A Phase-Shift Three-Phase Bidirectional Series Resonant DC/DC Converter R. Mirzahosseini , F. Tahami Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran
[email protected],
[email protected]
Abstract— The demand for bidirectional DC/DC converters is increasing because of their potential capability for sustainable energy conversion systems where the power is to be captured from or stored in energy storage elements. This paper presents a three-phase series resonant bidirectional converter which is suitable for high power applications. The converter consists of two three-phase full bridge networks, the resonant tanks and a high frequency three-phase transformer. All switches can work either in ZVS or ZCS mode. The voltage drop on leakage inductances is minimized by choosing the switching frequency close to the resonant frequency, allowing high efficient power conversion. The phasor analysis is applied to the circuit to develop a dynamic model for designing the control system. Simulation and experimental results are used to verify proper operation of the proposed converter and the obtained dynamic model. Index Terms— Bidirectional converter, Soft switching, resonant converter, averaged modeling.
I.
INTRODUCTION
Sustainable energy conversion systems need to interface an energy storage element such as a battery or an ultra capacitor bank. Usually renewable energy systems like fuel cell vehicles (FCVs) and stand alone photovoltaic (PV) distributed generation use these energy storages through bidirectional dc-dc converters. Increasing power density and achieving higher efficiency are challenging issues in power conversion market. To achieve these goals, it is necessary to reduce power losses, overall system size, and weight. The most popular approach for increased power density is increasing the switching frequency, which reduces the size of magnetic components and filter elements. High frequency (HF) isolated bidirectional converters are attractive due to their low cost, high power density, isolation and higher voltage gain to avoid series connection of batteries or ultra capacitors [1]. The zero-voltage-switching (ZVS) topologies that enable high-frequency switching are becoming popular thanks to extremely low switching losses, and low devices stress. Among many topologies, the phase-shifted ZVS full-bridge is widely used for medium or high power applications since it allows all switches to operate at ZVS by effective use the parasitic capacitance of power switches and the leakage inductances of transformers without an additional auxiliary switch. The dual active bridge (DAB) converter was
978-1-61284-972-0/11/$26.00 ©2011 IEEE
proposed in [2] for bidirectional dc-dc power conversion. It uses the transformer leakage inductance as an energy transfer element and the power flow is controlled with the phase shift angle between two active bridges. A three-phase version for high power applications was suggested in [3]. Interleaving nature of three-phase version results in higher power capability and much lower current ripple in dc ports which leads to lower filter capacitance. In these converters the net power is inversely proportional to the link impedance at switching frequency [4]. The link impedance is equal or more than the impedance of leakage inductance of the transformer, so increasing the switching frequency for a certain leakage inductance will limit the maximum power transfer capability of the converter. Furthermore, the ZVS range is very narrow and the freewheeling current consumes high circulating energy [5]. The resonant converters are attracting increased attention as they are able to maintain high efficiency for increased switching frequency. Resonant converters process power in a sinusoidal manner and the switching devices are softly commutated therefore, the switching losses and noise can be dramatically reduced. Dual bridge series resonant converter (DBSRC) proposed in [6] with fixed frequency operation can operate in higher frequency due to the impedance cancellation between the resonant link capacitor and inductance. This converter is controlled by the phase shift between the input and output bridges. Another bidirectional resonant converter with switching frequency control is suggested in [7]. In this paper a three-phase resonant fixed frequency phase shift control converter is proposed which is a suitable high frequency candidate for high power applications. The converter can be designed so that all switches operate in ZVS; therefore utilizing MOSFETs with high switching frequency is possible. A control oriented linear model from control signal to output current based on phasor analysis is developed. It is shown that the converter small signal model has two poles near imaginary axis in s plane which should be considered both in converter design and control system. The converter operation and the developed model are verified by simulation and experimental results.
1137
iP1 v1A v1B v1C
V1
iP 2
Power Flow CA
LA
CB
LB
CC
LC
v2 A v2 B v2C
1:n
Phase Shift
ϕ
Primary
V2
Secondary
Fig. 1. Phase-shift three-phase bidirectional series resonant converter
II.
STEADY STATE ANALYSIS
i1A
A. Principle of operation Fig. 1 shows the proposed three-phase bidirectional series resonant converter. The converter consists of dual threephase active bridges, a high frequency three-phase transformer connected in Y/Y, and three series resonant tanks. Each leg in each three-phase bridge is operating at duty cycle of d=0.5. Therefore, the phase voltages of the primary windings are six-step waveforms with 2π / 3 phase displacement, as shown in Fig. 2. The same voltage waveforms are induced in the secondary side. The six-step waveform has lower high frequency harmonic contents compared to the square wave voltage produced in single phase inverters. The first harmonic after the fundamental component is the 5th one. When the tank network contains a high-Q resonance at or near the switching frequency and a low-pass characteristic at higher frequencies, the resonant tank responds primarily to the fundamental component of the switch waveform and has negligible
v1A
jLωS
1 jCωS
v2′ A
Fig. 3. Equivalent per phase circuit
response at the harmonic frequencies, and then the tank waveforms are well approximated by their fundamental components and the harmonics can be ignored. The fundamental harmonic component of phase voltage can be obtained by: 1 2π 2 (1) V 11A = ∫ v 1A .sin(ωs t ) d ωs t = V 1 π 0 π where v 1A is the voltage of the primary side phase A. Fig. 3 shows the per phase equivalent circuit. Since the converter operates in fixed switching frequency, the resonant tank inductance and capacitor can be modeled with fixed impedances. Total impedance in switching frequency can be expressed by: 1 ) (2) Z = j (L ωs − C ωs where C is the resonant tank capacitor, L is the total transformer leakage inductance referred to the primary side and ωs is the switching frequency. In converters containing MOSFETs and diodes, zerovoltage switching mitigates the switching loss otherwise caused by diode recovered charge and semiconductor output capacitance. Zero-current switching can mitigate the switching loss caused by current tailing in IGBTs and by stray inductances. In the majority of applications, where diode recovered charge and semiconductor output capacitances are the dominant sources of PWM switching loss, zero-voltage switching is preferred [8]. To obtain ZVS operation for all switches, a switching frequency above the resonance is applied:
Fig. 2. Waveforms of the primary phase voltages
1138
ωs > ωr 1 ωr = LC
ωs = 2π f s
V11
ϕ
(3)
As harmonics of the switching frequency are neglected, the tank waveforms are assumed to be purely sinusoidal and can be express by phasors as follow:
{ } v ′ (t ) = real {V e } i (t ) = real {I e } v 11 (t )
= real
1 j ωs t 1
V21
V11
V2′
(b)
(a)
(4)
I11
3 P = real {V 11I 11*} 2 6 V 1V 2′ sin(ϕ ) = 2 π 1 Lωs − C ωs
V121
ϕ
The and are phasors that represent the primary voltage, the secondary voltage referred to the primary side and the primary current respectively. Fig. 4(a) shows phasor diagram of bridge voltages and resonant tank current. As the converter is operating above resonance, the resonant tank presents an effective inductive load to the switches, so the current I11 lags the tank voltage V121 . The power flow direction can be either positive or negative depending on the phase shift between two bridges ϕ . If switches are assumed to be lossless then the power delivered from the primary port to the secondary port is equal to power flowed from primary three-phase bridge to secondary threephase bridge and it can be find from:
where
V21
1 j ωs t 2
1 1
quantities V11 ,
I11
V 11e j ωs t
1 2
V121
V21
(5)
Fig. 4.(a) Voltage and current phasors for positive and negative phase shift (b) Voltage and current waveforms
Fig. 5. Phase A primary and secondary legs
TABLE I SOFTSWITCHING BOUNDARIES FOR ABOVE RESONANNCE SWITCHING FREQUENCY Phasor Diagram V2
V1 ϕ2
primary.
B. Soft switching conditions Depends on switching frequency and voltage ratings, MOSFET’s or IGBT’s may be used in converters [9]. The proposed converter can operate both in ZVS or ZCS condition for all the switches. Fig. 5 shows primary and secondary legs for phase A. At the switching instant when the primary upper switch S1 A is going to be turned on and the primary lower switch S1' A is going to be turned off, if the phase A resonant current i1 A is negative then by adding adequate dead-time, the S1 A body diode will conduct and this switch can be turned on at zero voltage. At the next switching instant in this leg when the lower switch S1' A is going to be turned on, positive i1 A will result in ZVS for this switch. The ZVS condition for the secondary side switches depends on the current i1 A in a reverse manner, since the current direction is into the bridge. Generally, if current i1 A lags v1 A , the primary leg switches S1 A and S1' A operate in ZVS. On the other hand if i1 A leads v1 A , the switches will operate in ZCS. The soft switching condition for the secondary side is reversed. The tank current phase is influenced by phase shift ϕ and the voltage ratio M is defined by:
ZVS
I1
is the secondary side dc voltage reflected to
Port1
Port 2 ZCS
condition M < cos(ϕ )
V1
ϕ2
V2
ZVS
I1
boundary ZCS
M = cos(ϕ )
V1
ϕ2
V2
I1
cos(ϕ ) < M
1 cos(ϕ )
V1
ϕ2
V2
V1 ϕ2
V2
I1
I1
1 cos(ϕ )
ZVS
ZCS
M =
V2 . nV1
(6)
Table I shows soft switching conditions for the converter operating at above resonant frequency where the resonant tank behavior is inductive. When resonant current is in phase with secondary bridge voltage then:
1139
G iL
+
G iC
jLωs
Ls
G vL
G+ vC
−
−
(a )
(7)
The secondary bridge is in boundary of ZCS since resonance current is in phase with voltage and the primary bridge works in ZVS condition because the current is lagging with respect to the primary bridge voltage. For larger values of M the secondary bridge can work in ZVS condition. There is a region where both primary and secondary bridges can operate in ZVS conditions. If the converter is designed to work below resonance, there would be a region where both bridges will work in ZCS conditions. III.
DYNAMIC ANALYSIS
Since the tank inductor current and the tank capacitor voltage are sinusoidal and have zero average, state space averaging method cannot be used to build a dynamic model [1]. One way to derive a dynamic model is to use (5) to calculate average current injected to ports. In this method the switch network is assumed to be lossless. Fig. 6 shows the averaged model in which the converter has been modeled by two current sources that are controlled by phase shift between two bridges [10] and are obtained by: P I P1 = V1 V 2′ 6 sin(ϕ ) = 2 π 1 L ωs − C ωs = K 1 sin(ϕ ) (8)
I P′ 2 = =
P V2 6
π
2
V1 L ωs −
= K 2 sin(ϕ )
1
G x (t ) = Re(xe j ωs t ) = Re(Ae j (ωs t +ϕ ) ) = A cos(ωs t + ϕ ).
Phasor transform of resonant tank capacitor and inductor are obtained by (12) through (15). di d G v L (t ) = L L = Re{L (i L e j ωs t )} (12) dt dt which means that: G G di L j ωs t G j ωs t v Le e =L + jL ωs i L e j ωs t (13) dt G G di G v L = L L + jL ωs i L . (14) dt Similarly for the capacitor phasor transform is G G dv G (15) iC = C C + jCωs vC . dt Voltage and currents now can be perturbed to get phasor transform small signal models of capacitor and inductor. Since switching frequency is kept constant there is no perturbation in switching frequency. G Gˆ G G G Gˆ d (I L + i L ) ˆ (V L +v L ) = L (17) + jLωs (I L + i L ) G dt G ˆ G Gˆ G G d (V C +v C ) (I C + i C ) = L (18) + jC ωs (V C +vˆC ) dt Phasor transform equivalent circuits for both ac and dc modeling of inductors and capacitors are shown in Fig. 7 [12]. To model switching network consider Fig 8. As mentioned earlier only fundamental harmonic component is considered. The fundamental component of the primary voltage and the secondary voltage referred to the primary and their corresponding phasors for phase A can be expressed by: 2V v 1A1 (t ) = 1 cos(ωs t ) (20) 2V v 2′ A 1 (t ) = 2 cos(ωs t + ϕ ) nπ 2V 1 G v 1A 1 =
(9)
Since the switching frequency is constant, all the tank currents and voltages are oscillating at constant frequency. Hence, the concept of the phasor analysis can be used, where the signals are all DC at steady state [11]. The phasor of x (t ) is a complex value defined by: G x = Ae jϕ (10) where
(b)
π
sin(ϕ )
C ωs
1 jCωs
Fig. 7. Phasor transform ac and dc equivalent circuit (a) Inductor equivalent circuit (b).Capacitor equivalent circuit
Fig. 6. Average model of the converter
M = cos(ϕ ).
1 Cs
π
(21)
(22)
2V G v 2′ A 1 = 2 e j ϕ . (23) nπ The primary side output voltage is considered as phase reference. To obtain the small signal model and control to output transfer function, the phase shift ϕ and the phasor of output voltage secondary switch network output voltage G v 2 A 1 (t ) are perturbed and linearized as follows:
(11)
1140
e
+j
2V2
e
jΦ
ϕˆ .
V1
(24)
+ −
π π The primary and secondary switch networks can be modeled by dc to phasor transformers with turns ratios defined by: G v 1A 1 2 σ1 = = (25) π GV 1 v 2 jΦ e . σ 2 = 2A 1 = (26) V2 πn
(a )
1 V1 3
where f P is the difference of switching and resonance frequency. Finally, the dynamic response of the converter can be calculated using by the equivalent circuit shown in Fig. 9. In average model shown in Fig .6 the current source are considered as constant gains which are given by (8) and (9). In this section it was shown that transfer function from control signal to output power and consequently to current sources in Fig .6 has poles at ± j ωP . So the transfer functions of the current sources can be expressed by: k1 ϕˆ iˆP 1 = (34) s 2ζ s + ( )2 1+
ωP
ωP
V
Secondary switch network
Primary switch network
According to (24) through (26) and using Fig. 7, phasor transformed model of the converter is represented in Fig 9. The transfer function from the control signal ϕ to tank G current phasor i 1A is expressed as follows: 1 (27) Z (s ) = Ls + Cs Gˆ 2V ϕˆ (28) i 1A (s ) = − j 2 e j Φ nπ Z (s + j ωs ) Since the three phase switch network is assumed to be lossless, power drawn from primary can be written as: G G p = v 1i P 1 = 3Re{v 1A i 1A ∗ }. (29)
v1 A
v2′ A
2 V1 3
ϕ
ωt
(b ) Fig. 8. (a) Switch networks (b) Switch networks output voltages
iP 1 V1 +−
G i1 A
Ls jLω s
G v1 A1
−
σ1 =
1 / jCω s j 2V2 e jΦϕˆ
πn
1 / Cs
+
The primary side Gdc voltage v 1 does not change. As mentioned earlier v 1A has zero phase, so (29) can be perturbed as follows: Gˆ G 6V P + pˆ = V 1 (I P 1 + iˆP 1 ) = 1 Re{(I 1A + i 1A )} (30) π Gˆ 6 (31) iˆP 1 = Re{i 1A } π Gˆ Gˆ 6 i 1A (s ) + i 1∗A (−s ) ˆ i P 1 (s ) = . (32) 2 π According to (28) and (32) the dc port current transfer function iˆP 1 (s ) is proportional to a transfer function that is obtained by shifting the admittance transfer function by + j ωs and mirroring it. It can be shown that the poles of the power transfer function are moved to j ωP = ± j (ωs − ωr ) f P = fs −f r (33)
+ −
Resonance tank
+ −
=
jΦ
Transformer &
iP 2
G v′2 A1
+
π
2V2
v2 A
v1 A
G 2V G V2 A1 + vˆ2 A1 = 2 e j (Φ+ϕˆ )
−
2 π
+ V − 2
2 σ = e jΦ 2 πn
Fig. 9. ac and dc equivalent circuit of the converter
Fig. 10 Voltage control loop
iˆP′ 2 =
k2 ϕˆ s 2 2ζ s +( ) 1+
(35)
ωP ωP where ζ is damping ratio which is determined by the transformer winding resistor and the MOSFET ON resistance. The obtained linear control oriented model is suitable for designing controllers based on linear control theory, e.g. PI controllers. IV.
CONTROL SCHEME
Fig. 10 shows the output voltage control loop. Block C is the controller and P is converter transfer function. The converter transfer function is given by (35). This fact that in phase shift dual active bridge series resonant converters there are resonant poles at frequency equal to difference of switching and resonance frequency should be considered both in design of the converter and the control system. 1141
Exciting these poles increases conduction losses and effects ZVS conditions obtained for steady state operation. One effective way to control the converter is to shape the control loop such that the bandwidth of control signal ϕˆ be less than f P . For example a controller consisting of a gain and a first order filter given by (36) can be used.
C (s ) =
V.
K 1+
s
ωc
