**2. Modes of operation of the converter**

The circuit diagram of the LCC transistor resonant DC/DC converter under investigation is shown in figure 1. It consists of an inverter (controllable switches constructed on base of the transistors *Q*1÷*Q*4 with freewheeling diodes *D*1÷*D*4), a resonant circuit (*L*, С), a matching transformer *Tr*, an uncontrollable rectifier (*D*5÷*D*8), capacitive input and output filters (*CF*<sup>1</sup> и *CF*2) and a load resistor (*R*0). The snubber capacitors (*C*1*÷C*4) are connected with the transistors in parallel.

The output power of the converter is controlled by changing the operating frequency, which is higher than the resonant frequency of the resonant circuit.

It is assumed that all the elements in the converter circuit (except for the matching transformer) are ideal, and the pulsations of the input and output voltages can be neglected.

Fig. 1. Circuit diagram of the LCC transistor DC/DC converter

All snubber capacitors *C*1÷*C*4 are equivalent in practice to just a single capacitor *CS* (dotted line in fig.1), connected in parallel to the output of the inverter. The capacity of the capacitor *CS* is equal to the capacity of each of the snubber capacitors *C*1÷*C*4.

The matching transformer *Tr* is shown in fig.1 together with its simplified equivalent circuit under the condition that the magnetizing current of the transformer is negligible with respect to the current in the resonant circuit. Then this transformer comprises both the full leakage inductance *LS* and the natural capacity of the windings *С*0, reduced to the primary winding, as well as an ideal transformer with its transformation ratio equal to *k*.

The leakage inductance *LS* is connected in series with the inductance of the resonant circuit *L* and can be regarded as part of it. The natural capacity *C*0 takes into account the capacity between the windings and the different layers in each winding of the matching transformer. *C*0 can has an essential value, especially with stepping up transformers (Liu et al., 2009).

Together with the capacity *С*0 the resonant circuit becomes a circuit of the third order (*L*, *C* and *С*0), while the converter could be regarded as LCC resonant DC/DC converter with a capacitive output filter.

The parasitic parameters of the matching transformer – leakage inductance and natural capacity of the windings – should be taken into account only at high voltages and high operating frequencies of the converter. At voltages lower than 1000 V and frequencies lower

The circuit diagram of the LCC transistor resonant DC/DC converter under investigation is shown in figure 1. It consists of an inverter (controllable switches constructed on base of the transistors *Q*1÷*Q*4 with freewheeling diodes *D*1÷*D*4), a resonant circuit (*L*, С), a matching transformer *Tr*, an uncontrollable rectifier (*D*5÷*D*8), capacitive input and output filters (*CF*<sup>1</sup> и *CF*2) and a load resistor (*R*0). The snubber capacitors (*C*1*÷C*4) are connected with the

The output power of the converter is controlled by changing the operating frequency, which

It is assumed that all the elements in the converter circuit (except for the matching transformer) are ideal, and the pulsations of the input and output voltages can be neglected.

All snubber capacitors *C*1÷*C*4 are equivalent in practice to just a single capacitor *CS* (dotted line in fig.1), connected in parallel to the output of the inverter. The capacity of the capacitor

The matching transformer *Tr* is shown in fig.1 together with its simplified equivalent circuit under the condition that the magnetizing current of the transformer is negligible with respect to the current in the resonant circuit. Then this transformer comprises both the full leakage inductance *LS* and the natural capacity of the windings *С*0, reduced to the primary

The leakage inductance *LS* is connected in series with the inductance of the resonant circuit *L* and can be regarded as part of it. The natural capacity *C*0 takes into account the capacity between the windings and the different layers in each winding of the matching transformer. *C*0 can has an essential value, especially with stepping up transformers (Liu et al., 2009). Together with the capacity *С*0 the resonant circuit becomes a circuit of the third order (*L*, *C* and *С*0), while the converter could be regarded as LCC resonant DC/DC converter with a

The parasitic parameters of the matching transformer – leakage inductance and natural capacity of the windings – should be taken into account only at high voltages and high operating frequencies of the converter. At voltages lower than 1000 V and frequencies lower

winding, as well as an ideal transformer with its transformation ratio equal to *k*.

**2. Modes of operation of the converter** 

is higher than the resonant frequency of the resonant circuit.

Fig. 1. Circuit diagram of the LCC transistor DC/DC converter

*CS* is equal to the capacity of each of the snubber capacitors *C*1÷*C*4.

transistors in parallel.

capacitive output filter.

than 100 kHz they can be neglected, and the capacitor *С*0 should be placed additionally (Liu et al., 2009).

Because of the availability of the capacitor *CS* , the commutations in the output voltage of the inverter (*ua*) are not instantaneous. They start with switching off the transistors *Q*1/*Q*3 or *Q*2/*Q*4 and end up when the equivalent snubber capacitor is recharged from *+Ud* to *–Ud* or backwards and the freewheeling diodes *D*2/*D*4 or *D*1/*D*3 start conducting. In practice the capacitors *С*2 and *С*4 discharge from *+Ud* to 0, while *С*1 and *С*3 recharge from 0 to*+Ud* or backwards. During these commutations, any of the transistors and freewheeling diodes of the inverter does not conduct and the current flowed through the resonant circuit is closed through the capacitor *СS*.

Because of the availability of the capacitor *C*0, the commutations in the input voltage of the rectifier (*ub*) are not instantaneous either. They start when the diode pairs (*D*5/*D*7 or *D*6/*D*8) stop conducting at the moments of setting the current to zero through the resonant circuit and end up with the other diode pair (*D*6/*D*<sup>8</sup> или *D*5/*D*7) start conducting, when the capacitor *С*0 recharges from *+kU*0 to *–kU*0 or backwards. During these commutations, any of the diodes of the rectifier does not conduct and the current flowed through the resonant circuit is closed through the capacitor *С*0.

The condition for natural switching on of the controllable switches at zero voltage (ZVS) is fulfilled if the equivalent snubber capacitor *СS* always manages to recharge from *+Ud* to *–Ud* or backwards. At modes, close to no-load, the recharging of *СS* is possible due to the availability of the capacitor *C*0. It ensures the flow of current through the resonant circuit, even when the diodes of the rectifier do not conduct.

When the load and the operating frequency are deeply changed, three different operation modes of the converter can be observed.

It is characteristic for the first mode that the commutations in the rectifier occur entirely in the intervals for conducting of the transistors in the inverter. This mode *is the main operation mode* of the converter. It is observed at comparatively small values of the load resistor *R*0.

At the second mode the commutation in the rectifier ends during the commutation in the inverter, i.e., the rectifier diodes start conducting when both the transistors and the freewheeling diodes of the inverter are closed. This is *the medial operation mode* and it is only observed in a narrow zone, defined by the change of the load resistor value which is however not immediate to no-load.

At modes, which are very close to no-load the third case is observed. The commutations in the rectifier now complete after the ones in the inverter, i.e. the rectifier diodes start conducting after the conduction beginning of the corresponding inverter's freewheeling diodes. This mode is *the boundary operation mode* with respect to no-load.

#### **3. Analysis of the converter**

In order to obtain general results, it is necessary to normalize all quantities characterizing the converter's state. The following quantities are included into relative units:

*C Cd xU uU* = = ′ - Voltage of the capacitor *С*;

*d* 0 *<sup>i</sup> y I U Z* = =′ - Current in the resonant circuit; *Ud U*<sup>0</sup> = *kU*<sup>0</sup> ′ - Output voltage;

Study of LCC Resonant Transistor DC / DC Converter with Capacitive Output Filter 115

Fig. 3. Trajectory of the depicting point at the main mode of the converter operation.

Fig. 4. Waveforms of the voltages and currents at the main operation mode of the converter

0 0 <sup>0</sup> *U Z <sup>I</sup> <sup>k</sup> <sup>I</sup> d* ′ = - Output current;

*UCm* =*UCm Ud* ′ - Maximum voltage of the capacitor *С*;

ν = ω ω0 - Distraction of the resonant circuit,

where ω is the operating frequency and ω<sup>0</sup> = 1 *LC* and *Z LC* <sup>0</sup> = are the resonant frequency and the characteristic impedance of the resonant circuit *L*-*C* correspondingly.

#### **3.1 Analysis at the main operation mode of the converter**

Considering the influence of the capacitors *СS* and *С*0, the main operation mode of the converter can be divided into eight consecutive intervals, whose equivalent circuits are shown in fig. 2. By the trajectory of the depicting point in the state plane ( ) ; *<sup>C</sup> x U*= = ′ ′ *y I* , shown in fig. 3, the converter's work is also illustrated, as well as by the waveform diagrams in fig.4.

The following four centers of circle arcs, constituting the trajectory of the depicting point, correspond to the respective intervals of conduction by the transistors and freewheeling diodes in the inverter: interval 1: *Q*1/*Q*3 - ( ) <sup>0</sup> 1 ;0 −*U*′ ; interval 3: *D*2/*D*4 - ( ) <sup>0</sup> − −1 ;0 *U*′ ; interval 5: *Q*2/*Q*4 - ( ) <sup>0</sup> − +1 ;0 *U*′ ; interval 7: *D*1/*D*3 - ( ) <sup>0</sup> 1 ;0 +*U*′ .

The intervals 2 and 6 correspond to the commutations in the inverter. The capacitors *С* and *СS* then are connected in series and the sinusoidal quantities have angular frequency of <sup>0</sup> <sup>1</sup> ω 1 = *LCE* ′ where *C CC C C ES S* <sup>1</sup> = + ( ) . For the time intervals 2 and 6 the input current *id* is equal to zero. These pauses in the form of the input current *id* (fig. 4) are the cause for increasing the maximum current value through the transistors but they do not influence the form of the output characteristics of the converter.

Fig. 2. Equivalent circuits at the main operation mode of the converter.

The intervals 4 and 8 correspond to the commutations in the rectifier. The capacitors *С* and *С*0 are then connected in series and the sinusoidal quantities have angular frequency of <sup>0</sup> <sup>2</sup> ω 1 = *LCE* ′′ where *C CC C C <sup>E</sup>*20 0 = + ( ) . For the time intervals 4 and 8, the output current *i0* is equal to zero. Pauses occur in the form of the output current *i0*, decreasing its average value by *Δ* <sup>0</sup>*I* (fig. 4) and essentially influence the form of the output characteristics of the converter.

where ω is the operating frequency and ω<sup>0</sup> = 1 *LC* and *Z LC* <sup>0</sup> = are the resonant frequency and the characteristic impedance of the resonant circuit *L*-*C* correspondingly.

Considering the influence of the capacitors *СS* and *С*0, the main operation mode of the converter can be divided into eight consecutive intervals, whose equivalent circuits are shown in fig. 2. By the trajectory of the depicting point in the state plane ( ) ; *<sup>C</sup> x U*= = ′ ′ *y I* , shown in fig. 3, the converter's work is also illustrated, as well as by the waveform diagrams

The following four centers of circle arcs, constituting the trajectory of the depicting point, correspond to the respective intervals of conduction by the transistors and freewheeling diodes in the inverter: interval 1: *Q*1/*Q*3 - ( ) <sup>0</sup> 1 ;0 −*U*′ ; interval 3: *D*2/*D*4 - ( ) <sup>0</sup> − −1 ;0 *U*′ ;

The intervals 2 and 6 correspond to the commutations in the inverter. The capacitors *С* and *СS* then are connected in series and the sinusoidal quantities have angular frequency of <sup>0</sup> <sup>1</sup> ω 1 = *LCE* ′ where *C CC C C ES S* <sup>1</sup> = + ( ) . For the time intervals 2 and 6 the input current *id* is equal to zero. These pauses in the form of the input current *id* (fig. 4) are the cause for increasing the maximum current value through the transistors but they do not influence the

The intervals 4 and 8 correspond to the commutations in the rectifier. The capacitors *С* and *С*0 are then connected in series and the sinusoidal quantities have angular frequency of <sup>0</sup> <sup>2</sup> ω 1 = *LCE* ′′ where *C CC C C <sup>E</sup>*20 0 = + ( ) . For the time intervals 4 and 8, the output current *i0* is equal to zero. Pauses occur in the form of the output current *i0*, decreasing its average value by *Δ* <sup>0</sup>*I* (fig. 4) and essentially influence the form of the output characteristics of the

0 0 <sup>0</sup> *U Z <sup>I</sup> <sup>k</sup> <sup>I</sup> d*

in fig.4.

converter.

′ = - Output current;

*UCm* =*UCm Ud* ′ - Maximum voltage of the capacitor *С*; ν = ω ω0 - Distraction of the resonant circuit,

**3.1 Analysis at the main operation mode of the converter** 

interval 5: *Q*2/*Q*4 - ( ) <sup>0</sup> − +1 ;0 *U*′ ; interval 7: *D*1/*D*3 - ( ) <sup>0</sup> 1 ;0 +*U*′ .

Fig. 2. Equivalent circuits at the main operation mode of the converter.

form of the output characteristics of the converter.

Fig. 3. Trajectory of the depicting point at the main mode of the converter operation.

Fig. 4. Waveforms of the voltages and currents at the main operation mode of the converter

Study of LCC Resonant Transistor DC / DC Converter with Capacitive Output Filter 117

the commutating capacitor C in relative units for each interval of converter operation. The

( )

*U aU UU aU a*

− + − + +⋅ ′ ′ ′′ ′

2 20 0 20 1 0

( )

′ ′′ ′ − + −⋅

*U UU aU a*

*y U aU UU aU a U*

( )

20 20

*Cm*

*aU U aU*

+ −+ ′′ ′

4 1

*Cm Cm*

*Cm Cm*

<sup>3</sup> <sup>2</sup>

*Cm Cm*

*Cm Cm*

2

( )

( )

*U UU aU a U*

For converters with only two reactive elements (*L* and *C*) in the resonant circuit the expression for its output current 0*I*′ is known from (Al Haddad et al., 1986; Cheron, 1989):

The LCC converter under consideration has three reactive elements in its resonant circuit (*L*, С и *C*0). From fig.4 it can be seen that its output current 0*I*′ decreases by the value

<sup>0</sup> <sup>0</sup> ( ) 2 0 2 2 *I U I U aU*

01 2 3 4 ( ) *tttt*

2 1

1 1

For the times of the four intervals at the main mode of operation of the converter within a

0 20 10

 *xU xU* = − − +− − +− ′ ′

πν

ν*Cm*

π

ν

ω

where the times *t*1÷*t*4 represent the durations of the different stages – from 1 to 4.

*y y t arctg arctg*

0 1 2 *I U* ′ ′ = ν*C m*

⋅ + − ++ + ′ ′′ ′ ′

2 0 20 1

0 20 1 0 2 2

π

= ⋅− + + − − + − + ′ ′ ′′ ′ ′

20 0 20 1

2

2

2 2 2

′ , *UCm*′ , 1*a* and 2 *a* :

*y aU U aU* 1 20 20 4 1 ( ) *Cm* = −+ ′′ ′ (13)

<sup>2</sup> *x UU aU a* 2 0 20 1 *Cm* = −− ′′ ′ (14)

<sup>2</sup> *x UU aU a* 3 0 20 1 *Cm* = −+ ′′ ′ (16)

*x U* <sup>4</sup> *Cm* = ′ (18)

<sup>4</sup> *y* = 0 (19)

 π*Cm* ′′ ′ ′ ′ = −Δ = − (21)

= +++ , (22)

(20)

(15)

(17)

(23a)

<sup>1</sup> 2 0 2 *x U aU Cm* =− + ′ ′ (12)

expressions for the coordinates are in function of *U*<sup>0</sup>

*y*

0 20 Δ = *I aU* ′ ′ 2 ν

π:

The following equation is known:

half-cycle the following equations hold:

1

1

ω

=

It has been proved in (Cheron, 1989; Bankov, 2009) that in the state plane (fig. 3) the points, corresponding to the beginning (p.*М*2) and the end (p.*М*3) of the commutation in the inverter belong to the same arc with its centre in point ( ) <sup>0</sup> −*U*′ ;0 . It can be proved the same way that the points, corresponding to the beginning (p.*М*8) and the end (p.*М*1) of the commutation in the rectifier belong to an arc with its centre in point ( ) 1;0 . It is important to note that only the end points are of importance on these arcs. The central angles of these arcs do not matter either, because as during the commutations in the inverter and rectifier the electric quantities change correspondingly with angular frequencies ω<sup>0</sup> ′ and ω<sup>0</sup> ′′ , not with ω<sup>0</sup> .

The following designations are made:

$$a\_1 = \mathbb{C}\_S / \mathbb{C} \qquad \qquad n\_1 = \sqrt{(a\_1 + 1) / a\_1} \tag{1}$$

$$a\_2 = \mathbb{C}\_0 / \mathbb{C} \qquad \qquad n\_2 = \sqrt{(a\_2 + 1) / a\_2} \tag{2}$$

$$m\_3 = \sqrt{1 + 1/a\_1 + 1/a\_2} \tag{3}$$

For the state plane shown in fig, 3 the following dependencies are valid:

$$\left(\left(\mathbf{x}\_1 - \mathbf{1} + \mathbf{U}\_0'\right)^2 + y\_1^2 = \left(\mathbf{x}\_2 - \mathbf{1} + \mathbf{U}\_0'\right)^2 + y\_2^2\tag{4}$$

$$\left(\mathbf{x}\_2 + \mathbf{U}\_0'\right)^2 + y\_2^2 = \left(\mathbf{x}\_3 + \mathbf{U}\_0'\right)^2 + y\_3^2\tag{5}$$

$$\left(\left(\mathbf{x}\_3 + \mathbf{1} + \mathbf{U}\_0'\right)^2 + y\_3^2\right) = \left(\mathbf{x}\_4 + \mathbf{1} + \mathbf{U}\_0'\right)^2 + y\_4^2\tag{6}$$

$$\left(\left(\mathbf{x}\_4+\mathbf{1}\right)^2+y\_4^2\right)+y\_4^2=\left(\mathbf{x}\_5+\mathbf{1}\right)^2+y\_5^2\tag{7}$$

From the existing symmetry with respect to the origin of the coordinate system of the state plane it follows:

$$
\infty\_5 = -\infty\_1 \tag{8}
$$

$$y\_5 = -y\_1\tag{9}$$

During the commutations in the inverter and rectifier, the voltages of the capacitors *СS* and *С*0 change correspondingly by the values 2*Ud* and 0 2*kU* , and the voltage of the commutating capacitor *C* changes respectively by the values 1 2 *<sup>d</sup> a U* and 2 0 2*a kU* . Consequently:

$$\mathbf{x}\_3 = \mathbf{x}\_2 + \mathbf{2}a\_1 \tag{10}$$

$$
\infty\_{\mathbb{S}} = \infty\_{4} - 2a\_{2} \mathrm{l} \mathrm{l}\_{0}' \tag{11}
$$

The equations (4)÷(11) allow for calculating the coordinates of the points *М*1÷*М*4 in the state plane, which are the starting values of the current through the inductor L and the voltage of

It has been proved in (Cheron, 1989; Bankov, 2009) that in the state plane (fig. 3) the points, corresponding to the beginning (p.*М*2) and the end (p.*М*3) of the commutation in the inverter belong to the same arc with its centre in point ( ) <sup>0</sup> −*U*′ ;0 . It can be proved the same way that the points, corresponding to the beginning (p.*М*8) and the end (p.*М*1) of the commutation in the rectifier belong to an arc with its centre in point ( ) 1;0 . It is important to note that only the end points are of importance on these arcs. The central angles of these arcs do not matter either, because as during the commutations in the inverter and rectifier the electric quantities change

′ and ω<sup>0</sup>

( )( ) 2 2 2 2

() () 2 2 2 2

( )( ) 2 2 2 2

( ) ( ) <sup>2</sup> 2 2 <sup>2</sup>

From the existing symmetry with respect to the origin of the coordinate system of the state

During the commutations in the inverter and rectifier, the voltages of the capacitors *СS* and *С*0 change correspondingly by the values 2*Ud* and 0 2*kU* , and the voltage of the commutating capacitor *C* changes respectively by the values 1 2 *<sup>d</sup> a U* and 2 0 2*a kU* .

The equations (4)÷(11) allow for calculating the coordinates of the points *М*1÷*М*4 in the state plane, which are the starting values of the current through the inductor L and the voltage of

For the state plane shown in fig, 3 the following dependencies are valid:

′′ , not with ω<sup>0</sup> .

<sup>1</sup> *<sup>S</sup> a CC* = *na a* 11 1 = + ( ) 1 (1)

2 0 *a CC* = *na a* 22 2 = + ( ) 1 (2)

1 0 12 0 2 *x U* −+ + = −+ + 1 1 ′ ′ *y x U y* (4)

3 0 34 0 4 *x U* ++ + = ++ + 1 1 ′ ′ *y x U y* (6)

20 2 30 3 *x U*+ += + + ′ ′ *y x U y* (5)

4 45 5 *x* + += + + 1 1 *y x y* (7)

5 1 *x x* = − (8)

5 1 *y* = −*y* (9)

32 1 *xx a* = + 2 (10)

5 4 20 *x x aU* = − 2 ′ (11)

3 12 *n aa* =+ + 11 1 (3)

correspondingly with angular frequencies ω<sup>0</sup>

The following designations are made:

plane it follows:

Consequently:

the commutating capacitor C in relative units for each interval of converter operation. The expressions for the coordinates are in function of *U*<sup>0</sup> ′ , *UCm*′ , 1*a* and 2 *a* :

$$\infty\_1 = -\mathcal{U}\_{\text{Cm}}' + 2a\_2 \mathcal{U}\_0' \tag{12}$$

$$y\_1 = \sqrt{4a\_2 \mathcal{U}\_0' \left(\mathcal{U}\_{\text{Cm}}' - a\_2 \mathcal{U}\_0' + 1\right)}\tag{13}$$

$$\propto \mathbf{x}\_2 = \mathbf{U}\_0' \mathbf{U}\_{\text{Cm}}' - a\_2 \mathbf{U}\_0'^2 - a\_1 \tag{14}$$

$$y\_2 = \begin{cases} \left( -\mathcal{U}\_{\text{Cm}}' + 2a\_2 \mathcal{U}\_0' - \mathcal{U}\_0' \mathcal{U}\_{\text{Cm}}' + a\_2 \mathcal{U}\_0'^2 + a\_1 \right) \cdot \\ \left( -\mathcal{U}\_{\text{Cm}}' + 2a\_2 \mathcal{U}\_0' + \mathcal{U}\_0' \mathcal{U}\_{\text{Cm}}' - a\_2 \mathcal{U}\_0'^2 - a\_1 + 2\mathcal{U}\_0' - 2 \right) + \\ + 4a\_2 \mathcal{U}\_0' \left( \mathcal{U}\_{\text{Cm}}' - a\_2 \mathcal{U}\_0' + 1 \right) \end{cases} \tag{15}$$

$$\propto\_3 = \mathbb{U}\_0' \mathbb{U}\_{\mathbb{C}m}' - a\_2 \mathbb{U}\_0'^2 + a\_1 \tag{16}$$

$$y\_3 = \begin{cases} \mathsf{U}'\_{\mathrm{Cm}} - \mathsf{U}'\_0 \mathsf{U}'\_{\mathrm{Cm}} + a\_2 \mathsf{U}'\_0^2 - a\_1 \Big) \cdot \\ \left( \mathsf{U}'\_{\mathrm{Cm}} + \mathsf{U}'\_0 \mathsf{U}'\_{\mathrm{Cm}} - a\_2 \mathsf{U}'\_0^2 + a\_1 + 2\mathsf{U}'\_0 + 2 \right) \end{cases} \tag{17}$$

$$\propto\_{4} = \mathbb{L}I\_{\text{Cm}}^{\prime} \tag{18}$$

$$y\_4 = 0\tag{19}$$

For converters with only two reactive elements (*L* and *C*) in the resonant circuit the expression for its output current 0*I*′ is known from (Al Haddad et al., 1986; Cheron, 1989):

$$\mathbf{I}\_0' = \mathbf{2} \nu \mathbf{L} \mathbf{I}\_{\mathbb{C}1m}' / \pi \tag{20}$$

The LCC converter under consideration has three reactive elements in its resonant circuit (*L*, С и *C*0). From fig.4 it can be seen that its output current 0*I*′ decreases by the value 0 20 Δ = *I aU* ′ ′ 2 ν π:

$$
\Delta I\_0' = \mathcal{D}\nu \mathcal{U}\_{\text{Cm}}' / \pi - \Delta I\_0' = \mathcal{D}\nu \left( \mathcal{U}\_{\text{Cm}}' - a\_2 \mathcal{U}\_0' \right) / \pi \tag{21}
$$

The following equation is known:

$$\frac{\pi}{\nu} = \alpha\_0 \left( t\_1 + t\_2 + t\_3 + t\_4 \right) \tag{22}$$

where the times *t*1÷*t*4 represent the durations of the different stages – from 1 to 4. For the times of the four intervals at the main mode of operation of the converter within a half-cycle the following equations hold:

$$t\_1 = \frac{1}{a\_0} \left( \operatorname{arctg} \frac{y\_2}{-x\_2 + 1 - \mathcal{U}\_0'} - \operatorname{arctg} \frac{y\_1}{-x\_1 + 1 - \mathcal{U}\_0'} \right) \tag{23a}$$

Study of LCC Resonant Transistor DC / DC Converter with Capacitive Output Filter 119

<sup>0</sup> <sup>2</sup> ω 1 = *LCE* ′′ , where *C CC C C <sup>E</sup>*20 0 = + ( ) , for stages 1, 3, 5 and 7;

ω<sup>0</sup> = 1 *LCE*<sup>3</sup> ′′′ , where *C CC C CC CC C C ES S S* 3 0 00 = ++ ( ) , for stages 2 and 6.

Fig. 5. Equivalent circuits at the boundary operation mode of the converter

*C d <sup>E</sup> xu U* = ; 0

In this case the representation in the state plane becomes complex and requires the use of two state planes (fig.6). One of them is ( ) ; *<sup>C</sup> x U*= = ′ ′ *y I* and it is used for presenting stages 4

Fig. 6. Trajectory of the depicting point at the boundary mode of operation of the converter

*d E*2 *i <sup>y</sup> U LC* = .

ω<sup>0</sup> = 1 *LC* for stages 4 and 8;

and 8, the other is ( ) 0 0 *x y*; , where:

0

2

$$\begin{array}{ll}\mbox{at} & \mathbf{x}\_{2} \leq \mathbf{1} - \mathsf{U}\_{0}^{\prime} \quad \mbox{and} & \mathbf{x}\_{1} \leq \mathbf{1} - \mathsf{U}\_{0}^{\prime} \\\\ \mathbf{x}\_{1} = \frac{1}{a\_{0}} \Big(\boldsymbol{\pi} - \operatorname{arctg}\frac{\boldsymbol{y}\_{2}}{\boldsymbol{x}\_{2} - \mathbf{1} + \mathsf{U}\_{0}^{\prime}} - \operatorname{arctg}\frac{\boldsymbol{y}\_{1}}{-\boldsymbol{x}\_{1} + \mathbf{1} - \mathsf{U}\_{0}^{\prime}}\Big) \\\\ & \mbox{at} & \mathbf{x}\_{2} \geq \mathbf{1} - \mathsf{U}\_{0}^{\prime} \quad \mbox{and} & \mathbf{x}\_{1} \leq \mathbf{1} - \mathsf{U}\_{0}^{\prime} \end{array} \tag{23b}$$

$$t\_1 = \frac{1}{a\_0} \left( \operatorname{arcctg} \frac{y\_2}{-\chi\_2 + 1 - \mathcal{U}\_0'} + \operatorname{arcctg} \frac{y\_1}{\chi\_1 - 1 + \mathcal{U}\_0'} \right) \tag{23c}$$

at 2 0 *x U* ≥ −1 ′ and 1 0 *x U* ≥ −1 ′

$$t\_2 = \frac{1}{n\_1 a\_0} \left( \operatorname{arctg} \frac{n\_1 y\_2}{x\_2 - 1 + \mathcal{U}\_0'} - \operatorname{arctg} \frac{n\_1 y\_3}{x\_3 + 1 + \mathcal{U}\_0'} \right) \tag{24a}$$

$$\text{at} \qquad \qquad \qquad \mathbf{x}\_2 \ge \mathbf{1} - \mathbf{U}'\_0$$

$$t\_2 = \frac{1}{n\_1 a\_0} \left( \operatorname{arctg} \frac{n\_1 y\_2}{x\_2 - 1 + \mathcal{U}\_0'} - \operatorname{arctg} \frac{n\_1 y\_3}{x\_3 + 1 + \mathcal{U}\_0'} \right) \tag{24b}$$

$$\text{at} \qquad \qquad x\_2 \le 1 - \mathcal{U}\_0'$$

$$t\_3 = \frac{1}{a\_0} \operatorname{arcctg} \frac{y\_3}{x\_3 + 1 + \mathcal{U}\_0'} \tag{25}$$

$$t\_4 = \frac{1}{n\_2 a\_0} \left( \operatorname{arctg} \frac{n\_2 y\_1}{-\mathbf{x}\_1 + \mathbf{1} - \mathbf{U}\_0'} \right) \tag{26a}$$

at 1 0 *x U* ≤ −1 ′

$$t\_4 = \frac{1}{n\_2 a\_0} \left(\pi - \operatorname{arclg} \frac{n\_2 y\_1}{x\_1 - 1 + \mathcal{U}\_0'}\right) \tag{26b}$$
 
$$\text{at} \qquad x\_1 \ge 1 - \mathcal{U}\_0'$$

It should be taken into consideration that for stages 1 and 3 the electric quantities change with angular frequency ω<sup>0</sup> , while for stages 2 and 4 – the angular frequencies are respectively ω ω ′ 0 10 = *n* and ω ω ′′0 20 = *n* .

#### **3.2 Analysis at the boundary operation mode of the converter**

At this mode, the operation of the converter for a cycle can be divided into eight consecutive stages (intervals), whose equivalent circuits are shown in fig. 5. It makes impression that the sinusoidal quantities in the different equivalent circuits have three different angular frequencies:

at 2 0 *x U* ≤ −1 ′ and 1 0 *x U* ≤ −1 ′

*y y t arctg arctg xU xU*

at 2 0 *x U* ≥ −1 ′ and 1 0 *x U* ≤ −1 ′

*y y t arctg arctg*

0 20 10

0 20 10

10 2 0 3 0

10 2 0 3 0

03 0

20 1 0

2 0 1 0

 = − − + ′

 <sup>=</sup> − +− ′

*x U* <sup>=</sup> + + ′

 = − − + ′ ′ + +

 = − − + ′ ′ + +

 *xU xU* = + − +− −+ ′ ′

at 2 0 *x U* ≥ −1 ′ and 1 0 *x U* ≥ −1 ′

*n y n y t arctg arctg n xU xU*

at 2 0 *x U* ≥ −1 ′

*n y n y t arctg arctg n xU xU*

at 2 0 *x U* ≤ −1 ′

 =− − −+ − +− ′ ′

1

1

2

2

1

ω

π

1

ω

1

ω

1

ω

3

4

4

0 20 = *n* .

**3.2 Analysis at the boundary operation mode of the converter** 

0 10 = *n* and ω ω ′′

respectively ω ω ′

frequencies:

1

1

1

ω

ω

*<sup>y</sup> t arctg* ω

*n y t arctg n xU*

at 1 0 *x U* ≤ −1 ′

*n y <sup>t</sup> arctg n xU* π

at 1 0 *x U* ≥ −1 ′ It should be taken into consideration that for stages 1 and 3 the electric quantities change with angular frequency ω<sup>0</sup> , while for stages 2 and 4 – the angular frequencies are

At this mode, the operation of the converter for a cycle can be divided into eight consecutive stages (intervals), whose equivalent circuits are shown in fig. 5. It makes impression that the sinusoidal quantities in the different equivalent circuits have three different angular

2 1

(23b)

(23c)

(24a)

(24b)

(25)

(26a)

(26b)

1 1

2 1

1 1

1 2 1 3

1 2 1 3

1 1

3

2 1

1

2 1

1

1

1 1

ω<sup>0</sup> = 1 *LC* for stages 4 and 8; <sup>0</sup> <sup>2</sup> ω 1 = *LCE* ′′ , where *C CC C C <sup>E</sup>*20 0 = + ( ) , for stages 1, 3, 5 and 7; ω<sup>0</sup> = 1 *LCE*<sup>3</sup> ′′′ , where *C CC C CC CC C C ES S S* 3 0 00 = ++ ( ) , for stages 2 and 6.

Fig. 5. Equivalent circuits at the boundary operation mode of the converter

In this case the representation in the state plane becomes complex and requires the use of two state planes (fig.6). One of them is ( ) ; *<sup>C</sup> x U*= = ′ ′ *y I* and it is used for presenting stages 4 and 8, the other is ( ) 0 0 *x y*; , where:

Fig. 6. Trajectory of the depicting point at the boundary mode of operation of the converter

Study of LCC Resonant Transistor DC / DC Converter with Capacitive Output Filter 121

( ) ( )<sup>2</sup> 0 0 <sup>2</sup>

( )( 20 0 20 ) 0 2 3 1 2 2 <sup>1</sup> *U aU U aU Cm <sup>x</sup> a n a* ′ ′ ′′ − ++

() () () <sup>222</sup> 00 0 0

( ) <sup>0</sup>

( )( ) <sup>0</sup>

At the boundary operation mode the output current 0*I*′ is defined by expression (21) again,

For the times of the four intervals at the boundary operation mode of the converter within a

0 2

1

( ) ( )

0 0 3 4

1 1

4

0 20 ′′ = *n* , while for stages 2 and 4 the angular frequencies are

1

1 1

( ) ( )

3 0 0 2 0 3 4

 *x x* = −

0 40

 *x U* <sup>=</sup>

It should be taken into consideration that for stages 1 and 3 the electric quantities change by

*y y t arctg arctg*

*<sup>y</sup> t arctg*

1 1

0 0 3 2 3 3

0 0 3 2 3 3

0 2

1

where *t*1÷*t*4 represent the times of the different stages – from 1 to 4.

1 0 2 0 2

1 0 2 0 2

*<sup>y</sup> <sup>t</sup> arctg n x* π

 *x* <sup>=</sup> <sup>−</sup>

= − <sup>−</sup>

2 0 0 3 0 22 23

=− − − +

*n y n y <sup>t</sup> arctg arctg n nx nx*

2 0 0 3 0 2 2 2 3

 *nx n x* = − − +

1

ω

*n y n y t arctg arctg*

1

4

*n* ω

*<sup>y</sup> t arctg*

1

1

ω

*n* ω

half-cycle the following equations hold:

1

1

*n* ω

ω

ω

 ω

 ω0 30 ′′′ = *n* and ω<sup>0</sup> .

angular frequency

correspondingly

ω

π

2 2 1 1 *Cm o yU U x* = ++ − − ′ ′ (38)

= + (39)

32 2 3 *yx yx* = +− (40)

4 02 1 2 *xUU a Cm* =− + + ′ ′ (41)

<sup>2</sup> *x* ≤ 1 (43a)

<sup>2</sup> *x* ≥ 1 (43b)

<sup>2</sup> *x* ≤ 1 (44a)

<sup>2</sup> *x* ≥ 1 (44b)

at <sup>0</sup>

at <sup>0</sup>

+ + (45)

+ + ′ (46)

4 20 0 20 2 1 *y aU U aU UCm* = ++ − ′ ′ ′′ (42)

at <sup>0</sup>

at <sup>0</sup>

Stages 2 and 6 correspond to the commutations in the inverter.

The commutations in the rectifier begin in p. <sup>0</sup> *M*1 or p. <sup>0</sup> *M*5 and end in p. <sup>0</sup> *M*<sup>4</sup> or p. <sup>0</sup> *M*<sup>8</sup> , comprising stages 1,2 and 3 or 5,6 and 7.

The transistors conduct for the time of stages 1 and 5, the freewheeling diodes – for the time of stages 3, 4, 7 and 8, and the rectifier diodes – for the time of stages 4 and 8.

The following equations are obtained in correspondence with the trajectory of the depicting point for this mode of operation (fig.6):

$$\left(1-\mathbf{x}\_1^0\right)^2 = \left(1-\mathbf{x}\_2^0\right)^2 + \left(y\_2^0\right)^2\tag{27}$$

$$\left(\mathbf{x}\_2^0\right)^2 + \left(y\_2^0\right)^2 = \left(\mathbf{x}\_3^0\right)^2 + \left(y\_3^0\right)^2\tag{28}$$

$$\left(\mathbf{x}\_3^0 + \mathbf{1}\right)^2 + \left(y\_3^0\right)^2 = \left(\mathbf{x}\_4^0 + \mathbf{1}\right)^2 + \left(y\_4^0\right)^2\tag{29}$$

$$\left(\left(\mathbf{x}\_4 + \mathbf{1} + \mathbf{U}\_0'\right)^2 + \left(\mathbf{y}\_4\right)^2\right) = \left(\mathbf{1} + \mathbf{U}\_0' - \mathbf{x}\_1\right)^2\tag{30}$$

During the commutations in the inverter, the voltage of the capacitor *СЕ*2 changes by the value 2 2*UC C dS E* and consequently:

$$\mathbf{x}\_3^0 = \mathbf{x}\_2^0 + 2a\_1 n\_2^2 \tag{31}$$

The same way during the commutation in the rectifier, the voltage of the capacitor *СЕ*<sup>2</sup> changes by the value 00 2 2 *<sup>E</sup> kU C C* and consequently:

$$\mathbf{x}\_4^0 = \mathbf{x}\_1^0 + 2\mathcal{U}\_0' \left(\mathbf{1} + a\_2\right) \tag{32}$$

From the principle of continuity of the current through the inductor *L* and of the voltage in the capacitor *C* it follows:

$$\mathbf{x}\_4^0 = \mathbf{x}\_4 + \mathbf{L}\_0^{\prime} \tag{33}$$

$$
\Delta y\_4^0 = n\_2 y\_4 \tag{34}
$$

where ( ) ;*i i <sup>x</sup> <sup>y</sup>* and ( ) 0 0 ; *i i x y* are the coordinates of *Mi* and <sup>0</sup> *Mi* respectively. The equations (27)÷(34) allow for defining the coordinates of the points <sup>0</sup> *M*<sup>1</sup> ÷ <sup>0</sup> *M*4 in the state plane:

$$
\omega\_1^0 = -\mathcal{U}\_{\text{Cm}}' - \mathcal{U}\_0' \tag{35}
$$

$$y\_1^0 = 0\tag{36}$$

$$\alpha\_2^0 = \frac{\left(\mathcal{U}\_{\text{Cm}}' - a\_2 \mathcal{U}\_0'\right)\left(1 + \mathcal{U}\_0' + a\_2 \mathcal{U}\_0'\right)}{a\_2} - a\_1 n\_2^2 \tag{37}$$

The commutations in the rectifier begin in p. <sup>0</sup> *M*1 or p. <sup>0</sup> *M*5 and end in p. <sup>0</sup> *M*<sup>4</sup> or p. <sup>0</sup> *M*<sup>8</sup> ,

The transistors conduct for the time of stages 1 and 5, the freewheeling diodes – for the time

The following equations are obtained in correspondence with the trajectory of the depicting

( ) ( ) () 2 22 0 00

() () () () <sup>2222</sup> <sup>0000</sup>

( ) () ( ) () 22 22 0 00 0

( ) ( ) ( ) 2 2 <sup>2</sup>

During the commutations in the inverter, the voltage of the capacitor *СЕ*2 changes by the

00 2

The same way during the commutation in the rectifier, the voltage of the capacitor *СЕ*<sup>2</sup>

( ) 0 0

From the principle of continuity of the current through the inductor *L* and of the voltage in

The equations (27)÷(34) allow for defining the coordinates of the points <sup>0</sup> *M*<sup>1</sup> ÷ <sup>0</sup> *M*4 in the

0

( )( 20 0 20 ) 0 2 2 1 2 2 <sup>1</sup> *U aU U aU Cm <sup>x</sup> a n a*

0

where ( ) ;*i i <sup>x</sup> <sup>y</sup>* and ( ) 0 0 ; *i i x y* are the coordinates of *Mi* and <sup>0</sup> *Mi* respectively.

0

1 22 1 1 − =− + *x xy* (27)

<sup>2233</sup> *xyxy* +=+ (28)

3 34 4 *x yx y* ++ =++ 1 1 (29)

4 0 4 01 *x U* ++ + = + − 1 1 ′ ′ *y U x* (30)

3 2 12 *x x an* = + 2 (31)

41 0 2 *xx U a* =+ + 2 1 ′ (32)

44 0 *x xU* = + ′ (33)

4 24 *y* = *n y* (34)

<sup>0</sup> *xUU* 1 0 *Cm* =− − ′ ′ (35)

′ ′ ′′ − ++ = − (37)

<sup>1</sup> *y* = 0 (36)

of stages 3, 4, 7 and 8, and the rectifier diodes – for the time of stages 4 and 8.

Stages 2 and 6 correspond to the commutations in the inverter.

comprising stages 1,2 and 3 or 5,6 and 7.

point for this mode of operation (fig.6):

value 2 2*UC C dS E* and consequently:

the capacitor *C* it follows:

state plane:

changes by the value 00 2 2 *<sup>E</sup> kU C C* and consequently:

$$y\_2^0 = \sqrt{\left(\mathcal{U}\_{\complement m}' + 1 + \mathcal{U}\_o'\right)^2 - \left(1 - \mathcal{x}\_2^0\right)^2} \tag{38}$$

$$\mathbf{x}\_3^0 = \frac{\left(\mathbf{L}\_{\rm Cm}^{\prime} - a\_2 \mathbf{L}\_0^{\prime}\right)\left(\mathbf{1} + \mathbf{L}\_0^{\prime} + a\_2 \mathbf{L}\_0^{\prime}\right)}{a\_2} + a\_1 n\_2^2 \tag{39}$$

$$\mathbf{x}\_3^0 = \sqrt{\left(\mathbf{x}\_2^0\right)^2 + \left(\mathbf{y}\_2^0\right)^2 - \left(\mathbf{x}\_3^0\right)^2} \tag{40}$$

$$
\ln \mathbf{x}\_4^0 = -\mathcal{U}\_{\odot m}' + \mathcal{U}\_0' \left(1 + 2a\_2\right) \tag{41}
$$

$$y\_4^0 = 2\sqrt{(a\_2 \mathcal{U}\_0' + 1 + \mathcal{U}\_0') \left(a\_2 \mathcal{U}\_0' - \mathcal{U}\_{\mathbb{C}m}'\right)}\tag{42}$$

At the boundary operation mode the output current 0*I*′ is defined by expression (21) again, where *t*1÷*t*4 represent the times of the different stages – from 1 to 4.

For the times of the four intervals at the boundary operation mode of the converter within a half-cycle the following equations hold:

$$t\_1 = \frac{1}{n\_2 a\_0} \left( \operatorname{arctg} \frac{y\_2^0}{1 - \mathbf{x}\_2^0} \right) \qquad\qquad\qquad\text{at }\mathbf{x}\_2^0 \le 1\tag{43a}$$

$$t\_1 = \frac{1}{n\_2 a\_0} \left(\pi - \operatorname{arctg} \frac{y\_2^0}{x\_2^0 - 1} \right) \qquad\qquad\text{at } x\_2^0 \ge 1\tag{43b}$$

$$t\_2 = \frac{1}{n\_3 a\_0} \left( \pi - \operatorname{arctg} \frac{n\_3 y\_2^0}{n\_2 \left( 1 - \mathbf{x}\_2^0 \right)} - \operatorname{arctg} \frac{n\_3 y\_3^0}{n\_2 \left( 1 + \mathbf{x}\_3^0 \right)} \right) \qquad \text{at } x\_2^0 \le 1 \tag{44a}$$

$$t\_2 = \frac{1}{n\_3 a\_0} \left( \operatorname{arctg} \frac{n\_3 y\_2^0}{n\_2 \left(x\_2^0 - 1\right)} - \operatorname{arctg} \frac{n\_3 y\_3^0}{n\_2 \left(1 + x\_3^0\right)} \right) \qquad\qquad\text{at } x\_2^0 \ge 1\tag{44b}$$

$$t\_3 = \frac{1}{n\_2 a\_0} \left( \operatorname{arctg} \frac{y\_3^0}{1 + x\_3^0} - \operatorname{arctg} \frac{y\_4^0}{1 + x\_4^0} \right) \tag{45}$$

$$t\_4 = \frac{1}{a\theta\_0} \left( \operatorname{arctg} \frac{y\_4}{x\_4 + 1 + \mathcal{U}\_0'} \right) \tag{46}$$

It should be taken into consideration that for stages 1 and 3 the electric quantities change by angular frequencyω ω 0 20 ′′ = *n* , while for stages 2 and 4 the angular frequencies are correspondingly ω ω0 30 ′′′ = *n* and ω<sup>0</sup> .

Study of LCC Resonant Transistor DC / DC Converter with Capacitive Output Filter 123

a)

b)

Equations (12), (14) and (47) are substituted in condition (48), while equations (16) and (47)

<sup>+</sup> ′ ′ ≥ ⋅

1 20

1 20

*U*

*U*

1 *a aU <sup>I</sup>*

1 *a aU <sup>I</sup>*

Inequalities (50) и (51) enable with the possibility to draw the boundary curve *A* between the main and the medial modes of operation of the converter, as well as the border of the natural commutation – curve *L*3 (fig. 7-а) or curve *L*4 (fig. 7-b) in the plane of the output characteristics. It can be seen that the area of the main operation mode of the converter is limited within the boundary curves *А* and *L*3 or *L*4. The bigger the capacity of the capacitors

0

0

<sup>+</sup> ′ (50)

<sup>−</sup> ′ ′ ≥ ⋅ <sup>−</sup> ′ (51)

are substituted in condition (49). Then the inequalities (48) and (49) obtain the form:

2

ν

π

2

ν

π

0

0

Fig. 7. Output characteristics of the LCC transistor DC/DC converter

## **4. Output characteristics and boundary curves**

On the basis of the analysis results, equations for the output characteristics are obtained individually for both the main and the boundary modes of the converter operation. Besides, expressions for the boundary curves of the separate modes are also derived.

#### **4.1 Output characteristics and boundary curves at the main operation mode**

From equation (21) *UCm*′ is expressed in function of *U*<sup>0</sup> ′ , 0*I*′ , ν and 2 *a*

$$\mathcal{L}I'\_{\text{C.m.}} = \frac{\pi}{2\nu}I'\_0 + a\_2 \mathcal{U}'\_0 \tag{47}$$

By means of expression (47) *UCm*′ is eliminated from the equations (12)÷(18). After consecutive substitution of expressions (12)÷(18) in equations (23)÷(26) as well as of expressions (23)÷(26) in equation (22), an expression of the kind *U fI a a* 0 0 12 ′ ′ = ( ) ,, , ν is obtained. Its solution provides with the possibility to build the output characteristics of the converter in relative units at the main mode of operation and at regulation by means of changing the operating frequency. The output characteristics of the converter respectively for ν=1.2; 1.3; 1.4; 1.5; 1.8; 2.5; 3.0; 3.3165; 3,6 and *а*1=0.1; *а*2=0.2 as well as for ν=1.2; 1.3; 1.4; 1.5; 1.6; 1.8 and *а*1=0.1; *а*2=1.0 are shown in fig.7-a and 7-b.

The comparison of these characteristics to the known ones from (Al Haddad et al., 1986; Cheron, 1989) shows the entire influence of the capacitors *С<sup>S</sup>* и *С*0. It can be seen that the output characteristics become more vertical and the converter can be regarded to a great extent as a source of current, stable at operation even at short circuit. Besides, an area of operation is noticeable, in which 0 *U*′ > 1 .

At the main operation mode the commutations in the rectifier (stages 4 and 8) must always end before the commutations in the inverter have started. This is guaranteed if the following condition is fulfilled:

$$\mathbf{x}\_1 \le \mathbf{x}\_2 \tag{48}$$

In order to enable natural switching of the controllable switches at zero voltage (ZVS), the commutations in the inverter (stages 2 and 6) should always end before the current in the resonant circuit becomes zero. This is guaranteed if the following condition is fulfilled:

$$
\infty\_3 \le \mathcal{U}'\_{\text{Cu}} \tag{49}
$$

If the condition (49) is not fulfilled, then switching a pair of controllable switches off does not lead to natural switching the other pair of controllable switches on at zero voltage and then the converter stops working. It should be emphasized that these commutation mistakes do not lead to emergency modes and they are not dangerous to the converter. When it "misses", all the semiconductor switches stop conducting and the converter just stops working. This is one of the big advantages of the resonant converters working at frequencies higher than the resonant one.

On the basis of the analysis results, equations for the output characteristics are obtained individually for both the main and the boundary modes of the converter operation. Besides,

> 0 20 2 *U I aU Cm* π

ν

consecutive substitution of expressions (12)÷(18) in equations (23)÷(26) as well as of expressions (23)÷(26) in equation (22), an expression of the kind *U fI a a* 0 0 12

obtained. Its solution provides with the possibility to build the output characteristics of the converter in relative units at the main mode of operation and at regulation by means of changing the operating frequency. The output characteristics of the converter respectively for ν=1.2; 1.3; 1.4; 1.5; 1.8; 2.5; 3.0; 3.3165; 3,6 and *а*1=0.1; *а*2=0.2 as well as for ν=1.2; 1.3; 1.4;

The comparison of these characteristics to the known ones from (Al Haddad et al., 1986; Cheron, 1989) shows the entire influence of the capacitors *С<sup>S</sup>* и *С*0. It can be seen that the output characteristics become more vertical and the converter can be regarded to a great extent as a source of current, stable at operation even at short circuit. Besides, an area of

At the main operation mode the commutations in the rectifier (stages 4 and 8) must always end before the commutations in the inverter have started. This is guaranteed if the following

In order to enable natural switching of the controllable switches at zero voltage (ZVS), the commutations in the inverter (stages 2 and 6) should always end before the current in the resonant circuit becomes zero. This is guaranteed if the following condition is fulfilled:

If the condition (49) is not fulfilled, then switching a pair of controllable switches off does not lead to natural switching the other pair of controllable switches on at zero voltage and then the converter stops working. It should be emphasized that these commutation mistakes do not lead to emergency modes and they are not dangerous to the converter. When it "misses", all the semiconductor switches stop conducting and the converter just stops working. This is one of the big advantages of the resonant converters working at frequencies

′ , 0*I*′ , ν and 2 *a*

′ ′′ = + (47)

1 2 *x x* ≤ (48)

*x U* <sup>3</sup> *Cm* ≤ ′ (49)

′ ′ = ( ) ,, , ν

is

′ is eliminated from the equations (12)÷(18). After

expressions for the boundary curves of the separate modes are also derived.

′ is expressed in function of *U*<sup>0</sup>

**4.1 Output characteristics and boundary curves at the main operation mode** 

**4. Output characteristics and boundary curves** 

1.5; 1.6; 1.8 and *а*1=0.1; *а*2=1.0 are shown in fig.7-a and 7-b.

From equation (21) *UCm*

By means of expression (47) *UCm*

operation is noticeable, in which 0 *U*′ > 1 .

condition is fulfilled:

higher than the resonant one.

Fig. 7. Output characteristics of the LCC transistor DC/DC converter

Equations (12), (14) and (47) are substituted in condition (48), while equations (16) and (47) are substituted in condition (49). Then the inequalities (48) and (49) obtain the form:

$$I\_0' \geq \frac{2\nu}{\pi} \cdot \frac{a\_1 + a\_2 \mathcal{U}\_0'}{1 + \mathcal{U}\_0'} \tag{50}$$

$$I\_0' \geq \frac{2\nu}{\pi} \cdot \frac{a\_1 - a\_2 \mathcal{U}\_0'}{1 - \mathcal{U}\_0'} \tag{51}$$

Inequalities (50) и (51) enable with the possibility to draw the boundary curve *A* between the main and the medial modes of operation of the converter, as well as the border of the natural commutation – curve *L*3 (fig. 7-а) or curve *L*4 (fig. 7-b) in the plane of the output characteristics. It can be seen that the area of the main operation mode of the converter is limited within the boundary curves *А* and *L*3 or *L*4. The bigger the capacity of the capacitors

Study of LCC Resonant Transistor DC / DC Converter with Capacitive Output Filter 125

0 0

After substitution of equations (39), (41) и (47) in the inequality (52), the mentioned above

0

2

ν

′ <sup>−</sup> ′ ≤ ⋅

π

20 1

*U*

1 *aU a <sup>I</sup>*

Condition (53) gives the possibility to define the area of the boundary operation mode of the converter in the plane of the output characteristics (fig. 7-а and fig. 7-b). It is limited between the y-axis (the ordinate) and the boundary curve *B.* It can be seen that the converter stays absolutely fit for work at high-Ohm loads, including at a no-load mode. It is due mainly to the capacitor *С*0. With the increase in its capacity (increase of *a*2) the area of the boundary

At this operation mode, the diodes of the rectifier start conducting during the commutation in the inverter. The equivalent circuits, corresponding to this mode for a cycle, are shown in

fig.9. In this case, the sinusoidal quantities have four different angular frequencies:

<sup>0</sup> <sup>3</sup> ω 1 = *LCE* ′′′ , where *C CC C CC CC C C ES S S* 3 0 00 = ++ ( ) , for stages 2 and 6.

Fig. 9. Equivalent circuits at the medial operation mode of the converter

and the smaller capacity of the capacitor *C*0.

Therefore, the analysis of the medial operation mode is considerably more complex. The area in the plane of the output characteristics, within which this mode appears, however, is completely defined by the boundary curves *A* and *B* for the main and the boundary modes respectively. Having in mind the monotonous character of the output characteristics for the other two modes, their building for the mode under consideration is possible through linear interpolation. It is shown in fig. 7-а for ν = 3.0; 3.3165 as well as in fig. 7-b for ν=1.5; 1.6; 1.8. The larger area of this mode corresponds to the higher capacity of the snubber capacitors *CS*

0

condition obtains the form:

operation mode can also be increased.

ω<sup>0</sup> = 1 *LC* for stages 4 and 8;

**5. Medial operation mode of the converter** 

<sup>0</sup> <sup>1</sup> ω 1 = *LCE* ′ , where *C CC C C ES S* <sup>1</sup> = + ( ) , for stages 3 and 7; <sup>0</sup> <sup>2</sup> ω 1 = *LCE* ′′ , where *C CC C C <sup>E</sup>*20 0 = + ( ) , for stages 1 and 5;

4 3 *x x* ≥ (52)

<sup>+</sup> ′ (53)

*СS* and *С*0, the smaller this area is. However, the increase of the snubber capacitors leads to a decrease in the commutation losses in the transistors as well as to limiting the electromagnetic interferences in the converter.

The expression (51) defines the borders, beyond which the converter stops working because of the breakage in the conditions for natural switching the controllable switches on at zero voltage (ZVS). Exemplary boundary curves have been drawn in the plane of the output characteristics (fig.8) at *а*1=0.10. Four values have been chosen for the other parameter: *а*2 = 0.05; 0.1; 0.2 and 1.0. When the capacity of the capacitor *С*0 is smaller or equal to that of the snubber capacitors *CS* ( ) 1 2 *a a* ≥ , then the converter is fit for work in the area between the curve *L*1 or *L*2 and the x-axis (the abscissa). Only the main operation mode of the converter is possible in this area. The increase in the load resistance or in the operating frequency leads to stopping the operation of the converter before it has accomplished a transition towards the medial and the boundary modes of work.

When *C*0 has a higher value than the value of *CS* ( ) 1 2 *a a* < then the boundary curve of the area of converter operation with ZVS is displaced upward (curve *L*3 or *L*4). It is possible now to achieve even a no-load mode.

Fig. 8. Borders of the converter operation capability

#### **4.2 Output characteristics and boundary curves at the boundary operation mode**

Applying expression (47) for equations (35)÷(42) *UCm*′ is eliminated. After that, by a consecutive substitution of expressions (35)÷(42) in equations (43)÷(45) as well as of expressions (43)÷(46) in equation (22), a dependence of the kind *U*0 0 12 ′ ′ = *f* ( ) *I aa* ,, , ν is obtained. Its solving enables with a possibility to build the outer (output) characteristics of the converter in relative units at the boundary operation mode under consideration and at regulation by changing the operating frequency. Such characteristics are shown in fig. 7-а for ν = 3.0; 3.3165; 3.6 and *а*1=0.1; *а*2=0.2 and in fig. 7-b for ν =1.5; 1.6; 1.8 and *а*1=0.1; *а*2=1.0. At the boundary operation mode, the diodes of the rectifier have to start conducting after opening the freewheeling diodes of the inverter. This is guaranteed if the following condition is fulfilled:

$$\mathbf{x}\_4^0 \succeq \mathbf{x}\_3^0 \tag{52}$$

After substitution of equations (39), (41) и (47) in the inequality (52), the mentioned above condition obtains the form:

$$\mathcal{U}\_0' \le \frac{2\nu}{\pi} \cdot \frac{a\_2 \mathcal{U}\_0' - a\_1}{1 + \mathcal{U}\_0'} \tag{53}$$

Condition (53) gives the possibility to define the area of the boundary operation mode of the converter in the plane of the output characteristics (fig. 7-а and fig. 7-b). It is limited between the y-axis (the ordinate) and the boundary curve *B.* It can be seen that the converter stays absolutely fit for work at high-Ohm loads, including at a no-load mode. It is due mainly to the capacitor *С*0. With the increase in its capacity (increase of *a*2) the area of the boundary operation mode can also be increased.
