Direct-current transformer circuit topology based on tandem three-phase bridge circuit and control method of direct-current transformer circuit topology

Abstract

The invention relates to a direct-current transformer circuit topology based on a series three-phase bridge circuit and a control method thereof. A high-voltage side power unit comprises N three-phase series half-bridge circuits which are sequentially connected in series; the low-voltage side power unit comprises N three-phase parallel half-bridge circuits which correspond to the three-phase series half-bridge circuits and are sequentially connected in parallel, each three-phase series half-bridge circuit is connected with the corresponding three-phase parallel half-bridge circuit through a three-phase isolation transformer to form N groups of three-phase bridge modules, and meanwhile, the N groups of three-phase bridge modules are connected with the N groups of three-phase isolation transformers. A symmetrical duty ratio and phase shift hybrid control strategy is adopted to realize bidirectional flow of energy and soft switching in a full range, and intra-module and inter-module phase shift control is adopted to realize voltage sharing of a series side, reduce current ripples of a parallel side, reduce backflow power and improve the efficiency of the converter.

Description

Technical Field

[0001] The present invention relates to a DC transformer circuit topology based on a series-connected three-phase bridge circuit and a control method thereof, belonging to the technical field of transformers. Background Art

[0002] As a key device connecting different DC buses, a DC transformer needs to undertake one or several functions such as bus voltage control, electrical isolation between different buses, two-way power flow scheduling, and fault blocking. On the other hand, the DC transformer also serves as the DC / DC stage in a power electronic transformer to complete voltage conversion and electrical isolation. However, currently limited by the technical levels of semiconductor devices and high-frequency magnetic components, existing DC transformers have problems such as large volume and high cost, and their transmission efficiency and reliability are also difficult to compare with power frequency transformers in traditional AC distribution networks, which indirectly hinders the further popularization and application of DC distribution networks. And as one of the possible scenarios for the future application of DC distribution technology, the problem of tight land use puts higher requirements on the volume and power density of DC transformers. Therefore, on the basis of applying the same voltage-resistant devices, proposing a new topology to reduce the number of power conversion units of the DC transformer, increase the capacity of the power conversion units, and improve the overall power density of the DC transformer is the current research focus of the industry. Summary of the Invention

[0003] The purpose of the present invention is to provide a DC transformer circuit topology based on a series-connected three-phase bridge circuit and a control method thereof to solve the problems of large floor area and low power density of DC transformers.

[0004] To achieve the above purpose, on the one hand, the present invention proposes a DC transformer circuit topology based on a series-connected three-phase bridge circuit, including a high-voltage side power unit, a low-voltage side power unit, and an isolation transformer unit connecting the high-voltage side power unit and the low-voltage side power unit.

[0005] The high-voltage side power unit includes N series-connected three-phase series half-bridge circuits in sequence; the low-voltage side power unit includes N three-phase parallel half-bridge circuits corresponding to and connected in parallel with each three-phase series half-bridge circuit in sequence; wherein, each three-phase series half-bridge circuit is connected to its corresponding three-phase parallel half-bridge circuit through a three-phase isolation transformer to form N groups of three-phase bridge modules.

[0006] The three-phase series half-bridge circuit includes a first arm of phase A, a first arm of phase B, and a first arm of phase C connected in series; each first arm of a phase includes two switching tubes connected in series, with complementary states and symmetric duty cycles.

[0007] The three-phase parallel half-bridge circuit includes a second arm of phase A, a second arm of phase B, and a second arm of phase C connected in parallel; each second arm of a phase includes two switching tubes connected in series, with complementary states and symmetric duty cycles.

[0008] Further, the connection points of the first arm of each phase are connected in series with a DC-blocking capacitor and an inductor, and after the series connection, they are connected to the input terminal of the corresponding phase of the primary side of the three-phase isolation transformer;

[0009] The connection points of the second arm of each phase are connected to the input terminal of the corresponding phase of the secondary side of the three-phase isolation transformer.

[0010] Further, the first arm of each phase is connected in parallel with the bus capacitor of its corresponding phase.

[0011] Further, a bus capacitor is connected in parallel at the output terminal of the three-phase parallel half-bridge circuit.

[0012] Further, the switching tube is also connected in parallel with a diode.

[0013] On the other hand, the present invention also proposes a DC transformer circuit topology control method based on a series-connected three-phase bridge circuit, which performs the following steps:

[0014] Set the duty cycle of the switching tube of each arm in each module to a symmetric duty cycle;

[0015] Preset the phase difference of the drive pulses between the current module and the next module, so that the drive pulses of the next module lag behind the drive pulses of the current module;

[0016] For each module, preset the phase shift angle for lagging the low-voltage side switching tube behind the high-voltage side switching tube in the current module;

[0017] Based on the phase shift angles of all modules, input PWM pulses to all three-phase bridge modules with the phase difference to adjust the on and off of the switching tubes in each module.

[0018] Further, the phase difference is set to 360° / N, where N is the total number of three-phase bridge modules.

[0019] Further, the phase shift angle is calculated by the following formula:

[0020]

[0021] Wherein, is the phase shift angle of the i-th phase (i = a, b, c); T s is the switching period; is the switching period of the i-th phase.

[0022] Further, the value range of the phase shift angle is 0 ≤ φ < π / 3 or π / 3 ≤ φ < 2π / 3.

[0023] The beneficial effects of the present invention are as follows: a high-voltage side power unit, a low-voltage side power unit, and an isolation transformer unit connecting the high-voltage side power unit and the low-voltage side power unit. The high-voltage side power unit includes N serially connected three-phase series half-bridge circuits in sequence; the low-voltage side power unit includes N three-phase parallel half-bridge circuits corresponding to and connected in parallel in sequence with the respective three-phase series half-bridge circuits; wherein, each three-phase series half-bridge circuit is connected to its corresponding three-phase parallel half-bridge circuit through a three-phase isolation transformer to form N sets of three-phase bridge modules; the three-phase series half-bridge circuit includes a phase-A first bridge arm, a phase-B first bridge arm, and a phase-C first bridge arm connected in series; each first bridge arm includes two switching tubes connected in series, with complementary states and symmetric duty cycles; the three-phase parallel half-bridge circuit includes a phase-A second bridge arm, a phase-B second bridge arm, and a phase-C second bridge arm connected in parallel; each second bridge arm includes two switching tubes connected in series, with complementary states and symmetric duty cycles. At the same time, a symmetric duty cycle and phase-shift hybrid control strategy is adopted to achieve bidirectional energy flow and soft switching within the full range, and phase-shift control within the module and between modules is used to achieve voltage sharing on the series side, reduce the current ripple on the parallel side, reduce the reflux power, and improve the converter efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0024] Figure 1 is a schematic structural diagram of a DC transformer circuit topology based on a series-connected three-phase bridge circuit proposed by the present invention;

[0025] Figure 2 is a schematic structural diagram of a set of three-phase bridge modules in a DC transformer circuit topology based on a series-connected three-phase bridge circuit proposed by the present invention;

[0026] Figure 3 is a working waveform diagram of some components according to different phase-shift angle value ranges in the actual working state of a DC transformer for a DC transformer circuit topology control method based on a series-connected three-phase bridge circuit proposed by the present invention;

[0027] Figure 4 is corresponding to the actual working state of a DC transformer for a DC transformer circuit topology control method based on a series-connected three-phase bridge circuit proposed by the present invention Figure 3 schematic diagram of the current and on / off states of the corresponding components;

[0028] Reference Signs:

[0029] Q1-Q6, S1-S6: switching tubes; Ca, Cb, Cc, C LV , Co, Cda, Cdb, Cdc: bus capacitors; Tra, Trb, Trc: three-phase windings corresponding to the high-frequency isolation transformer; La, Lb, Lc: three-phase transmission inductors;

[0030] At the same time, Figure 4Including: (a), (b), (c), (d), (e) and (f), wherein, (a) is a schematic diagram of the component current and on / off state in the time range [t0, t2]; (b) is a schematic diagram of the component current and on / off state in the time range [t2, t3]; (c) is a schematic diagram of the component current and on / off state in the time range [t3, t5]; (d) is a schematic diagram of the component current and on / off state in the time range [t5, t6]; (e) is a schematic diagram of the component current and on / off state in the time range [t6, t8]; (f) is a schematic diagram of the component current and on / off state in the time range [t8, t9]. Detailed implementation manner

[0031] To make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0032] The inventive concept of the present invention lies in: aiming at the problems of large floor area and low power density of DC transformers in the existing urban distribution network system, a DC transformer based on a series-connected three-phase bridge circuit with a high topological voltage ratio designed on the basis of the same power devices and the input port voltage increased by 3 times is realized, so as to reduce the number of power modules to 2 / 3 under the same conditions, effectively improve the power density of the DC transformer, and at the same time adopt a symmetric duty cycle and phase-shifted hybrid control method to realize the bidirectional energy flow and full-range soft switching of the DC transformer.

[0033] Circuit topology embodiment 1:

[0034] On the one hand, as Figure 1 shown, it is a schematic structural diagram of a DC transformer circuit topology based on a series-connected three-phase bridge circuit proposed by the present invention. Among them, a high-voltage side power unit, a low-voltage side power unit and an isolation transformer unit connecting the high-voltage side power unit and the low-voltage side power unit. Specifically:

[0035] The high-voltage side power unit (Vin) includes N series-connected three-phase series half-bridge circuits in sequence; the low-voltage side power unit (Vo) includes N three-phase parallel half-bridge circuits corresponding to each three-phase series half-bridge circuit and connected in parallel in sequence; among them, each three-phase series half-bridge circuit is connected to its corresponding three-phase parallel half-bridge circuit through a three-phase isolation transformer to form N groups of three-phase bridge modules; here, N is an integer greater than or equal to 1; the isolation transformer is preferably a high-frequency isolation transformer in practical applications.

[0036] The three-phase series half-bridge circuit includes the first arm of phase A, the first arm of phase B, and the first arm of phase C that are connected in series with each other; each first arm of a phase includes two switching tubes that are connected in series, have complementary states, and have symmetric duty cycles; that is, switching tube Q1 and switching tube Q2 form the first arm of phase A, switching tube Q3 and switching tube Q4 form the first arm of phase B, and switching tube Q5 and switching tube Q6 form the first arm of phase C.

[0037] In the three-phase series half-bridge circuit, the connection point of each first arm of a phase is connected in series with a DC-blocking capacitor and an inductor, and after the series connection, it is connected to the input terminal of the corresponding phase of the primary side of the three-phase isolation transformer; that is, the connection point of switching tube Q1 and switching tube Q2 is connected in series with DC-blocking capacitor Cda and inductor La and then connected to the input terminal of phase A of the primary side of the high-frequency isolation transformer; the connection point of switching tube Q3 and switching tube Q4 is connected in series with DC-blocking capacitor Cdb and inductor Lb and then connected to the input terminal of phase B of the primary side of the high-frequency isolation transformer; the connection point of switching tube Q5 and switching tube Q6 is connected in series with DC-blocking capacitor Cdc and inductor Lc and then connected to the input terminal of phase C of the primary side of the high-frequency isolation transformer. At the same time, each first arm of a phase is connected in parallel with its corresponding phase bus capacitor; that is, bus capacitor Ca is connected in parallel with the first arm of phase A, Cb is connected in parallel with the first arm of phase B, and Cc is connected in parallel with the first arm of phase C, so that three bus capacitors (Ca, Cb, Cc) are connected in series at the input terminal of the three-phase series half-bridge circuit. At the same time, switching tubes Q1 - Q6 are also each connected in parallel with a diode.

[0038] The three-phase parallel half-bridge circuit includes the second arm of phase A, the second arm of phase B, and the second arm of phase C that are connected in parallel with each other; each second arm of a phase includes two switching tubes that are connected in series, have complementary states, and have symmetric duty cycles; that is, switching tube S1 and switching tube S2 form the second arm of phase A, switching tube S3 and switching tube S4 form the second arm of phase B, and switching tube S5 and switching tube S6 form the second arm of phase C.

[0039] In the three-phase parallel half-bridge circuit, the connection point of each second arm of a phase is connected to the input terminal of the corresponding phase of the secondary side of the three-phase isolation transformer; that is, the connection point of switching tube S1 and switching tube S2 is connected to the input terminal of phase A of the secondary side of the high-frequency isolation transformer, the connection point of switching tube S3 and switching tube S4 is connected to the input terminal of phase B of the secondary side of the high-frequency isolation transformer, and the connection point of switching tube S5 and switching tube S6 is connected to the input terminal of phase C of the secondary side of the high-frequency isolation transformer. At the same time, bus capacitors are connected in parallel at the output terminal of the three-phase parallel half-bridge circuit; that is, bus capacitor C LV is connected in parallel with the arms of phases A, B, and C so that C LV is connected in parallel with the arms of phases A, B, and C. At the same time, switching tubes S1 - S6 are also each connected in parallel with a diode.

[0040] Through the above embodiments, voltage conversion of high-voltage DC and low-voltage DC buses and bidirectional power transmission are carried out, so that the voltage level of the high-voltage side unit is relatively high, which can effectively reduce the number of power modules of the whole DC transformer and improve the power density.

[0041] Circuit topology embodiment 2:

[0042] As Figure 2 shown, it is a schematic structural diagram of a set of three-phase bridge modules in a DC transformer circuit topology based on a series-connected three-phase bridge circuit proposed by the present invention. Among them, V M is the total voltage of the high-voltage side; V L is the total voltage of the low-voltage side; Switching transistors Q1 - Q6 are switching power devices on the high-voltage side, and switching transistors S1 - S6 are switching power devices on the low-voltage side; moreover, diodes are built in the power devices; Ca, Cb, and Cc are series bus capacitors at the high-voltage side input, Co is a parallel bus capacitor at the low-voltage side output, Cda, Cdb, and Cdc are primary DC-blocking capacitors of the high-frequency isolation transformer, La, Lb, and Lc are three-phase transmission inductors, and Tra, Trb, and Trc are three-phase windings corresponding to the high-frequency isolation transformer.

[0043] Through the above embodiments, switching transistors on the corresponding high- and low-voltage sides are set in each phase of each module, and a high-frequency isolation transformer connecting the high- and low-voltage sides is set in each phase. It is the topological basis for realizing that on the basis of the same power devices, the input port voltage is increased by 3 times, and the number of power modules is reduced to 2 / 3 under the same conditions.

[0044] Method embodiment 1:

[0045] On the other hand, the present invention also proposes a control method for a DC transformer circuit topology based on a series-connected three-phase bridge circuit. Among them, the method includes steps S11 - S14. Specifically:

[0046] Step S11, set the duty cycle of the switching transistors of each bridge arm in each module to a symmetric duty cycle; here, the symmetric duty cycle means that in a periodic signal, the high-level duration is the same as the low-level duration, and it can be represented by a 50% duty cycle; at the same time, the states of the switching transistors of each bridge arm are complementary (the upper and lower switching transistors conduct complementarily), that is, each corresponding high-voltage side power unit and low-voltage side power unit satisfy: switching transistor Q1 is complementary to switching transistor Q2, switching transistor Q3 is complementary to switching transistor Q4, switching transistor Q5 is complementary to switching transistor Q6, switching transistor S1 is complementary to switching transistor S2, switching transistor S3 is complementary to switching transistor S4, and switching transistor S5 is complementary to switching transistor S6; in an actual application scenario, phase shift angles lagging by 120° are also set for switching transistors Q1, Q3, and Q5 respectively to achieve power loss between each module.

[0047] Step S12, preset the phase difference of the driving pulses between the current module and the subsequent module, so that the driving pulses of the subsequent module lag behind those of the current module; it should be noted that the phase difference is set to 360° / N, where N is the total number of three-phase bridge modules, that is, the phase of the driving PWM pulse of the subsequent module lags behind the phase of the previous module by 360° / N, and the adjustment is cycled continuously. For example: the driving PWM pulses of Q1 and S1 of the second module lag behind the driving PWM pulses of Q1 and S1 of the first module by 360° / N, and so on for the subsequent modules. At the same time, the PWM driving pulses of the remaining switching tubes of the second module to the Nth module are generated by the symmetric duty cycle and phase-shift control method to control the switching tubes to turn on and achieve power flow.

[0048] Step S13, for each module, preset the phase-shift angle for lagging the low-voltage side switching tube behind the high-voltage side switching tube in the current module; here, the phase-shift angle is calculated by the following formula:

[0049]

[0050] where, is the phase-shift angle of phase i (i = a, b, c); T s is the switching period; is the switching period of phase i.

[0051] At the same time, the value range of the phase-shift angle is 0 ≤ φ < π / 3 or π / 3 ≤ φ < 2π / 3, so that six working modes are formed between each module within each value range, and the three-phase transmitted power is adjusted by adjusting the three-phase phase-shift angles.

[0052] Step S14, based on the phase-shift angles of all modules, input PWM pulses to all three-phase bridge modules with the phase difference to adjust the on and off of the switching tubes in each module.

[0053] Through the above steps S11 - S14, the method of symmetric duty cycle and phase-shift mixing is achieved to ensure that the switching tubes of the high-voltage side power unit and the low-voltage side power unit can all achieve zero-voltage switching within the full load range, ensuring the zero-voltage turn-on of all switching tubes, which helps to reduce the switching loss of the DC transformer and improve the efficiency.

[0054] Method Embodiment 2:

[0055] Corresponding to Figure 3 (In the working waveform diagram of some components according to different phase-shift angle value ranges in the actual working state of the DC transformer for a DC transformer circuit topology control method based on a series-connected three-phase bridge circuit proposed by the present invention), analyze the working principle of the first half cycle when the phase-shift angle is in the range of 0 ≤ φ < π / 3, and obtain Figure 4Schematic diagram of the component current and on / off states, specifically:

[0056] 1) t0: Before the moment of t0, the primary side currents i pa , i pb and i pc flow through Q2, Q4, and Q5 respectively, and the secondary side currents i sa , i sb and i sc flow through the anti-parallel diodes of the switching tubes S2, S4, and S5 respectively. At the moment of t0, Q2 is turned off, and the current i pa immediately flows through Q1.

[0057] 2) Mode 1 [t0, t2]: As Q2 is turned off, the voltage v A’N changes from –V M / 9 to V M / 9, and the voltage of the phase A inductor La becomes (V M / 9 + KV L / 3). Therefore, the primary side current i pa of phase A rises linearly, while i pb and i pc decline.

[0058] 3) Mode 2 [t2, t3]: At the moment of t2, S2 is turned off, and the current i sa freewheels through the diode S1. At this time, turning on S1 can achieve ZVS (zero voltage switching) turn-on. In this mode, the voltage v an rises from –V L / 3 to V L / 3, the voltage of the phase A inductor La becomes (V M / 9 – KV L / 3), v Lb and v Lc are respectively (–2V M / 9 + 2KV L / 3) and (V M / 9 – KV L / 3).

[0059] 4) Mode 3 [t3, t5]: At the moment of t3, Q5 is turned off, and the current i pc freewheels through the diode Q6. At this time, turning on Q6 can achieve ZVS turn-on. The three-phase inductor voltages v La , v Lb , v Lc are respectively (2V M / 9 - KV L / 3), (–V M / 9 + 2KV L / 3), and (–V M / 9 - KV L / 3).

[0060] 5) Mode 4 [t5, t6]: Turn off S5 at t5, and current i sc immediately flows through S6. At this time, turning on S6 can achieve ZVS turn-on. In this mode, voltage v La changes to (2V M / 9 - 2KV L / 3), while v Lb and v Lc are both (-V M / 9 + KV L / 3). Similarly, if V M = 3KV L , i pa ~ i pc will remain unchanged in Mode 4.

[0061] 6) Mode 5 [t6, t8]: When turning off Q4 at t6, current i pb flows through Q3. At this time, turning on Q3 can achieve ZVS turn-on. In this mode, voltages v La , v Lb , v Lc become (V M / 9 - 2KV L / 3), (V M / 9 + KV L / 3) and (-2V M / 9 + KV L / 3) respectively, causing i pa and i pc to decrease, while i pb increases. At t7, i pb rises through zero, i pb transfers from Q3 to Q3, and at the same time, i sb transfers from S4 to S4.

[0062] 7) Mode 6 [t8, t9]: At t8, S4 is turned off, and i sb flows through diode S3. At this time, turning on S3 can achieve ZVS turn-on. As S4 is turned off, inductor voltages v La and v Lb both become (V M / 9 - KV L / 3), while v Lc is (-2V M / 9 + 2KV L / 3). If V M = 3KVL, i pa ~ i pc will remain unchanged in this mode.

[0063] In summary, the present invention has the following beneficial technical effects: 1. The present invention proposes a DC transformer topology based on a series-connected three-phase bridge circuit. The high-voltage side power unit adopts a three-phase series half-bridge topology, the low-voltage side power unit adopts a three-phase parallel half-bridge topology, and a high-frequency isolation transformer is used in the middle to achieve electrical isolation and voltage transformation, enabling bidirectional energy transmission. 2. The present invention proposes a symmetric duty cycle and phase-shifted hybrid control method to ensure that the switch cabinets of both the high-voltage side power unit and the low-voltage side power unit can achieve zero-voltage switching within the full load range. 3. The topology of the present invention can be extended to an ISOP topology with series-connected medium-voltage sides and parallel-connected low-voltage sides composed of N power units, which is suitable for high-voltage and large-capacity occasions in urban distribution systems. 4. Compared with the conventional full-bridge converter topology, this topology can achieve a three-fold increase in the high-voltage side voltage, a 2 / 3 reduction in the number of power modules, and a theoretically 30% increase in power density.

Claims

1. A DC transformer circuit topology based on a series three-phase bridge circuit, comprising a high-voltage side power unit, a low-voltage side power unit and an isolation transformer unit connecting the high-voltage side power unit and the low-voltage side power unit, characterized in that: The high-voltage side power unit includes N three-phase series half-bridge circuits connected in series in sequence; the low-voltage side power unit includes N three-phase parallel half-bridge circuits corresponding to each three-phase series half-bridge circuit and connected in parallel in sequence; wherein each three-phase series half-bridge circuit is connected to its corresponding three-phase parallel half-bridge circuit through a three-phase isolation transformer to form N groups of three-phase bridge modules; The three-phase series half-bridge circuit comprises a first bridge arm of phase A, a first bridge arm of phase B and a first bridge arm of phase C connected in series; the first bridge arm of each phase comprises two switch tubes connected in series, with complementary states and symmetrical duty cycles; The three-phase parallel half-bridge circuit includes a second bridge arm of phase A, a second bridge arm of phase B and a second bridge arm of phase C which are connected in parallel to each other; the second bridge arm of each phase includes two switching tubes which are connected in series with each other, have complementary states and symmetrical duty cycles.

2. The DC transformer circuit topology based on the series three-phase bridge circuit according to claim 1 is characterized in that: A connection point of the first bridge arm of each phase is connected in series with a DC blocking capacitor and an inductor, and after the series connection, is connected to the input end of the primary side of the three-phase isolation transformer and the corresponding phase; The connection point of the second bridge arm of each phase is connected to the input end of the corresponding phase of the secondary side of the three-phase isolation transformer.

3. The DC transformer circuit topology based on the series three-phase bridge circuit according to claim 2 is characterized in that: The first bridge arm of each phase is connected in parallel with the bus capacitor of its corresponding phase.

4. The DC transformer circuit topology based on the series three-phase bridge circuit according to claim 2 is characterized in that: A bus capacitor is connected in parallel to the output end of the three-phase parallel half-bridge circuit.

5. The DC transformer circuit topology based on a series three-phase bridge circuit according to any one of claims 1 to 4, characterized in that: The switch tube is also connected in parallel with the diode.

6. A DC transformer circuit topology control method based on a series three-phase bridge circuit, characterized in that: Follow these steps: The duty cycle of the switch tube of each bridge arm in each module is set to a symmetrical duty cycle; Presetting the phase difference of the driving pulse between the current module and the next module so that the driving pulse of the next module lags behind the driving pulse of the current module; For each module, a phase shift angle is preset to lag the low-voltage side switch tube in the current module behind the high-voltage side switch tube; Based on the phase shift angles of all modules, PWM pulses are input to all three-phase bridge modules with the phase difference to adjust the on and off of the switch tubes in each module.

7. The DC transformer circuit topology control method based on a series three-phase bridge circuit according to claim 6, characterized in that: The phase difference is set to 360° / N, where N is the total number of three-phase bridge modules.

8. The DC transformer circuit topology control method based on a series three-phase bridge circuit according to claim 6, characterized in that: The phase shift angle is calculated by the following formula: in, is the phase shift angle of phase i (i=a, b, c); T s is the switching cycle; is the switching period of phase i.

9. The DC transformer circuit topology control method based on a series three-phase bridge circuit according to any one of claims 6 to 8, characterized in that: The phase shift angle range is 0≤ φ < π / 3 or π / 3≤ φ <2 π / 3.

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