*3.2.4. Full-bridge converter with controlled at secondary side*

The configuration shown in Figure 17 corresponds to a secondary side controlled converter with a full-bridge inverter on primary side and phase-controlled rectifier on secondary side. The switches on primary side are operated by complementary gating scheme with fixed duty ratio. The switches on secondary side are controlled to produce phase difference between primary and secondary side voltages of HF transformer to control the output voltage with load and input line voltage variations.

The main characteristics of this configuration can be summarized as follows:

**1.** Control is easy and simple.

**Figure 15.** Series Resonant converter with inductive output filter.

be summarized as follows:

344 New Developments in Renewable Energy

**Figure 16.** PWM full-bridge converter.

*3.2.3. PWM full-bridge converter with inductive output filter*

**2.** High voltage stress on rectifier diodes on secondary side.

conduction losses, which decreases converter efficiency.

**3.** Parasitic ringing at the secondary side of transformer.

required to keep conduction losses lower.

The configuration shown in Figure 16 is Phase-shifted full bridge converter with inductive output filter. This is most widely used soft-switched configuration for high power applications. This constant frequency converter features ZVS of the primary switches with relatively small circulating circuit. The ZVS is achieved by filter inductance, transformer leakage inductance, snubber capacitance and parasitic junction capacitances of switches. The control of the output voltage at constant frequency is achieved by phase shift technique. Its main characteristics can

**1.** The major limitation of this configuration is loss of duty cycle on the secondary side.

**4.** Large inductor may increase ZVS range but needs a transformer turns-ratio (Np/Ns) to be decreased at the same time, which increases primary side current, causing large

**5.** A good compromise between the transformer turns-ratio and leakage inductance is


**Figure 17.** PWM full-bridge converter.

#### *3.2.5. Current-fed two-inductor boost converter*

Figure 18 shows the current-fed two-inductor boost converter, which is a dual of voltage-fed half-bridge converter configuration. This topology requires a very small turns ratio trans‐ former with only two switches used on the primary side. The turn off of the primary switches is smooth since voltage across the switches is sinusoidal. Also the same for turn on and turn off of rectifier diodes. However, the peak and average current through rectifier diodes is high and this is very difficult to achieve ZVS at variable line input and load condition. This topology is a good selection for constant-input and constant-output voltage applications.

**Figure 20.** Electrical circuit of the SRC.

low pass filter in the secondary (Lf

current, iL are assumed to be purely sinusoidal.

as represented by Figure 22 and equation 14.

the load, respectively.

*3.4.1. Voltage*

The operation of the converter can be described as follows: the voltage supplied by the fuel cell stack, which is typically low must be converted to a high and constant level, for example; 48 V or 400 VDC in order to be able to feed an electric vehicle or to be sent to the grid through an inverter. The HF transformer is a step-up voltage transformer, which also serves as galvanic isolation between the high and low voltage levels of the circuits. The waveforms of the voltage and current in the LC series resonant circuit in the primary side of the transformer are sinusoidal. Selecting appropriate values for the Lr and Cr components, the resonant frequency of the circuit is established. Then, the DC voltage of the fuel cell is firstly inverted in the primary side of the HF transformer, being rectified on the secondary side. The low pass filter in the primary side (LPEM, CPEM) allows at protecting the PEM fuel cell from the ripple current and voltage produced by the converter, and also allows the storage of energy in the DC bus. The

Methodology of Designing Power Converters for Fuel Cell Based Systems: A Resonant Approach

http://dx.doi.org/10.5772/54674

347

The series-resonant inverter as part of the topology of Figure 21 is constitute by a controlled

The analysis of this circuit by sinusoidal approximation neglects the harmonics of the switch‐ ing frequency and the tank waveforms, namely the capacitor voltage, VC and the inductance

In Figure 21 the output voltage of the full-bridge inverter, which in turn is the input voltage of the series resonant circuit, is a square-waveform, wherein in the first half period [0-π] the IGBTs Q1 and Q3 conduct, while in the second half period conduct IGBTs Q2 and Q4 [π - 2π]

full-bridge and a series resonant circuit with natural frequency *fr* and impedance Zr.

) allows at reducing the ripples of current and voltage to

, Cf

**3.4. Mathematical analysis of the series resonant converter (SRC)**

**Figure 18.** Current-fed two-inductor boost converter.

#### *3.2.6. Bidirectional current–fed converter*

The configuration of Figure 19 corresponds to a soft-switched bidirectional current-fed boost converter. The active switches in the both sides of the transformer make this converter bidirectional. Normally the Mosfets are used in the low voltage side and the IGBTs are used in the high voltage side. This configuration can be in push-pull, half-bridge or full-bridge, with the full-bridge considered one of the best topology choices for fuel cell applications.

**Figure 19.** Soft-switched bidirectional current-fed boost converter.

#### **3.3. Selection of converter topology and operation**

Considering the analysis made previously a full-bridge series-resonant inverter, followed by a high frequency transformer and a rectifier, composes the topology selected. Additionally, two low-pass filters are included, one in the primary side which is used to protect the PEM of high ripple-current and another in the secondary side, to improve the quality of the energy supplied by the power system to the load or the grid. Figure 20 below represents electrical scheme of its topology, which is also called a series resonant converter (SRC).

**Figure 20.** Electrical circuit of the SRC.

**Figure 18.** Current-fed two-inductor boost converter.

**Figure 19.** Soft-switched bidirectional current-fed boost converter.

**3.3. Selection of converter topology and operation**

The configuration of Figure 19 corresponds to a soft-switched bidirectional current-fed boost converter. The active switches in the both sides of the transformer make this converter bidirectional. Normally the Mosfets are used in the low voltage side and the IGBTs are used in the high voltage side. This configuration can be in push-pull, half-bridge or full-bridge, with

Considering the analysis made previously a full-bridge series-resonant inverter, followed by a high frequency transformer and a rectifier, composes the topology selected. Additionally, two low-pass filters are included, one in the primary side which is used to protect the PEM of high ripple-current and another in the secondary side, to improve the quality of the energy supplied by the power system to the load or the grid. Figure 20 below represents electrical

scheme of its topology, which is also called a series resonant converter (SRC).

the full-bridge considered one of the best topology choices for fuel cell applications.

*3.2.6. Bidirectional current–fed converter*

346 New Developments in Renewable Energy

The operation of the converter can be described as follows: the voltage supplied by the fuel cell stack, which is typically low must be converted to a high and constant level, for example; 48 V or 400 VDC in order to be able to feed an electric vehicle or to be sent to the grid through an inverter. The HF transformer is a step-up voltage transformer, which also serves as galvanic isolation between the high and low voltage levels of the circuits. The waveforms of the voltage and current in the LC series resonant circuit in the primary side of the transformer are sinusoidal. Selecting appropriate values for the Lr and Cr components, the resonant frequency of the circuit is established. Then, the DC voltage of the fuel cell is firstly inverted in the primary side of the HF transformer, being rectified on the secondary side. The low pass filter in the primary side (LPEM, CPEM) allows at protecting the PEM fuel cell from the ripple current and voltage produced by the converter, and also allows the storage of energy in the DC bus. The low pass filter in the secondary (Lf , Cf ) allows at reducing the ripples of current and voltage to the load, respectively.

### **3.4. Mathematical analysis of the series resonant converter (SRC)**

The series-resonant inverter as part of the topology of Figure 21 is constitute by a controlled full-bridge and a series resonant circuit with natural frequency *fr* and impedance Zr.

The analysis of this circuit by sinusoidal approximation neglects the harmonics of the switch‐ ing frequency and the tank waveforms, namely the capacitor voltage, VC and the inductance current, iL are assumed to be purely sinusoidal.

### *3.4.1. Voltage*

In Figure 21 the output voltage of the full-bridge inverter, which in turn is the input voltage of the series resonant circuit, is a square-waveform, wherein in the first half period [0-π] the IGBTs Q1 and Q3 conduct, while in the second half period conduct IGBTs Q2 and Q4 [π - 2π] as represented by Figure 22 and equation 14.

tank rings sinusoidal and is(t) is well approximated by a sinusoid waveform as represented

Im sin , for 0 0, for 2

ï+ ´ -F < < <sup>í</sup> <sup>ï</sup> < < <sup>î</sup>

( )

A simple equivalent circuit similar to that represented in Figure 23, whose operation can be analyzed by a second-order differential equation as follows, can perform the analysis of the

Applying the Kirchoff loop equation (sum of voltages around loop equals zero), to the RLC

*diLr Lr VCr RI V*

circuit of Figure 23 the equation of the series resonant circuit is the follow.

*dt*

<sup>ï</sup> < < <sup>í</sup> ï- ´ -F < < <sup>î</sup>

Im sin , for 2

*ax wt wt*

0, for 0

*ax wt wt*

*wt*

*wt*

p

Methodology of Designing Power Converters for Fuel Cell Based Systems: A Resonant Approach

p

p

(15)

349

http://dx.doi.org/10.5772/54674

(16)

 p

p

 p

+ += (17)

( ) 1 3

by the equations 15 and 16.

SRC.

*iQ iQ*

2 4

= = ì

*iQ iQ*

**Figure 23.** Series resonant circuit with a resistive load.

= = ì

**Figure 21.** Electrical equivalent circuit of the SRC.

**Figure 22.** Output voltage in a full-bridge inverter.

$$\begin{cases} +V\_{\mathbb{F}\epsilon^{\prime}} \text{ for } 0 < \mathsf{wt} \le \pi \\ -V\_{\mathbb{F}\epsilon^{\prime}} \text{ for } \pi < \mathsf{wt} \le 2\pi \end{cases} \tag{14}$$

#### *3.4.2. Current*

The current is (t) is equal to the output current +is (t) in the first half period [0-π], and is its inverse -is (t), in the second half period [π - 2π]. Under the conditions described above, the tank rings sinusoidal and is(t) is well approximated by a sinusoid waveform as represented by the equations 15 and 16.

$$\begin{aligned} \mathbf{iQ1} = \mathbf{iQ3} = \begin{cases} \\ + \text{Im} \, a \mathbf{x} \times \sin \left( wt - \Phi \right) , \text{ for } 0 < wt < \pi \\ 0 , \qquad \text{for } \ \pi < wt < 2\pi \end{cases} \end{aligned} \tag{15}$$

$$\begin{aligned} \text{iQ2} = \text{iQ4} = \begin{cases} 0, & \text{for } 0 < wt < \pi \\ -\text{Im}\,ax \times \sin\left(wt - \Phi\right), & \text{for } \pi < wt < 2\pi \end{cases} \end{aligned} \tag{16}$$

A simple equivalent circuit similar to that represented in Figure 23, whose operation can be analyzed by a second-order differential equation as follows, can perform the analysis of the SRC.

**Figure 23.** Series resonant circuit with a resistive load.

**Figure 22.** Output voltage in a full-bridge inverter.

**Figure 21.** Electrical equivalent circuit of the SRC.

348 New Developments in Renewable Energy

*3.4.2. Current*

, for 0 wt , for wt 2

ìï+ <£

ï- <£ î

p

The current is (t) is equal to the output current +is (t) in the first half period [0-π], and is its inverse -is (t), in the second half period [π - 2π]. Under the conditions described above, the

p

 p (14)

*Fc Fc V V*

í

Applying the Kirchoff loop equation (sum of voltages around loop equals zero), to the RLC circuit of Figure 23 the equation of the series resonant circuit is the follow.

$$Lr\frac{d\bar{u}Lr}{dt} + V\mathcal{C}r + RI = \overline{V} \tag{17}$$

And the input impedance Z of this circuit is expressed by equation 18 with*R* =*Z*cos(*θ*) and X=*Z*sin(*θ*).

$$\overline{Z} = R + j\left(w\,L\_r - \frac{1}{w\,\mathrm{Cr}}\right) \tag{18}$$

#### *3.4.3. Angular frequency and reactance*

If the reactance of the resonant circuit becomes zero with XLr = XCr then Z=R and the circuit operates at resonant conditions that is, for fs=fr. This is characterized by the angular frequency wr and characteristic impedance Zr as defined by equations 19 and 20 follow.

$$
\pi w\_r = 2\pi f\_r = \frac{1}{L\_r \mathbb{C}\_r} \tag{19}
$$

$$Z\_r = \sqrt{\frac{L\_r}{C\_r}}\tag{20}$$

**Figure 24.** Amplitude of the current in the series resonant circuit

Methodology of Designing Power Converters for Fuel Cell Based Systems: A Resonant Approach

http://dx.doi.org/10.5772/54674

351

**Figure 25.** Amplitude of the power in the series resonant circuit

#### *3.4.4. Quality factor*

The quality factor Q measures the "goodness" or the quality of the circuit. This is the ratio between the power stored in the reactance and the power dissipated in the resistance of the resonant circuit. The quality factor Q, can be defined expressed by the equation 21 or equiva‐ lently by equation 22.

$$Q = \frac{w\_r \times Lr}{R} = \frac{1}{w\_r \times C\_r \times R} = \frac{Z\_r}{R} \tag{21}$$

$$Q = \frac{1}{R} \times \sqrt{\frac{L\_r}{C\_r}} \tag{22}$$

#### *3.4.5. Normalized amplitudes of the current and power*

Normalized amplitudes of the current and output power in the series resonant circuit as a function of fs/fr and R/Zr =1/Q are represented in figures 24 and 25, respectively. The maximum current value in the circuit occurs at the resonance frequency and with low load resistance. It is also observed that at resonant frequency fr if the load R→ 0Ω then the peak current I→∝ and the system can de destroyed. This analysis leads to the conclusion that it can't ever work with a short circuit in the load side. The maximum output power occurs at the resonance frequency and with low load resistance.

**Figure 24.** Amplitude of the current in the series resonant circuit

And the input impedance Z of this circuit is expressed by equation 18 with*R* =*Z*cos(*θ*) and

æ ö =+ - ç ÷

If the reactance of the resonant circuit becomes zero with XLr = XCr then Z=R and the circuit operates at resonant conditions that is, for fs=fr. This is characterized by the angular frequency

*r r*

*r*

*r L*

The quality factor Q measures the "goodness" or the quality of the circuit. This is the ratio between the power stored in the reactance and the power dissipated in the resistance of the resonant circuit. The quality factor Q, can be defined expressed by the equation 21 or equiva‐

> 1 *r r r r w Lr <sup>Z</sup> <sup>Q</sup> R wCR R*

> > 1 *<sup>r</sup>*

*r*

Normalized amplitudes of the current and output power in the series resonant circuit as a function of fs/fr and R/Zr =1/Q are represented in figures 24 and 25, respectively. The maximum current value in the circuit occurs at the resonance frequency and with low load resistance. It is also observed that at resonant frequency fr if the load R→ 0Ω then the peak current I→∝ and the system can de destroyed. This analysis leads to the conclusion that it can't ever work with a short circuit in the load side. The maximum output power occurs at the resonance

è ø (18)

(19)

*<sup>C</sup>* <sup>=</sup> (20)

´ == = ´ ´ (21)

*<sup>L</sup> <sup>Q</sup> R C* = ´ (22)

<sup>1</sup> *Z R j wLr wCr*

wr and characteristic impedance Zr as defined by equations 19 and 20 follow.

<sup>1</sup> <sup>2</sup> *r r*

*w f L C* = = p

*r*

*Z*

X=*Z*sin(*θ*).

350 New Developments in Renewable Energy

*3.4.4. Quality factor*

lently by equation 22.

*3.4.5. Normalized amplitudes of the current and power*

frequency and with low load resistance.

*3.4.3. Angular frequency and reactance*

**Figure 25.** Amplitude of the power in the series resonant circuit
