CoupMag Expert | Princeton Power Electronics

Operating Point

Interleaving terms are derived from the duty ratio, phase count, and turns per winding.

Duty Ratio $D$

Number of Phases $M$

Turns per Winding $N$

Interleaving Boosting Inductance $1/\delta$ --

Number of Overlapped Phases $k$ --

Interleaving Ripple Compression $\delta$ --

Design Parameters

Use reluctance in H^-1 and inductance in H. Flux-per-current outputs are reported in Wb/A.

Inductance Dual Model

$\mathcal{R}_L$ H^-1

$\mathcal{R}_C$ H^-1

$\beta = \mathcal{R}_C/\mathcal{R}_L$ --

Inductance Matrix Model

$L_S$ H

$L_M$ H

$\alpha = -L_M/L_S$ --

Multiwinding Transformer Model

$L_l$ H

$L_\mu$ H

$\rho = L_\mu/L_l$ --

Model Explorer

Static equations remain visible beside the live numerical results for each modeling view.

$$ N^2 \begin{bmatrix} \frac{di_1}{dt} \\ \frac{di_2}{dt} \\ \vdots \\ \frac{di_M}{dt} \end{bmatrix} = \begin{bmatrix} \mathcal{R}_L + \mathcal{R}_C & \mathcal{R}_C & \ldots & \mathcal{R}_C \\ \mathcal{R}_C & \mathcal{R}_L + \mathcal{R}_C & \ldots & \mathcal{R}_C \\ \vdots & \vdots & \ddots & \vdots \\ \mathcal{R}_C & \ldots & \mathcal{R}_C & \mathcal{R}_L + \mathcal{R}_C \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_M \end{bmatrix} $$

Model Parameters
$\mathcal{R}_L$	$\mathcal{R}_L$	--
$\mathcal{R}_C$	$\mathcal{R}_C$	--
$L_l$	$L_l = \frac{N^2}{\mathcal{R}_L + M\mathcal{R}_C}$	--
$L_\mu$	$L_\mu = \frac{N^2(M-1)\mathcal{R}_C}{\mathcal{R}_L(\mathcal{R}_L + M\mathcal{R}_C)}$	--
$L_S$	$L_S = \frac{N^2(\mathcal{R}_L + (M-1)\mathcal{R}_C)}{\mathcal{R}_L(\mathcal{R}_L + M\mathcal{R}_C)}$	--
$L_M$	$L_M = \frac{-N^2\mathcal{R}_C}{\mathcal{R}_L(\mathcal{R}_L + M\mathcal{R}_C)}$	--
$L_L$	$L_L = \frac{1}{\mathcal{R}_L}$	--
$L_C$	$L_C = \frac{1}{\mathcal{R}_C}$	--
$L_L^*$	$L_L^* = \frac{N^2(\mathcal{R}_L + (M-1)\mathcal{R}_C)}{\mathcal{R}_L(\mathcal{R}_L + M\mathcal{R}_C)}$	--
$L_C^*$	$L_C^* = \frac{N^2}{\mathcal{R}_L/M + \mathcal{R}_C}$	--

Converter Quantities
$L_{oss}$	$\frac{(1-D)DMN^2}{(\mathcal{R}_L + M\mathcal{R}_C)(k + 1 - DM)(DM - k)}$	--
$L_{pss}$	$\frac{N^2(1-D)}{-\frac{k^2\mathcal{R}_C}{DM}-\frac{k\mathcal{R}_C}{DM}+2k\mathcal{R}_C-DM\mathcal{R}_C+\mathcal{R}_C-D\mathcal{R}_L+\mathcal{R}_L}$	--
$L_{otr}$	$\frac{N^2}{M(\mathcal{R}_L + M\mathcal{R}_C)}$	--
$L_{ptr}$	$\frac{N^2}{\mathcal{R}_L + M\mathcal{R}_C}$	--
$L_{ptr}/L_{pss}$	$\frac{-\frac{k^2\beta}{DM}-\frac{k\beta}{DM}+2k\beta-DM\beta+\beta-D+1}{(1-D)(1+M\beta)}$	--
$\Phi_{L,DC}/I_{out}$	$\frac{N}{M(\mathcal{R}_L + M\mathcal{R}_C)}$	--
$\Phi_{C,DC}/I_{out}$	$\frac{N}{\mathcal{R}_L + M\mathcal{R}_C}$	--

$$ \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_M \end{bmatrix} = \begin{bmatrix} L_S & L_M & \ldots & L_M \\ L_M & L_S & \ldots & L_M \\ \vdots & \vdots & \ddots & \vdots \\ L_M & \ldots & L_M & L_S \end{bmatrix} \begin{bmatrix} \frac{di_1}{dt} \\ \frac{di_2}{dt} \\ \vdots \\ \frac{di_M}{dt} \end{bmatrix} $$

Model Parameters
$\mathcal{R}_L$	$\mathcal{R}_L = \frac{N^2}{L_S - L_M}$	--
$\mathcal{R}_C$	$\mathcal{R}_C = \frac{-N^2L_M}{(L_S - L_M)(L_S + (M-1)L_M)}$	--
$L_l$	$L_l = L_S + (M-1)L_M$	--
$L_\mu$	$L_\mu = -(M-1)L_M$	--
$L_S$	$L_S$	--
$L_M$	$L_M$	--
$L_L$	$L_L = \frac{1}{\mathcal{R}_L}$	--
$L_C$	$L_C = \frac{1}{\mathcal{R}_C}$	--
$L_L^*$	$L_L^* = L_S$	--
$L_C^*$	$L_C^* = M(L_S + (M-1)L_M)$	--

Converter Quantities
$L_{oss}$	$\frac{(1-D)DM(L_S + (M-1)L_M)}{(DM-k)(1+k-DM)}$	--
$L_{pss}$	$\frac{(L_S-L_M)(L_S+(M-1)L_M)}{L_S + \left(M-2k-2+\frac{k(k+1)}{MD}+\frac{MD(M-2k-1)+k(k+1)}{M(1-D)}\right)L_M}$	--
$L_{otr}$	$\frac{L_S + (M-1)L_M}{M}$	--
$L_{ptr}$	$L_S + (M-1)L_M$	--
$L_{ptr}/L_{pss}$	$\frac{1-\left(M-2k-2+\frac{k(k+1)}{MD}+\frac{MD(M-2k-1)+k(k+1)}{M(1-D)}\right)\alpha}{1+\alpha}$	--
$\Phi_{L,DC}/I_{out}$	$\frac{L_S+(M-1)L_M}{MN}$	--
$\Phi_{C,DC}/I_{out}$	$\frac{L_S+(M-1)L_M}{N}$	--

$$ \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_M \end{bmatrix} = \begin{bmatrix} L_\mu + L_l & -\frac{1}{M-1}L_\mu & \ldots & -\frac{1}{M-1}L_\mu \\ -\frac{1}{M-1}L_\mu & L_\mu + L_l & \ldots & -\frac{1}{M-1}L_\mu \\ \vdots & \vdots & \ddots & \vdots \\ -\frac{1}{M-1}L_\mu & -\frac{1}{M-1}L_\mu & \ldots & L_\mu + L_l \end{bmatrix} \begin{bmatrix} \frac{di_1}{dt} \\ \frac{di_2}{dt} \\ \vdots \\ \frac{di_M}{dt} \end{bmatrix} $$

Multiwinding transformer lumped circuit model

Model Parameters
$\mathcal{R}_L$	$\mathcal{R}_L = \frac{N^2(M-1)}{(M-1)L_l + ML_\mu}$	--
$\mathcal{R}_C$	$\mathcal{R}_C = \frac{N^2L_\mu}{L_l((M-1)L_l + ML_\mu)}$	--
$L_l$	$L_l$	--
$L_\mu$	$L_\mu$	--
$L_S$	$L_S = L_\mu + L_l$	--
$L_M$	$L_M = -\frac{1}{M-1}L_\mu$	--
$L_L$	$L_L = \frac{1}{\mathcal{R}_L}$	--
$L_C$	$L_C = \frac{1}{\mathcal{R}_C}$	--
$L_L^*$	$L_L^* = L_\mu + L_l$	--
$L_C^*$	$L_C^* = ML_l$	--

Converter Quantities
$L_{oss}$	$\frac{(1-D)DML_l}{(DM-k)(1+k-DM)}$	--
$L_{pss}$	$\frac{DM(1-D)((M-1)L_l + ML_\mu)L_l}{DM(1-D)(M-1)L_l + (DM(1-DM)-k^2-k+2DMk)L_\mu}$	--
$L_{otr}$	$\frac{L_l}{M}$	--
$L_{ptr}$	$L_l$	--
$L_{ptr}/L_{pss}$	$\frac{DM(1-D)(M-1)+(DM(1-DM)-k^2-k+2DMk)\rho}{DM(1-D)(M-1+M\rho)}$	--
$\Phi_{L,DC}/I_{out}$	$\frac{L_l}{MN}$	--
$\Phi_{C,DC}/I_{out}$	$\frac{L_l}{N}$	--

\(\mathcal{R}_L\)	\(\mathcal{R}_L\)	--
\(\mathcal{R}_C\)	\(\mathcal{R}_C\)	--
\(L_l\)	\(L_l = \frac{N^2}{\mathcal{R}_L + M\mathcal{R}_C}\)	--
\(L_\mu\)	\(L_\mu = \frac{N^2(M-1)\mathcal{R}_C}{\mathcal{R}_L(\mathcal{R}_L + M\mathcal{R}_C)}\)	--
\(L_S\)	\(L_S = \frac{N^2(\mathcal{R}_L + (M-1)\mathcal{R}_C)}{\mathcal{R}_L(\mathcal{R}_L + M\mathcal{R}_C)}\)	--
\(L_M\)	\(L_M = \frac{-N^2\mathcal{R}_C}{\mathcal{R}_L(\mathcal{R}_L + M\mathcal{R}_C)}\)	--
\(L_L\)	\(L_L = \frac{1}{\mathcal{R}_L}\)	--
\(L_C\)	\(L_C = \frac{1}{\mathcal{R}_C}\)	--
\(L_L^*\)	\(L_L^* = \frac{N^2(\mathcal{R}_L + (M-1)\mathcal{R}_C)}{\mathcal{R}_L(\mathcal{R}_L + M\mathcal{R}_C)}\)	--
\(L_C^*\)	\(L_C^* = \frac{N^2}{\mathcal{R}_L/M + \mathcal{R}_C}\)	--

\(L_{oss}\)	\(\frac{(1-D)DMN^2}{(\mathcal{R}_L + M\mathcal{R}_C)(k + 1 - DM)(DM - k)}\)	--
\(L_{pss}\)	\(\frac{N^2(1-D)}{-\frac{k^2\mathcal{R}_C}{DM}-\frac{k\mathcal{R}_C}{DM}+2k\mathcal{R}_C-DM\mathcal{R}_C+\mathcal{R}_C-D\mathcal{R}_L+\mathcal{R}_L}\)	--
\(L_{otr}\)	\(\frac{N^2}{M(\mathcal{R}_L + M\mathcal{R}_C)}\)	--
\(L_{ptr}\)	\(\frac{N^2}{\mathcal{R}_L + M\mathcal{R}_C}\)	--
\(L_{ptr}/L_{pss}\)	\(\frac{-\frac{k^2\beta}{DM}-\frac{k\beta}{DM}+2k\beta-DM\beta+\beta-D+1}{(1-D)(1+M\beta)}\)	--
\(\Phi_{L,DC}/I_{out}\)	\(\frac{N}{M(\mathcal{R}_L + M\mathcal{R}_C)}\)	--
\(\Phi_{C,DC}/I_{out}\)	\(\frac{N}{\mathcal{R}_L + M\mathcal{R}_C}\)	--

\(\mathcal{R}_L\)	\(\mathcal{R}_L = \frac{N^2}{L_S - L_M}\)	--
\(\mathcal{R}_C\)	\(\mathcal{R}_C = \frac{-N^2L_M}{(L_S - L_M)(L_S + (M-1)L_M)}\)	--
\(L_l\)	\(L_l = L_S + (M-1)L_M\)	--
\(L_\mu\)	\(L_\mu = -(M-1)L_M\)	--
\(L_S\)	\(L_S\)	--
\(L_M\)	\(L_M\)	--
\(L_L\)	\(L_L = \frac{1}{\mathcal{R}_L}\)	--
\(L_C\)	\(L_C = \frac{1}{\mathcal{R}_C}\)	--
\(L_L^*\)	\(L_L^* = L_S\)	--
\(L_C^*\)	\(L_C^* = M(L_S + (M-1)L_M)\)	--

\(L_{oss}\)	\(\frac{(1-D)DM(L_S + (M-1)L_M)}{(DM-k)(1+k-DM)}\)	--
\(L_{pss}\)	\(\frac{(L_S-L_M)(L_S+(M-1)L_M)}{L_S + \left(M-2k-2+\frac{k(k+1)}{MD}+\frac{MD(M-2k-1)+k(k+1)}{M(1-D)}\right)L_M}\)	--
\(L_{otr}\)	\(\frac{L_S + (M-1)L_M}{M}\)	--
\(L_{ptr}\)	\(L_S + (M-1)L_M\)	--
\(L_{ptr}/L_{pss}\)	\(\frac{1-\left(M-2k-2+\frac{k(k+1)}{MD}+\frac{MD(M-2k-1)+k(k+1)}{M(1-D)}\right)\alpha}{1+\alpha}\)	--
\(\Phi_{L,DC}/I_{out}\)	\(\frac{L_S+(M-1)L_M}{MN}\)	--
\(\Phi_{C,DC}/I_{out}\)	\(\frac{L_S+(M-1)L_M}{N}\)	--

\(\mathcal{R}_L\)	\(\mathcal{R}_L = \frac{N^2(M-1)}{(M-1)L_l + ML_\mu}\)	--
\(\mathcal{R}_C\)	\(\mathcal{R}_C = \frac{N^2L_\mu}{L_l((M-1)L_l + ML_\mu)}\)	--
\(L_l\)	\(L_l\)	--
\(L_\mu\)	\(L_\mu\)	--
\(L_S\)	\(L_S = L_\mu + L_l\)	--
\(L_M\)	\(L_M = -\frac{1}{M-1}L_\mu\)	--
\(L_L\)	\(L_L = \frac{1}{\mathcal{R}_L}\)	--
\(L_C\)	\(L_C = \frac{1}{\mathcal{R}_C}\)	--
\(L_L^*\)	\(L_L^* = L_\mu + L_l\)	--
\(L_C^*\)	\(L_C^* = ML_l\)	--