CN1545194A - Cascaded Bidirectional DC-DC Converter - Google Patents
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Abstract
Description
技术领域technical field
本发明的级联式双向DC-DC变换器属电力电子变换器。The cascaded bidirectional DC-DC converter of the invention belongs to a power electronic converter.
背景技术Background technique
隔离式双向DC-DC变换器的传统方案有以下三种:There are three traditional solutions for isolated bidirectional DC-DC converters:
1.移相式双向变换技术1. Phase-shifting bidirectional conversion technology
这种技术是利用变压器漏电感(或外接电感),通过桥臂移相来完成双向能量传输。它利用漏电感来传输能量,限制了这种技术在大功率的应用场合的应用。This technology uses transformer leakage inductance (or external inductance) to complete bidirectional energy transmission through bridge arm phase shifting. It uses leakage inductance to transfer energy, limiting the application of this technology in high-power applications.
2.电压型-电流型组合式双向变换技术2. Voltage type-current type combined bidirectional conversion technology
这种技术即为电压型-电流型组合式拓扑,它的主要缺陷在于开关器件上承受的由变压器漏电感造成的电压尖峰很大。这个缺陷是由于这种技术中含有的电流型拓扑造成的,限制了这种技术的应用。This technology is the combined voltage mode-current mode topology, and its main drawback is that the voltage spikes on the switching devices caused by the leakage inductance of the transformer are large. This drawback is due to the current mode topology involved in this technology, which limits the application of this technology.
3.反激式双向变换技术3. Flyback bidirectional conversion technology
这种技术利用单端反激变换器为主拓扑,它的主要缺陷在于:利用耦合电感(反激变压器)来传输能量,限制了其在大功率场合的应用。This technology uses a single-ended flyback converter as the main topology. Its main defect is that it uses a coupled inductor (flyback transformer) to transmit energy, which limits its application in high-power applications.
发明内容Contents of the invention
本发明的目的在于针对上述现有技术存在的缺陷,研制一种能克服现有缺陷的,能实现全部开关管的零电压开关和电气隔离,具有两部分相对独立,可独立实现控制且电路简单的级联式双向DC-DC变换器。The purpose of the present invention is to aim at the above-mentioned defects in the prior art, and develop a method that can overcome the existing defects, realize zero-voltage switching and electrical isolation of all switching tubes, have two parts that are relatively independent, can independently realize control, and have a simple circuit cascaded bidirectional DC-DC converter.
本发明级联式双向DC-DC变换器,由相对独立的不隔离双向DC-DC变换器与高频直流变压器两部分相互连接成级联结构式电路。其中,不隔离双向DC-DC变换器可以为双向升压式(Boost)变换器,双向降压式(Buck)变换器,双向升降压式(Buck/Boost)变换器等拓扑;直流变压器由高频逆变/整流电路连于高频变压器原边,其副边与高频整流/逆变电路相连,其中高频整流/逆变单元可以是全波整流/推挽逆变,全桥整流/全桥逆变,半桥整流/半桥逆变,推挽正激整流/推挽正激逆变等结构构成,同样高频逆变/整流单元也可以是推挽逆变/全波整流,全桥逆变/全桥整流,半桥逆变/半桥整流,推挽正激逆变/推挽正激整流等结构。The cascaded bidirectional DC-DC converter of the present invention is composed of a relatively independent non-isolated bidirectional DC-DC converter and a high-frequency DC transformer connected to each other to form a cascaded structural circuit. Among them, the non-isolated bidirectional DC-DC converter can be a bidirectional step-up (Boost) converter, a bidirectional step-down (Buck) converter, a bidirectional step-down (Buck/Boost) converter and other topologies; the DC transformer consists of The high-frequency inverter/rectifier circuit is connected to the primary side of the high-frequency transformer, and its secondary side is connected to the high-frequency rectifier/inverter circuit. The high-frequency rectifier/inverter unit can be full-wave rectifier/push-pull inverter, full-bridge rectifier /Full-bridge inverter, half-bridge rectifier/half-bridge inverter, push-pull forward rectification/push-pull forward inverter and other structures, the same high-frequency inverter/rectifier unit can also be push-pull inverter/full-wave rectifier , Full-bridge inverter/full-bridge rectifier, half-bridge inverter/half-bridge rectifier, push-pull forward inverter/push-pull forward rectifier and other structures.
附图说明Description of drawings
图1.级联式双向DC-DC变换器的结构框图。其中图(a)为不隔离双向变换器和高频直流变压器级联,图(b)为高频直流变压器和不隔离双向变换器级联。Figure 1. Block diagram of a cascaded bidirectional DC-DC converter. Among them, figure (a) is a cascaded connection of a non-isolated bidirectional converter and a high-frequency DC transformer, and figure (b) is a cascaded connection of a high-frequency DC transformer and a non-isolated bidirectional converter.
图2.不隔离双向DC-DC变换器的三种典型电路结构主电路原理图。其中图(c)是双向升压式变换器主电路原理图,图(d)是双向降压式变换器主电路原理图,图(e)是双向升降压式变换器主电路原理图。Figure 2. Schematic diagram of the main circuit of three typical circuit structures of non-isolated bidirectional DC-DC converters. Figure (c) is a schematic diagram of the main circuit of a bidirectional boost converter, Figure (d) is a schematic diagram of the main circuit of a bidirectional buck converter, and Figure (e) is a schematic diagram of the main circuit of a bidirectional buck converter.
图3.高频直流变压器构成主电路原理图。Figure 3. Schematic diagram of the main circuit composed of a high-frequency DC transformer.
图4.直流变压器构成单元示意图。其中图(f)为推挽电路,图(g)为半桥电路,图(h)为全桥电路,图(i)为推挽正激电路,图(j)为双管正激电路,图(k)为有源箝位正激电路,图(1)为不对称半桥电路。Figure 4. Schematic diagram of the constituent units of a DC transformer. Among them, picture (f) is a push-pull circuit, picture (g) is a half-bridge circuit, picture (h) is a full-bridge circuit, picture (i) is a push-pull forward circuit, picture (j) is a two-tube forward circuit, Figure (k) is an active clamp forward circuit, and Figure (1) is an asymmetrical half-bridge circuit.
图5.级联式双向DC-DC变换器的两种典型主电路原理图。其中,图(m)为不隔离降压式(Buck)双向DC-DC变换器+全桥逆变/全桥整流和全桥整流/全桥逆变构成的直流变压器级联组成的级联式双向DC-DC变换器主电路原理图;图(n)为不隔离升压式(Boost)双向DC-DC变换器+推挽逆变/全波整流和全桥整流/全桥逆变构成的直流变压器级联组成的级联式双向DC-DC变换器主电路原理图。Figure 5. Two typical main circuit schematics of cascaded bidirectional DC-DC converters. Among them, the figure (m) is a cascaded type composed of non-isolated step-down (Buck) bidirectional DC-DC converter + full-bridge inverter/full-bridge rectifier and full-bridge rectifier/full-bridge inverter composed of cascaded DC transformers Schematic diagram of the main circuit of the bidirectional DC-DC converter; Figure (n) is a non-isolated boost (Boost) bidirectional DC-DC converter + push-pull inverter/full-wave rectifier and full-bridge rectifier/full-bridge inverter. Schematic diagram of the main circuit of the cascaded bidirectional DC-DC converter composed of cascaded DC transformers.
图6.不隔离双向升压式DC-DC变换器主电路原理图。Figure 6. Schematic diagram of the main circuit of the non-isolated bidirectional step-up DC-DC converter.
图7.不隔离双向升压式(Boost)DC-DC变换器电感电流等电流电压波形。Figure 7. Non-isolated bidirectional boost (Boost) DC-DC converter inductor current and other current and voltage waveforms.
图8.双向升压式DC-DC变换器等效模态图(采用电感电流iLIII交变工作方式)。Figure 8. Equivalent modal diagram of a bidirectional step-up DC-DC converter (using the alternating working mode of inductor current iLIII).
图9.采用推挽电路和全桥电路构成的直流变压器主电路原理图。Figure 9. Schematic diagram of the main circuit of a DC transformer composed of a push-pull circuit and a full-bridge circuit.
图10.直流变压器工作原理波形。Figure 10. DC transformer working principle waveform.
图11.级联式双向DC-DC变换器的串并联结构主电路原理图。Figure 11. Schematic diagram of the main circuit of the series-parallel structure of the cascaded bidirectional DC-DC converter.
具体实施方式Detailed ways
图1是级联式双向DC-DC变换器的结构框图,由图1可知,级联式双向DC-DC变换器是由不隔离双向DC-DC变换器与高频直流变压器相互连接而成。即不隔离双向变换器和高频直流变压器级联,或者反过来,高频直流变压器和不隔离双向变换器级联。Figure 1 is a structural block diagram of a cascaded bidirectional DC-DC converter. It can be seen from Figure 1 that the cascaded bidirectional DC-DC converter is formed by connecting a non-isolated bidirectional DC-DC converter and a high-frequency DC transformer. That is, the non-isolated bidirectional converter is cascaded with the high-frequency DC transformer, or conversely, the high-frequency DC transformer is cascaded with the non-isolated bidirectional converter.
不隔离双向DC-DC变换器可为图2中的图(c)所示的双向升压式变换器,图(d)所示的双向降压式变换器电路和图(e)所示的双向升降压式变换器等双向变换拓扑。The non-isolated bidirectional DC-DC converter can be a bidirectional boost converter shown in Figure (c) in Figure 2, a bidirectional buck converter circuit shown in Figure (d) and a bidirectional buck converter circuit shown in Figure (e) Bidirectional conversion topologies such as bidirectional buck-boost converters.
高频直流变压器的组成如图3所示,即高频逆变/整流电路连于高频变压器原边,高频变压器副边连于高频整流/逆变电路所构成。其中高频逆变/整流单元和高频整流/逆变单元可以采用如图4所示的几种结构形式的电路。高频逆变/整流单元和高频整流/逆变单元所连接的高频变压器的耦合关系,可以根据需要任意选择配合。The composition of the high-frequency DC transformer is shown in Figure 3, that is, the high-frequency inverter/rectifier circuit is connected to the primary side of the high-frequency transformer, and the secondary side of the high-frequency transformer is connected to the high-frequency rectifier/inverter circuit. Among them, the high-frequency inverter/rectifier unit and the high-frequency rectifier/inverter unit can adopt circuits of several structural forms as shown in FIG. 4 . The coupling relationship between the high-frequency inverter/rectifier unit and the high-frequency transformer connected to the high-frequency rectifier/inverter unit can be arbitrarily selected and coordinated as required.
采用图1所示的结构框图,图2所示的几种不隔离双向DC-DC变换拓扑,图3所示的高频直流变压器的构成示意图和图4所示的直流变压器构成单元,可以构成不同结构形式的级联式双向DC-DC变换器拓扑。图5所示的就是两种形式的拓扑示意图。其中,图(m)为不隔离降压式(Buck)双向DC-DC变换器+全桥逆变/全桥整流和全桥整流/全桥逆变构成的直流变压器级联组成的级联式双向DC-DC变换器示意图;图(n)为不隔离升压式(Boost)双向DC-DC变换器+推挽逆变/全波整流和全桥整流/全桥逆变构成的直流变压器级联组成的级联式双向DC-DC变换器示意图。下面以图(n)所示拓扑为例,说明其工作原理。Using the structural block diagram shown in Figure 1, several non-isolated bidirectional DC-DC conversion topologies shown in Figure 2, the schematic diagram of the high-frequency DC transformer shown in Figure 3 and the DC transformer components shown in Figure 4, it can be constructed Cascaded bidirectional DC-DC converter topologies with different structures. Figure 5 shows the topology diagrams of the two forms. Among them, the figure (m) is a cascaded type composed of non-isolated step-down (Buck) bidirectional DC-DC converter + full-bridge inverter/full-bridge rectifier and full-bridge rectifier/full-bridge inverter composed of cascaded DC transformers Schematic diagram of a bidirectional DC-DC converter; Figure (n) is a DC transformer stage composed of a non-isolated boost (Boost) bidirectional DC-DC converter + push-pull inverter/full-wave rectifier and full-bridge rectifier/full-bridge inverter A schematic diagram of a cascaded bidirectional DC-DC converter composed of cascades. Take the topology shown in Figure (n) as an example to illustrate its working principle.
不隔离升压式(Boost)双向DC-DC变换器工作原理Working principle of non-isolated boost (Boost) bidirectional DC-DC converter
级联式双向DC-DC变换器由两部分构成,这两部分的控制电路可以相互独立,因此可以分别说明其工作原理。The cascaded bidirectional DC-DC converter consists of two parts, and the control circuits of these two parts can be independent of each other, so their working principles can be explained separately.
把升压式(Boost)变换器的单向开关换成双向开关就构成了升压式(Boost)双向DC-DC变换器。图6为不隔离升压式(Boost)双向DC-DC变换器,图中以功率场效应管(Power MOSFET)代表双向开关。其中双向开关为具有双向电流流动能力的有源可控开关,包括Power MOSFET,IGBT,CoolMOS,或将来发明的类似开关,也可以是具有同样作用的复合开关结构。V1、V2为直流电压源或直流有源负载。在升压式(Boost)双向DC-DC变换器中,电感L有三种工作模式,图7为开关管控制信号和电感电流波形。A boost bidirectional DC-DC converter is formed by replacing the unidirectional switch of the boost converter with a bidirectional switch. Figure 6 is a non-isolated boost (Boost) bidirectional DC-DC converter, in which a power MOSFET is used to represent a bidirectional switch. The bidirectional switch is an active controllable switch with bidirectional current flow capability, including Power MOSFET, IGBT, CoolMOS, or a similar switch invented in the future, or a composite switch structure with the same function. V 1 and V 2 are DC voltage sources or DC active loads. In the boost type (Boost) bidirectional DC-DC converter, the inductor L has three working modes. Figure 7 shows the control signal of the switching tube and the waveform of the inductor current.
在第一种工作模式下,电感电流(iLI)恒大于零,能量由电压源V1流向电压V2侧,是常规的升压式(Boost)变换器。开关管S1导通期间,电压V1加在电感L1上,电感电流增加,开关管S1关断后,二极管D2导通续流,电感L1上的电压为:V2-V1,电感电流减小。设开关管S1的导通占空比为D,由稳态工作时电感L1上的正负“伏秒”面积相等可得:In the first working mode, the inductor current (i LI ) is always greater than zero, and the energy flows from the voltage source V 1 to the voltage V 2 side, which is a conventional boost converter. During the conduction period of the switch tube S1 , the voltage V1 is applied to the inductor L1 , and the inductor current increases. After the switch tube S1 is turned off, the diode D2 conducts freewheeling, and the voltage on the inductor L1 is: V 2 -V 1 , the inductor current decreases. Assuming that the conduction duty cycle of the switch tube S1 is D, the positive and negative "volt-second" areas on the inductor L1 are equal in steady state operation:
第二种工作模式下,电感电流(iLII)恒小于零,能量由电压源V2流向电压V1侧,是常规的降压式(Buck)变换器。开关管S2导通期间,电感L1上的电压为:V2-V1,电感电流增加,开关管S2关断后,二极管D1导通续流,电压V1加在电感L1上,电感电流减小。设开关管S2的导通占空比为(1-D),由稳态工作时L1上的正负“伏秒”面积相等可得:In the second working mode, the inductor current (i LII ) is always less than zero, and the energy flows from the voltage source V 2 to the voltage V 1 side, which is a conventional step-down (Buck) converter. During the conduction period of the switch tube S 2 , the voltage on the inductor L 1 is: V 2 -V 1 , and the inductor current increases. After the switch tube S 2 is turned off, the diode D 1 conducts freewheeling, and the voltage V 1 is added to the inductor L 1 , the inductor current decreases. Assuming that the conduction duty cycle of the switching tube S2 is (1-D), the positive and negative "volt-second" areas on L1 are equal in steady state operation:
V1=(1-D)V2 V 1 =(1-D)V 2
第三种工作模式下,电感电流过零,当电感电流的平均值大于0时,能量由电压源V1流向电压V2源,而电感电流的平均值小于0时,能量由电压源V2流向电压源V1。能量由电压源V1流向电压源V2时有6个工作模态。等效电路如图8所示。In the third working mode, the inductor current crosses zero. When the average value of the inductor current is greater than 0, the energy flows from the voltage source V 1 to the voltage V 2 source. When the average value of the inductor current is less than 0, the energy flows from the voltage source V 2 flow to voltage source V 1 . There are 6 working modes when energy flows from voltage source V 1 to voltage source V 2 . The equivalent circuit is shown in Figure 8.
[t0-t1]t0时刻前,二极管D1续流导通,t0时刻,开关管S1零电压导通。iL在电压V1的作用下增大。[t 0 -t 1 ] Before time t 0 , diode D 1 conducts freewheeling, and at time t 0 , switching tube S 1 conducts with zero voltage. i L increases under the action of voltage V1 .
[t1-t2]t1时刻,电感电流过零,并经开关管S1增大,t2时刻电感电流iL达到最大值iLmax。[t 1 -t 2 ] At time t 1 , the inductor current crosses zero and increases through the switch tube S 1 , and the inductor current i L reaches the maximum value i Lmax at time t 2 .
[t2-t3]t2时刻,开关管S1关断,二极管D2自然续流,电感电流iL在电压V2-V1作用下减小。[t 2 -t 3 ] At time t 2 , the switch tube S 1 is turned off, the diode D 2 continues to flow naturally, and the inductor current i L decreases under the action of the voltage V 2 -V 1 .
[t3-t4]t3时刻,开关管S2零电压导通,S2为同步整流工作方式。电感电流通过S2给电容C和负载提供能量。[t 3 -t 4 ] At time t 3 , switch tube S 2 is turned on with zero voltage, and S 2 is in synchronous rectification mode. The inductor current provides energy to the capacitor C and the load through S2 .
[t4-t5]t4时刻,电感电流在电压V2-V1的作用下过零,通过开关管S2反向流动。[t 4 -t 5 ] At time t 4 , the inductor current crosses zero under the action of the voltage V 2 -V 1 and flows in reverse through the switch tube S 2 .
[t5-t6]t5时刻,开关管S2关断,电感电流达到最小值iLmin,此后二极管D1续流,t6时刻零电压开通S1,开始下一个开关周期。[t 5 -t 6 ] At time t 5 , switch S 2 is turned off, and the inductor current reaches the minimum value i Lmin , after which diode D 1 freewheels, and at time t 6 S 1 is turned on with zero voltage to start the next switching cycle.
功率流反向时,工作模态是类似的,这里从略。When the power flow is reversed, the working mode is similar, which is omitted here.
在这种工作模式下,设开关管S1的导通占空比为D,由稳态工作时电感L1上的正负“伏秒”面积相等同样可以得到:V1=(1-D)V2。In this mode of operation, assuming that the conduction duty cycle of the switch tube S1 is D, the positive and negative "volt-second" areas on the inductor L1 during steady-state operation are equal, which can also be obtained: V 1 =(1-D )V 2 .
在这种工作模式下开关管S1、S2都是零电压开通(ZVS),开关损耗小;二极管D1、D2都是自然导通和关断的,没有反向恢复问题。因此通过合理设计电感,使电路工作在第三种工作模式,可以实现电路的可靠高效工作。In this working mode, the switch tubes S 1 and S 2 are zero-voltage turn-on (ZVS), and the switching loss is small; the diodes D 1 and D 2 are naturally turned on and off, and there is no reverse recovery problem. Therefore, by rationally designing the inductor to make the circuit work in the third working mode, reliable and efficient operation of the circuit can be realized.
直流变压器工作原理Working principle of DC transformer
直流变压器能将一种直流电压变换成与之成正比的另一种或多种直流电压。通常直流变压器的开关管工作在固定占空比下,完成电压变换功能,不进行调压控制。包括单向和双向两种类型。在双向直流变压器中,所用的开关管均为双向开关。下面以图9所示拓扑为例说明其工作原理。在图9中,Llk为变压器的等效漏电感,V1、V2为直流电压源或直流有源负载。A DC transformer can transform a DC voltage into another or more DC voltages proportional to it. Usually, the switching tube of the DC transformer works under a fixed duty ratio to complete the voltage conversion function without voltage regulation control. Including one-way and two-way two types. In the bidirectional DC transformer, the switching tubes used are bidirectional switches. The following uses the topology shown in FIG. 9 as an example to illustrate its working principle. In Fig. 9, L lk is the equivalent leakage inductance of the transformer, and V 1 and V 2 are DC voltage sources or DC active loads.
在图9所示拓扑中,开关管S3(S5、S8)和开关管S4(S6、S7)互补导通(留有死区,以避免直通),分别导通半个开关周期,这样等效占空比为1,故直流变压器的变比就等于高频变压器的变比。图10为工作原理波形,其中Vgs3(Vgs5、Vgs8)、Vgs4(Vgs6、Vgs7)为开关管控制信号,V4、V5为加在等效漏电感Llk两侧的电压波形,VLlk为加在等效漏电感上的电压波形,iLlk为等效漏电感中的电流。整个开关周期有6个开关模态,下面详细说明其工作原理。In the topology shown in Figure 9, the switching tubes S 3 (S 5 , S 8 ) and the switching tubes S 4 (S 6 , S 7 ) are turned on complementary (with a dead zone to avoid direct connection), and half of the The switching cycle, so the equivalent duty cycle is 1, so the transformation ratio of the DC transformer is equal to the transformation ratio of the high frequency transformer. Figure 10 is the waveform of the working principle, where V gs3 (V gs5 , V gs8 ), V gs4 (V gs6 , V gs7 ) are the switching tube control signals, and V 4 and V 5 are the voltages added to both sides of the equivalent leakage inductance L lk Voltage waveform, V Llk is the voltage waveform added to the equivalent leakage inductance, and i Llk is the current in the equivalent leakage inductance. There are 6 switching modes in the whole switching cycle, and its working principle is explained in detail below.
[t0-t1]在t0时刻以前,变压器Tr磁复位,二极管D3处于续流状态,二极管D5(D8)和二极管D6(D7)同时导通换流。t0时刻,开关管S3零电压(ZVS)导通,漏感电流继续以Vm/Llk的斜率增长(设VLlk的幅值为Vm),开关管S5(S8)也由于二极管D5(D8)续流可以零电压(ZVS)导通。[t 0 -t 1 ] Before time t 0 , transformer T r is magnetically reset, diode D 3 is in freewheeling state, and diode D 5 (D 8 ) and diode D 6 (D 7 ) are turned on and commutated at the same time. At time t 0 , switch S 3 is turned on with zero voltage (ZVS), and the leakage inductance current continues to increase at the slope of V m /L lk (assuming that the amplitude of V Llk is V m ), and switch S 5 (S 8 ) also Due to the diode D 5 (D 8 ) freewheeling can conduct at zero voltage (ZVS).
[t1-t2]t1时刻,漏感电流iLlk达到负载电流值,电感Llk上施加的电压为0,功率向电压V3侧传输,直到t2时刻开关管S3关断。[t 1 -t 2 ] At time t 1 , the leakage inductance current i Llk reaches the load current value, the voltage applied to the inductor L lk is 0, and the power is transmitted to the voltage V 3 side until the switch tube S 3 is turned off at time t 2 .
[t2-t3]t2时刻开关管S3在其输出结电容的缓冲下软关断,同时D4开始导通给变压器磁复位。二极管D6(D7)和二极管D5(D8)同时导通换流,因此开关管S5(S8)软关断。t3时刻,S4在二极管D4续流时零电压(ZVS)开通,开关管S6(S7)也在二极管D6(D7)续流时零电压(ZVS)开通,变换器进入下半个开关周期。下半周期的工作和上半周期的工作情形是对称的,这里从略。[t 2 -t 3 ] At time t 2 , the switch tube S 3 is softly turned off under the buffer of its output junction capacitance, and at the same time D 4 starts conducting to reset the transformer magnetically. Diode D 6 (D 7 ) and diode D 5 (D 8 ) conduct commutation at the same time, so switch tube S 5 (S 8 ) is softly turned off. At time t3 , S 4 is turned on at zero voltage (ZVS) when diode D 4 is freewheeling, and switch S 6 (S 7 ) is also turned on at zero voltage (ZVS) when diode D 6 (D 7 ) is freewheeling, and the converter enters the second half of the switching cycle. The work of the second half cycle is symmetrical to that of the first half cycle, which is omitted here.
图11所示的是本发明串并联结构的级联式双向DC-DC变换器的组成框图。其中,可将不隔离双向DC-DC变换器组成并联或交错并联的结构形式;将高频直流变压器组成并联、串联或多路输出/输入的结构形式;如图11中的图(o)、图(p)和图(q)所示。FIG. 11 is a block diagram of a cascaded bidirectional DC-DC converter with a series-parallel structure according to the present invention. Among them, non-isolated bidirectional DC-DC converters can be composed of parallel or interleaved parallel structures; high-frequency DC transformers can be composed of parallel, series or multiple output/input structures; as shown in Figure 11 (o), Figure (p) and Figure (q) shown.
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