Modelling and dynamical analysis of a DC–DC converter with coupled inductors

Abstract

The large-signal averaged model of a coupled-inductor double-boost converter is developed and analysed in this paper. Due to the large current fluctuations, the differential system is deduced by averaging the circuit equations of the operation modes over a switching period. Generic expressions that permit to calculate the current commutation intervals as function of the averaged state variables are also found to complete the model. Resistive losses are introduced into an equivalent averaged circuit leading to a more realistic scenario. The proposed state-space model is used for studying voltage conversion ratios, transients and frequency-domain responses of the converter as well as for designing a control loop that regulates the output voltage. Numerical simulations and experimental measurements corroborate the obtained results.

Appendices

Appendix A

The solution of set of linear equations defined in Subsection 3.2 is given by

$$\begin{aligned} Ip_{11}= & {} \dfrac{d_1 T[L_2 V_\mathrm{dc}-M (\overline{v_{C1}}-\overline{v_{C2}})]}{L_1 L_2 - M^2}, \\ Ip_{12}= & {} \dfrac{\hbox {d} T}{L_1} V_\mathrm{dc}+\dfrac{d_1 T M [M V_\mathrm{dc}-L_1 (\overline{v_{C1}}-\overline{v_{C2}})]}{L_1 (L_1 L_2 - M^2)},\\ Ip_{21}= & {} \dfrac{d_1 T[M V_\mathrm{dc}-L_1 (\overline{v_{C1}}-\overline{v_{C2}})]}{L_1 L_2 - M^2},\\ Ip_{22}= & {} \dfrac{d_1 M (L_1+M) T [M V_\mathrm{dc}-L_1 (\overline{v_{C1}}-\overline{v_{C2}})]}{L_1 (L_2+M) (L_1 L_2 - M^2)}\\&+\dfrac{T [d (L_1+M) V_\mathrm{dc}+d_2 L_1 (V_\mathrm{dc}-\overline{v_{C2}})]}{L_1 (L_2+M)} \end{aligned}$$

Notice that $Ip_{12}$ is equivalent to (6) since both expressions arise in the same set of equations (Table 2). But, additional relations given by the average of the currents are used here to find $d_1$ and $d_2$ values.

Appendix B

Considering that $L_{1M}= L_1 + M$, $L_{2M}= L_2 + M$, $L_p=L_1+L_2$, $L_n=L_1-L_2$, $L_{d}= d L_2 + M$ and $L_\alpha =M[\alpha +L_1(L_2+M)]$, the polynomials that permit to calculate $d_1$ and $d_2$ as a function of the average state-space variables are

$$\begin{aligned} P_1= & {} L_d V_\mathrm{dc}-(1-d) M \overline{v_{C2}},\\ P_2= & {} L_\alpha T [-M V_\mathrm{dc}+L_1 (\overline{v_{C1}} - \overline{v_{C2}})],\\ P_3= & {} [(1-d) M^2 L_p + 2 L_1^2 L_2 L_d] V_\mathrm{dc}\\&-L_1 L_\alpha \overline{v_{C1}}+L_1 (3 L_\alpha -4 L_1 M L_{2M})\overline{v_{C2}},\\ P_4= & {} \{(1-d) M(M^4+L_2 L_\alpha )\\&-L_1 L_2 L_d[3(1-d) M^2+ 2 L_1L_d]\}V_\mathrm{dc}^2\\&+L_1 L_\alpha (2L_d-L_{2M}) V_\mathrm{dc}\overline{v_{C1}}\\&+2 (1-d) L_1 M L_\alpha \overline{v_{C2}}(\overline{v_{C2}}-\overline{v_{C1}})\\&+L_1\{L_{2M} L_\alpha +4 M [(1-d)^2 M^2 L_2\\&-d L_1 L_2(L_d+M)-L_1 M^2]\}V_\mathrm{dc}\overline{v_{C2}},\\ P_5= & {} - [2 M L_{1M} L_p + 2 d(L_2 -L_d)(L_1^2 L_2 + M^3) \\&+ d L_1 M (L_2^2 - M^2)]V_\mathrm{dc}^2\\&+L_1 [2 M L_\alpha +2 d L_2 (1-d)M^2\\&-d L_1 (2 d L_2 M +L_2^2 + 3 M^2)] V_\mathrm{dc} \overline{v_{C1}}\\&+d L_1^2 M^2 V_\mathrm{dc} \overline{v_{C2}}\\&+\{L_1^2 L_2[(1 - 2 d) L_d+M (1-8d+4 d^2)]\\&-2(1-d)M[M L_1(2 L_1+3 L_d)-L_\alpha ]\}V_\mathrm{dc} \overline{v_{C2}} \\&-2 (1 - d) L_1\{ M[(1-d) M L_{1M} + L_1 L_{2M}]\overline{v_{C1}}\\&-[L_\alpha -L_1 M L_d+(1 - d) M^3]\overline{v_{C2}}\}\overline{v_{C2}},\\ P_6= & {} (2 L_1 L_2 L_d +M^2 L_{2M}) V_\mathrm{dc}^2\\&- L_1 M (L_{2M} + 2 L_d) V_\mathrm{dc} \overline{v_{C1}}\\&- L_1 M (L_{2M} - 4 L_d) V_\mathrm{dc} \overline{v_{C2}} \\&-2 (1 - d) L_1 M^2 \overline{v_{C2}} (\overline{v_{C2}} - \overline{v_{C1}}), \\ P_7= & {} -M (L_{2M} - 2 d L_2)^2 V_\mathrm{dc}^3\\&+[L_1 L_{2M}^2 - 4 (1 - d) M^2 L_d]V_\mathrm{dc}^2 \overline{v_{C1}}\\&+4 (1-d)(L_\alpha -L_1 M L_{2M}-2L_1 M L_d) V_\mathrm{dc} \overline{v_{C2}}\overline{v_{C1}}\\&-4 (1-d)(L_\alpha -3 L_1 ML_d) V_\mathrm{dc}\overline{v_{C2}}^2 \\&-(L_{2M} - 2 L_d) [4 (1- d) M^2 \\&+ L_1 (L_{2M} - 2 L_d)]V_\mathrm{dc}^2 \overline{v_{C2}}\\&+ 4 (1 - d)^2 L_1 M^2 \overline{v_{C2}}^2 (\overline{v_{C1}}-\overline{v_{C2}}),\\ P_8= & {} (M L_{1M}+d L_n) [L_pV_\mathrm{dc}+L_1 M (2 \overline{v_{C2}}-\overline{v_{C1}})]\\&-L_1^2 M L_{2M} \overline{v_{C1}}. \end{aligned}$$