*2.1.1 Circuit state (Q1 = ON/OFF) and state space equation*

For circuit averaging, it is necessary to determine the circuit's ON/OFF states. When mathematically modeling the state of a circuit, the state equation and the following output equation are used:

**Figure 1.** *Single-phase boost-type DC-DC converter.*

*Power Balance Mode Control for Boost-Type DC-DC Converter DOI: http://dx.doi.org/10.5772/intechopen.82787*

gain-crossover frequency (which determines the high-frequency response) due to

On the other hand, control of switching converter using sliding mode control (SMC) has been studied [5–9]. Sliding mode control has high robustness and is resistant to influences by plant fluctuations. However, the control system has a

In this research, we developed power balance mode control (PBMC), which is a new control method that incorporates SMC concept into CMC [10]. In the PBMC approach, the input voltage and the output current are incorporated into the control system as in the conventional control method, and new control items are added by calculation. As a result, the performance of the control system can be greatly improved, when compared with the conventional control method. Furthermore, since the added control items are constituted by four arithmetic operations,

In this study, a single-phase boost-type DC-DC converter was used as a plant. **Figure 1** shows the circuit diagram of the plant. To obtain the transfer function of this plant, a modeling method called the state-space averaging method was used. In this section, various transfer functions used for designing the control system of the

**2.1 Derivation of the transfer function model using the state-space averaging**

can be set to either ON or OFF. Therefore, the state-space averaging method [11–13], which averages the circuit by a duty ratio, was used. The derivation for obtaining the transfer function of the switching converter using the state-space

The switching converter is a time-varying circuit in which the state of the circuit

For circuit averaging, it is necessary to determine the circuit's ON/OFF states. When mathematically modeling the state of a circuit, the state equation and the

problem that it is very complicated compared with VMC and CMC.

**2. Transfer functions of the boost-type DC-DC converter**

the presence of this unstable zero.

*Control Theory in Engineering*

implementation is also very easy.

DC-DC converter are described.

averaging method is shown below.

following output equation are used:

*Single-phase boost-type DC-DC converter.*

*2.1.1 Circuit state (Q1 = ON/OFF) and state space equation*

**method**

**Figure 1.**

**250**

$$\begin{cases} \frac{d\mathbf{x}(t)}{dt} = \mathbf{A}\mathbf{x}(t) + \mathbf{b}\mathbf{u}(t) \\ \mathbf{y}(t) = \mathbf{c}\mathbf{x}(t) + \mathbf{d}\mathbf{u}(t) \end{cases} \tag{1}$$

where *u***(***t***)**, input vector; *x***(***t***)**, state vector; *y*(*t*), output vector; *A*, state matrix; *b*, input matrix; *c*, output matrix; *d*, direct matrix.

With respect to the circuit shown in **Figure 2**, the state equation and the output equation are expressed using the following equations:

$$\begin{cases} \frac{d}{dt} \begin{bmatrix} i\_{\mathcal{L}}(t) \\ v\_{\mathcal{C}}(t) \end{bmatrix} = \begin{bmatrix} a\_{11} & a\_{12} \\ a\_{21} & a\_{22} \end{bmatrix} \begin{bmatrix} i\_{\mathcal{L}}(t) \\ v\_{\mathcal{C}}(t) \end{bmatrix} + \begin{bmatrix} b\_{11} & b\_{12} \\ b\_{21} & b\_{22} \end{bmatrix} \begin{bmatrix} V\_{i}(t) \\ I\_{o}(t) \end{bmatrix} \\ v\_{o}(t) = \begin{bmatrix} c\_{11} & c\_{12} \end{bmatrix} \begin{bmatrix} i\_{\mathcal{L}}(t) \\ v\_{\mathcal{C}}(t) \end{bmatrix} + \begin{bmatrix} d\_{11} & d\_{12} \end{bmatrix} \begin{bmatrix} V\_{i}(t) \\ I\_{o}(t) \end{bmatrix} \end{cases} \tag{2}$$

In Eq. (2), the inductor current and capacitor voltage comprise the state vector, while the input voltage and the output current comprise the input vector. **Figure 2** shows the equivalent circuit for the ON and OFF states of the switch Q1.

#### *2.1.2 Circuit state averaging using duty ratio*

When the state of a circuit is averaged over one switching period using the duty ratio, the state equation and the output equation are given as follows:

$$\begin{cases} \frac{d\overline{\mathbf{x}}(t)}{dt} = (D\mathbf{A}\_{\text{on}} + D'\mathbf{A}\_{\text{off}}) \cdot \overline{\mathbf{x}}(t) + (D\mathbf{b}\_{\text{on}} + D'\mathbf{b}\_{\text{off}}) \cdot \overline{\mathbf{u}}(t) \\\ \overline{v}\_{\text{o}}(t) = (D\mathbf{c}\_{\text{on}} + D'\mathbf{c}\_{\text{off}}) \cdot \overline{\mathbf{x}}(t) + (D\mathbf{d}\_{\text{on}} + D'\mathbf{d}\_{\text{off}}) \cdot \overline{\mathbf{u}}(t) \\\ D' = \mathbf{1} - D \end{cases} \tag{3}$$

Here *D* and *D*<sup>0</sup> represent the time ratio of the ON and OFF periods in one switching cycle, respectively. The switching converter averages the circuit state by the duty ratio, and it can be regarded as a linear time-invariant system for frequencies lower than the switching frequency. Additional characterization includes static characteristic analysis, steady-state dynamic analysis, and Laplace transforms. Although the transfer function of the switching converter can be derived using the above, in this study the following derivation is omitted. The transfer function of the

**Figure 2.** *Equivalent circuits for the ON and OFF states. (a) Switch Q1: ON; (b) switch Q1: OFF.*

single-phase boost-type DC-DC converter is shown in Eq. (4). The transfer functions derived using the state-space averaging method include output impedance and audio susceptibility. In this chapter, the most important transfer function is described in the control system design of the switching converter.

The ratio per switching cycle of this relationship is the duty ratio. When a small disturbance Δ*V*c(*s*) occurs in the control voltage *V*c, a small disturbance Δ*D*(*s*) is generated in the duty ratio *D* in the steady state. The relationship between these is equal to the slope of the sawtooth wave in one switching cycle. Accordingly, when the amplitude of the sawtooth wave is *V*p-p, the transfer function of the PWM gain

*Power Balance Mode Control for Boost-Type DC-DC Converter*

*DOI: http://dx.doi.org/10.5772/intechopen.82787*

*<sup>F</sup>*<sup>m</sup> <sup>¼</sup> *<sup>Δ</sup>D s*ð Þ

*<sup>Δ</sup>V*Cð Þ*<sup>s</sup>* <sup>¼</sup> <sup>1</sup>

From Eq. (5), when the amplitude of the sawtooth wave is *V*p-p = 1 V, the PWM

When current and voltage are used for feedback directly, the sensor gain can be

In this section, voltage mode control (VMC) and current mode control (CMC)

**Figure 4** shows the block diagram of the VMC. As shown, the control loop is configured to maintain a constant output voltage. The loop transfer function *G*loop(*s*) of the VMC is given in Eq. (6). This control system is the simplest feedback

However, there is a long phase lag due to the second-order lag system 1/*P*(*s*) in the plant *G*vd(*s*). Furthermore, due to the RHP-zero, there is a phase delay of up to �270° at the plant *G*vd(*s*). Therefore, it is necessary to design a compensator for

In addition, there is a gain peak owing to the LC resonance. As a result, large overshoots or undershoots can occur in the inductor current and the output voltage following sudden changes such as load changes. In particular, the peak inductor

*Gloop*ðÞ¼ *s Gcv*ðÞ� *s Fm* � *Gvd*ðÞ� *s Kv* (6)

neglected. However, when the voltage is high, it is necessary to lower it to the voltage value that can be provided to the controller. In addition, when inputting the current value to the controller, it is necessary to convert it into voltage. Therefore, when designing a control system, it is necessary to consider various sensor gains. In this chapter, the voltage gain is denoted by *K*<sup>v</sup> and the current gain is denoted by *K*i.

**3. Conventional control methods for the DC-DC converter**

are compared to the power balance mode control (PBMC).

*<sup>V</sup>*p‐<sup>p</sup>

(5)

*F*<sup>m</sup> is expressed by Eq. (5).

gain *F*<sup>m</sup> can be neglected.

**2.3 Sensor gain:** *K***<sup>v</sup> and** *K***<sup>i</sup>**

**3.1 Voltage mode control (VMC)**

improving such a long phase delay.

system.

**Figure 4.**

**253**

*Voltage mode control.*

$$\begin{cases} \begin{aligned} \text{G}\_{\text{id}(s)} &= \frac{\Delta I\_{\text{L}}(s)}{\Delta D(s)} = \frac{K\_{\text{dc.}}}{P(s)} \left( 1 + \frac{s}{oo\_{\text{o}}} \right) \\\ \text{G}\_{\text{vd}(s)} &= \frac{\Delta V\_{\text{o}}(s)}{\Delta D(s)} = \frac{K\_{\text{dc.}}}{P(s)} \left( 1 + \frac{s}{oo\_{\text{ev}}} \right) \left( 1 - \frac{s}{o\_{\text{ch}}} \right) \end{aligned} \\\ \begin{aligned} \text{K}\_{\text{dc.}\_{i}} &= \frac{I\_{\text{o}}}{D'} \quad K\_{\text{dc.}\_{i}} = \frac{V\_{i}}{D'^{2}} \\\ \frac{1}{P(s)} &= \frac{o\_{\text{n}}^{2}}{s^{2} + 2\zeta a\_{\text{n}}s + o\_{\text{n}}^{2}} = \frac{1}{\left( \frac{s}{o\_{\text{n}}} \right)^{2} + \frac{2\zeta}{o\_{\text{n}}}s + 1} \\\ \text{\text{\textdegree}} &= \frac{r\_{\text{L}} + D'r\_{\text{C}}}{2D'} \sqrt{\frac{C\_{\text{o}}}{L}} \\\ \text{o}\_{\text{n}} &= \frac{D'}{\sqrt{LC\_{\text{o}}}} \quad o\_{\text{o}} = \frac{I\_{\text{o}}}{C\_{\text{o}}V\_{\text{o}}} \quad o\_{\text{exf}} = \frac{1}{C\_{\text{o}}r\_{\text{C}}} \quad o\_{\text{thp}} = \frac{D'V\_{\text{i}}}{LI\_{\text{o}}} \end{aligned} \end{cases} \tag{4}$$

where *G*id(*s*), transfer function of the duty ratio to the inductor current; *G*vd(*s*), transfer function of the duty ratio to the output voltage; *K*dc\_i, DC gain of *G*id(*s*); *K*dc\_v, DC gain of *G*vd(*s*); 1/*P*(*s*), second-order lag system; *ζ*, damping factor; *ω*n, resonance frequency; *ω*0, zero frequency of load of the boost-type DC-DC converter; *ω*esr, ESR zero frequency of the output smoothing capacitor; *ω*rhp, right half plane (RHP) zero frequency.
