Half-Bridge Bidirectional Buck Boost Converter with an Opportunity for Zero Voltage Switching Franz M. Hanser Institute for Energy Systems The University of Edinburgh Edinburgh, UK [email protected] Abstract— A study of critical information (maximum current and voltage estimation for the used components, mean values for semiconductor parts and an assessment of expected power losses) for a half-bridge bidirectional buck boost DC-DC converter equipped with an LC filter is carried out. A practical model from this converter is derived and a transfer function from the duty cycle and from the voltages (input to the output) is calculated via a linearization method. A signal flow graph of the converter can be obtained. To decrease the power losses, the concept of zero voltage switching is used and explained. Keywords— Battery chargers, push-pull converter, buck boost DC-DC converter.

used as a starting point. If the converter operates in stationary (steady) state the voltage-time-areas of the inductors are equal (in terms of absolute values). This means that the average of the voltages across the inductors is zero and the graphs as shown in Fig. 2 can be drawn. It is also assumed that the capacitors are large and the voltages are constant for at least one commutation cycle. Fig. 2 (a, b) presents the voltage across and the current through the impedance L1. Of course, this current increase to time. This depends on the value of the impedance L1 and the voltage U1. Fig. 2 (c, d) presents the voltage across and the current through L2, which only acts as a filter connected to the output.

I. INTRODUCTION The half-bridge bidirectional converter (Fig.1) presented in this paper is an inverting step-down or step-up (buckboost) converter from the input to the output voltages (U1/ U2). Its function is described here for the first case. A halfbridge in push-pull mode is used for numerous converter structures. Developments are shown in [1] and a review of existing structures is given in [2] and [3]. This assembly is analyzed as a PWM structure. Hard switching technology is applied. Dimensioning of parts, a linearized model (and also a non-linear model) of the transfer function of this converter feeding a resistive load is derived. The output LC filter was taken into consideration for the transfer function but is not necessary for the main function of the converter. The capacitor C2 (of the LC output filter) is connected directly from input to the output.

Fig. 2. Voltages across (a, b) and current (c, d) through the inductors L1 and L2.

For this reason, the voltage across L2 is nearly zero and hence the current through L2 is constant and equals the output current. According to the previously introduced equality formula of the voltage-time-areas in a non-transient state, the transformation relationship between U2 (output voltage) as function of U1 (input voltage) and d (duty ratio) from Fig. 2 (a) can be derived as follows: d ⋅ U1 = (1 − d ) ⋅ U 2

The voltage transformation factor M can be written as:

Fig.1. Converter structure.

II. BASIC ANALYZES Idealized components are used for the basic analysis of the inverter (no parasitic impedances and switching losses are neglegted). The converter is treated in continuous mode and in steady state. The voltages across the inductors are

U2 d =M = U1 1− d

978-1-4799-7993-6/15/$31.00 ©2015 IEEE

with d ∈ [0,1)

(2)

Fig. 3 presents this factor M (rate of Voltageoutput / Voltageinput) versus the duty cycle d. This converter can be treated as an inverting step-down (buck) and also as a stepup (boost) converter. 4 3,5 3 2,5 M 2 1,5 1 0,5 0

value of the current flowing through the component is important to obtain power losses during the on-state. The ripple current is not considered for the sake of simplicity. For the switch S1 we get approximately:

Function

(I

S1

= I D 2 = d ⋅ I L1 )

I S1, rms = I L1 d

(5.a)

and for the diode D2:

0

0,2

0,4

0,6

0,8

1

and:

d Fig. 3. Transformation factor M (U1/U2) versus duty cycle d.

⎛ ⎞ ⎜⎜ I D 2 = (1 − d ) ⋅ I L1 ⎟⎟ ⎝ ⎠

(5.b)

I D 2,rms = I L1 1 − d

(5.c)

Fig. 4. Currents through capacitors iC1 (a) and iC2 (b) versus time.

Fig. 4 shows the current through the capacitors C1 and C2 in dependence of the duty cycle. The next step is to calculate dependences between load current, input current, and currents through the two inductors. This is especially important for getting dimensioning information. By setting input energy equal to the output energy, the current through L1 can be calculated as follows: U 1 ⋅ I L 1 ⋅ d ⋅ T = U 2 ⋅ I LA ⋅ T with

d ⋅ T ⋅ U 1 ⋅ I L1 I L1 =

d ⋅U 1 1− d d = U1 ⋅ ⋅ T ⋅ I LA 1− d

U2 =

(3)

1 ⋅ I LA 1− d

The currents through and the voltages across the active and the passive switches (Fig.1) are calculated in the next step. The semiconductor components can be chosen after calculation of the maximum ratings (US1max, US2max, IS1max and IS2max). Fig. 1 shows that the current through the semiconductor components is based on the inductor current iL1 (current through S1 during Ton and through D2 during time Toff) as can also be seen in Fig. 5. The maximum currents for the semiconductor components can be calculated as: I S 1 max = I D 2 max = IˆL1

(4)

The mean and the rms values are especially important to obtain the losses during the on-state. In this case only a pure resistor is assumed for the on state behavior. Only the rms

Fig. 5. Voltage drop across (a, b); current flowing through semiconductor components (c, d).

Fig. 5. (c, d.) presents the current flowing through active switches in the circuit. This input current, which is a pulse current, is the same as the current through S1 => i1 = iS1. The voltage across the semiconductor components are especially important to get dimensioning information of this circuit. Based on the on-state of this circuit (S1 closed and S2 open), the voltage drop across component S2 can be obtained as:

U S 2,max = U 2 + U1

(6)

The same result occurs for the other state (S1 open and S2 closed):

U S1,max = U 1 + U 2

(7)

Hence, the value for the voltage for the semiconductor components must be chosen higher than the voltage occurring on the input. Fig. 5 (a, b). shows this relation graphically. Continuous operation mode and synchronous rectification are used to calculate an approximation of the efficiency of this bidirectional DC-DC converter. Based on

rD as the on-resistance of the semiconductor components the conduction losses are calculate as: 2

2

PVS 1 = I S 1,rms ⋅ rds1 = d ⋅ I L1 ⋅ rDS 1 PVS 2 = I D 2,rms ⋅ rds 2 = (1 − d ) ⋅ I L1 ⋅ rDS 2 2

2

(8)

Assuming that (rDS1 = rDS2 = rD ) the efficiency of this circuit can roughly be calculated as: ( I La ⋅ U 2 ) η= = PV + ( I La ⋅ U 2 ) with I L1 =

I La 1− d

( I La ⋅ R L ⋅ I La ) ( I La ⋅ R L ⋅ I La ) + I La

η≈

2

⎛ 1 ⎞ ⋅ rD ⋅ ⎜ ⎟ ⎝1− d ⎠

=η =

RL ⎛ 1 ⎞ R L + rD ⋅ ⎜ ⎟ ⎝1− d ⎠

( I La ⋅ U 2 ) ≈ PV + ( I La ⋅ U 2 )

( I La ⋅ U 2 ) ≈ ≈ 1 1 2 2 ( I La ⋅ U 2 ) + I La ⋅ rDS 1 ⋅ d ⋅ + I La ⋅ rDS 2 ⋅ (1 − d ) ⋅ 1− d 1− d with rDS 1 = rDS 2 = rD

(9)

( I La ⋅ U 2 ) ⎛ I 2 ⋅r ⎞ ( I La ⋅ U 2 ) + ⎜⎜ La D ⎟⎟ ⎝ 1− d ⎠ with U 2 = RL ⋅ I La

III. DERIVATION OF THE DYNAMIC MODEL To shorten the calculation path and for simplification the following assumptions have been made: switches are assumed to be ideal and the influence of serial or parallel connected parasitic impedances (resistances or reactances) to any component of the circuit are omitted. For any switch mode converter operating in continuous mode, two describing differential equation systems can be developed. The first one for the closed active switch. After that the second one for the open switch can be calculated. The two systems can be combined. The next step is to linearize this combined system of nonlinear differential equations. This is done at its steady state operating point and therefore linear control theory [2] can be applied. During the interval Ton, the following equations hold:

duC1 −i L 2 = dt C1

diL1 − uC1 = dt L1

(10.a) (10.b) (10.c)

(11.a)

diL 2 uC1 + U1 − uC 2 (11.b) = dt L2 duC1 iL1 − iL 2 (11.c) = dt C1 duC 2 iL 2 + (U1 − uC 2 ) / RL (11.d) = dt C2 Equations (10) and (11) describe the system behavior. The obtained system time constants have to be large compared to the commutation cycle. The two sets of equations can then be combined by a weighed sum of the differential equations [2]. By defining the duty ratio as: d=

≈

diL1 U1 = dt L1 diL 2 uC1 + U1 − uC 2 = dt L2

u iL1 + iL 2 − C 2 duC 2 RL (10.d) = dt C2 During the interval Toff, the corresponding equations are:

Ton S1

(12)

T

If the two sets are weighted with (12) and combined this leads to: ⎡ ⎢ 0 ⎛ i L1 ⎞ ⎢ ⎜ ⎟ ⎢ 0 d ⎜ iL2 ⎟ ⎢ = dt ⎜ u C1 ⎟ ⎢⎢1 − d ⎜ ⎟ ⎜ u ⎟ ⎢ C1 ⎝ C2 ⎠ ⎢ ⎢ 0 ⎣

0 0 1 C1 1 C2

−

d −1 L1 1 L2 0 0

⎤ ⎛ d ⎥ ⎥ ⎛ i ⎞ ⎜ L1 1 ⎥ ⎜ L1 ⎟ ⎜ − ⎜ 1 L2 ⎥ ⎜ i L 2 ⎟ ⎜ ⋅ + L2 ⎥ ⎜ ⎟ 0 ⎥ ⎜ u C1 ⎟ ⎜ 0 ⎜ ⎜ ⎟ ⎥ ⎝ uC 2 ⎠ ⎜ 1 1 ⎥ ⎜ R ⋅C − ⎝ 2 R ⋅ C 2 ⎥⎦ 0

⎞ ⎟ ⎟ ⎟ ⎟ ⋅ u1 ⎟ ⎟ ⎟ ⎟ ⎠

(13) which is a nonlinear differential equation with respect to the duty cycle d. This calculated system of equations describes, the dynamic behavior of the idealized converter correctly in the average. A general view of the dynamic behavior of this assembly can be obtained quickly. The superimposed ripple is not important for qualifying the dynamic behavior. This noticeable ripple appears in the coils. However, the voltage ripple occurring across the capacitors is small. For instance in a switched mode power supply (SMPS) a smooth DC voltage is desired and sufficiently large capacitors need to be chosen. The converter model might be used as a large-signal model as well because no linearization regarding the signal values is calculated. IV. MODEL FOR CONTROL

A. Linearization Equation (13) represents the dynamic behavior of the converter. It is a nonlinear weighted matrix differential

equation with respect to the duty cycle. A linearization is necessary to apply linear control theory. Capital letters are used to introduce operating point values where as small letters are used for disturbance around the converter operating point: ∧

a bidirectional converter. It is suitable for automotive applications (e.g. e-bikes and cars), DC-DC coupling and battery chargers. The voltages are impressed to the circuit thus a control variable has to be chosen that is a current. For example, the current through the inductor L1 which is calculated as follows:

iL1 = I L10 + i L1

I L 2 (s) D(s) U1 ( s ) = 0

(18)

s ⋅ I L 10 10 ⋅ ( D 0 − 1 ) − − L1 ⋅ L 2 ⋅ C 1 C1 ⋅ L2 = ⎡ ( D 0 − 1 )2 ⎤ 1 s3 + s ⋅ ⎢ + ⎥ L C L C ⋅ ⋅ 1 1 2 2 ⎣ ⎦

(19)

GIL 2 D ( s ) =

∧

iL 2 = I L 20 + i L 2 ∧

uC1 = U C10 + u C1

U C 2 ( s) =0

(14)

∧

uC 2 = U C 20 + u C 2

I L 2 (s ) D (s )

U

C. Signal flow chart The transfer function can also be determined by using the signal flow graph. From equation (17) and with U1(s) = 0 we can write the matrix form:

∧

u1 = U10 + u1 ∧

d = D0 + d The linearized converter model is calculated based on: D0 −1 U10 ⎤ ⎤ ⎡ D0 ⎡ 0 0 ⎥ ⎢ L ⎢ 0 L1 L1 ⎥ ∧ ⎥ ⎥⎛ ⎞ ⎢ 1 ⎛ ⎞ ⎢ i i 1 1 ⎥ ⎜ L1 ⎟ ⎢ 1 ⎜ L1 ⎟ ⎢ ⎥ ∧ (15) ∧ ∧ 0 0 0 − ⎥ ⎛⎜u1 ⎞⎟ d ⎜ i L2 ⎟ ⎢ L2 L2 ⎥ ⎜ i L2 ⎟ ⎢ L2 ⎜ ∧ ⎟ =⎢ ⎥⋅ ∧ ⎥ ⋅⎜ ∧ ⎟ + ⎢ D I 1 − − 1 dt⎜uC1 ⎟ ⎢ 0 L10 ⎜ d ⎟ ⎥⎝ ⎠ ⎥ ⎜uC1 ⎟ ⎢ 0 0 0 − ⎜⎜ ∧ ⎟⎟ ⎢ C1 C1 C1 ⎥ ⎥ ⎜⎜ ∧ ⎟⎟ ⎢ u u C 2 2 C ⎝ ⎠ ⎢ ⎥ 1 1 ⎥⎝ ⎠ ⎢ 1 0 − 0 ⎥ ⎥ ⎢ ⎢ 0 C R C R C ⋅ 2 2⎦ ⎦ ⎣ L 2 ⎣ ∧

⎛ I L1 ( s) ⎞ ⎡ 0 ⎟ ⎢ ⎜ ⎜ I L 2 (s) ⎟ ⎢ 0 [ s] ⋅ ⎜ = U ( s) ⎟ ⎢ A ⎜ C1 ⎟ ⎢ 31 ⎜U ( s ) ⎟ ⎝ C2 ⎠ ⎣ 0

0 0

A13 A23

A32

0

A42

0

0 ⎤ ⎛ I L1 ( s) ⎞ ⎛ B12 ⎞ ⎟ ⎜ ⎜ ⎟ A24 ⎥⎥ ⎜ I L 2 ( s ) ⎟ ⎜ 0 ⎟ + ⋅ D( s) ⋅ 0 ⎥ ⎜ U C1 ( s) ⎟ ⎜ B32 ⎟ ⎟ ⎜ ⎜ ⎟ ⎥ A44 ⎦ ⎜⎝U C 2 ( s) ⎟⎠ ⎜⎝ 0 ⎟⎠

(20) with [s] representing the identity matrix with s in the diagonal. Considering U2 =UC2 the signal flow graph can be obtained from Fig. 6:

B. Transfer Functions Equation (15) describes the linearised converter and its important transfer function: GIL 2 D ( s) =

I L 2 ( s) D(s)

(16)

Equation (16) is most important for the design of the converter controller considering the input voltage which is another input variable cannot be used for control. This is due to the fact that U1 is impressed to the circuit. Presumably, the input voltage acts as a disturbance input to the control system. A fourth order transfer function can now be written caused by the four storage elements in this circuit. Rewriting the state equations (15) by applying the Laplace transformation leads to: ⎡ 0 ⎢ s ⎢ ⎢ 0 s ⎢ ⎢ D −1 1 ⎢ 0 C1 ⎢ C1 1 ⎢ − ⎢ 0 C 2 ⎣

1 − D0 L1 1 − L2 s 0

⎡ D0 ⎤ ⎢ L ⎥ ⎥ ⎛ I (s ) ⎞ ⎢ 1 1 ⎥ ⎜ L1 ⎟ ⎢ 1 L2 ⎥ ⎜ I L 2 (s ) ⎟ ⎢ L2 ⎥.⎜ U (s ) ⎟ = ⎢ 0 ⎥ ⎜ C1 ⎟ ⎢ 0 ⎥ ⎜⎝U C 2 (s )⎟⎠ ⎢ 1 ⎥ ⎢ 1 s+ ⎢R C R ⋅ C2 ⎥⎦ ⎣ L 2 0

U10 ⎤ L1 ⎥ ⎥ (17) 0 ⎥ ⎥ ⎛U1 (s )⎞ ⎟ ⋅⎜ − I L10 ⎥ ⎜⎝ D(s ) ⎟⎠ ⎥ C1 ⎥ ⎥ 0 ⎥ ⎦

and for U1(s) = 0 we get the transfer function written in the appendix. It can be seen that the converter can be used as

Fig. 6. Signal flow graph.

With Mason’s law, it is very simple to calculate the transfer ratio. The forward paths (a path is defined as a set of consecutive, codirectional branches as long as no node is encountered more than once as we move in the graph from the independent variable to the output or state variable under quest, the value is the product of all branches from the starting node to the end node) are: 1 (21.a) F1 = 2 ⋅ A23 ⋅ A31 ⋅ A42 ⋅ B12 s

F2 =

1 ⋅ A23 ⋅ A42 ⋅ B32 s

(21.b)

the loops (a loop is the product of the branches encountered by making a round trip if we move from a node backwards to the original node in direction of the arrow):

1 L1 = ⋅ A44 s

1 L2 = 2 ⋅ A13 A31 s 1 ⋅ A23 A32 s2 1 L4 = 2 ⋅ A24 A42 s L3 =

(22. a) (22. b) (22. c) (22. d)

L1 and L2, L2 and L4, L1 and L3, are non-touching loops. Hence, we get the sum of all second order loops (L1 ⋅ L2 + L1 ⋅ L3 + L2 ⋅ L4 ) and with Mason’s rule we get: F1 + F2 (23) GU 2 D ( s ) = 1 − (L1 + L2 + L3 + L4 ) + (L1 ⋅ L2 + L1 ⋅ L3 + L2 ⋅ L4 )

The result is shown in the appendix (A). The calculation of the transfer function is an alternative way to the solution of the matrix equation (17).

diL1 U1 = dt L1 diL 2 =0 dt

(24.a) (24.b)

C. State 2 S1 opens at t1 and the current commutates into the capacitors C11 and C22 Fig. 8 (b) charging the upper capacitor C11 and discharging the lower capacitor C22. The voltage across C1 and C2 is nearly constant and the current through L1 is constant. Therefore only one state variable is necessary to describe the transient process. If C11 and C22 are chosen equal the current through C11 is given by:

iL1 2 For the transient part of the voltage we can write: iC11 = iC 22 =

C11 ⋅

du C11 i L1 i = → u C11 = L1 ⋅ t dt 2 C11 ⋅ 2

(25)

(26)

The voltage across C11 (uC11) increases until UC2 is reached and the diode of S2 starts to conduct the current at time t2 when D2 turns on both capacitors C11, C22 are now effect less. The whole process can be obtained from Fig. 9.

V. LOW LOSS SWITCHING

A. Principle A reduction in losses can be achieved by applying a zero voltage switching concept (ZVS) to the converter. For this concept the converter from Fig. 1 is extended. The used halfbridge module is extended with two capacitors C11 and C22 as can be seen in Fig. 7. For the analysis of the switching behaviour as can be obtained from Fig. 8, we transform the filter capacitors into voltage sources (the voltage across C1 and C2 is nearly constant during one period of the switching frequency) with zero voltage across C11 and UC2 across C22 for S1 closed. Further for S2 closed and S1 open we have zero voltage across C22 and UC2 across C11. The transient behaviour for the voltages of C11 and C22 can be described by a differential equation. The differential equation involves only C11, assuming that C11 is equal to C22.

Fig.8. Equivalent circuits for state 1 (a) and 2 (b).

tr iL1

t u C11 U1 + U2

t u C22

U1 + U2

Fig.7. Converter circuit with ZVS opportunity.

B. State 1 When the upper switch S1 is closed as seen in Fig. 8 (a), the current through the inductance L1 changes according to:

t0;

t1; t2;

I

t

t3; t4;

II

III

IV

Fig. 9. Current through L1, and voltages across the snubber capacitors.

D. State 3 Now, no losses occur by closing S2. The current through the inductor (iL1) decreases, reaches zero and a negative value can be obtained starting at t3. The energy stored in the inductor must be so high that if S2 opens at t3, capacitor C22 again can be charged to a value reaching UC2 and C11 is discharged to zero voltage at t4. S1 can be closed again without losses. The transient process between t3 and t4 can be described as follows: ⎡ 0 d ⎛ i L1 ⎞ ⎢ ⎜ ⎟=⎢ dt ⎜⎝ u C11 ⎟⎠ ⎢ 1 ⎢⎣ 2 ⋅ C11

−

1⎤ ⎡1⎤ L1 ⎥ ⎛ i L1 ⎞ ⎢ ⎥ ⎟⎟ + L1 ⋅ u1 ⎥ ⋅ ⎜⎜ ⎢ ⎥ 0 ⎥ ⎝ u C11 ⎠ ⎣ 0 ⎦ ⎥⎦

2

(C + C 22 ) ⋅ (U1 + U 2 ) 2 L1 ⋅ I tr = 11 2 2 C11 + C 22 I tr = ⋅ (U1 + U 2 ) L1

WL1 =

⇒ tr =

I tr 2 ⋅U 2

⋅ L1 = (C11 + C 22 ) ⋅ L1 ⋅

U1 + U 2 2 ⋅U2

(30)

This is the first possible moment for switching on S2. Now, the current iL1(t) through the inductance L1 can be calculated as follows:

(27) iL1 (t ) =

U 2 ⋅ 2 L1 ⋅ C11 L1

⎛ ⎞ ⎛ ⎞ (C11 + C 22 ) ⋅ L1 ⋅ (U 1 + U 2 ) 1 1 ⋅ sin⎜ ⋅t ⎟ − ⋅ cos⎜ ⋅t⎟ ⎜ 2L ⋅ C ⎟ ⎜ 2L ⋅ C ⎟ ⋅ L 2 1 1 11 1 11 ⎝ ⎠ ⎝ ⎠

(31)

we get for the charging process starting from zero: uC11 (t ) = U1 +

⎛ ⎞ ⎛ ⎞ iL1 (0) ⋅ 2 ⋅ L1 ⋅ C11 1 1 ⋅ sin⎜ ⋅ t ⎟ + (uC11 (0) − U1 ) ⋅ cos⎜ ⋅t⎟ ⎜ 2⋅ L ⋅C ⎟ ⎜ 2⋅ L ⋅C ⎟ 2 ⋅ C11 1 11 1 11 ⎝ ⎠ ⎝ ⎠

(28. a) iL1 (t ) =

(U1 − uC11 (0) ) ⋅ L1

2 L1 ⋅ C11

⎛ ⎞ ⎛ ⎞ 1 1 ⋅ sin ⎜ ⋅ t ⎟ + iL1 (0) ⋅ cos⎜ ⋅t⎟ ⎜ 2L ⋅ C ⎟ ⎜ 2L ⋅ C ⎟ 1 11 1 11 ⎝ ⎠ ⎝ ⎠

(28. b) and the initial conditions: iL1 (0) < 0; u c11 (0) = U 1 + U 2

And from t2 to t3: di L1 (0) U U = − 2 ; iL1 (0) = − 2 ⋅ t r dt L1 L1

where t r is the " reverse time" (Fig. 9) : iL1 (tr ) = − I tr

(29)

VI. CONCLUSION A standout amongst the aspects of the investigated converter is the prefabricated half-bridge module and a halfbridge driver which can easily be implemented into any circuit or assembly. This is particularly helpful for higher current levels. The converter is especially suitable for battery chargers solar power applications and electric drives which are increasingly popular nowadays. However, another aspect is that the converter can easily be used for power control and signal applications such as coupling of two voltage links, in uninterruptible power supplies and two quadrant choppers for DC drives. The use of the zero voltage switching concept is another interesting aspect for further developments because of the low losses. This concept offers an opportunity for the future to decrease power losses for, instance in photo voltaic applications or DC drives. It is essential that the control is done in pseudo discontinuous mode or by variable frequency [4]. REFERENCES [1] [2]

The initial condition iL1(tr) can be calculated using the energy stored in the inductance:

[3] [4]

F.A. Himmelstoss, and F.C. Zach: “Bidirectional Converters”, Austrian patent, filed 1992.3.13, AT 399 625 B N. Mohan, T. Undeland and W.P. Robbins, Power Electronics Converters, Applications, and Design, 3rd ed., New York: John Wiley & Sons, 2003. F. Himmelstoss: “Analysis and Comparison of Half-bridge Bidirectional Dc-Dc Converters”, IEEE PESC’94, pp.922-928. F.A. Himmelstoss: “Control Concept for the Reduction of Switching Losses of Bidirectional Converters”, Austrian patent, filed 2001.07.16. AT 412371B.

APPENDIX

(A)

used as a starting point. If the converter operates in stationary (steady) state the voltage-time-areas of the inductors are equal (in terms of absolute values). This means that the average of the voltages across the inductors is zero and the graphs as shown in Fig. 2 can be drawn. It is also assumed that the capacitors are large and the voltages are constant for at least one commutation cycle. Fig. 2 (a, b) presents the voltage across and the current through the impedance L1. Of course, this current increase to time. This depends on the value of the impedance L1 and the voltage U1. Fig. 2 (c, d) presents the voltage across and the current through L2, which only acts as a filter connected to the output.

I. INTRODUCTION The half-bridge bidirectional converter (Fig.1) presented in this paper is an inverting step-down or step-up (buckboost) converter from the input to the output voltages (U1/ U2). Its function is described here for the first case. A halfbridge in push-pull mode is used for numerous converter structures. Developments are shown in [1] and a review of existing structures is given in [2] and [3]. This assembly is analyzed as a PWM structure. Hard switching technology is applied. Dimensioning of parts, a linearized model (and also a non-linear model) of the transfer function of this converter feeding a resistive load is derived. The output LC filter was taken into consideration for the transfer function but is not necessary for the main function of the converter. The capacitor C2 (of the LC output filter) is connected directly from input to the output.

Fig. 2. Voltages across (a, b) and current (c, d) through the inductors L1 and L2.

For this reason, the voltage across L2 is nearly zero and hence the current through L2 is constant and equals the output current. According to the previously introduced equality formula of the voltage-time-areas in a non-transient state, the transformation relationship between U2 (output voltage) as function of U1 (input voltage) and d (duty ratio) from Fig. 2 (a) can be derived as follows: d ⋅ U1 = (1 − d ) ⋅ U 2

The voltage transformation factor M can be written as:

Fig.1. Converter structure.

II. BASIC ANALYZES Idealized components are used for the basic analysis of the inverter (no parasitic impedances and switching losses are neglegted). The converter is treated in continuous mode and in steady state. The voltages across the inductors are

U2 d =M = U1 1− d

978-1-4799-7993-6/15/$31.00 ©2015 IEEE

with d ∈ [0,1)

(2)

Fig. 3 presents this factor M (rate of Voltageoutput / Voltageinput) versus the duty cycle d. This converter can be treated as an inverting step-down (buck) and also as a stepup (boost) converter. 4 3,5 3 2,5 M 2 1,5 1 0,5 0

value of the current flowing through the component is important to obtain power losses during the on-state. The ripple current is not considered for the sake of simplicity. For the switch S1 we get approximately:

Function

(I

S1

= I D 2 = d ⋅ I L1 )

I S1, rms = I L1 d

(5.a)

and for the diode D2:

0

0,2

0,4

0,6

0,8

1

and:

d Fig. 3. Transformation factor M (U1/U2) versus duty cycle d.

⎛ ⎞ ⎜⎜ I D 2 = (1 − d ) ⋅ I L1 ⎟⎟ ⎝ ⎠

(5.b)

I D 2,rms = I L1 1 − d

(5.c)

Fig. 4. Currents through capacitors iC1 (a) and iC2 (b) versus time.

Fig. 4 shows the current through the capacitors C1 and C2 in dependence of the duty cycle. The next step is to calculate dependences between load current, input current, and currents through the two inductors. This is especially important for getting dimensioning information. By setting input energy equal to the output energy, the current through L1 can be calculated as follows: U 1 ⋅ I L 1 ⋅ d ⋅ T = U 2 ⋅ I LA ⋅ T with

d ⋅ T ⋅ U 1 ⋅ I L1 I L1 =

d ⋅U 1 1− d d = U1 ⋅ ⋅ T ⋅ I LA 1− d

U2 =

(3)

1 ⋅ I LA 1− d

The currents through and the voltages across the active and the passive switches (Fig.1) are calculated in the next step. The semiconductor components can be chosen after calculation of the maximum ratings (US1max, US2max, IS1max and IS2max). Fig. 1 shows that the current through the semiconductor components is based on the inductor current iL1 (current through S1 during Ton and through D2 during time Toff) as can also be seen in Fig. 5. The maximum currents for the semiconductor components can be calculated as: I S 1 max = I D 2 max = IˆL1

(4)

The mean and the rms values are especially important to obtain the losses during the on-state. In this case only a pure resistor is assumed for the on state behavior. Only the rms

Fig. 5. Voltage drop across (a, b); current flowing through semiconductor components (c, d).

Fig. 5. (c, d.) presents the current flowing through active switches in the circuit. This input current, which is a pulse current, is the same as the current through S1 => i1 = iS1. The voltage across the semiconductor components are especially important to get dimensioning information of this circuit. Based on the on-state of this circuit (S1 closed and S2 open), the voltage drop across component S2 can be obtained as:

U S 2,max = U 2 + U1

(6)

The same result occurs for the other state (S1 open and S2 closed):

U S1,max = U 1 + U 2

(7)

Hence, the value for the voltage for the semiconductor components must be chosen higher than the voltage occurring on the input. Fig. 5 (a, b). shows this relation graphically. Continuous operation mode and synchronous rectification are used to calculate an approximation of the efficiency of this bidirectional DC-DC converter. Based on

rD as the on-resistance of the semiconductor components the conduction losses are calculate as: 2

2

PVS 1 = I S 1,rms ⋅ rds1 = d ⋅ I L1 ⋅ rDS 1 PVS 2 = I D 2,rms ⋅ rds 2 = (1 − d ) ⋅ I L1 ⋅ rDS 2 2

2

(8)

Assuming that (rDS1 = rDS2 = rD ) the efficiency of this circuit can roughly be calculated as: ( I La ⋅ U 2 ) η= = PV + ( I La ⋅ U 2 ) with I L1 =

I La 1− d

( I La ⋅ R L ⋅ I La ) ( I La ⋅ R L ⋅ I La ) + I La

η≈

2

⎛ 1 ⎞ ⋅ rD ⋅ ⎜ ⎟ ⎝1− d ⎠

=η =

RL ⎛ 1 ⎞ R L + rD ⋅ ⎜ ⎟ ⎝1− d ⎠

( I La ⋅ U 2 ) ≈ PV + ( I La ⋅ U 2 )

( I La ⋅ U 2 ) ≈ ≈ 1 1 2 2 ( I La ⋅ U 2 ) + I La ⋅ rDS 1 ⋅ d ⋅ + I La ⋅ rDS 2 ⋅ (1 − d ) ⋅ 1− d 1− d with rDS 1 = rDS 2 = rD

(9)

( I La ⋅ U 2 ) ⎛ I 2 ⋅r ⎞ ( I La ⋅ U 2 ) + ⎜⎜ La D ⎟⎟ ⎝ 1− d ⎠ with U 2 = RL ⋅ I La

III. DERIVATION OF THE DYNAMIC MODEL To shorten the calculation path and for simplification the following assumptions have been made: switches are assumed to be ideal and the influence of serial or parallel connected parasitic impedances (resistances or reactances) to any component of the circuit are omitted. For any switch mode converter operating in continuous mode, two describing differential equation systems can be developed. The first one for the closed active switch. After that the second one for the open switch can be calculated. The two systems can be combined. The next step is to linearize this combined system of nonlinear differential equations. This is done at its steady state operating point and therefore linear control theory [2] can be applied. During the interval Ton, the following equations hold:

duC1 −i L 2 = dt C1

diL1 − uC1 = dt L1

(10.a) (10.b) (10.c)

(11.a)

diL 2 uC1 + U1 − uC 2 (11.b) = dt L2 duC1 iL1 − iL 2 (11.c) = dt C1 duC 2 iL 2 + (U1 − uC 2 ) / RL (11.d) = dt C2 Equations (10) and (11) describe the system behavior. The obtained system time constants have to be large compared to the commutation cycle. The two sets of equations can then be combined by a weighed sum of the differential equations [2]. By defining the duty ratio as: d=

≈

diL1 U1 = dt L1 diL 2 uC1 + U1 − uC 2 = dt L2

u iL1 + iL 2 − C 2 duC 2 RL (10.d) = dt C2 During the interval Toff, the corresponding equations are:

Ton S1

(12)

T

If the two sets are weighted with (12) and combined this leads to: ⎡ ⎢ 0 ⎛ i L1 ⎞ ⎢ ⎜ ⎟ ⎢ 0 d ⎜ iL2 ⎟ ⎢ = dt ⎜ u C1 ⎟ ⎢⎢1 − d ⎜ ⎟ ⎜ u ⎟ ⎢ C1 ⎝ C2 ⎠ ⎢ ⎢ 0 ⎣

0 0 1 C1 1 C2

−

d −1 L1 1 L2 0 0

⎤ ⎛ d ⎥ ⎥ ⎛ i ⎞ ⎜ L1 1 ⎥ ⎜ L1 ⎟ ⎜ − ⎜ 1 L2 ⎥ ⎜ i L 2 ⎟ ⎜ ⋅ + L2 ⎥ ⎜ ⎟ 0 ⎥ ⎜ u C1 ⎟ ⎜ 0 ⎜ ⎜ ⎟ ⎥ ⎝ uC 2 ⎠ ⎜ 1 1 ⎥ ⎜ R ⋅C − ⎝ 2 R ⋅ C 2 ⎥⎦ 0

⎞ ⎟ ⎟ ⎟ ⎟ ⋅ u1 ⎟ ⎟ ⎟ ⎟ ⎠

(13) which is a nonlinear differential equation with respect to the duty cycle d. This calculated system of equations describes, the dynamic behavior of the idealized converter correctly in the average. A general view of the dynamic behavior of this assembly can be obtained quickly. The superimposed ripple is not important for qualifying the dynamic behavior. This noticeable ripple appears in the coils. However, the voltage ripple occurring across the capacitors is small. For instance in a switched mode power supply (SMPS) a smooth DC voltage is desired and sufficiently large capacitors need to be chosen. The converter model might be used as a large-signal model as well because no linearization regarding the signal values is calculated. IV. MODEL FOR CONTROL

A. Linearization Equation (13) represents the dynamic behavior of the converter. It is a nonlinear weighted matrix differential

equation with respect to the duty cycle. A linearization is necessary to apply linear control theory. Capital letters are used to introduce operating point values where as small letters are used for disturbance around the converter operating point: ∧

a bidirectional converter. It is suitable for automotive applications (e.g. e-bikes and cars), DC-DC coupling and battery chargers. The voltages are impressed to the circuit thus a control variable has to be chosen that is a current. For example, the current through the inductor L1 which is calculated as follows:

iL1 = I L10 + i L1

I L 2 (s) D(s) U1 ( s ) = 0

(18)

s ⋅ I L 10 10 ⋅ ( D 0 − 1 ) − − L1 ⋅ L 2 ⋅ C 1 C1 ⋅ L2 = ⎡ ( D 0 − 1 )2 ⎤ 1 s3 + s ⋅ ⎢ + ⎥ L C L C ⋅ ⋅ 1 1 2 2 ⎣ ⎦

(19)

GIL 2 D ( s ) =

∧

iL 2 = I L 20 + i L 2 ∧

uC1 = U C10 + u C1

U C 2 ( s) =0

(14)

∧

uC 2 = U C 20 + u C 2

I L 2 (s ) D (s )

U

C. Signal flow chart The transfer function can also be determined by using the signal flow graph. From equation (17) and with U1(s) = 0 we can write the matrix form:

∧

u1 = U10 + u1 ∧

d = D0 + d The linearized converter model is calculated based on: D0 −1 U10 ⎤ ⎤ ⎡ D0 ⎡ 0 0 ⎥ ⎢ L ⎢ 0 L1 L1 ⎥ ∧ ⎥ ⎥⎛ ⎞ ⎢ 1 ⎛ ⎞ ⎢ i i 1 1 ⎥ ⎜ L1 ⎟ ⎢ 1 ⎜ L1 ⎟ ⎢ ⎥ ∧ (15) ∧ ∧ 0 0 0 − ⎥ ⎛⎜u1 ⎞⎟ d ⎜ i L2 ⎟ ⎢ L2 L2 ⎥ ⎜ i L2 ⎟ ⎢ L2 ⎜ ∧ ⎟ =⎢ ⎥⋅ ∧ ⎥ ⋅⎜ ∧ ⎟ + ⎢ D I 1 − − 1 dt⎜uC1 ⎟ ⎢ 0 L10 ⎜ d ⎟ ⎥⎝ ⎠ ⎥ ⎜uC1 ⎟ ⎢ 0 0 0 − ⎜⎜ ∧ ⎟⎟ ⎢ C1 C1 C1 ⎥ ⎥ ⎜⎜ ∧ ⎟⎟ ⎢ u u C 2 2 C ⎝ ⎠ ⎢ ⎥ 1 1 ⎥⎝ ⎠ ⎢ 1 0 − 0 ⎥ ⎥ ⎢ ⎢ 0 C R C R C ⋅ 2 2⎦ ⎦ ⎣ L 2 ⎣ ∧

⎛ I L1 ( s) ⎞ ⎡ 0 ⎟ ⎢ ⎜ ⎜ I L 2 (s) ⎟ ⎢ 0 [ s] ⋅ ⎜ = U ( s) ⎟ ⎢ A ⎜ C1 ⎟ ⎢ 31 ⎜U ( s ) ⎟ ⎝ C2 ⎠ ⎣ 0

0 0

A13 A23

A32

0

A42

0

0 ⎤ ⎛ I L1 ( s) ⎞ ⎛ B12 ⎞ ⎟ ⎜ ⎜ ⎟ A24 ⎥⎥ ⎜ I L 2 ( s ) ⎟ ⎜ 0 ⎟ + ⋅ D( s) ⋅ 0 ⎥ ⎜ U C1 ( s) ⎟ ⎜ B32 ⎟ ⎟ ⎜ ⎜ ⎟ ⎥ A44 ⎦ ⎜⎝U C 2 ( s) ⎟⎠ ⎜⎝ 0 ⎟⎠

(20) with [s] representing the identity matrix with s in the diagonal. Considering U2 =UC2 the signal flow graph can be obtained from Fig. 6:

B. Transfer Functions Equation (15) describes the linearised converter and its important transfer function: GIL 2 D ( s) =

I L 2 ( s) D(s)

(16)

Equation (16) is most important for the design of the converter controller considering the input voltage which is another input variable cannot be used for control. This is due to the fact that U1 is impressed to the circuit. Presumably, the input voltage acts as a disturbance input to the control system. A fourth order transfer function can now be written caused by the four storage elements in this circuit. Rewriting the state equations (15) by applying the Laplace transformation leads to: ⎡ 0 ⎢ s ⎢ ⎢ 0 s ⎢ ⎢ D −1 1 ⎢ 0 C1 ⎢ C1 1 ⎢ − ⎢ 0 C 2 ⎣

1 − D0 L1 1 − L2 s 0

⎡ D0 ⎤ ⎢ L ⎥ ⎥ ⎛ I (s ) ⎞ ⎢ 1 1 ⎥ ⎜ L1 ⎟ ⎢ 1 L2 ⎥ ⎜ I L 2 (s ) ⎟ ⎢ L2 ⎥.⎜ U (s ) ⎟ = ⎢ 0 ⎥ ⎜ C1 ⎟ ⎢ 0 ⎥ ⎜⎝U C 2 (s )⎟⎠ ⎢ 1 ⎥ ⎢ 1 s+ ⎢R C R ⋅ C2 ⎥⎦ ⎣ L 2 0

U10 ⎤ L1 ⎥ ⎥ (17) 0 ⎥ ⎥ ⎛U1 (s )⎞ ⎟ ⋅⎜ − I L10 ⎥ ⎜⎝ D(s ) ⎟⎠ ⎥ C1 ⎥ ⎥ 0 ⎥ ⎦

and for U1(s) = 0 we get the transfer function written in the appendix. It can be seen that the converter can be used as

Fig. 6. Signal flow graph.

With Mason’s law, it is very simple to calculate the transfer ratio. The forward paths (a path is defined as a set of consecutive, codirectional branches as long as no node is encountered more than once as we move in the graph from the independent variable to the output or state variable under quest, the value is the product of all branches from the starting node to the end node) are: 1 (21.a) F1 = 2 ⋅ A23 ⋅ A31 ⋅ A42 ⋅ B12 s

F2 =

1 ⋅ A23 ⋅ A42 ⋅ B32 s

(21.b)

the loops (a loop is the product of the branches encountered by making a round trip if we move from a node backwards to the original node in direction of the arrow):

1 L1 = ⋅ A44 s

1 L2 = 2 ⋅ A13 A31 s 1 ⋅ A23 A32 s2 1 L4 = 2 ⋅ A24 A42 s L3 =

(22. a) (22. b) (22. c) (22. d)

L1 and L2, L2 and L4, L1 and L3, are non-touching loops. Hence, we get the sum of all second order loops (L1 ⋅ L2 + L1 ⋅ L3 + L2 ⋅ L4 ) and with Mason’s rule we get: F1 + F2 (23) GU 2 D ( s ) = 1 − (L1 + L2 + L3 + L4 ) + (L1 ⋅ L2 + L1 ⋅ L3 + L2 ⋅ L4 )

The result is shown in the appendix (A). The calculation of the transfer function is an alternative way to the solution of the matrix equation (17).

diL1 U1 = dt L1 diL 2 =0 dt

(24.a) (24.b)

C. State 2 S1 opens at t1 and the current commutates into the capacitors C11 and C22 Fig. 8 (b) charging the upper capacitor C11 and discharging the lower capacitor C22. The voltage across C1 and C2 is nearly constant and the current through L1 is constant. Therefore only one state variable is necessary to describe the transient process. If C11 and C22 are chosen equal the current through C11 is given by:

iL1 2 For the transient part of the voltage we can write: iC11 = iC 22 =

C11 ⋅

du C11 i L1 i = → u C11 = L1 ⋅ t dt 2 C11 ⋅ 2

(25)

(26)

The voltage across C11 (uC11) increases until UC2 is reached and the diode of S2 starts to conduct the current at time t2 when D2 turns on both capacitors C11, C22 are now effect less. The whole process can be obtained from Fig. 9.

V. LOW LOSS SWITCHING

A. Principle A reduction in losses can be achieved by applying a zero voltage switching concept (ZVS) to the converter. For this concept the converter from Fig. 1 is extended. The used halfbridge module is extended with two capacitors C11 and C22 as can be seen in Fig. 7. For the analysis of the switching behaviour as can be obtained from Fig. 8, we transform the filter capacitors into voltage sources (the voltage across C1 and C2 is nearly constant during one period of the switching frequency) with zero voltage across C11 and UC2 across C22 for S1 closed. Further for S2 closed and S1 open we have zero voltage across C22 and UC2 across C11. The transient behaviour for the voltages of C11 and C22 can be described by a differential equation. The differential equation involves only C11, assuming that C11 is equal to C22.

Fig.8. Equivalent circuits for state 1 (a) and 2 (b).

tr iL1

t u C11 U1 + U2

t u C22

U1 + U2

Fig.7. Converter circuit with ZVS opportunity.

B. State 1 When the upper switch S1 is closed as seen in Fig. 8 (a), the current through the inductance L1 changes according to:

t0;

t1; t2;

I

t

t3; t4;

II

III

IV

Fig. 9. Current through L1, and voltages across the snubber capacitors.

D. State 3 Now, no losses occur by closing S2. The current through the inductor (iL1) decreases, reaches zero and a negative value can be obtained starting at t3. The energy stored in the inductor must be so high that if S2 opens at t3, capacitor C22 again can be charged to a value reaching UC2 and C11 is discharged to zero voltage at t4. S1 can be closed again without losses. The transient process between t3 and t4 can be described as follows: ⎡ 0 d ⎛ i L1 ⎞ ⎢ ⎜ ⎟=⎢ dt ⎜⎝ u C11 ⎟⎠ ⎢ 1 ⎢⎣ 2 ⋅ C11

−

1⎤ ⎡1⎤ L1 ⎥ ⎛ i L1 ⎞ ⎢ ⎥ ⎟⎟ + L1 ⋅ u1 ⎥ ⋅ ⎜⎜ ⎢ ⎥ 0 ⎥ ⎝ u C11 ⎠ ⎣ 0 ⎦ ⎥⎦

2

(C + C 22 ) ⋅ (U1 + U 2 ) 2 L1 ⋅ I tr = 11 2 2 C11 + C 22 I tr = ⋅ (U1 + U 2 ) L1

WL1 =

⇒ tr =

I tr 2 ⋅U 2

⋅ L1 = (C11 + C 22 ) ⋅ L1 ⋅

U1 + U 2 2 ⋅U2

(30)

This is the first possible moment for switching on S2. Now, the current iL1(t) through the inductance L1 can be calculated as follows:

(27) iL1 (t ) =

U 2 ⋅ 2 L1 ⋅ C11 L1

⎛ ⎞ ⎛ ⎞ (C11 + C 22 ) ⋅ L1 ⋅ (U 1 + U 2 ) 1 1 ⋅ sin⎜ ⋅t ⎟ − ⋅ cos⎜ ⋅t⎟ ⎜ 2L ⋅ C ⎟ ⎜ 2L ⋅ C ⎟ ⋅ L 2 1 1 11 1 11 ⎝ ⎠ ⎝ ⎠

(31)

we get for the charging process starting from zero: uC11 (t ) = U1 +

⎛ ⎞ ⎛ ⎞ iL1 (0) ⋅ 2 ⋅ L1 ⋅ C11 1 1 ⋅ sin⎜ ⋅ t ⎟ + (uC11 (0) − U1 ) ⋅ cos⎜ ⋅t⎟ ⎜ 2⋅ L ⋅C ⎟ ⎜ 2⋅ L ⋅C ⎟ 2 ⋅ C11 1 11 1 11 ⎝ ⎠ ⎝ ⎠

(28. a) iL1 (t ) =

(U1 − uC11 (0) ) ⋅ L1

2 L1 ⋅ C11

⎛ ⎞ ⎛ ⎞ 1 1 ⋅ sin ⎜ ⋅ t ⎟ + iL1 (0) ⋅ cos⎜ ⋅t⎟ ⎜ 2L ⋅ C ⎟ ⎜ 2L ⋅ C ⎟ 1 11 1 11 ⎝ ⎠ ⎝ ⎠

(28. b) and the initial conditions: iL1 (0) < 0; u c11 (0) = U 1 + U 2

And from t2 to t3: di L1 (0) U U = − 2 ; iL1 (0) = − 2 ⋅ t r dt L1 L1

where t r is the " reverse time" (Fig. 9) : iL1 (tr ) = − I tr

(29)

VI. CONCLUSION A standout amongst the aspects of the investigated converter is the prefabricated half-bridge module and a halfbridge driver which can easily be implemented into any circuit or assembly. This is particularly helpful for higher current levels. The converter is especially suitable for battery chargers solar power applications and electric drives which are increasingly popular nowadays. However, another aspect is that the converter can easily be used for power control and signal applications such as coupling of two voltage links, in uninterruptible power supplies and two quadrant choppers for DC drives. The use of the zero voltage switching concept is another interesting aspect for further developments because of the low losses. This concept offers an opportunity for the future to decrease power losses for, instance in photo voltaic applications or DC drives. It is essential that the control is done in pseudo discontinuous mode or by variable frequency [4]. REFERENCES [1] [2]

The initial condition iL1(tr) can be calculated using the energy stored in the inductance:

[3] [4]

F.A. Himmelstoss, and F.C. Zach: “Bidirectional Converters”, Austrian patent, filed 1992.3.13, AT 399 625 B N. Mohan, T. Undeland and W.P. Robbins, Power Electronics Converters, Applications, and Design, 3rd ed., New York: John Wiley & Sons, 2003. F. Himmelstoss: “Analysis and Comparison of Half-bridge Bidirectional Dc-Dc Converters”, IEEE PESC’94, pp.922-928. F.A. Himmelstoss: “Control Concept for the Reduction of Switching Losses of Bidirectional Converters”, Austrian patent, filed 2001.07.16. AT 412371B.

APPENDIX

(A)