}1&0\\0&\frac{R}{R+R_c}\end{bmatrix} [10RcRR+RcRR+Rc]\begin{bmatrix}1&0\\\frac{R_cR}{R+R_c}&\frac{R}{R+R_c}\end{bmatrix}
Buck-Boost [000−1(R+Rc)C]\begin{bmatrix}0&0\\0&-\frac{1}{(R+R_c)C}\end{bmatrix} [−RcR(R+Rc)LR(R+Rc)L−R(R+Rc)C−1(R+Rc)C]\begin{bmatrix}-\frac{R_cR}{(R+R_c)L}&\frac{R}{(R+R_c)L}\\-\frac{R}{(R+R_c)C}&-\frac{1}{(R+R_c)C}\end{bmatrix} [1L0]\begin{bmatrix}\frac{1}{L}\\0\end{bmatrix} [00]\begin{bmatrix}0\\0\end{bmatrix} [100−RR+Rc]\begin{bmatrix}1&0\\0&-\frac{R}{R+R_c}\end{bmatrix} [00RcRR+Rc−RR+Rc]\begin{bmatrix}0&0\\\frac{R_cR}{R+R_c}&-\frac{R}{R+R_c}\end{bmatrix}

ESR of capacitor and inductor

[i˙Lu˙C]=A[iLuC]+BVg\begin{bmatrix}\dot{i}_L\\\dot{u}_C\end{bmatrix}=A\begin{bmatrix}i_L\\u_C\end{bmatrix}+BV_g

[igvo]=C[iLuC]\begin{bmatrix}i_g\\v_o\end{bmatrix}=C\begin{bmatrix}i_L\\u_C\end{bmatrix}

Converter A1A_1 A2A_2 B1B_1 B2B_2 C1C_1 C2C_2
Buck [−RcR(R+Rc)L−RlL−R(R+Rc)LR(R+Rc)C−1(R+Rc)C]\begin{bmatrix}-\frac{R_cR}{(R+R_c)L}-\frac{R_l}{L}&-\frac{R}{(R+R_c)L}\\\frac{R}{(R+R_c)C}&-\frac{1}{(R+R_c)C}\end{bmatrix} [−RcR(R+Rc)L−RlL−R(R+Rc)LR(R+Rc)C−1(R+Rc)C]\begin{bmatrix}-\frac{R_cR}{(R+R_c)L}-\frac{R_l}{L}&-\frac{R}{(R+R_c)L}\\\frac{R}{(R+R_c)C}&-\frac{1}{(R+R_c)C}\end{bmatrix} [1L0]\begin{bmatrix}\frac{1}{L}\\0\end{bmatrix} [00]\begin{bmatrix}0\\0\end{bmatrix} [10RcRR+RcRR+Rc]\begin{bmatrix}1&0\\\frac{R_cR}{R+R_c}&\frac{R}{R+R_c}\end{bmatrix} [00RcRR+RcRR+Rc]\begin{bmatrix}0&0\\\frac{R_cR}{R+R_c}&\frac{R}{R+R_c}\end{bmatrix}
Boost [−RlL00−1(R+Rc)C]\begin{bmatrix}-\frac{R_l}{L}&0\\0&-\frac{1}{(R+R_c)C}\end{bmatrix} [−RcR(R+Rc)L−RlL−R(R+Rc)LR(R+Rc)C−1(R+Rc)C]\begin{bmatrix}-\frac{R_cR}{(R+R_c)L}-\frac{R_l}{L}&-\frac{R}{(R+R_c)L}\\\frac{R}{(R+R_c)C}&-\frac{1}{(R+R_c)C}\end{bmatrix} [1L0]\begin{bmatrix}\frac{1}{L}\\0\end{bmatrix} [1L0]\begin{bmatrix}\frac{1}{L}\\0\end{bmatrix} [100RR+Rc]\begin{bmatrix}1&0\\0&\frac{R}{R+R_c}\end{bmatrix} [10RcRR+RcRR+Rc]\begin{bmatrix}1&0\\\frac{R_cR}{R+R_c}&\frac{R}{R+R_c}\end{bmatrix}
Buck-Boost [−RlL00−1(R+Rc)C]\begin{bmatrix}-\frac{R_l}{L}&0\\0&-\frac{1}{(R+R_c)C}\end{bmatrix} [−RcR(R+Rc)L−RlLR(R+Rc)L−R(R+Rc)C−1(R+Rc)C]\begin{bmatrix}-\frac{R_cR}{(R+R_c)L}-\frac{R_l}{L}&\frac{R}{(R+R_c)L}\\-\frac{R}{(R+R_c)C}&-\frac{1}{(R+R_c)C}\end{bmatrix} [1L0]\begin{bmatrix}\frac{1}{L}\\0\end{bmatrix} [00]\begin{bmatrix}0\\0\end{bmatrix} [100−RR+Rc]\begin{bmatrix}1&0\\0&-\frac{R}{R+R_c}\end{bmatrix} [00RcRR+Rc−RR+Rc]\begin{bmatrix}0&0\\\frac{R_cR}{R+R_c}&-\frac{R}{R+R_c}\end{bmatrix}

All-considered

[i˙Lu˙C]=A[iLuC]+B[VgVD]\begin{bmatrix}\dot{i}_L\\\dot{u}_C\end{bmatrix}=A\begin{bmatrix}i_L\\u_C\end{bmatrix}+B\begin{bmatrix}V_g\\V_D\end{bmatrix}

[igvo]=C[iLuC]\begin{bmatrix}i_g\\v_o\end{bmatrix}=C\begin{bmatrix}i_L\\u_C\end{bmatrix}

Converter A1A_1 A2A_2 B1B_1 B2B_2 C1C_1 C2C_2
Buck [−RcR(R+Rc)L−RlL−RonL−R(R+Rc)LR(R+Rc)C−1(R+Rc)C]\begin{bmatrix}-\frac{R_cR}{(R+R_c)L}-\frac{R_l}{L}-\frac{R_{on}}{L}&-\frac{R}{(R+R_c)L}\\\frac{R}{(R+R_c)C}&-\frac{1}{(R+R_c)C}\end{bmatrix} [−RcR(R+Rc)L−RlL−R(R+Rc)LR(R+Rc)C−1(R+Rc)C]\begin{bmatrix}-\frac{R_cR}{(R+R_c)L}-\frac{R_l}{L}&-\frac{R}{(R+R_c)L}\\\frac{R}{(R+R_c)C}&-\frac{1}{(R+R_c)C}\end{bmatrix} [1L000]\begin{bmatrix}\frac{1}{L}&0\\0&0\end{bmatrix} [0−1L00]\begin{bmatrix}0&-\frac{1}{L}\\0&0\end{bmatrix} [10RcRR+RcRR+Rc]\begin{bmatrix}1&0\\\frac{R_cR}{R+R_c}&\frac{R}{R+R_c}\end{bmatrix} [00RcRR+RcRR+Rc]\begin{bmatrix}0&0\\\frac{R_cR}{R+R_c}&\frac{R}{R+R_c}\end{bmatrix}
Boost [−RlL−RonL00−1(R+Rc)C]\begin{bmatrix}-\frac{R_l}{L}-\frac{R_{on}}{L}&0\\0&-\frac{1}{(R+R_c)C}\end{bmatrix} [−RcR(R+Rc)L−RlL−R(R+Rc)LR(R+Rc)C−1(R+Rc)C]\begin{bmatrix}-\frac{R_cR}{(R+R_c)L}-\frac{R_l}{L}&-\frac{R}{(R+R_c)L}\\\frac{R}{(R+R_c)C}&-\frac{1}{(R+R_c)C}\end{bmatrix} [1L000]\begin{bmatrix}\frac{1}{L}&0\\0&0\end{bmatrix} [1L−1L00]\begin{bmatrix}\frac{1}{L}&-\frac{1}{L}\\0&0\end{bmatrix} [100RR+Rc]\begin{bmatrix}1&0\\0&\frac{R}{R+R_c}\end{bmatrix} [10RcRR+RcRR+Rc]\begin{bmatrix}1&0\\\frac{R_cR}{R+R_c}&\frac{R}{R+R_c}\end{bmatrix}
Buck-Boost [−RlL−RonL00−1(R+Rc)C]\begin{bmatrix}-\frac{R_l}{L}-\frac{R_{on}}{L}&0\\0&-\frac{1}{(R+R_c)C}\end{bmatrix} [−RcR(R+Rc)L−RlLR(R+Rc)L−R(R+Rc)C−1(R+Rc)C]\begin{bmatrix}-\frac{R_cR}{(R+R_c)L}-\frac{R_l}{L}&\frac{R}{(R+R_c)L}\\-\frac{R}{(R+R_c)C}&-\frac{1}{(R+R_c)C}\end{bmatrix} [1L000]\begin{bmatrix}\frac{1}{L}&0\\0&0\end{bmatrix} [0−1L00]\begin{bmatrix}0&-\frac{1}{L}\\0&0\end{bmatrix} [100−RR+Rc]\begin{bmatrix}1&0\\0&-\frac{R}{R+R_c}\end{bmatrix} [00RcRR+Rc−RR+Rc]\begin{bmatrix}0&0\\\frac{R_cR}{R+R_c}&-\frac{R}{R+R_c}\end{bmatrix}

Steady State Values

X=−A−1BVgX=-A^{-1}BV_g

Transfer Functions

General

x^d^=(sI−A)−1[(A1−A2)X+(B1−B2)Vg]\frac{\hat{x}}{\hat{d}}=(sI-A)^{-1}[(A_1-A_2)X+(B_1-B_2)V_g]

vo^d^=Cx^d^+(C1−C2)X\frac{\hat{v_o}}{\hat{d}}=C\frac{\hat{x}}{\hat{d}}+(C_1-C_2)X

x^vg^=(sI−A)−1⋅B\frac{\hat{x}}{\hat{v_g}}=(sI-A)^{-1}\cdot B

vo^vg^=C⋅(sI−A)−1⋅B\frac{\hat{v_o}}{\hat{v_g}}=C\cdot(sI-A)^{-1}\cdot B

Buck

v~oi~L=R(1+sCRc)1+sC(R+Rc)\frac{\tilde{v}_o}{\tilde{i}_L}=\frac{R(1+sCR_c)}{1+sC(R+R_c)}

The v~oi~L\frac{\tilde{v}_o}{\tilde{i}_L} transfer function is the parallel of output resistor and capacitor.