Introduction

This paper presents an analysis of a Buck-Boost converter with unidirectional power flow, featuring two active and two passive switches. Both the ideal scenario and a quasi-static loss analysis are investigated, demonstrating that under certain load and loss conditions, the transfer function can exhibit a negative slope. This phenomenon can be critical, potentially leading to a blocking operating point and component destruction. In this context, a quasi-static loss analysis implies that diode reverse recovery losses (i.e., reverse recovery charge) and MOSFET gate charge injection losses are neglected. In systems utilizing fast-recovery diodes, these losses are highly negligible, which also applies to the transistors under specific load and gate-drive conditions.

Non inverting Buck-Boost

The non-inverting buck-boost topology with two active switches is shown below:

In this topology, the two switching elements of the Buck and Boost topologies have been combined, preserving the energy-storage elements L and C, respectively. The switch parameters \(\delta_1\) and \(\delta_2\) are dimensionless quantities ranging from 0 to 1, representing the fraction of the total period during which the respective switches are closed. Therefore, this system presents two degrees of freedom; in principle, there is no restriction preventing them from being turned on and off at will within the total switching period.

Consequently, there are 4 possible \((S1,S2)\) states: Out of the 4 possible states, state \(E_2\) is prohibited because it is a useless state: assuming ideal components, it neither stores energy nor releases it to the load. When considering lossy elements, this state merely introduces dissipation losses across the various components without contributing to any significant energy transfer.

This imposes the following constraint: \(\delta_2 \leq \delta_1\)

State space analysis

Static analysis:

Let us define the load voltage as the variable \(x_1\) and the current as \(x_2\). The state-space analysis is formulated as follows:

\[\begin{equation} \begin{pmatrix} \dot{x}_1\\ \dot{x}_2 \end{pmatrix} = E_0(1-\delta_1)+E_1(\delta_1-\delta_2)+E_3\delta_2 \end{equation}\]

State \(E_0\) \[\begin{equation} \begin{pmatrix} \dot{x}_1\\ \dot{x}_2 \end{pmatrix} = \begin{pmatrix} -\frac{1}{RC} & \frac{1}{C}\\ -\frac{1}{L} & 0\\ \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix} \end{equation}\]

State \(E_1\) \[\begin{equation} \begin{pmatrix} \dot{x}_1\\ \dot{x}_2 \end{pmatrix} = \begin{pmatrix} -\frac{1}{RC} & \frac{1}{C}\\ -\frac{1}{L} & 0\\ \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix}+ \begin{pmatrix} 0\\ \frac{V_i}{L} \end{pmatrix} \end{equation}\]

State \(E_3\) \[\begin{equation} \begin{pmatrix} \dot{x}_1\\ \dot{x}_2 \end{pmatrix} = \begin{pmatrix} -\frac{1}{RC} & 0\\ 0 & 0\\ \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix}+ \begin{pmatrix} 0\\ \frac{V_i}{L} \end{pmatrix} \end{equation}\]

Yielding the averaged model¹:

\[\begin{equation} \begin{pmatrix} \dot{x}_1\\ \dot{x}_2 \end{pmatrix} = \begin{pmatrix} -\frac{1}{RC} & \frac{1-\delta_2}{C}\\ -\frac{1-\delta_2}{L} & 0\\ \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix}+ \begin{pmatrix} 0\\ \frac{V_i \delta_1}{L} \end{pmatrix} \end{equation}\]

Considering the equilibrium points of the dynamic system, \(\dot{x}_1 = 0\) and \(\dot{x}_2 = 0\):

\[\begin{equation} \boxed{x_1 = V_i\frac{\delta_1}{1-\delta_2}} \end{equation}\]

\[\begin{equation} \boxed{x_2 = V_i\frac{\delta_1}{R\cdot(1-\delta_2)^2}} \end{equation}\]

Dynamic Analysis:

Revisiting the dynamic equations of the boost stage, the DC term \(\frac{V_i}{L}\) is now replaced by \(\frac{V_i\delta_1}{L}\) and decomposed into two components according to small-signal perturbation theory: \(\frac{V_i(\delta_1+\hat{\delta}_1)}{L}\). Analyzing this in the Laplace domain yields:

\[\begin{equation} \left\{ \begin{matrix} s\hat{X}_1(s) =& -\frac{X_1(s)}{RC}+ \left(\frac{1-\delta_2}{C}\right)\hat{X}_2(s) - \frac{X_2(s)}{R} \hat{\delta}_2(s)\\ s\hat{X}_2(s) =& -\frac{1-\delta_2}{L}\hat{X}_1(s) + \frac{X_1}{L}\hat{\delta}_2(s)+ \frac{V_i}{L}\hat{\delta}_1(s) \end{matrix}\right. \end{equation}\]

Solving for the voltage, we have:

\[\begin{equation} \hat{X}_1(s) = \frac{\frac{1}{LC}}{s^2+ \left(\frac{1}{RC}\right)s + \frac{(1-\delta_2)^2}{LC}} \cdot \left[ (1-\delta_2)V_i\hat{\delta}_1(s)+ \left((1-\delta_2)X_1-X_2Ls \right)\hat{\delta}_2(s) \right] \end{equation}\]

Substituting the equilibrium points for \(X_1\) and \(X_2\) finally yields:

\[\begin{equation} \boxed{ \hat{X}_1(s) = \frac{\frac{1}{LC}V_i}{s^2+ \left(\frac{1}{RC}\right)s + \frac{(1-\delta_2)^2}{LC}} \cdot \begin{pmatrix} (1-\delta_2)\\ \delta_1 \left(1-\frac{Ls}{R(1-\delta_2)^2}\right) \end{pmatrix}^T \cdot \begin{pmatrix} \hat{\delta}_1(s)\\ \hat{\delta}_2(s) \end{pmatrix} } \end{equation}\]

Loss Analysis

Analysis considering losses is essential as it enables the quantification of conduction power losses, the generated heat, and consequently, the temperature rise, as well as the operating limits for \(\delta_1\) and \(\delta_2\). Furthermore, it provides insights into the switching requirements.

The following lossy model serves as the starting point, where each loss resistance is associated with a specific active or passive component. It should be noted that within this model, the diodes could be replaced by MOSFETs under synchronous switching, and the required operational limits would remain valid.

Once again, the state variables are defined as \(x_1=V_C\) and \(x_2=i_l\). Furthermore, since the output voltage is of interest, it must be related to the state variable \(x_1\), given that \(V_C \neq V_R\) in the lossy case. The resulting state-space model is given by:

\[\begin{equation} \begin{matrix} \dot{x} = & Ax+Bu\\ y = & Cx\\ \end{matrix} \end{equation}\]

Next, each of the states is analyzed, and the C matrix is also determined.

State \(E_0\)

With both \(\delta_1\) and \(\delta_2\) open, the system yields:

\[\begin{equation} \begin{pmatrix} \dot{x}_1 \\ \dot{x}_2 \end{pmatrix} = \begin{pmatrix} -\frac{1}{C(R+r_C)} & \frac{R}{C(R+r_C)}\\ -\frac{R}{L(R+r_C)} & -\frac{r_{D1}+r_{D2}+r_l+r_C||R}{L}\\ \end{pmatrix} \end{equation}\]

Furthermore, matrix C is obtained by superposition as a linear combination of \(i_L\) and \(V_C\), accounting for its Thévenin impedance \(r_C\):

\[\begin{equation} C = \begin{pmatrix} \frac{R}{R+r_C} & \frac{r_CR}{R+r_C} \end{pmatrix} \end{equation}\]

State \(E_1\)

With \(\delta_1\) closed and \(\delta_2\) open, we have:

\[\begin{equation} \begin{pmatrix} \dot{x}_1 \\ \dot{x}_2 \end{pmatrix} = \begin{pmatrix} -\frac{1}{C(R+r_C)} & \frac{R}{C(R+r_C)}\\ -\frac{R}{L(R+r_C)} & -\frac{r_{D1}+r_{D2}+r_l+r_C||R}{L}\\ \end{pmatrix} + \begin{pmatrix} \frac{V_i}{L} \end{pmatrix} \end{equation}\]

And the C matrix remains the same as previous state: \[\begin{equation} C = \begin{pmatrix} \frac{R}{R+r_C} & \frac{r_CR}{R+r_C} \end{pmatrix} \end{equation}\]

State \(E_3\)

Con \(\delta_1\) y \(\delta_2\) cerrados, tenemos:

\[\begin{equation} \begin{pmatrix} \dot{x}_1 \\ \dot{x}_2 \end{pmatrix} = \begin{pmatrix} -\frac{1}{C(R+r_C)} & \frac{R}{C(R+r_C)}\\ -\frac{R}{L(R+r_C)} & -\frac{r_{D1}+r_{D2}+r_l+r_C||R}{L}\\ \end{pmatrix} + \begin{pmatrix} \frac{V_i}{L} \end{pmatrix} \end{equation}\] \[\begin{equation} C = \begin{pmatrix} \frac{R}{R+r_C} & 0 \end{pmatrix} \end{equation}\]

Average loss model:

\[\begin{equation} A= \begin{pmatrix} -\frac{1}{C(R+r_C)} & \frac{R(1-\delta_2)}{C(R+r_C)} \\ -\frac{R(1-\delta_2)}{L(R+r_C)} & -\frac{1}{L}\left( r_{D1}(1-\delta_1)+r_{D2}(1-\delta_2)+r_l+ (r_C||R)(1-\delta_2)+r_{T1}\delta_1+r_{T2}\delta_2 \right) \end{pmatrix} \label{eq:matriz_A_perdidas} \end{equation}\] \[\begin{equation} \large B=\begin{pmatrix} 0\\ \frac{V_i\delta_1}{L} \end{pmatrix} \end{equation}\] \[\begin{equation} \large C = \begin{pmatrix} \frac{R}{R+r_C} & \frac{Rr_C(1-\delta_2)}{R+r_C} \end{pmatrix} \end{equation}\]

Equilibrium Points with Losses

From average loss model and the equation let’s take in \(a_{22}\):

\[\begin{equation} \large \varphi = r_{D1}\bar{\delta}_1+r_{D2}\bar{\delta}_2+(r_C||R)\bar{\delta}_2 + \delta_1 r_{T1}+ \delta_2 r_{T2} \label{eq:cambiovarphi} \end{equation}\] With \[\begin{gather*} \bar{\delta}_1 = 1-\delta_1\\ \bar{\delta}_2 = 1-\delta_2 \end{gather*}\] [eq:matriz_A_perdidas] We have:

\[\begin{equation} \large A= \begin{pmatrix} -\frac{1}{C(R+r_C)} & \frac{R(1-\delta_2)}{C(R+r_C)} \\ -\frac{R(1-\delta_2)}{L(R+r_C)} & -\frac{\varphi}{L} \end{pmatrix} \label{eq:matriz_A_perdidas_simplif} \end{equation}\] From this, the steady-state operating points for the lossy converter are obtained:

\[\begin{equation} \large \begin{pmatrix} -\frac{1}{C(R+r_C)} & \frac{R(1-\delta_2)}{C(R+r_C)} \\ -\frac{R(1-\delta_2)}{L(R+r_C)} & -\frac{\varphi}{L} \end{pmatrix} \cdot \begin{pmatrix} \dot{x}_1\\ \dot{x}_2 \end{pmatrix} + \begin{pmatrix} 0\\ \frac{\delta_1}{L} \end{pmatrix}V_i = \begin{pmatrix} 0\\ 0 \end{pmatrix} \end{equation}\]

We have as solutions: \[\begin{gather} \large \boxed{X_1 = \frac{V_i\delta_1\bar{\delta}_2R(R+r_C)}{(R\bar{\delta}_2)^2 + \varphi(R+r_C)}}\\ \large \boxed{X_2 = \frac{V_i\delta_1(R+r_C)}{(R\bar{\delta}_2)^2 + \varphi(R+r_C)}}\\ \large \boxed{V_o = \frac{V_i\delta_1\bar{\delta}_2R(1+r_C)}{(R\bar{\delta}_2)^2 + \varphi(R+r_C)}}\label{eq:vo_conperdidas} \end{gather}\]

When compared to the ideal transfer function, [eq:vo_conperdidas] takes the following form:

A forbidden operating region exists because within it, the sign of the slope reverses, completely altering the system dynamics. Consequently, it is crucial to characterize the losses of practical components and limit the modulation range, which depends entirely on the voltage conversion ratio.

Calculation of the Maximum Duty Cycle for Given Losses:

This calculation can be carried out by rewriting [eq:cambiovarphi] for algebraic convenience as follows:

\[\begin{gather} \large \varphi = K\bar{\delta}_2+m\\ K = r_{D2}+R||r_C-r_{T2}\\ m=r_{D1}\bar{\delta_1}+r_{T1}\delta_1+r_l+r_{T2} \end{gather}\] Parameters \(K\) and \(m\) are referred to as loss parameters because they are associated with the system losses. Then, its maximum value is calculated from [eq:vo_conperdidas]:

\[\begin{equation} \frac{\partial V_o}{\partial \delta_2} = 0 \end{equation}\]

With the following result: \[\begin{gather} \large \boxed{ V_{oMAX} = \frac{Vi\delta_1}{2\sqrt{\frac{m}{R+r_C}+\frac{K}{R}}} }\label{eq:vomax}\\ \large\boxed{ \delta_{2MAX} = 1-\frac{1}{R}\sqrt{m(R+r_C)}}\label{eq:delta2max} \end{gather}\]

Conclusion

In this work, we have obtained both the steady-state and dynamic state-space models by employing the state-space averaging technique. We have also demonstrated the slope inversion of the transfer function under certain load and quasi-static loss conditions. In a forthcoming paper, we will present a practical application of this phenomenon via simulation and subsequent experimental laboratory measurements to validate the proposed theory using controlled losses.

References

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1. Note that \((1-\delta_1)+(\delta_1-\delta_2)+\delta_2 = 1\)↩︎