Input Parameters Duty Ratio ($D$) Number of Phases ($M$) Number of Turns per Winding ($N$)
Derived Parameters Interleaving Boosting Inductance ($1/\delta$) Number of Overlaped Phases ($k$) Interleaving Ripple Compression ($\delta$)
Method Name Inductance Dual Model Inductance Matrix Model Multiwinding Transformer Model
Design Parameters $$\mathcal{R}_L$$ $$L_S$$ $$L_l$$
$$\mathcal{R}_C$$ $$L_M$$ $$L_\mu$$
$$\beta = \frac{\mathcal{R}_C}{\mathcal{R}_L}$$ $$\alpha = -\frac{L_M}{L_S}$$ $$\rho = -\frac{L_\mu}{L_l}$$
Description Matrix $$N^2 \begin{bmatrix} \frac{di_1}{dt} \\ \frac{di_2}{dt} \\ \vdots \\ \frac{di_M}{dt} \\ \end{bmatrix} = \begin{bmatrix} \mathcal{R}_L + \mathcal{R}_C & \mathcal{R}_C & \ldots & \mathcal{R}_C \\ \mathcal{R}_C & \mathcal{R}_L + \mathcal{R}_C & \ldots & \mathcal{R}_C \\ \vdots & \vdots & \ddots & \vdots \\ \mathcal{R}_C & \ldots & \mathcal{R}_C & \mathcal{R}_L + \mathcal{R}_C \\ \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_M \\ \end{bmatrix}$$ $$\begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_M \\ \end{bmatrix} = \begin{bmatrix} L_S & L_M & \ldots & L_M \\ L_M & L_S & \ldots & L_M \\ \vdots & \vdots & \ddots & \vdots \\ L_M & \ldots & L_M & L_S \\ \end{bmatrix} \begin{bmatrix} \frac{di_1}{dt} \\ \frac{di_2}{dt} \\ \vdots \\ \frac{di_M}{dt} \\ \end{bmatrix}$$ $$\begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_M \\ \end{bmatrix} = \begin{bmatrix} L_\mu + L_l & -\frac{1}{M-1} L_\mu & \ldots & -\frac{1}{M-1} L_\mu \\ -\frac{1}{M-1} L_\mu & L_\mu + L_l & \ldots & -\frac{1}{M-1} L_\mu \\ \vdots & \vdots & \ddots & \vdots \\ -\frac{1}{M-1} L_\mu & -\frac{1}{M-1} L_\mu & \ldots & L\mu + L_l \\ \end{bmatrix} \begin{bmatrix} \frac{di_1}{dt} \\ \frac{di_2}{dt} \\ \vdots \\ \frac{di_M}{dt} \\ \end{bmatrix}$$
Lumped Circuit Model
Model Parameters (Units: $\mathcal{R}$ ~ 1/Henry, $L$ ~ Henry, $\Phi$ ~ Webber) $$\mathcal{R}_L$$ $$\mathcal{R}_L = \frac{N^2}{L_S - L_M}$$ $$\mathcal{R}_L = \frac{N^2(M-1)}{(M-1)L_l + ML_\mu}$$
$$\mathcal{R}_C$$ $$\mathcal{R}_C = \frac{-N^2 L_M}{(L_S - L_M)(L_S + (M-1)L_M)}$$ $$\mathcal{R}_C = \frac{-N^2 L_\mu}{L_l((M-1)L_l + M L_\mu)}$$
$$L_l = \frac{N^2}{\mathcal{R}_L + M \mathcal{R}_C}$$ $$L_l = L_S + (M-1) L_M$$ $$L_l$$
$$L_\mu = \frac{N^2 (M-1) \mathcal{R}_C}{\mathcal{R}_L (\mathcal{R}_L + M \mathcal{R}_C)}$$ $$L_\mu = -(M-1) L_M$$ $$L_\mu$$
$$L_S = \frac{N^2(\mathcal{R}_L + (M-1) \mathcal{R}_C)}{\mathcal{R}_L (\mathcal{R}_L + M \mathcal{R}_C)}$$ $$L_S$$ $$L_S = L_\mu + L_l$$
$$L_M = \frac{-N^2 \mathcal{R}_C}{\mathcal{R}_L(\mathcal{R}_L + M \mathcal{R}_C)}$$ $$L_M$$ $$L_M = -\frac{1}{M-1} L_\mu$$
$$L_L = \frac{1}{\mathcal{R}_L}$$ $$L_L = \frac{1}{\mathcal{R}_L}$$ $$L_L = \frac{1}{\mathcal{R}_L}$$
$$L_C = \frac{1}{\mathcal{R}_C}$$ $$L_C = \frac{1}{\mathcal{R}_C}$$ $$L_C = \frac{1}{\mathcal{R}_C}$$
$$L_L^*= \frac{N^2(\mathcal{R}_L + (M-1) \mathcal{R}_C)}{\mathcal{R}_L(\mathcal{R}_L + M \mathcal{R}_C)}$$ $$L_L^*= L_S$$ $$L_L^*= L_\mu+L_l$$
$$L_C^*= \frac{N^2}{\frac{\mathcal{R}_L}{M} + \mathcal{R}_C}$$ $$L_C^*= M(L_S+(M-1)L_M)$$ $$L_C^*= ML_l$$
$L_{oss}$ $$\frac{(1-D)DMN^2}{(\mathcal{R}_L + M \mathcal{R}_C)(k + 1 - DM)(DM - k)}$$ $$\frac{(1-D)DM(L_S + L_M(M-1))}{(DM - k)(1 + k - DM)}$$ $$\frac{(1-D)DML_l}{(DM-k)(1 + k - DM)}$$
$L_{pss}$ $$\frac{N^2(1-D)}{-\frac{k^2\mathcal{R}_C}{DM}-\frac{k\mathcal{R}_C}{DM} + 2k\mathcal{R}_C - DM\mathcal{R}_C + \mathcal{R}_C - DR_L + R_L}$$ $$\frac{(L_S - L_M)(L_S + (M-1)L_M)}{L_S + ((M-2k-2) + \frac{k(k+1)}{MD} + \frac{MD(M-2k-1)+k(k+1)}{M(1-D)} L_M)}$$ $$\frac{DM(1-D)((M-1)L_l + ML_\mu)L_l}{DM(1-D)(M-1)L_l + (DM(1-DM)-k^2-k+2DMk)L_\mu}$$
$L_{otr}$ $$\frac{N^2}{M(\mathcal{R}_L + M \mathcal{R}_C)}$$ $$\frac{L_S + (M-1) L_M}{M}$$ $$\frac{L_l}{M}$$
$L_{ptr}$ $$\frac{N^2}{\mathcal{R}_L + M \mathcal{R}_C}$$ $$L_S + (M-1) L_M$$ $$L_l$$
$L_{ptr}/L_{pss}$ $$\frac{-\frac{k^2\beta}{DM} - \frac{k\beta}{DM} + 2k\beta - DM \beta + \beta - D + 1}{(1-D)(1 + M\beta)}$$ $$\frac{1 - ((M-2k-2) + \frac{k(k+1)}{MD} + \frac{MD(M-2k-1) + k(k+1)}{M(1-D)})\alpha}{1+\alpha}$$ $$\frac{DM(1-D)(M-1) + (DM(1-DM)-k^2-k+2DMk)\rho}{DM(1-D)(M-1+M_\rho)}$$
$\Phi_{L,DC}/I_{out}$ $$\frac{N}{M(\mathcal{R}_L + M \mathcal{R}_C)}$$ $$\frac{L_S+(M-1)L_M}{MN}$$ $$\frac{L_l}{MN}$$
$\Phi_{C,DC}/I_{out}$ $$\frac{N}{\mathcal{R}_L + M \mathcal{R}_C}$$ $$\frac{L_S+(M-1)L_M}{N}$$ $$\frac{L_l}{N}$$

Usage Count:

Designed by Seungjae Ryan Lee, under the supervision of Haoran Li, Daniel Zhou and Minjie Chen