2.2. Steady-State Mathematical Model under Three Modulation Strategies
The working waveform of the series-resonant DAB converter in EPS modulation is shown in
Figure 3, where
uA,
uB,
uC and
uD are the drive signals of the four bridge arms,
uAB and
uCD are the midpoint voltages of the arms of two active bridges and
iL is current flowing through the inductance.
The converter has 8 operating modes per cycle under EPS phase-shift control, of which the operating modes from t0 to t4 are completely symmetrical with t5 to t8, and only the first four are analyzed.
- (1)
t0–t1 phase
The t0 moment is the initial moment, before this state, S2 and D4 are on, while the resonant current is negative, at the t0 moment, S2 is disconnected and the driving signal reaches S1; but since the current cannot be changed suddenly, the resonant slot current is still negative, so D1 and D4 are on. The secondary side of the transformer D5 and D8 are on, the voltage direction applied to both ends of the resonant slot is opposite to the current flow, resulting in the absolute value of current gradually decreasing, and when it decreases to 0, the energy is fed back from the resonant tank to the current side, and D1, D4, D5 and D8 are on in this stage.
- (2)
t1–t2 stage
At the moment of t1, the trigger pulse reaches S7, the current is naturally transferred to S7, S6 conduction, the primary state of the transformer remains unchanged, and the circulation path of the secondary side becomes S7, S6 This state continues until the voltage of the secondary side is reduced to the transformer induction voltage and this stage ends. In this stage, D1, D4, S7 and S6 are in the conduction state.
- (3)
t2–t3 phase
At the moment of t2, the absolute value of resonant current decreases to 0. S1 and S4 realize zero voltage switching; the resonant current is about to increase positively, and the transformer secondary current flow path becomes D7 and D6. S1, S4, D7 and D6 are on in this stage.
- (4)
t3–t4 phase
At moment t4, S4 is disconnected. At this time the inductor current is positive, causing the current path to switch from S4 to D3, with S1 and D3 on for the high-voltage side, and D5 and D8 on for the low-voltage side. At the same time, the resonant tank current starts to decrease gradually from the peak due to the voltage applied to both ends of the resonant tank and the opposite direction of the resonant current flow. In this stage S1, D7, D3 and D6 are on.
According to the steady-state working waveform combined with fundamental wave analysis, the time-domain expressions of the two fundamental wave voltages in
Figure 2 are shown in Equations (1) and (2):
In order to calculate the complex power using the phase volume method, the phase volumes of the fundamental voltages
u1(
ωt) and
u2(
ωt) are defined here as
and
, and the expressions for both are shown in Equations (3) and (4):
The phasor of the fundamental wave component
iL of resonant current is defined as
, and its expression is shown in Equation (5):
According to
Figure 2, the complex power transmitted by the AB bridge from the input side can be expressed as Equation (6):
By solving this, the active power and reactive power expressions of which can be obtained as Equations (7) and (8), respectively.
Taking
as the power reference value, the expressions of the normalized value of active power
PEPS,pu and reactive power
QEPS,pu under EPS modulation are shown in Equations (9) and (10):
The working waveform of series-resonant DAB converter under DPS modulation is shown in
Figure 4.
The converter operating modes are analyzed according to the steady-state operating waveform, which is divided into eight operating modes for each cycle under DPS modulation, of which the operating modes from t0 to t4 are completely symmetrical with t5 to t8.
- (1)
Phase 1: t0–t1
At the moment of t0, the converter transitions from mode 8 to mode 1, S1 and S4 are on, the inductor current is negative at this time, diodes D5 and D8 are on, and switching tube S1 achieves zero voltage turn-on. The transformer secondary side switching tubes S6 and S7 are in continuous conduction. Power flows from the inductor to the secondary side.
- (2)
Phase 2: t1–t2
Transformer primary side full-bridge switching tubes S1 and S4 continue to conduct, transformer secondary side switching tubes S6 and S7 conduct. t1 moment inductor current comes to over zero, and both sides of the transformer at the same time charge the inductor current.
- (3)
Stage 3: t2–t3
Transformer primary side full-bridge switching tube S2 is on, transformer secondary side switching tubes S6 and S7 are continuously on. The inductor current charges the primary side and the power flows from the inductor to the primary side.
- (4)
Stage 4: t3–t4
Transformer primary side full-bridge switching tube S2 is on, transformer secondary side switching tube S5 is on. The inductor current charges the secondary side.
Similarly, the time-domain expressions of the two fundamental voltages under DPS modulation can be derived as shown in Equations (11) and (12):
The same power reference values as for EPS modulation are chosen. The derived expressions for the normalized value of active power
PDPS,pu and reactive power
QDPS,pu under DPS modulation are given in (13) and (14) below:
The working waveform of the series-resonant DAB converter under TPS modulation is shown in
Figure 5.
The model is mainly divided into 10 working states, of which the first 1–5 working states and 6–10 working states are symmetrical, so only the first 5 working states are analyzed.
- (1)
Phase 1: t0–t1
At the moment of t0, the converter transitions from mode 10 to mode 1, S2 turns off, the inductor current is negative at this time, diode D1 conducts, and switching tube S1 achieves zero voltage turn-on. Switching tubes S6 and S7 on the secondary side of the transformer are continuously on. Power flows from the inductor to the secondary side.
- (2)
Phase 2: t1–t2
At moment t1, the primary side of the full-bridge switching tube S3 off, at this time the inductor current is negative, diode D4 is in natural conduction, switching tube S4 achieves zero-voltage opening, the transformer secondary side switching tubes S6 and S7 continue to conduct.
- (3)
Phase 3: t2–t3
Transformer primary side full-bridge switching tube S1, S4 continue to conduct, transformer secondary side switching tubes S6 and S7 continue to conduct. t2 moment inductor current comes to over the zero point and both sides of the transformer at the same time charge the inductor current.
- (4)
Phase 4: t3–t4
At moment t3, the secondary side switching tube S6 off, S5 voltage discharge at both ends to 0, diode D5 natural conduction, switching tube S5 to achieve zero voltage open, the transformer primary side switching tube S1, S4 continue to conduct, to charge the inductor.
- (5)
Phase V: t4–t5
At moment t4, the secondary side of the full-bridge switching tube S7 off, the voltage at both ends of S8 discharge to 0, diode D8 natural conduction, switching tube S8 to achieve zero voltage open, the transformer primary side S1, S4 continue to conduct.
Similarly, the time domain expressions of the two fundamental wave voltages under TPS modulation can be derived (15) and (16) shown:
The same power reference values as for EPS modulation are chosen. The expressions for the normalized value of active power
PTPS,pu and reactive power
QTPS,pu under TPS modulation are derived as follows (17) and (18).