# Research on the Comparison Analyses of Three-Phase Discrete and Integrated LC Filters in Three-Phase PV Inverter

Authors: Ying Jiang; Jianguo Li; Sanbo Pan; Xi Zhang; Peng Hu; Haiyan Zhang
Journal: International Journal of Photoenergy
 (2013)

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## Abstract

In three-phase photovoltaic (PV) system, three-phase filter inductors are important part for the output electrical power quality. The comparison analyses of three-phase discrete filter inductors and two kinds of three-phase integrated filter inductors in three-phase PV inverter are proposed. Firstly, the three-phase PV inverter operation with discrete filter inductors is analyzed, and the design of discrete filter inductors is given; then operation of the three-phase PV inverter with three-phase integrated five-limb magnetic core filter inductors is analyzed, the design of integrated filter inductors with five-limb magnetic core is given, then the operation of three-phase PV inverter with three-phase integrated three-limb magnetic core filter inductors is analyzed, and the design of integrated filter inductors with three limbs magnetic core is given. The conclusion of comparison between three-phase discrete filter inductors and two kinds of three-phase integrated filter inductors is done; it means that the three-phase discrete filter inductors can achieve better output electrical power quality with lower power density and three-phase integrated filter inductors can achieve higher power density with lower output electrical power quality. Finally, the experiment results are given to compare the volume and filter effect of three kinds of filter inductors in three-phase PV inverter.


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## Body

## 1. Introduction

Photovoltaic (PV) system is becoming increasingly more interesting among all renewable energy [1–4]. Three-phase inverter is wildly used in PV system, and the output electrical power quality is a very important part for the inverter no matter for the grid or the local load [5–7]. In order to achieve better output electrical power quality, the switching harmonics generated by the power inverters must be properly filtered out reaching a compromise between the cost and the weight of the filter and a fulfillment of international standards. Therefore, the PV inverter requires an output low-pass filter to interface with the grid or local load. Nowadays, L filter, LC filter, and LCL filter are wildly used in PV system [8–10]. For L filter, a single inductor L filter is popular and simple to use. Due to causing a long-time response, the system dynamics is poor, but the L filter has the lowest cost among the three filters. The LC filter is very commonly used with most applications, and its output electrical power quality is better than L filter. For LCL filter, it has the best output electrical power quality, but the stability of the system may be compromised by the resonances of the LCL filter, so that several authors have recently studied these issues [11–13]. For example, in [14–16] some control techniques to actively damp the resonances of the filter have been presented.The demands for PV inverter are not only the good electrical performance but also the high power density, high reliability, and low cost. Among the L filter, LC filter, and LCL filter, most papers pay attention to the electrical performance, and most applications are the single phase PV inverter [17–21]. There are seldom papers of filters about the combination of the electrical performance and the power density, especially in the three-phase PV inverter. However, the three-phase filters are very important factor for the electrical performance, power density, and cost. Therefore, this paper takes the most used LC filter, for example, to research the relation between the output electrical power quality and the volume of three-phase inductors with discrete magnetic core and integrated magnetic core in three-phase PV inverter.In this paper, the comparison analyses of three-phase discrete filter inductors and two kinds of three-phase integrated filter inductors of LC filter in three-phase PV inverter are proposed, and the design of three-phase is analyzed including the method how to integrate the three-phase filter inductors into five-limb magnetic core and three-limb magnetic core. The main works are done as below: according to the volume of filter inductor from large to small, the operation of three-phase PV inverter with discrete magnetic filter inductor is analyzed firstly, and the design of magnetic core is given; then according to the deduced mathematic expression formula of filter inductor with five-limb magnetic core, the operation of three-phase PV inverter with five-limb magnetic core filter inductor is analyzed, and the design of inductance and magnetic core is given, and then according to the deduced mathematic expression formula of filter inductor with three-limb magnetic core, the operation of three-phase PV inverter with three-limb magnetic core filter inductor is analyzed, and the design of inductance and magnetic core is given. Finally, the experiment results are given to compare the volume and filter effect of three kinds of filter inductors.


## 2. The Three-Phase Discrete Filter Inductors in Three-Phase Inverter

### 2.1. The Operational Principle of Three-Phase Discrete Filter Inductors in Three-Phase LC Filter

The three-phase discrete filter inductors in three-phase inverter are shown in Figure1. Every filter inductor has its own magnetic core. Therefore, the volumes of inductors are large and the costs are high. Before the analyses, several assumptions are made as follows:(1)
the input voltage source isVin which come from the PV panel;(2)
switchesQ1–6 are ideal, diode D1–6 is the antiparallel diode of Q1–6, and the voltages Va, Vb, and Vc are the output voltage of three bridge arms;(3)
Lf1, Lf2, and Lf3 are three filter inductors, C1, C2, and C3 are three filter capacitors, and VLf1, VLf2, and VLf3 are voltages of three inductors;(4)
Ra, Rb, and Rc are three-phase resistance load, ea, eb, and ec are three-phase voltage, ia, ib, and ic are three-phase current, and eo is the voltage of the three-phase load neutral point;(5)
the switch signals can be defined as follows:(1)Sa={1Q1is  turned  on,Q4is  turned  off;0Q4is  turned  on,Q1is  turned  off;Sb={1Q3is  turned  on,Q6is  turned  off;0Q6is  turned  on,Q3is  turned  off;Sc={1Q5is  turned  on,Q2is  turned  off;0Q2is  turned  on,Q5is  turned  off.Figure 1
The three-phase discrete filter inductors in three-phase inverter.According to Figure1, VLf1, VLf2, and VLf3 can be described as follows:
(2)VLf1=Va-ea-eo=Lf1diadt,VLf2=Vb-eb-eo=Lf2dibdt,VLf3=Vc-ec-eo=Lf3dicdt.In three-phase system, the amplitude of current and voltage of every phase are the same, and the 120° phase difference between every two phase, and then it can be got as(3)ea+eb+ec=0,ia+ib+ic=0.According to (2) and (3), neutral point voltage eo can be got as follows:
(4)eo=Va+Vb+Vc3.Also,Va, Vb, and Vc can be described as follows:
(5)Va=SaVin,Vb=SbVin,Vc=ScVin.Therefore, finallyVLf1, VLf2, and VLf3 can be simplified as follows:
(6)VLf1=Lf1diadt=2Sa-Sb-Sc3Vin-ea,VLf2=Lf2dibdt=2Sb-Sa-Sc3Vin-eb,VLf3=Lf3dicdt=2Sc-Sa-Sb3Vin-ec.The above formula is the basic expression of three-phase discrete inductors. The main functions of three-phase inductors are to reduce the harmonic by limiting the ripple current.


### 2.2. The SPWM Control Method of Three-Phase Inverter

In this paper, the sine-wave pulse-width modulation (SPWM) control method is adopted. The SPWM control method is widely used in the inverter due to its simple and easy realizing, and it is has been a mature technology. Though there are some advantaged control methods such as space voltage vector pulse-width modulation (SVPWM), the SPWM control method is enough to verify the basic function of LC filter, and the SPWM control method is shown in Figure2. Through the comparison between the high frequency triangular wave signal uo and three-phase line frequency sine wave signal uga, ugb, and ugc, the control signals uQ1–6 for switches Q1–6 are got.Figure 2
The three-phase SPWM control method.


### 2.3. The Ripple Current of Three-Phase Discrete Filter Inductors

In order to get better output electrical power quality, smaller ripple current of inductor should be achieved. Besides, the ripple current also decides the volume and the value of inductor. Therefore, the ripple current of filter inductor is an important parameter.According to (6), take the ripple current of Lf1, for example, to analyze, it can be got that
(7)VLf1=Lf1ΔiaΔt=2Sa-Sb-Sc3Vdc-eaΔia is the ripple current of Lf1, and it can be got as follows:
(8)Δia=(2Sa-Sb-Sc3Vdc-ea)DonTLf1,
where T is cycle and Don is duty cycle when the ripple current increases. Since the current ripple directly decides the harmonic, the ripple current Δia is always selected as the (10–20)% of the max value of ia. Based on the selected ripple current Δia, the value of inductor Lf1 can also be selected. Since the three-phase inductors have three discrete magnetic cores, every inductor does not affect each other; then the demanded filter effect can be achieved and the design of inductor is not very complicated. The three-phase discrete inductors can achieve wide application, but the volume, weight, and cost are their disadvantage.


### 2.4. The Max Flux Density of Discrete Filter Inductors

In order to design the magnetic core of the discrete filter inductor, the flux density should be analyzed. Take theLf1 for example, to analyze, and the other Lf2 and Lf3 have the same analyzed method. The Lf1 filter inductor with magnetic core is shown in Figure 3(a). ϕa is the flux generated by Lf1 winding; ℜ01 are the magnetic resistance of the middle limb, and ℜ02 are the magnetic resistance of the other two limbs; NLf1 is the number of turns of Lf1 winding. According to the magnetic circuit which is shown in Figure 3(b), ϕa can be described as follows:
(9)ϕa=NLf1iaℜ01+ℜ02//ℜ02=2NLf1ia2ℜ01+ℜ02.Three-phase discrete inductor: (a) magnetic core, (b) magnetic circuit.
(a)
(b)Therefore, the voltage of inductorsVLf1 can be described as follows:
(10)VLf1=NLf1dϕadt.According to (10), VLf1 can be described as follows:
(11)VLf1=Lf1diadt,
where Lf1 is self-inductor and can be described as follows:
(12)Lf1=2NLf122ℜ01+ℜ02.The magnetic resistance can be got as follows:(13)ℜ0i=lgiμ0Ai,i=1,2,
where lgi and Ai are, respectively, the air gap and cross section area of ℜ0i, μ0 is air permeability, and for the EE type magnetic core, the cross section area of middle limb is two times of that of the other two limbs. Then, when make the air gap of three limbs are the same, it can be got that 2ℜ01=ℜ02.The three-phase currentia can be described as follows:
(14)ia=2Imsin(ωt),
where Im is the effective value of current.Therefore, whenϕa achieves its max value ϕamax, the max flux density Bamax of middle limb can be described as follows:
(15)Bamax=ϕamaxA2=Lf12ImA2NLf1=2ImNLf1u0lg2.Then, the magnetic core can be selected according to theBamax.


## 2.1. The Operational Principle of Three-Phase Discrete Filter Inductors in Three-Phase LC Filter

The three-phase discrete filter inductors in three-phase inverter are shown in Figure1. Every filter inductor has its own magnetic core. Therefore, the volumes of inductors are large and the costs are high. Before the analyses, several assumptions are made as follows:(1)
the input voltage source isVin which come from the PV panel;(2)
switchesQ1–6 are ideal, diode D1–6 is the antiparallel diode of Q1–6, and the voltages Va, Vb, and Vc are the output voltage of three bridge arms;(3)
Lf1, Lf2, and Lf3 are three filter inductors, C1, C2, and C3 are three filter capacitors, and VLf1, VLf2, and VLf3 are voltages of three inductors;(4)
Ra, Rb, and Rc are three-phase resistance load, ea, eb, and ec are three-phase voltage, ia, ib, and ic are three-phase current, and eo is the voltage of the three-phase load neutral point;(5)
the switch signals can be defined as follows:(1)Sa={1Q1is  turned  on,Q4is  turned  off;0Q4is  turned  on,Q1is  turned  off;Sb={1Q3is  turned  on,Q6is  turned  off;0Q6is  turned  on,Q3is  turned  off;Sc={1Q5is  turned  on,Q2is  turned  off;0Q2is  turned  on,Q5is  turned  off.Figure 1
The three-phase discrete filter inductors in three-phase inverter.According to Figure1, VLf1, VLf2, and VLf3 can be described as follows:
(2)VLf1=Va-ea-eo=Lf1diadt,VLf2=Vb-eb-eo=Lf2dibdt,VLf3=Vc-ec-eo=Lf3dicdt.In three-phase system, the amplitude of current and voltage of every phase are the same, and the 120° phase difference between every two phase, and then it can be got as(3)ea+eb+ec=0,ia+ib+ic=0.According to (2) and (3), neutral point voltage eo can be got as follows:
(4)eo=Va+Vb+Vc3.Also,Va, Vb, and Vc can be described as follows:
(5)Va=SaVin,Vb=SbVin,Vc=ScVin.Therefore, finallyVLf1, VLf2, and VLf3 can be simplified as follows:
(6)VLf1=Lf1diadt=2Sa-Sb-Sc3Vin-ea,VLf2=Lf2dibdt=2Sb-Sa-Sc3Vin-eb,VLf3=Lf3dicdt=2Sc-Sa-Sb3Vin-ec.The above formula is the basic expression of three-phase discrete inductors. The main functions of three-phase inductors are to reduce the harmonic by limiting the ripple current.


## 2.2. The SPWM Control Method of Three-Phase Inverter

In this paper, the sine-wave pulse-width modulation (SPWM) control method is adopted. The SPWM control method is widely used in the inverter due to its simple and easy realizing, and it is has been a mature technology. Though there are some advantaged control methods such as space voltage vector pulse-width modulation (SVPWM), the SPWM control method is enough to verify the basic function of LC filter, and the SPWM control method is shown in Figure2. Through the comparison between the high frequency triangular wave signal uo and three-phase line frequency sine wave signal uga, ugb, and ugc, the control signals uQ1–6 for switches Q1–6 are got.Figure 2
The three-phase SPWM control method.


## 2.3. The Ripple Current of Three-Phase Discrete Filter Inductors

In order to get better output electrical power quality, smaller ripple current of inductor should be achieved. Besides, the ripple current also decides the volume and the value of inductor. Therefore, the ripple current of filter inductor is an important parameter.According to (6), take the ripple current of Lf1, for example, to analyze, it can be got that
(7)VLf1=Lf1ΔiaΔt=2Sa-Sb-Sc3Vdc-eaΔia is the ripple current of Lf1, and it can be got as follows:
(8)Δia=(2Sa-Sb-Sc3Vdc-ea)DonTLf1,
where T is cycle and Don is duty cycle when the ripple current increases. Since the current ripple directly decides the harmonic, the ripple current Δia is always selected as the (10–20)% of the max value of ia. Based on the selected ripple current Δia, the value of inductor Lf1 can also be selected. Since the three-phase inductors have three discrete magnetic cores, every inductor does not affect each other; then the demanded filter effect can be achieved and the design of inductor is not very complicated. The three-phase discrete inductors can achieve wide application, but the volume, weight, and cost are their disadvantage.


## 2.4. The Max Flux Density of Discrete Filter Inductors

In order to design the magnetic core of the discrete filter inductor, the flux density should be analyzed. Take theLf1 for example, to analyze, and the other Lf2 and Lf3 have the same analyzed method. The Lf1 filter inductor with magnetic core is shown in Figure 3(a). ϕa is the flux generated by Lf1 winding; ℜ01 are the magnetic resistance of the middle limb, and ℜ02 are the magnetic resistance of the other two limbs; NLf1 is the number of turns of Lf1 winding. According to the magnetic circuit which is shown in Figure 3(b), ϕa can be described as follows:
(9)ϕa=NLf1iaℜ01+ℜ02//ℜ02=2NLf1ia2ℜ01+ℜ02.Three-phase discrete inductor: (a) magnetic core, (b) magnetic circuit.
(a)
(b)Therefore, the voltage of inductorsVLf1 can be described as follows:
(10)VLf1=NLf1dϕadt.According to (10), VLf1 can be described as follows:
(11)VLf1=Lf1diadt,
where Lf1 is self-inductor and can be described as follows:
(12)Lf1=2NLf122ℜ01+ℜ02.The magnetic resistance can be got as follows:(13)ℜ0i=lgiμ0Ai,i=1,2,
where lgi and Ai are, respectively, the air gap and cross section area of ℜ0i, μ0 is air permeability, and for the EE type magnetic core, the cross section area of middle limb is two times of that of the other two limbs. Then, when make the air gap of three limbs are the same, it can be got that 2ℜ01=ℜ02.The three-phase currentia can be described as follows:
(14)ia=2Imsin(ωt),
where Im is the effective value of current.Therefore, whenϕa achieves its max value ϕamax, the max flux density Bamax of middle limb can be described as follows:
(15)Bamax=ϕamaxA2=Lf12ImA2NLf1=2ImNLf1u0lg2.Then, the magnetic core can be selected according to theBamax.


## 3. The Three-Phase Integrated Filter Inductors with One Five-Limb Magnetic Core in Three-Phase Inverter

The three-phase integrated filter inductors with one five-limb magnetic core in three-phase inverter are shown in Figure4. The three filter inductors have one magnetic core with five limbs, and the magnetic core consists of two EE magnetic cores. The three filter inductors are on the middle three limbs, and the cross section area of three middle limbs is twice than that of the other two limbs.Figure 4
The three-phase integrated filter inductors with five magnetic limbs in three-phase inverter.


### 3.1. The Expression of Three-Phase Integrated Filter Inductors with Five Magnetic Limbs

The three-phase integrated filter inductors with five magnetic limbs are shown in Figure5(a). ϕa, ϕb, and ϕc are, respectively, the fluxes generated by Lf1 winding, Lf2 winding, and Lf3 winding; ℜ01 are the magnetic resistance of three middle limbs, and ℜ02 are the magnetic resistance of the other two limbs; NLf1, NLf2, and NLf3 are, respectively, the number of turns of Lf1 winding, Lf2 winding, and Lf3 winding. According to the magnetic circuit which is shown in Figure 5(b), ϕa is the flux generated by Lf1 winding, ϕab is the flux generated by Lf1 winding flows through Lf2 winding, and ϕac is the flux generated by Lf1 winding flows through Lf3 winding, and they can be described as follows:
(16)ϕa=NLf1iaℜ01+ℜ01//ℜ01//ℜ02//ℜ02=2NLf1ia(ℜ01+ℜ02)2ℜ012+3ℜ01ℜ02,ϕab=ϕac=2NLf1ia(ℜ01+ℜ02)2ℜ012+3ℜ01ℜ02ℜ022(ℜ01+ℜ02)=NLf1iaℜ022ℜ012+3ℜ01ℜ02ϕb is the flux generated by Lf2 winding, ϕba is the flux generated by Lf2 winding flows through Lf1 winding, and ϕbc is the flux generated by Lf2 winding flows through Lf3 winding, and they can be described as follows:
(17)ϕb=NLf2ibℜ01+ℜ01//ℜ01//ℜ02//ℜ02=2NLf2ib(ℜ01+ℜ02)2ℜ012+3ℜ01ℜ02,ϕba=ϕbc=2NLf2ib(ℜ01+ℜ02)2ℜ012+3ℜ01ℜ02ℜ022(ℜ01+ℜ02)=NLf2ibℜ022ℜ012+3ℜ01ℜ02ϕc is the flux generated by Lf3 winding, ϕca is the flux generated by Lf3 winding flows through Lf1 winding, and ϕcb is the flux generated by Lf3 winding flows through Lf2 winding, and they can be described as follows:
(18)ϕc=NLf3icℜ01+ℜ01//ℜ01//ℜ02//ℜ02=2NLf3ic(ℜ01+ℜ02)2ℜ012+3ℜ01ℜ02,ϕca=ϕcb=2NLf3ic(ℜ01+ℜ02)2ℜ012+3ℜ01ℜ02ℜ022(ℜ01+ℜ02)=NLf3icℜ022ℜ012+3ℜ01ℜ02.Three-phase integrated inductors with five magnetic limbs: (a) three-phase magnetic core with five limbs, (b) magnetic circuit.
(a)
(b)Every total flux of inductor winding consists of three parts: one is the self-flux and the others are generated by the other two inductor windings. Therefore, the voltages of inductorsVLf1, VLf2, and VLf3 can be described as follows:
(19)VLf1=NLf1ddt(ϕa+ϕba+ϕca),VLf2=NLf2ddt(ϕb+ϕab+ϕcb),VLf3=NLf3ddt(ϕc+ϕac+ϕbc).According to (16), (17), (18), and (19), VLf1, VLf2, and VLf3 also can be described as follows:
(20)(VLf1VLf2VLf3)=(Lf1MabMacMabLf2MbcMacMbcLf3)(diadtdibdtdicdt),
where Lf1, Lf2, and Lf3 are self-inductor and can be described as follows:
(21)Lf1=2NLf12(ℜ01+ℜ02)2ℜ012+3ℜ01ℜ02,Lf2=2NLf22(ℜ01+ℜ02)2ℜ012+3ℜ01ℜ02,Lf3=2NLf32(ℜ01+ℜ02)2ℜ012+3ℜ01ℜ02.Mab, Mac, and Mbc are mutual inductance and can be described as follows:
(22)Mab=NLf1NLf2ℜ022ℜ012+3ℜ01ℜ02,Mac=NLf1NLf3ℜ022ℜ012+3ℜ01ℜ02,Mbc=NLf2NLf3ℜ022ℜ012+3ℜ01ℜ02.In order to make the three-phases balanced, the number of turns of three-phase inductor is the same; it means thatNLf1=NLf2=NLf3. Therefore, it can be got that Lf1=Lf2=Lf3=Lf, and Mab=Mac=Mbc=M. Then, finally the expression of three-phase integrated inductors with five magnetic limbs can be described as follows:
(23)(VLf1VLf2VLf3)=(LfMMMLfMMMLf)(diadtdibdtdicdt).Based on (23), it can be got that the three-phase inductors couple with each other, and then the current ripple can be analyzed.


### 3.2. The Ripple Current of Three-Phase Integrated Filter Inductors with Five-Limb Magnetic Core

In order to achieve the demanded filter effect of three-phase integrated filter inductors with five magnetic limbs, the current ripples should be analyzed. According to (23), VLf1,VLf2, and VLf3 can be described as follows:
(24)VLf1=Va-ea-eo=Lf1diadt+Mdibdt+Mdicdt,VLf2=Vb-eb-eo=Lf2dibdt+Mdiadt+Mdicdt,VLf3=Vc-ec-eo=Lf3dicdt+Mdibdt+Mdiadt.Furthermore, it can be simplified as follows:(25)VLf1=2Sa-Sb-Sc3Vin-ea=Lf1diadt+Mdibdt+Mdicdt,VLf2=2Sb-Sa-Sc3Vin-eb=Lf2dibdt+Mdiadt+Mdicdt,VLf3=2Sc-Sa-Sb3Vin-ec=Lf3dicdt+Mdibdt+Mdiadt.Take the ripple currentΔia of Lf1, for example, to analyze, it can be got that
(26)Δia=(2Sa-Sb-Sc3Vdc-ea-Mdibdt-Mdicdt)×DonTLf1.It can be got that the ripple currentΔia are influenced by switching signals Sa, Sb, and Sc which are decided by control method, and the mutual inductors and currents of the other two phases. Since the ripple currents are coupled with each other, it is complicated to design the ideal value of ripple current for fulfilling the demand of harmonic. In order to reduce the influence by other two phases, the mutual inductors can be eliminated. According to (22), the mutual inductor is described as follows:
(27)M=NLf2ℜ022ℜ012+3ℜ01ℜ02.The magnetic resistance can be got as follows:(28)ℜ0i=lgiμ0Ai,i=1,2,
where lgi and Ai are, respectively, the air gap and cross section area of ℜ0i and μ0 is air permeability.In order to reduce the mutual inductor, according to (27), when the magnetic resistance ℜ02 is zero, the mutual inductor M can be eliminated. Therefore, the air gap lg2 of ℜ02 can be adjusted to zero to reduce the mutual inductor. Then, (26) can be simplified as follows:
(29)Δia=(2Sa-Sb-Sc3Vdc-ea)×DonTLf1.Based on (22), it can be got that the ripple currents of three-phase integrated filter inductors with five limbs are the same as those of three-phase discrete filter inductors. It means that the two kinds of three-phase inductors have the same filter effect, but the volume of integrated filter inductors with five limbs is less than that of discrete filter inductors.


### 3.3. The Max Flux Density of Integrated Filter Inductors with Five Magnetic Limbs

In order to design the magnetic core of the three-phase integrated filter inductors with five limbs, the flux density of every limb should be analyzed. There are five limbs of the magnetic core of integrated filter inductor, and the middle three limbs have the windings of three-phase filter inductors. In order to reduce the mutual inductor, the middle three limbs have the same air gaps; the other two limbs have no air gaps; it means that the middle three limbs have the magnetic resistances and the other two limbs have no magnetic resistances. Therefore, the fluxes generated by the three inductor windings of middle three limbs all flow through the other two limbs and donot flow through each other. Taking the fluxϕa generated by Lf1 winding of second limb for example, ϕa can be got as follows:
(30)ϕa=Lf1iaNLf1.Since the magnetic resistanceℜ02 is zero, and according to (28), then Lf1 can be got as follows:
(31)Lf1=2NLf12(ℜ01+ℜ02)2ℜ012+3ℜ01ℜ02=NLf12ℜ01=NLf12u0A2lg2.The three-phase currentia, ib, and ic can be described as follows:
(32)ia=2Imsin(ωt),ib=2Imsin(ωt+2π3),ic=2Imsin(ωt-2π3),
where Im is the effective value of current.Therefore, whenϕa achieves its max value ϕamax, the max flux density B2max of second limb can be described as follows:
(33)B2max=ϕamaxA2=Lf12ImA2NLf1=2ImNLf1u0lg2.Also, the max flux densitiesB3max and B4max of the third and fourth limbs can be described as follows:
(34)B3max=2ImNLf2u0lg3,B4max=2ImNLf3u0lg4.The fluxes of the first and fifth limbs consist of three parts of fluxes which are generated by the middle three limbs. Taking the fluxϕ1 of the first limb, for example, ϕ1 consists of ϕa1 generated by Lf1 winding, ϕb1 generated by Lf2 winding, and ϕc1 generated by Lf3 winding. ϕa1, ϕb1, and ϕc1 can be described as follows:
(35)ϕa1=NLf1iaℜ012ℜ012+3ℜ01ℜ02,ϕb1=NLf2ibℜ012ℜ012+3ℜ01ℜ02,ϕc1=NLf3icℜ012ℜ012+3ℜ01ℜ02.Therefore,ϕ1 can be got as follows:
(36)ϕ1=ϕa1+ϕb1+ϕc1=NLfℜ01(ia+ib+ic)2ℜ012+3ℜ01ℜ02.Since in three-phase system,ia+ib+ic=0. Therefore, the total flux ϕ1 is zero. The main functions of the first and fifth limbs are to adjust the mutual inductors.


## 3.1. The Expression of Three-Phase Integrated Filter Inductors with Five Magnetic Limbs

The three-phase integrated filter inductors with five magnetic limbs are shown in Figure5(a). ϕa, ϕb, and ϕc are, respectively, the fluxes generated by Lf1 winding, Lf2 winding, and Lf3 winding; ℜ01 are the magnetic resistance of three middle limbs, and ℜ02 are the magnetic resistance of the other two limbs; NLf1, NLf2, and NLf3 are, respectively, the number of turns of Lf1 winding, Lf2 winding, and Lf3 winding. According to the magnetic circuit which is shown in Figure 5(b), ϕa is the flux generated by Lf1 winding, ϕab is the flux generated by Lf1 winding flows through Lf2 winding, and ϕac is the flux generated by Lf1 winding flows through Lf3 winding, and they can be described as follows:
(16)ϕa=NLf1iaℜ01+ℜ01//ℜ01//ℜ02//ℜ02=2NLf1ia(ℜ01+ℜ02)2ℜ012+3ℜ01ℜ02,ϕab=ϕac=2NLf1ia(ℜ01+ℜ02)2ℜ012+3ℜ01ℜ02ℜ022(ℜ01+ℜ02)=NLf1iaℜ022ℜ012+3ℜ01ℜ02ϕb is the flux generated by Lf2 winding, ϕba is the flux generated by Lf2 winding flows through Lf1 winding, and ϕbc is the flux generated by Lf2 winding flows through Lf3 winding, and they can be described as follows:
(17)ϕb=NLf2ibℜ01+ℜ01//ℜ01//ℜ02//ℜ02=2NLf2ib(ℜ01+ℜ02)2ℜ012+3ℜ01ℜ02,ϕba=ϕbc=2NLf2ib(ℜ01+ℜ02)2ℜ012+3ℜ01ℜ02ℜ022(ℜ01+ℜ02)=NLf2ibℜ022ℜ012+3ℜ01ℜ02ϕc is the flux generated by Lf3 winding, ϕca is the flux generated by Lf3 winding flows through Lf1 winding, and ϕcb is the flux generated by Lf3 winding flows through Lf2 winding, and they can be described as follows:
(18)ϕc=NLf3icℜ01+ℜ01//ℜ01//ℜ02//ℜ02=2NLf3ic(ℜ01+ℜ02)2ℜ012+3ℜ01ℜ02,ϕca=ϕcb=2NLf3ic(ℜ01+ℜ02)2ℜ012+3ℜ01ℜ02ℜ022(ℜ01+ℜ02)=NLf3icℜ022ℜ012+3ℜ01ℜ02.Three-phase integrated inductors with five magnetic limbs: (a) three-phase magnetic core with five limbs, (b) magnetic circuit.
(a)
(b)Every total flux of inductor winding consists of three parts: one is the self-flux and the others are generated by the other two inductor windings. Therefore, the voltages of inductorsVLf1, VLf2, and VLf3 can be described as follows:
(19)VLf1=NLf1ddt(ϕa+ϕba+ϕca),VLf2=NLf2ddt(ϕb+ϕab+ϕcb),VLf3=NLf3ddt(ϕc+ϕac+ϕbc).According to (16), (17), (18), and (19), VLf1, VLf2, and VLf3 also can be described as follows:
(20)(VLf1VLf2VLf3)=(Lf1MabMacMabLf2MbcMacMbcLf3)(diadtdibdtdicdt),
where Lf1, Lf2, and Lf3 are self-inductor and can be described as follows:
(21)Lf1=2NLf12(ℜ01+ℜ02)2ℜ012+3ℜ01ℜ02,Lf2=2NLf22(ℜ01+ℜ02)2ℜ012+3ℜ01ℜ02,Lf3=2NLf32(ℜ01+ℜ02)2ℜ012+3ℜ01ℜ02.Mab, Mac, and Mbc are mutual inductance and can be described as follows:
(22)Mab=NLf1NLf2ℜ022ℜ012+3ℜ01ℜ02,Mac=NLf1NLf3ℜ022ℜ012+3ℜ01ℜ02,Mbc=NLf2NLf3ℜ022ℜ012+3ℜ01ℜ02.In order to make the three-phases balanced, the number of turns of three-phase inductor is the same; it means thatNLf1=NLf2=NLf3. Therefore, it can be got that Lf1=Lf2=Lf3=Lf, and Mab=Mac=Mbc=M. Then, finally the expression of three-phase integrated inductors with five magnetic limbs can be described as follows:
(23)(VLf1VLf2VLf3)=(LfMMMLfMMMLf)(diadtdibdtdicdt).Based on (23), it can be got that the three-phase inductors couple with each other, and then the current ripple can be analyzed.


## 3.2. The Ripple Current of Three-Phase Integrated Filter Inductors with Five-Limb Magnetic Core

In order to achieve the demanded filter effect of three-phase integrated filter inductors with five magnetic limbs, the current ripples should be analyzed. According to (23), VLf1,VLf2, and VLf3 can be described as follows:
(24)VLf1=Va-ea-eo=Lf1diadt+Mdibdt+Mdicdt,VLf2=Vb-eb-eo=Lf2dibdt+Mdiadt+Mdicdt,VLf3=Vc-ec-eo=Lf3dicdt+Mdibdt+Mdiadt.Furthermore, it can be simplified as follows:(25)VLf1=2Sa-Sb-Sc3Vin-ea=Lf1diadt+Mdibdt+Mdicdt,VLf2=2Sb-Sa-Sc3Vin-eb=Lf2dibdt+Mdiadt+Mdicdt,VLf3=2Sc-Sa-Sb3Vin-ec=Lf3dicdt+Mdibdt+Mdiadt.Take the ripple currentΔia of Lf1, for example, to analyze, it can be got that
(26)Δia=(2Sa-Sb-Sc3Vdc-ea-Mdibdt-Mdicdt)×DonTLf1.It can be got that the ripple currentΔia are influenced by switching signals Sa, Sb, and Sc which are decided by control method, and the mutual inductors and currents of the other two phases. Since the ripple currents are coupled with each other, it is complicated to design the ideal value of ripple current for fulfilling the demand of harmonic. In order to reduce the influence by other two phases, the mutual inductors can be eliminated. According to (22), the mutual inductor is described as follows:
(27)M=NLf2ℜ022ℜ012+3ℜ01ℜ02.The magnetic resistance can be got as follows:(28)ℜ0i=lgiμ0Ai,i=1,2,
where lgi and Ai are, respectively, the air gap and cross section area of ℜ0i and μ0 is air permeability.In order to reduce the mutual inductor, according to (27), when the magnetic resistance ℜ02 is zero, the mutual inductor M can be eliminated. Therefore, the air gap lg2 of ℜ02 can be adjusted to zero to reduce the mutual inductor. Then, (26) can be simplified as follows:
(29)Δia=(2Sa-Sb-Sc3Vdc-ea)×DonTLf1.Based on (22), it can be got that the ripple currents of three-phase integrated filter inductors with five limbs are the same as those of three-phase discrete filter inductors. It means that the two kinds of three-phase inductors have the same filter effect, but the volume of integrated filter inductors with five limbs is less than that of discrete filter inductors.


## 3.3. The Max Flux Density of Integrated Filter Inductors with Five Magnetic Limbs

In order to design the magnetic core of the three-phase integrated filter inductors with five limbs, the flux density of every limb should be analyzed. There are five limbs of the magnetic core of integrated filter inductor, and the middle three limbs have the windings of three-phase filter inductors. In order to reduce the mutual inductor, the middle three limbs have the same air gaps; the other two limbs have no air gaps; it means that the middle three limbs have the magnetic resistances and the other two limbs have no magnetic resistances. Therefore, the fluxes generated by the three inductor windings of middle three limbs all flow through the other two limbs and donot flow through each other. Taking the fluxϕa generated by Lf1 winding of second limb for example, ϕa can be got as follows:
(30)ϕa=Lf1iaNLf1.Since the magnetic resistanceℜ02 is zero, and according to (28), then Lf1 can be got as follows:
(31)Lf1=2NLf12(ℜ01+ℜ02)2ℜ012+3ℜ01ℜ02=NLf12ℜ01=NLf12u0A2lg2.The three-phase currentia, ib, and ic can be described as follows:
(32)ia=2Imsin(ωt),ib=2Imsin(ωt+2π3),ic=2Imsin(ωt-2π3),
where Im is the effective value of current.Therefore, whenϕa achieves its max value ϕamax, the max flux density B2max of second limb can be described as follows:
(33)B2max=ϕamaxA2=Lf12ImA2NLf1=2ImNLf1u0lg2.Also, the max flux densitiesB3max and B4max of the third and fourth limbs can be described as follows:
(34)B3max=2ImNLf2u0lg3,B4max=2ImNLf3u0lg4.The fluxes of the first and fifth limbs consist of three parts of fluxes which are generated by the middle three limbs. Taking the fluxϕ1 of the first limb, for example, ϕ1 consists of ϕa1 generated by Lf1 winding, ϕb1 generated by Lf2 winding, and ϕc1 generated by Lf3 winding. ϕa1, ϕb1, and ϕc1 can be described as follows:
(35)ϕa1=NLf1iaℜ012ℜ012+3ℜ01ℜ02,ϕb1=NLf2ibℜ012ℜ012+3ℜ01ℜ02,ϕc1=NLf3icℜ012ℜ012+3ℜ01ℜ02.Therefore,ϕ1 can be got as follows:
(36)ϕ1=ϕa1+ϕb1+ϕc1=NLfℜ01(ia+ib+ic)2ℜ012+3ℜ01ℜ02.Since in three-phase system,ia+ib+ic=0. Therefore, the total flux ϕ1 is zero. The main functions of the first and fifth limbs are to adjust the mutual inductors.


## 4. The Integrated Three-Phase Filter Inductors with Five-Limb Magnetic Core in Three-Phase Inverter

In order to further decrease the volume of three-phase filter inductor, the five magnetic limbs can be simplified to three magnetic limbs which are shown in Figure6. In Figure 6, one EE magnetic core is adopted, and the cross section area of middle limb is twice than that of the other two limbs. Since, compared with the above two kinds of three-phase filter inductors, this kind of three-phase integrated filter inductor can achieve the least volume.Figure 6
The three-phase integrated filter inductors with three magnetic limbs in three-phase inverter.


### 4.1. The Expression of Three-Phase Integrated Filter Inductors with Three Magnetic Limbs

The integrated three-phase filter inductors with three-limb magnetic core are shown in Figure7(a). ϕa, ϕb, and ϕc are, respectively, the fluxes generated by Lf1 winding, Lf2 winding, and Lf3 winding; ℜ1, ℜ2, and ℜ3 are, respectively, the magnetic resistance of three limbs. According to the magnetic circuit which is shown in Figure 7(b), ϕa is the flux generated by Lf1 winding, ϕab is the flux generated by Lf1 winding flows through Lf2 winding, and ϕac is the flux generated by Lf1 winding flows through Lf3 winding, and they can be described as follows:
(37)ϕa=NLf1iaℜ1+ℜ2//ℜ3=NLf1ia(ℜ3+ℜ2)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3,ϕab=NLf1ia(ℜ3+ℜ2)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3ℜ3ℜ3+ℜ2=NLf1iaℜ3ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3,ϕac=NLf1ia(ℜ3+ℜ2)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3ℜ2ℜ3+ℜ2=NLf1iaℜ2ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3.Three-phase integrated inductors with five magnetic limbs: (a) three-phase magnetic core with five limbs, (b) magnetic circuit.
(a)
(b)Also,ϕb is the flux generated by Lf2 winding, ϕba is the flux generated by Lf2 winding flows through Lf1 winding, and ϕbc is the flux generated by Lf2 winding flows through Lf3 winding, and they can be described as follows:
(38)ϕb=NLf2iLf2ℜ2+ℜ1//ℜ3=NLf2iLf2(ℜ1+ℜ3)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3,ϕba=NLf2ib(ℜ1+ℜ3)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3ℜ3ℜ1+ℜ3=NLf2ibℜ3ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3,ϕbc=NLf2ib(ℜ1+ℜ3)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3ℜ2ℜ1+ℜ3=NLf2ibℜ1ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3ϕc is the flux generated by Lf3 winding, ϕca is the flux generated by Lf3 winding flows through Lf1 winding, and ϕcb is the flux generated by Lf3 winding flows through Lf2 winding, and they can be described as follows:
(39)ϕc=NLf3icℜ3+ℜ1//ℜ2=NLf2ic(ℜ1+ℜ2)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3,ϕca=NLf3ic(ℜ1+ℜ2)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3ℜ2ℜ1+ℜ2=NLf3icℜ2ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3,ϕcb=NLf3ic(ℜ1+ℜ2)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3ℜ1ℜ1+ℜ2=NLf3icℜ1ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3.Furthermore,VLf1, VLf2, and VLf3 can be represented as follows:
(40)VLf1=NLf1ddt(ϕa+ϕba+ϕca),VLf2=NLf2ddt(ϕb+ϕab+ϕcb),VLf3=NLf3ddt(ϕc+ϕac+ϕbc).Based on the above formulas, the expression of integrated three-phase filter inductors with three magnetic limbs can be represented as follows:(41)(VLf1VLf2VLf3)=(Lf1MabMacMabLf2MbcMacMbcLf3)(diadtdibdtdicdt),
where Lf1, Lf2, and Lf3 can be represented as follows:
(42)Lf1=NLf12(ℜ3+ℜ2)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3,Lf2=NLf22(ℜ1+ℜ3)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3,Lf3=NLf32(ℜ1+ℜ2)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3.Mab, Mac, and Mbc are mutual inductors and can be represented as follows:
(43)Mab=NLf1NLf2ℜ3ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3,Mac=NLf1NLf3ℜ2ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3,Mbc=NLf2NLf3ℜ1ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3.Based on (43), it can be got that the three inductors couple with each other, and then the current ripple can be analyzed.


### 4.2. The Ripple Current of Three-Phase Integrated Filter Inductors with Three-Limb Magnetic Core

According to (43), VLf1, VLf2, and VLf3 can be described as follows:
(44)VLf1=Va-ea-eo=Lf1diadt+Mabdibdt+Macdicdt,VLf2=Vb-eb-eo=Lf2dibdt+Mabdiadt+Mbcdicdt,VLf3=Vc-ec-eo=Lf3dicdt+Mbcdibdt+Macdiadt.Based on (42) and (43), the self-inductors and mutual inductors are decided by magnetic resistors. The magnetic resistor can be got as follows:
(45)ℜi=lgiμ0Ai,i=1,2,3,
where lgi and Ai, are respectively, the air gap and cross section area of ℜ0i and μ0 is air permeability.Since one EE magnetic core is adopted in this paper, the cross section area of middle limb is twice than that of the other two limbs. Then, it makes the self-inductors and mutual inductors not the same. In order to make the three-phases balanced, the air gaps can be adjusted to make the magnetic resistors of three limbs the same. Therefore, it can be got thatlg2=2lg1=2lg3. Furthermore, when NLf1=NLf2=NLf3, it can be got that Lf1=Lf2=Lf3=Lf and Mab=Mac=Mbc=M. Therefore, (44) can be simplified as follows:
(46)VLf1=2Sa-Sb-Sc3Vin-ea=Lfdiadt+Mdibdt+Mdicdt,VLf2=2Sb-Sa-Sc3Vin-eb=Lfdibdt+Mdiadt+Mdicdt,VLf3=2Sc-Sa-Sb3Vin-ec=Lfdicdt+Mdibdt+Mdiadt.In order to analyze conveniently, take the one-phase ripple currentΔia of Lf1, for example, to analyze, it can be got that
(47)Δia=(2Sa-Sb-Sc3Vdc-ea-Mdibdt-Mdicdt)×DonTLf1.In three-phase system, the three-phase currentia, ib, and ic can be described as follows:
(48)ia=2Imsin(ωt),ib=2Imsin(ωt+2π3),ic=2Imsin(ωt-2π3),
where Im is the effective value of current and ω is the line angular frequency.According to (48), the ripple current Δia can be simplified as follows:
(49)Δia=(2Sa-Sb-Sc3Vdc-ea+2MImωcos(ωt))×DonTLf1.Compared with the ripple current of discrete inductors and integrated inductors with five limbs, it can be got that the ripple current with three limbs is influenced by additional2MImωcos(ωt). Therefore, mutual inductor M, current Im, and angular frequency ω should be designed to fulfill the demand of current ripple. Then the value of 2MImωcos(ωt) can be designed to be less than 10% ((2Sa-Sb-Sc)Vin/3-ea), and then the influence of 2MImωcos(ωt) can be omitted. When the three-phase integrated inductors with three limbs operate in high frequency, large current, and large mutual inductor, the influence cannot be omitted and the harmonic cannot meet the demand. Therefore, when the value of 2MImωcos(ωt) can be omitted, three-phase integrated inductors with three limbs can save more volume and achieve better power density.


### 4.3. The Max Flux Density of Integrated Filter Inductors with Three Magnetic Limbs

Since there are only three magnetic limbs in one magnetic core, the three-phase fluxes are coupled with each other. Take the first limb, for example, which is shown in Figure7(a), the total flux ϕ1 of first limb consists of the flux ϕa generated byLf1winding, the flux ϕba generated by Lf2 winding flows through the first limb, and flux ϕca generated by Lf3 winding flows through the first limb; then it can be got that
(50)ϕ1=ϕa+ϕba+ϕca.The total flux can be further simplified as follows:(51)ϕ1=1NLf1(Lf1-M)2Imsin(ωt).Since the three-phase magnetic resistors and the number of turns of three-phase inductor are the same, it can be got thatℜ1=ℜ2=ℜ3=ℜ0, andNLf1=NLf2=NLf3=NLf; thenLf1 and M can be described as follows:
(52)Lf1=2NLf2u0A13lg1,M=NLf2u0A13lg1.Whenϕ1 achieves its max value ϕ1max, the max flux density B1max of second limb can be described as follows:
(53)B1max=ϕ1maxA1=NLfu02Im3lg1.Also the max flux densityB2max of second limb and the max flux density B3max of third limb can be described as follows:
(54)B2max=NLfu02Im3lg2,B3max=NLfu02Im3lg3.


## 4.1. The Expression of Three-Phase Integrated Filter Inductors with Three Magnetic Limbs

The integrated three-phase filter inductors with three-limb magnetic core are shown in Figure7(a). ϕa, ϕb, and ϕc are, respectively, the fluxes generated by Lf1 winding, Lf2 winding, and Lf3 winding; ℜ1, ℜ2, and ℜ3 are, respectively, the magnetic resistance of three limbs. According to the magnetic circuit which is shown in Figure 7(b), ϕa is the flux generated by Lf1 winding, ϕab is the flux generated by Lf1 winding flows through Lf2 winding, and ϕac is the flux generated by Lf1 winding flows through Lf3 winding, and they can be described as follows:
(37)ϕa=NLf1iaℜ1+ℜ2//ℜ3=NLf1ia(ℜ3+ℜ2)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3,ϕab=NLf1ia(ℜ3+ℜ2)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3ℜ3ℜ3+ℜ2=NLf1iaℜ3ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3,ϕac=NLf1ia(ℜ3+ℜ2)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3ℜ2ℜ3+ℜ2=NLf1iaℜ2ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3.Three-phase integrated inductors with five magnetic limbs: (a) three-phase magnetic core with five limbs, (b) magnetic circuit.
(a)
(b)Also,ϕb is the flux generated by Lf2 winding, ϕba is the flux generated by Lf2 winding flows through Lf1 winding, and ϕbc is the flux generated by Lf2 winding flows through Lf3 winding, and they can be described as follows:
(38)ϕb=NLf2iLf2ℜ2+ℜ1//ℜ3=NLf2iLf2(ℜ1+ℜ3)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3,ϕba=NLf2ib(ℜ1+ℜ3)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3ℜ3ℜ1+ℜ3=NLf2ibℜ3ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3,ϕbc=NLf2ib(ℜ1+ℜ3)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3ℜ2ℜ1+ℜ3=NLf2ibℜ1ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3ϕc is the flux generated by Lf3 winding, ϕca is the flux generated by Lf3 winding flows through Lf1 winding, and ϕcb is the flux generated by Lf3 winding flows through Lf2 winding, and they can be described as follows:
(39)ϕc=NLf3icℜ3+ℜ1//ℜ2=NLf2ic(ℜ1+ℜ2)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3,ϕca=NLf3ic(ℜ1+ℜ2)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3ℜ2ℜ1+ℜ2=NLf3icℜ2ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3,ϕcb=NLf3ic(ℜ1+ℜ2)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3ℜ1ℜ1+ℜ2=NLf3icℜ1ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3.Furthermore,VLf1, VLf2, and VLf3 can be represented as follows:
(40)VLf1=NLf1ddt(ϕa+ϕba+ϕca),VLf2=NLf2ddt(ϕb+ϕab+ϕcb),VLf3=NLf3ddt(ϕc+ϕac+ϕbc).Based on the above formulas, the expression of integrated three-phase filter inductors with three magnetic limbs can be represented as follows:(41)(VLf1VLf2VLf3)=(Lf1MabMacMabLf2MbcMacMbcLf3)(diadtdibdtdicdt),
where Lf1, Lf2, and Lf3 can be represented as follows:
(42)Lf1=NLf12(ℜ3+ℜ2)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3,Lf2=NLf22(ℜ1+ℜ3)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3,Lf3=NLf32(ℜ1+ℜ2)ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3.Mab, Mac, and Mbc are mutual inductors and can be represented as follows:
(43)Mab=NLf1NLf2ℜ3ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3,Mac=NLf1NLf3ℜ2ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3,Mbc=NLf2NLf3ℜ1ℜ1ℜ2+ℜ1ℜ3+ℜ2ℜ3.Based on (43), it can be got that the three inductors couple with each other, and then the current ripple can be analyzed.


## 4.2. The Ripple Current of Three-Phase Integrated Filter Inductors with Three-Limb Magnetic Core

According to (43), VLf1, VLf2, and VLf3 can be described as follows:
(44)VLf1=Va-ea-eo=Lf1diadt+Mabdibdt+Macdicdt,VLf2=Vb-eb-eo=Lf2dibdt+Mabdiadt+Mbcdicdt,VLf3=Vc-ec-eo=Lf3dicdt+Mbcdibdt+Macdiadt.Based on (42) and (43), the self-inductors and mutual inductors are decided by magnetic resistors. The magnetic resistor can be got as follows:
(45)ℜi=lgiμ0Ai,i=1,2,3,
where lgi and Ai, are respectively, the air gap and cross section area of ℜ0i and μ0 is air permeability.Since one EE magnetic core is adopted in this paper, the cross section area of middle limb is twice than that of the other two limbs. Then, it makes the self-inductors and mutual inductors not the same. In order to make the three-phases balanced, the air gaps can be adjusted to make the magnetic resistors of three limbs the same. Therefore, it can be got thatlg2=2lg1=2lg3. Furthermore, when NLf1=NLf2=NLf3, it can be got that Lf1=Lf2=Lf3=Lf and Mab=Mac=Mbc=M. Therefore, (44) can be simplified as follows:
(46)VLf1=2Sa-Sb-Sc3Vin-ea=Lfdiadt+Mdibdt+Mdicdt,VLf2=2Sb-Sa-Sc3Vin-eb=Lfdibdt+Mdiadt+Mdicdt,VLf3=2Sc-Sa-Sb3Vin-ec=Lfdicdt+Mdibdt+Mdiadt.In order to analyze conveniently, take the one-phase ripple currentΔia of Lf1, for example, to analyze, it can be got that
(47)Δia=(2Sa-Sb-Sc3Vdc-ea-Mdibdt-Mdicdt)×DonTLf1.In three-phase system, the three-phase currentia, ib, and ic can be described as follows:
(48)ia=2Imsin(ωt),ib=2Imsin(ωt+2π3),ic=2Imsin(ωt-2π3),
where Im is the effective value of current and ω is the line angular frequency.According to (48), the ripple current Δia can be simplified as follows:
(49)Δia=(2Sa-Sb-Sc3Vdc-ea+2MImωcos(ωt))×DonTLf1.Compared with the ripple current of discrete inductors and integrated inductors with five limbs, it can be got that the ripple current with three limbs is influenced by additional2MImωcos(ωt). Therefore, mutual inductor M, current Im, and angular frequency ω should be designed to fulfill the demand of current ripple. Then the value of 2MImωcos(ωt) can be designed to be less than 10% ((2Sa-Sb-Sc)Vin/3-ea), and then the influence of 2MImωcos(ωt) can be omitted. When the three-phase integrated inductors with three limbs operate in high frequency, large current, and large mutual inductor, the influence cannot be omitted and the harmonic cannot meet the demand. Therefore, when the value of 2MImωcos(ωt) can be omitted, three-phase integrated inductors with three limbs can save more volume and achieve better power density.


## 4.3. The Max Flux Density of Integrated Filter Inductors with Three Magnetic Limbs

Since there are only three magnetic limbs in one magnetic core, the three-phase fluxes are coupled with each other. Take the first limb, for example, which is shown in Figure7(a), the total flux ϕ1 of first limb consists of the flux ϕa generated byLf1winding, the flux ϕba generated by Lf2 winding flows through the first limb, and flux ϕca generated by Lf3 winding flows through the first limb; then it can be got that
(50)ϕ1=ϕa+ϕba+ϕca.The total flux can be further simplified as follows:(51)ϕ1=1NLf1(Lf1-M)2Imsin(ωt).Since the three-phase magnetic resistors and the number of turns of three-phase inductor are the same, it can be got thatℜ1=ℜ2=ℜ3=ℜ0, andNLf1=NLf2=NLf3=NLf; thenLf1 and M can be described as follows:
(52)Lf1=2NLf2u0A13lg1,M=NLf2u0A13lg1.Whenϕ1 achieves its max value ϕ1max, the max flux density B1max of second limb can be described as follows:
(53)B1max=ϕ1maxA1=NLfu02Im3lg1.Also the max flux densityB2max of second limb and the max flux density B3max of third limb can be described as follows:
(54)B2max=NLfu02Im3lg2,B3max=NLfu02Im3lg3.


## 5. The Comparison between Discrete Filter Inductors and Integrated Filter Inductors

According to the above analyses, the comparison of the discrete filter inductors and integrated filter inductors is shown in Table1. Then, based on the different demands of the PV inverter, different LC filter can be selected to fulfill the demands for the power density and the electrical performance.Table 1
The comparison of the discrete filter inductors and integrated filter inductors.
The volume of magnetic core
The cost
The three phase coupling
Current ripple influenced by the other two phases
Discrete filter inductor
Large
Large
Decoupling
Not influenced
Integrated filter inductor (five limbs)
Medium
Medium
Can be adjusted to decoupling
Can be adjusted to not influenced
Integrated filter inductor (three limbs)
Small
Small
Coupling
Influenced


## 6. Experimental Results

The parameters of three-phase inverter are shown as follows: the input voltage isVin = 700 VDC, the three-phase output voltages are ea=eb=ec=220VAC, the load frequency is 50 Hz, load power is 3 kW, three filter capacitors are 9μF, the control method is SPWM, switching frequency is 15 kHz, the max duty cycle Don max is 0.85, and the max value of every phase current is 7A. In this paper, three-phase discrete filter inductors, three-phase integrated filter inductors with five magnetic limbs, and three-phase integrated filter inductors with three magnetic limbs are, respectively, adopted to fulfill the demanded filter effect. In particular, the Three-phase integrated filter inductors can save more volume and achieve more power density.


### 6.1. The Volume Comparison of Three Kinds of Three-Phase Filter Inductors

In this paper, all the magnetic cores of three kinds of three-phase filter inductors are composed by power ferrite EE96/48/32 whose saturated flux density is 0.5 T and the cross section area of middle limb is 1024 mm2. The parameters of three kinds of filter inductors are shown in Table 2, the three-phase discrete filter inductors adopt three power ferrite EE96/48/32 cores, the three-phase integrated filter inductors with five limbs adopt two power ferrite EE96/48/32 cores, and the three-phase integrated filter inductors with three limbs adopt one power ferrite EE96/48/32 core. The self-inductance of three kinds of three filter inductors is 3.8 mH, and the number of turns is 80. For the three-phase integrated filter inductors with three limbs; the cross section area of middle limb is twice than that of the other two limbs; then the air gap of middle limb is twice than that of the other two limbs to make the three-phases balanced. According to the analyses of flux density, it can be calculated that the max flux densities of three discrete filter inductors are 0.33 T, the max flux densities of three middle limbs of integrated filter inductors with five limbs are 0.33 T and that of the other two limbs are zero, and the max flux densities of middle limb of integrated filter inductors with three limbs are 0.17 T and that of the other two limbs are 0.33 T.Table 2
The comparison of the discrete magnetics, five-limb magnetics, and three-limb magnetics.
The number of magnetic core
The cross section area of middle limb
Air gap distribution
Inductance
The number of turns
The max flux density
Discrete filter inductor
Three EE96/48/32
1024 mm2
The air gap of middle limb is 2.0 mm; the air gaps of the other limbs are zero
Self-inductance is 3.8 mH
80
0.33 T
Integrated filter inductor (five limbs)
Two EE96/48/32
1024 mm2
The air gaps of middle three limbs are 2.0 mm; the air gaps of the other two limbs are zero
Self-inductance is 3.8 mH
80
0.33 T
Integrated filter inductor (three limbs)
One EE96/48/32
1024 mm2
The air gap of middle limb is 1.4 mm; the air gaps of the other limbs are 0.7 mm
Self-inductance is 3.8 mH; mutual inductance is 1.9 mH
80
Middle limb is 0.17 T; the other two limbs are 0.33 TThe magnetic cores of the discrete filter inductors, integrated filter inductors with five limbs and three limbs magnetics are shown in Figure8. In Figure 8(a), three power ferrite EE96/48/32 cores are adopted in three-phase discrete filter inductors, and compared with the other two kinds of three-phase filter inductors, the volume of the three-phase discrete filter inductors is the largest. In Figure 8(b), three power ferrite EE96/48/32 cores are adopted in three-phase integrated filter inductors with five magnetic limbs; the middle three limbs have three-phase inductor’s windings; the air gaps of the other two limbs can be adjusted to eliminate the three-phase flux coupling; therefore, this kind of three-phase integrated filter inductors can achieve wild application and the volume is smaller than that of the three-phase discrete filter inductors. In Figure 8(c), one power ferrite EE96/48/32 cores is adopted in three-phase integrated filter inductors with three magnetic limbs, and compared with the other two kinds of three-phase filter inductors, all the three limbs have three-phase inductor’s windings and the three-phase fluxes couple with each other, and then the application area is a little limited due to the coupling, but the advantage is that this kind of three-phase integrated filter inductors achieves the minimum volume.The comparison of discrete magnetics, five-limb magnetics, and three-limb magnetics: (a) discrete magnetic cores, (b) integrated magnetic core with five limbs, and (c) integrated magnetic core with three limbs.
(a)
(b)
(c)


### 6.2. The Filter Effect Comparison of Three Kinds of Three-Phase Filter Inductors

The SPWM control method is adopted in the three-phase inverter due to its simple and easy realizing, and it can be easily used to testify the three kinds of LC filters. The switching process of switchQ1 and the control voltages uQ1 and uQ4 for the switch Q1 and Q4 are shown in Figure 9. The other control voltages are similar to uQ1 and uQ4.The waveforms of switching process and control voltages: (a) the switching process of switchQ1, (b) the control voltages for the switch Q1 and Q4. The voltage measured in CH2 by the voltage transformer; the proportion is 1 V = 100 V; in CH1, 1 V = 1 A; the time per DIV is 100 ns. In CH1, the voltage per DIV is 7.5 V; in CH2, the voltage per DIV is 7.5 V; the time per DIV is 16.6 us.
(a)
(b)Furthermore, in order to testify the filter effects of the three kinds of three-phase filter inductors, the current THD (total harmonic distortion) of three kinds of three-phase filter inductors is compared. The three-phase load currents of three-phase discrete filter inductors are shown in Figure10, and the THD of A phase, B phase, and C phase is, respectively, 4.5%, 4.7%, and 4.1%, and all can achieve the demand. The three-phase load currents of three-phase integrated filter inductors with five limbs are shown in Figure 11, and the THD of A phase, B phase, and C phase is, respectively, 4.4%, 4.6%, and 4.4%, and all can achieve the demand. The three-phase load currents of three-phase integrated filter inductors with three limbs are shown in Figure 12, and the THD of A phase, B phase, and C phase is, respectively, 4.5%, 4.4%, and 4.1%, and all can achieve the demand. Therefore, all three kinds of three-phase filter inductors can fulfill the demand of the current THD. For the low and medium power application, the three-phase coupling in the integrated filter inductors with three limbs can be omitted, and then the integrated filter inductors with three limbs can be chosen due to the minimum volume. Take A phase current of three-phase integrated filter inductors with three limbs, for example, based on formula (42), 2MImωcos(ωt) is caused by coupling, and its max value is2MImω. In this paper, M=1.9 mH, Im=5A, ω=2πf=314, and then it can be calculated that 2MImω=4.2V. Compared with Vin (700 VDC) and eas(220 VAC), the coupling voltage2MImω=4.2V is very small and can be omitted. But for the large power application, load current Im is large and the coupling cannot be omitted; then it is difficult to achieve the ideal filter effect. Therefore, for the large power application, the integrated filter inductors with five limbs can be chosen. The coupling of the integrated filter inductors with five limbs can be eliminated by adjusting the air gap of two side limbs, and then the filter effect is the same as the discrete filter inductors and the volume is less than the discrete filter inductors.The three-phase load current with discrete filter inductor: (a) the current of A phase and B phase, (b) the current of A phase and C phase. In CH1, the current comes from the current transformer and the proportion is 1 V = 5 A; it means that the current per DIV is 5 A; in CH2, the current per DIV is 5 A; the time per DIV is 5 ms.
(a)
(b)The three-phase load current with integrated filter inductor with five limbs: (a) the current of A phase and B phase, (b) the current of A phase and C phase. In CH1, the current comes from the current transformer and the proportion is 1 V = 5 A; it means that the current per DIV is 5 A; in CH2, the current per DIV is 5 A; the time per DIV is 5 ms.
(a)
(b)The three-phase load current with integrated filter inductor with three limbs: (a) the current of A phase and B phase, (b) the current of A phase and C phase. In CH1, the current comes from the current transformer and the proportion is 1 V = 5 A; it means that the current per DIV is 5 A; in CH2, the current per DIV is 5 A; the time per DIV is 5 ms.
(a)
(b)


## 6.1. The Volume Comparison of Three Kinds of Three-Phase Filter Inductors

In this paper, all the magnetic cores of three kinds of three-phase filter inductors are composed by power ferrite EE96/48/32 whose saturated flux density is 0.5 T and the cross section area of middle limb is 1024 mm2. The parameters of three kinds of filter inductors are shown in Table 2, the three-phase discrete filter inductors adopt three power ferrite EE96/48/32 cores, the three-phase integrated filter inductors with five limbs adopt two power ferrite EE96/48/32 cores, and the three-phase integrated filter inductors with three limbs adopt one power ferrite EE96/48/32 core. The self-inductance of three kinds of three filter inductors is 3.8 mH, and the number of turns is 80. For the three-phase integrated filter inductors with three limbs; the cross section area of middle limb is twice than that of the other two limbs; then the air gap of middle limb is twice than that of the other two limbs to make the three-phases balanced. According to the analyses of flux density, it can be calculated that the max flux densities of three discrete filter inductors are 0.33 T, the max flux densities of three middle limbs of integrated filter inductors with five limbs are 0.33 T and that of the other two limbs are zero, and the max flux densities of middle limb of integrated filter inductors with three limbs are 0.17 T and that of the other two limbs are 0.33 T.Table 2
The comparison of the discrete magnetics, five-limb magnetics, and three-limb magnetics.
The number of magnetic core
The cross section area of middle limb
Air gap distribution
Inductance
The number of turns
The max flux density
Discrete filter inductor
Three EE96/48/32
1024 mm2
The air gap of middle limb is 2.0 mm; the air gaps of the other limbs are zero
Self-inductance is 3.8 mH
80
0.33 T
Integrated filter inductor (five limbs)
Two EE96/48/32
1024 mm2
The air gaps of middle three limbs are 2.0 mm; the air gaps of the other two limbs are zero
Self-inductance is 3.8 mH
80
0.33 T
Integrated filter inductor (three limbs)
One EE96/48/32
1024 mm2
The air gap of middle limb is 1.4 mm; the air gaps of the other limbs are 0.7 mm
Self-inductance is 3.8 mH; mutual inductance is 1.9 mH
80
Middle limb is 0.17 T; the other two limbs are 0.33 TThe magnetic cores of the discrete filter inductors, integrated filter inductors with five limbs and three limbs magnetics are shown in Figure8. In Figure 8(a), three power ferrite EE96/48/32 cores are adopted in three-phase discrete filter inductors, and compared with the other two kinds of three-phase filter inductors, the volume of the three-phase discrete filter inductors is the largest. In Figure 8(b), three power ferrite EE96/48/32 cores are adopted in three-phase integrated filter inductors with five magnetic limbs; the middle three limbs have three-phase inductor’s windings; the air gaps of the other two limbs can be adjusted to eliminate the three-phase flux coupling; therefore, this kind of three-phase integrated filter inductors can achieve wild application and the volume is smaller than that of the three-phase discrete filter inductors. In Figure 8(c), one power ferrite EE96/48/32 cores is adopted in three-phase integrated filter inductors with three magnetic limbs, and compared with the other two kinds of three-phase filter inductors, all the three limbs have three-phase inductor’s windings and the three-phase fluxes couple with each other, and then the application area is a little limited due to the coupling, but the advantage is that this kind of three-phase integrated filter inductors achieves the minimum volume.The comparison of discrete magnetics, five-limb magnetics, and three-limb magnetics: (a) discrete magnetic cores, (b) integrated magnetic core with five limbs, and (c) integrated magnetic core with three limbs.
(a)
(b)
(c)


## 6.2. The Filter Effect Comparison of Three Kinds of Three-Phase Filter Inductors

The SPWM control method is adopted in the three-phase inverter due to its simple and easy realizing, and it can be easily used to testify the three kinds of LC filters. The switching process of switchQ1 and the control voltages uQ1 and uQ4 for the switch Q1 and Q4 are shown in Figure 9. The other control voltages are similar to uQ1 and uQ4.The waveforms of switching process and control voltages: (a) the switching process of switchQ1, (b) the control voltages for the switch Q1 and Q4. The voltage measured in CH2 by the voltage transformer; the proportion is 1 V = 100 V; in CH1, 1 V = 1 A; the time per DIV is 100 ns. In CH1, the voltage per DIV is 7.5 V; in CH2, the voltage per DIV is 7.5 V; the time per DIV is 16.6 us.
(a)
(b)Furthermore, in order to testify the filter effects of the three kinds of three-phase filter inductors, the current THD (total harmonic distortion) of three kinds of three-phase filter inductors is compared. The three-phase load currents of three-phase discrete filter inductors are shown in Figure10, and the THD of A phase, B phase, and C phase is, respectively, 4.5%, 4.7%, and 4.1%, and all can achieve the demand. The three-phase load currents of three-phase integrated filter inductors with five limbs are shown in Figure 11, and the THD of A phase, B phase, and C phase is, respectively, 4.4%, 4.6%, and 4.4%, and all can achieve the demand. The three-phase load currents of three-phase integrated filter inductors with three limbs are shown in Figure 12, and the THD of A phase, B phase, and C phase is, respectively, 4.5%, 4.4%, and 4.1%, and all can achieve the demand. Therefore, all three kinds of three-phase filter inductors can fulfill the demand of the current THD. For the low and medium power application, the three-phase coupling in the integrated filter inductors with three limbs can be omitted, and then the integrated filter inductors with three limbs can be chosen due to the minimum volume. Take A phase current of three-phase integrated filter inductors with three limbs, for example, based on formula (42), 2MImωcos(ωt) is caused by coupling, and its max value is2MImω. In this paper, M=1.9 mH, Im=5A, ω=2πf=314, and then it can be calculated that 2MImω=4.2V. Compared with Vin (700 VDC) and eas(220 VAC), the coupling voltage2MImω=4.2V is very small and can be omitted. But for the large power application, load current Im is large and the coupling cannot be omitted; then it is difficult to achieve the ideal filter effect. Therefore, for the large power application, the integrated filter inductors with five limbs can be chosen. The coupling of the integrated filter inductors with five limbs can be eliminated by adjusting the air gap of two side limbs, and then the filter effect is the same as the discrete filter inductors and the volume is less than the discrete filter inductors.The three-phase load current with discrete filter inductor: (a) the current of A phase and B phase, (b) the current of A phase and C phase. In CH1, the current comes from the current transformer and the proportion is 1 V = 5 A; it means that the current per DIV is 5 A; in CH2, the current per DIV is 5 A; the time per DIV is 5 ms.
(a)
(b)The three-phase load current with integrated filter inductor with five limbs: (a) the current of A phase and B phase, (b) the current of A phase and C phase. In CH1, the current comes from the current transformer and the proportion is 1 V = 5 A; it means that the current per DIV is 5 A; in CH2, the current per DIV is 5 A; the time per DIV is 5 ms.
(a)
(b)The three-phase load current with integrated filter inductor with three limbs: (a) the current of A phase and B phase, (b) the current of A phase and C phase. In CH1, the current comes from the current transformer and the proportion is 1 V = 5 A; it means that the current per DIV is 5 A; in CH2, the current per DIV is 5 A; the time per DIV is 5 ms.
(a)
(b)


## 7. Conclusion

In this paper, the comparison analyses of three-phase discrete filter inductors and two kinds of three-phase integrated filter inductors in three-phase PV inverter are proposed. The experiment results show that the volumes of three-phase integrated filter inductors are reduced obviously. Compared with three-phase discrete filter inductors, the volumes of the five-limb integrated filter inductors are reduced third, the three-limb integrated filter inductors are reduced two-thirds, and the filter effects of three kinds of filter inductors are almost the same; all can achieve the good filter effect. Based on the different demands of the PV inverter, different LC filter can be selected to fulfill the demands for the power density and the electrical performance. The analyzed method also can be used in three-phase LCL filter.



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*Source: 217023-2013-09-11.xml*