ANALYSIS AND SIMULATION OF A THREE-PHASE DELTA INVERTER FOR A STAR-CONNECTED RL LOAD

In this study, analysis and simulation of the electrical and mathematical model of a 3-phase and 3-switch delta inverter that fed a three-phase RL load were performed. First, the Matlab/Simulink model of the inverter was created. In this model, an RL load of R=10Ω and L=0.005 H was fed and various simulations were made. The simulation results were obtained using 3 step switching technique for different conduction modes. The delta inverter that has a simpler structure and less switching elements than conventional inverter structures can be easily realized with low-cost microcontrollers.

. Classification of the power converters.
Depending on the type of power transfer link used between them, this conversion is done in two different ways: i) Single-stage ac-ac conversion (direct link), ii) two-stage ac-dc-ac conversion (dc link) [3]. Both types of converters require the use of a large number of semiconductor switching elements and appropriate modulation techniques. In most motor drive systems used in industry, dc link converters that make two-phase conversion are used. These converters are made up of two parts. First, by rectifying the ac input voltage with the help of rectifier circuit, it is converted to the dc voltage. In the second step, with the help of an inverter circuit, this obtained dc voltage is converted to ac voltage at variable amplitude and frequency. According to the configuration of the used dc link, inverters are divided into two sections, voltage source inverters (VSI) and current source inverters (CSI).
The voltage source inverters have large filter capacitors that are at the input terminals and hold the dc link voltage constant. In these inverters, the adjustable output voltage and frequency are independent of the load current. On the other side, the current source inverters provide a more controlled load current with the help of large-value inductors connected in series to the input terminals. That is, the load current is more controlled compared to the voltage. Here, the inverter output voltage depends on the load impedance. Especially the voltage source inverters are widely used in academic studies and industrial applications. Delta inverter, which is from the family of voltage source inverter, was first proposed by Evans and colleagues in 1980 for variable speed drive systems fed from off-grid insulated batteries [12].
Although there are many studies related to the power converters given in Fig. 1 in the literature, the studies related to the delta inverter, which cannot take place in this classification, are limited. In these studies, inverter design and harmonic analysis of it [5,7], vector control application [6], the openloop control of the induction motor [4,16], comparison of the delta inverter with the bridge inverter [8], analysis of the switching techniques having 120° and 240° conduction modes [8], speed and torque estimation of the BDCM motor [9], vector control and performance analysis of the BACM motor [10], for large-scale PV systems, a 30 kW grid-connected delta inverter design and analysis of it [11], self-control application of the BDCM motor with FPGA [13,18], direct torque control of the induction motor with FPGA [14], direct torque control of the induction motor [12,15,22,23], control of the delta inverter with SVPWM [17,19,20], and comparison of RFOC and DTC techniques [21] were performed successfully.
In this study, the electrical and mathematical model of a 3-phase and 3-switch delta inverter that fed a star-connected RL load was analyzed and simulated in Matlab/Simulink environment. The simulation results obtained in different conduction modes were presented comparatively.

Structure of the Delta Inverter
Inverters are widely used in all areas where variable frequency and voltage are needed [24,25]. In its structure, there are 3 semiconductor switching elements and 3 insulated equivalent DC sources. A switching element is connected in series with each DC source to form a leg of the inverter.
The inverter output voltage is generated with the appropriate modulation of S1, S2 and S3 switches.
Each output line voltage is generated by two switching elements. In order for the delta inverter to be able to generate a nominal output voltage, only 2 switching elements must be OFF at any instant of "t" time. If one switching element is OFF and the other two are in the ON position, the inverter cannot generate the voltage and current required for the load. If all three switching elements are in OFF position in the same "t" instant, then the inverter is short circuited. Short-circuit currents at high-value occur and damage the inverter. To avoid short circuit, 3 switching elements must not be in the conduction at the same time.
For the switching elements, the state of being in the conduction is expressed with 1 and the state of being in cutoff is expressed with 0. Accordingly, SA, SB, SC switching functions can be written as follows, Among these, V1, V3, V5 vectors are called "low-amplitude", V2, V4, V6 vectors are called "highamplitude" voltage vectors. Fig. 3 shows the switching states corresponding to each of the 6 active voltage vectors generated clockwise at inverter output according to the being in conduction and cutoff states, respectively.
The V0 vector is obtained in the case of S1=S2=S3=0 where all switching elements are in cutoff and is called "zero" voltage vector. When the zero voltage vector is applied, the source currents and the external circuit currents become zero. Figure 3. Switching states.
The V7 vector is obtained in the case of S1=S2=S3=1 where all switching elements are in the conduction and is called the "short circuit" voltage vector. The V7 vector causes high-value short circuit currents to flow by making the inverter short circuit. Therefore, V7 voltage vector should not be applied in practice. The inverters output voltages, which are obtained depending on the switching states given in Fig. 3, are shown in Table I.

Delta Inverters and Star-Connected Load
In Fig. 4, the connection of a 3-phase star-connected RL load to the delta inverter is presented.
Here, the conversion coefficient matrix obtained by taking = 0 0 is given in Equation 8. This coefficient matrix will be used in later motor control applications.
VAn, VBn, VCn phase-neutral voltages and Vα, Vβ components of the V1-6 vectors that form these voltages are given in Table II.
The regions of the Vα and Vβ components forming the V1-6 vectors and their polarities are given in Table III.  The schematic representation of V1-6 vectors, whose polarities and regions are given in Table III, on the αβ plane is given in Fig. 5.

Modulation Technique
Electrical drives are widely used in control and automation systems. These drives can be examined in two different parts, the power and control sections. In the power section, switching power supplies, called controlled power supply, converters and inverters are used. On the other hand, the control section includes modulation techniques (MT) which change both the waveform and amplitude of the output voltage of these controlled power supplies at the desired intervals.
Many different modulation techniques have been developed for the control of electrical drive systems whose detailed classification is given in Fig. 1. The most commonly known of these techniques are: Six-step switching, Pulse Width Modulation, Sinusoidal PWM, space vector PWM, and Hysteresis Band switching technique.
At the end of the analysis conducted in this study, it has been observed that for a 3-phase delta inverter, depending on the switching combination of 3 IGBT, there are three different conduction modes. These are 120° conduction mode, 180° conduction mode, and 240° conduction mode. Each conduction mode requires a different modulation process. In a one-period duration, 120° conduction mode and 240° conduction mode take place within 3 steps. On the other hand, 180° conduction mode takes place within 6 steps. The voltage vectors that are active in each conduction mode are given in Table III. As can be seen from the Table III, whereas V1 ,V3 ,V5 voltage vectors with low amplitude are used in 120° conduction mode, V2 ,V4 ,V6 vectors with high amplitude are used in 240° conduction mode. In 180° conduction mode, all of the V1-V6 vectors are switched together. Depending on these active voltage vectors, the inverter shows different behavior in each conduction mode. In this way, different voltages with low and high amplitude can be obtained at the inverter output. In this regard, delta inverter can be used in many different applications. Using these three basic conduction modes, more advanced different modulation algorithms can also be developed.
For the 120°, 180° and 240° conduction modes, switching states, phase-neutral voltages that can be obtained from the inverter output, and phase-phase voltages are shown in Fig. 6. These graphs are obtained in the condition in which the load connected to the inverter output is star-connected.  For modulation, switching elements were switched by selecting 120°, 180° and 240° conduction modes. Fig. 8 shows the angular change graph of the modulation signal, from which these conduction modes are obtained, and the sectors. This modulation signal, which was obtained by the movement of the unit vector with 50 Hz scanning frequency in the 360° circular plane, was divided into 3 regions of 120-degree within a one-period of time. These regions are called sectors and different switches are switched sequentially in each sector. The change of this given modulation signal was carried out and simulations were performed primarily in 120° conduction mode by using sectors, and the graph for the obtained load current is given in Fig. 9. In this conduction mode, at any instant of "t" time, only a DC source and a switching element are on. Because the DC source voltage disperses equally on both phase windings, the phase currents have I/2 amplitude. Any two of the star-connected 3 phase windings are equal to each other in terms of amplitude, but they are in opposite directions. Since the third winding is not active, this winding current is zero. The graph of the load current obtained in 240° conduction mode is given in Fig. 10. In this conduction mode, two switching elements and two DC sources are active at any instant of "t" time.
Because both sources are connected in series, the total end voltage is as much as -2VDC and each winding voltage is as much as VDC. Winding currents have "I" amplitude. Any two of the 3 phase starconnected windings are equal to each other in terms of amplitude, but they are in opposite directions.
Since the first and second winding currents flow in reverse direction over the third winding, this winding current is zero. Fig. 11 shows the graphs belonging to the phase and line voltages obtained in 240° conduction mode. In terms of amplitude, the winding phase voltages are as much as +VDC and -VDC and line voltages are as much as = , = , = −2 , respectively.  The graphs of the load current obtained in 180° conduction mode are given in Fig. 12. This conduction mode is obtained by the use of 120° and 240° conduction modes together. Each phase current is formed by the synthesis of the phase currents with I/2 and I amplitude. Moreover, in Fig. 13, the Iha currents obtained from a_phase in each three conduction modes are given in a single graph. Figure 13. A phase currents obtained in 120°, 180°, 240° conduction modes.

Conclusion
In this study, analysis of the electrical and mathematical model of a three-phase three-switch delta inverter feeding a three-phase RL load was performed. By feeding an RL load having R=10Ω