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

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

*P s*ð Þ <sup>1</sup> <sup>þ</sup>

*P s*ð Þ <sup>1</sup> <sup>þ</sup>

*D*0<sup>2</sup>

*<sup>s</sup>*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*ζω*n*<sup>s</sup>* <sup>þ</sup> *<sup>ω</sup>*n<sup>2</sup> <sup>¼</sup> <sup>1</sup>

*C*o*V*<sup>o</sup>

*s ω*o � �

> *s ω*n � �<sup>2</sup>

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

Because the switching converter is controlled by the pulse width modulation (PWM) signal corresponding to the duty ratio, it is necessary to modulate the control signal from the compensator to the PWM signal. **Figure 3** shows the correspondence between the control signal and the PWM signal. In an analog circuit, a comparator is used for comparing the control signal *V*<sup>c</sup> to a sawtooth wave (or a triangular wave) *V*tri. Therefore, it is ON when *V*<sup>c</sup> > *V*tri and OFF when *V*<sup>c</sup> < *V*tri.

þ 2*ζ ω*n *s* þ 1

*<sup>ω</sup>*esr <sup>¼</sup> <sup>1</sup>

*C*o*r*<sup>C</sup>

*<sup>ω</sup>*rhp <sup>¼</sup> *<sup>D</sup>*<sup>0</sup>

*V*i *LI*<sup>o</sup>

(4)

*s ω*esr � �

<sup>1</sup> � *<sup>s</sup> ω*rhp � �

is described in the control system design of the switching converter.

*ΔI*Lð Þ*s <sup>Δ</sup>D s*ð Þ <sup>¼</sup> *<sup>K</sup>*dc\_i

*ΔV*oð Þ*s <sup>Δ</sup>D s*ð Þ <sup>¼</sup> *<sup>K</sup>*dc\_v

*r*C

*<sup>D</sup>*0<sup>2</sup> *<sup>K</sup>*dc\_v <sup>¼</sup> *<sup>V</sup>*<sup>i</sup>

2

ffiffiffiffiffi *C*o *L*

r

<sup>p</sup> *<sup>ω</sup>*<sup>o</sup> <sup>¼</sup> *<sup>I</sup>*<sup>o</sup>

*G*idðÞ¼ *s*

8

*Control Theory in Engineering*

>>>>>>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>>>>>>:

plane (RHP) zero frequency.

**Figure 3.**

**252**

*PWM modulation* F*m.*

**2.2 Pulse modulation gain:** *F***<sup>m</sup>**

*G*vdðÞ¼ *s*

*<sup>K</sup>*dc\_i <sup>¼</sup> *<sup>I</sup>*<sup>o</sup>

*<sup>ζ</sup>* <sup>¼</sup> *<sup>r</sup>*<sup>L</sup> <sup>þ</sup> *<sup>D</sup>*<sup>0</sup>

*<sup>ω</sup>*<sup>n</sup> <sup>¼</sup> *<sup>D</sup>*<sup>0</sup>

*P s*ð Þ <sup>¼</sup> *<sup>ω</sup>*<sup>n</sup>

2*D*<sup>0</sup>

ffiffiffiffiffiffiffiffi *LC*<sup>o</sup>

1

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

In this section, voltage mode control (VMC) and current mode control (CMC) are compared to the power balance mode control (PBMC).

#### **3.1 Voltage mode control (VMC)**

**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 system.

$$\mathbf{G}\_{loop}(\mathbf{s}) = \mathbf{G}\_{cv}(\mathbf{s}) \cdot \mathbf{F}\_m \cdot \mathbf{G}\_{vd}(\mathbf{s}) \cdot \mathbf{K}\_v \tag{6}$$

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 improving such a long phase delay.

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

**Figure 4.** *Voltage mode control.*

**Figure 5.** *Current mode control.*

current is remarkable, and when the overcurrent protection (OCP) operates, the DC-DC converter halts. For these reasons, VMC is typically not used in DC-DC converters.

#### **3.2 Current mode control (CMC)**

**Figure 5** shows the block diagram of the CMC. In the CMC, a control loop is added to the voltage control loop. The loop transfer function *G*loop(*s*) of the CMC is given in Eq. (7):

$$\mathbf{G\_{loop}}(\mathbf{s}) = \mathbf{G\_{cv}}(\mathbf{s}) \cdot \frac{F\_{\mathbf{m}}}{\mathbf{1} + F\_{\mathbf{m}} \cdot \mathbf{G\_{id}}(\mathbf{s}) \cdot \mathbf{K\_i}} \cdot \mathbf{G\_{vd}}(\mathbf{s}) \cdot \mathbf{K\_v} \tag{7}$$

Because the CMC also feeds back the inductor current, the duty ratio is finely adjusted. However, in the transient state, the inductor suppresses sudden changes

On the other hand, when the charge/discharge current of the output capacitor is used as the feedforward input, the charge/discharge current in the transient state rapidly changes depending on the capacitor. As a result, the duty ratio can be changed faster than for the CMC. Furthermore, when shifting from the transient state to the steady state, the average charge/discharge current becomes zero, and the influence of the feedforward input automatically decreases. Therefore, the feedforward input gain automatically becomes minimal during the transient and in

In addition, by appropriately designing the various sensor gains and compensators of this control system, it is possible to set an operation state called the sliding mode. It is known that the control system operating in this sliding mode is not affected by disturbances or plant fluctuations. Therefore, responsiveness and

Although this output capacitor current can be detected directly, equivalent series resistance (ESR) and equivalent series inductance (ESL) increase owing to the addition of a shunt resistance and a current transformer, which affects the control system and output voltage. In addition, in digital control systems, analog-to-digital conversion cannot be performed precisely owing to an increase in the noise associated with charging/discharging. On the other hand, it is possible to derive the charge/discharge current of the output capacitor without directly detecting it, by appropriately detecting the output current and the inductor current and performing the calculation. However, as the inductor current of the boost-type DC-DC converter flows only to the output side during the OFF period, the output current

Therefore, it is necessary to consider the control system corresponding to the step-up-type DC-DC converter considering output capacitor current detection and digital control. In the next section, we describe the PBMC with improved respon-

**Figure 7** shows the block diagram of the PBMC. First, various blocks are

in the current, and the system's responsiveness worsens.

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

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

*Buck-type DC-DC converter using sliding mode control.*

robustness can be improved by operating in sliding mode.

siveness and robustness for boost-type DC-DC converters.

• *G*cv(*s*): transfer function of voltage compensator

the steady state.

**Figure 6.**

differs from the inductor current.

**4.2 Power balance mode control**

• *K*vo: output voltage sensor gain

described.

**255**

From Eq. (7), the second-order lag system 1/*P*(*s*) in the transfer function of the plant is approximately canceled out. In addition, the peak of the gain near the resonance frequency disappears. As a result, no overshoots or undershoots of the inductor current occur following sudden changes such as load changes, and stable operation is ensured without reaching the OCP threshold. Therefore, stability and responsiveness of the control system are much better, compared with the VMC. Additional modes, not discussed in this chapter, include the peak current mode control (PCMC), which is based on the CMC, and the average current mode control (ACMC), which is used in power factor correction (PFC) converters.
