State and Space Vectors of the 5-Phase 2 -Level VSI †

: The paper proposes a general description system of the ﬁve-phase two-level inverter. The two base methods are presented and discussed. The ﬁrst one is based on the standard space vector transformation, while the other uses state vectors which enable the deﬁnition of the basic physical quantities of the inverter: current and voltage. The proposed notation system o ﬀ ers a general simpliﬁcation of vector identiﬁcation. It comprises a standardized proposal of notation and vector marking, which may be extremely useful for the speciﬁcation of inverter states. The described notation system makes it possible to reach correlation between state and space vectors. It presents space and state vectors using the same digits. These properties suggest that the proposed notation system is a useful mathematical tool and may be really suitable in designing control algorithms. This mathematical tool was veriﬁed during simulation tests performed with the use of the Simulation Platform for Power Electronics Systems—PLECS.


Introduction
A large number of electrical energy converters are applied in contemporary industry and the public area. Their performance as well as the level of power are continually intensely developed. The increasing range of novel requirements has a great influence on the development of inverters as well as the technology of power electronics components, especially power semiconductors. Innovative elements used in converters, e.g., current switching diodes and transistors can work at higher frequencies and lower power losses. For instance, the SiC (silicon carbon) transistors have better features, such as switch-on and off times as well as a significantly higher temperature range. As a result, they work faster than IGBTs (insulated gate bipolar transistor) and make it possible to diminish the overall dimensions of the converter. Generally, converters have to fulfil specific and diversified requirements concerning supply and conversion methods in possibly a large power range. The application area of inverters is very common and large. There are many industrial applications where inverters have to be equipped with quite different features. This implies diversified converter circuits and various conversion methods of electrical energy. In alternating current (AC) electrical machine and drive applications, the ability of speed regulation is crucial, so the electrical converter has to guarantee the related adjustment of output voltage and frequency as fundamental. Uninterruptible power supplies (UPS) or distributed power generation systems represent a good example of another requirement. There, the basic requirement is to generate 50 or 60 Hz stable sinusoidal voltage waveforms.
During the first period of power electronics evolution all applications were based on the three-phase supply voltage network. Three-phase two-level inverters became the most suitable devices used drives. These contemporary electric motors are often created as multiphase units. It is possible to meet 5, 6, 9 or even 11-phase motors [11]. The drives using these motors have favorable properties, so drives based on three-phase motors have slowly disappeared [12][13][14][15].
The higher phase number determines a new impact on research works concerning inverter circuits and control algorithms. More phases mean higher complexity of the converter circuit and a greater number of semiconductor devices. The number of vectors grows in the power scale in relation to the phase number. For a five-phase two-level inverter the number reaches 2 5 = 32. Such a number of accessible vectors requires using more sophisticated control methods and algorithms in comparison to standard control algorithms used in three-phase inverters. This is the reason to search for good mathematical tools and converter models. These tools should be useful in order to facilitate the composition of control algorithms that are easy to implement and which can guarantee fast converter performance.
In the literature of the subject one can find an enormous variety of proposals concerning converter models and their descriptions. Although there are many proposals relating to this, there is none that can be respected and accepted [16,17]. Additionally, it happens that the notational system is assumed accidentally or has no mnemonic features.
This contribution presents a significant development of the notation system idea proposed in the paper [18] during the ICREPQ'14 conference, and it might be the first step of a general system concerning multiphase inverters.
The paper presents a description system of the five-phase two-level inverter. The notational system provides a general standardization and simplification of vectors. It helps to identify voltage states. It contains a novel handy proposal of notation and stamping, which is helpful for a converter description as well as for developing control algorithms. This system of notation enables accomplishing a comparison and correlation between state and space vectors. It describes space and state vectors using the same numbers. The proposal described in the paper refers to the notation system designed for three-phase inverters [18]. It determines a significant extension of that system and was designed for five-phase inverters. The considerations presented there give on opportunity to make some generalizations and allow the writing of expressions circumscribing state and space vectors of the multiphase VSI (voltage source inverter). It is worth adding that the main rule of the presented system can be easily extended and adopted for multiphase and multilevel inverters. Figure 1 presents a simplified model of the five-phase two-level inverter. The model includes five two-state switches Ka, Kb, Kc, Kd, Ke, assigned to output phases a, b, c, d, e and connected to the voltage source U D . The negative pole of the voltage source U D is denoted as 0, while the positive pole as 1. Every switch can connect one pole of the voltage source to one phase output.

Five-Phase Two-Level Inverter Model
Accordingly, the switching states of the power electronic switches are also denoted by the numbers 1 and 0. The inverter state consisting states of all five keys can be described using a set of digits abcde where a, b, c, d, e = 0, 1. Of course, the set of five elements where elements can be denoted by digits 0 or 1 may have 32 variations that is: 00000, 00001, . . . 11111. It is useful and wide-spread in many publications to assume that these variations form binary numbers (numbers of the base-2 positional numeral system). As a result, each inverter state is determined by one five-digit binary number where the order of digits corresponds precisely to the order of phases abcde. By converting binary numbers to the decimal number system it is possible to denote all 32 inverter states by decimals: 0, 1 10 , . . . 31 10 , respectively. Thus, the state denoted as 16 10 (10000) 2 determines that the phase a is connected to the positive pole of the voltage source, while the phases b, c, d and e-to the negative pole. That is the model example presented in Figure 1. The indices 10 and 2 indicate the base of the number system. This principle which eliminates decimal numbers may be used in other notation systems. From the very beginning of the power electronics development the base-10 and base-2 number systems were applied to describe 2-level inverter states, although the rule of the precisely controlled relation "number-phase" was not always kept. The aforementioned discussion was cited here as this idea of a description might be used for other converters. The description method of multiphase or even multilevel inverters may be based on further development of this concept.

State-Vector of the Five-Phase Two-Level Inverter
The five-phase two-level voltage inverter can produce 32 different switching states. The consecutive states are denoted by the decimal index k = 0, 1, ... 31. Each selected k-state means that five phase-to-phase voltages are connected to load. These voltages form a matrix row assembled of five elements as a result defining the state vector of the VSI: Since + + + + = 0, then the state vector Vk is determined by four successive phase-to-phase voltages under the condition that x and y denote the neighboring phases.
In a five-phase two-level VSI, phase-to-phase voltages are designed as a potential difference between respective phase outputs. The switch Ka,b,c,d,e may connect the relevant phase output to the negative pole or positive pole of the voltage source UD according to the preferred vector .
The vector index k may be converted to the binary numeral system and written as: where: , , , , = 0,1 and = 0,1,2, . . .31. The binary expansion of the index k permits associating directly the phase output potential with the relevant binary symbol, so it makes it possible to determine output polar voltages. They are referenced to the negative pole of the voltage source : As a result, it permits defining the five-phase two-level VSI state vector by use of respective binary symbols of the index k. The transpose of the matrix , denoted as the vector , presents all five voltages: From the very beginning of the power electronics development the base-10 and base-2 number systems were applied to describe 2-level inverter states, although the rule of the precisely controlled relation "number-phase" was not always kept. The aforementioned discussion was cited here as this idea of a description might be used for other converters. The description method of multiphase or even multilevel inverters may be based on further development of this concept.

State-Vector of the Five-Phase Two-Level Inverter
The five-phase two-level voltage inverter can produce 32 different switching states. The consecutive states are denoted by the decimal index k = 0, 1, . . . 31. Each selected k-state means that five phase-to-phase voltages are connected to load. These voltages form a matrix row assembled of five elements as a result defining the state vector V k of the VSI: Since U abk + U bck + U cdk + U dek + U eak = 0, then the state vector V k is determined by four successive U xy phase-to-phase voltages under the condition that x and y denote the neighboring phases.
In a five-phase two-level VSI, phase-to-phase voltages are designed as a potential difference between respective phase outputs. The switch K a,b,c,d,e may connect the relevant phase output to the negative pole or positive pole of the voltage source U D according to the preferred vector V k .
The vector index k may be converted to the binary numeral system and written as: where: a k , b k , c k , d k , e k = 0, 1 and k = 0, 1, 2, . . . 31. The binary expansion of the index k permits associating directly the phase output potential with the relevant binary symbol, so it makes it possible to determine output polar voltages. They are referenced to the negative pole of the voltage source U D : Energies 2020, 13, 4385

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As a result, it permits defining the five-phase two-level VSI state vector by use of respective binary symbols of the index k. The transpose of the matrix V k , denoted as the vector V k , presents all five voltages: Since the symbols a k , b k , c k , d k , e k can assume only values 0 or 1, then the output phase-to-phase voltage, independently of the phase number, may assume only three values: 0, +U D , −U D .
However, possessing five phases it is possible to select another set of phase-to-phase voltages. In (4) every phase-to-phase voltage is determined by the neighboring phases. The row of the matrix is composed of five U x(x+1)k elements, where x = a, b, c, d, e and x + 1 denote the subsequent phase. It would be interesting to consider the set of phase-to-phase voltages created according to the rule that the phase-to-phase voltage is determined as U x(x+2)k . Then, the state vector could be defined as a matrix composed of one row of five elements: The definition of the five-phase two-level VSI state vector using respective binary symbols of the index k is the following: Figure 2 presents selected polar and phase-to-phase voltage: u a0 , u b0 , u c0 , u ab , u ac . It was assumed that all phase outputs are connected successively, every 72 • , for half a period to the positive pole of the voltage source U D . Evidently, waveforms and rms values of the u ab , u ac voltage are significantly diverse because they depend on the method of the definition: U x(x+1)k or U x(x+2)k . The relevant rms values are: U ab ≈ 0.63U D whereas U ac ≈ 0.89U D .
Energies 2020, 13, x FOR PEER REVIEW 5 of 22 Since the symbols , , , , can assume only values 0 or 1, then the output phase-to-phase voltage, independently of the phase number, may assume only three values: 0, +UD, −UD.
However, possessing five phases it is possible to select another set of phase-to-phase voltages. In (4) every phase-to-phase voltage is determined by the neighboring phases. The row of the matrix is composed of five ( ) elements, where = , , , , and + 1 denote the subsequent phase. It would be interesting to consider the set of phase-to-phase voltages created according to the rule that the phase-to-phase voltage is determined as ( ) . Then, the state vector could be defined as a matrix composed of one row of five elements: The definition of the five-phase two-level VSI state vector using respective binary symbols of the index k is the following:

Pentagon Connected Load of the Five-PhaseTwo-Level Inverter
The inverter state vector V k works on physical quantities and it does not need transformation to be applied for calculation of the current in the load. A model of the five-phase two-level inverter with a load connected in pentagon is presented in Figure 3. It is assumed that the load is symmetrical and every phase to phase load is equal to two-terminal networks: resistor R-inductance L-counter RMF connected in series. The inverter state is determined by the vector V k which means that five corresponding phase-to-phase voltages are connected to the load. Figure 3 presents the situation when the phase a is connected to the pole 1 (+U D ), while the remaining phases to the pole 0 (−U D ). So, the state vector is determined as V 16(10000) .

Pentagon Connected Load of the Five-PhaseTwo-Level Inverter
The inverter state vector works on physical quantities and it does not need transformation to be applied for calculation of the current in the load. A model of the five-phase two-level inverter with a load connected in pentagon is presented in Figure 3. It is assumed that the load is symmetrical and every phase to phase load is equal to two-terminal networks: resistor R-inductance L-counter RMF connected in series. The inverter state is determined by the vector which means that five corresponding phase-to-phase voltages are connected to the load. Figure 3 presents the situation when the phase a is connected to the pole 1 (+ ), while the remaining phases to the pole 0 (− ). So, the state vector is determined as  If the load remains in a pentagon connection, then suitable phase-to-phase voltages are equal to the voltage determined in (4). For each vector switched on in a time point t = tn the inverter model is reduced to the equivalent circuit described by the system of five equations. Every equation has the form: where x and y denote the adjacent phases: x ≠ y and x, y = a, b, c, d, e. The voltage is determined according to (4) where the time constant is = and initial current in the time point .
The expressions (7) and (8) define the dependence between phase-to-phase voltage and the current load using the suitable symbols of the vector index k binary expansion. If the load remains in a pentagon connection, then suitable phase-to-phase voltages are equal to the voltage determined in (4). For each vector V k switched on in a time point t = t n the inverter model is reduced to the equivalent circuit described by the system of five equations. Every equation has the form: where x and y denote the adjacent phases: x y and x, y = a, b, c, d, e. The voltage u xyk is determined Assuming that the counter EMF xy is constant in a time interval t tt n , t n+1 and equals E xy respectively, it is easier to resolve the equations and obtain expressions describing currents of a load (8). The symbol τ represents the time constant of one load leg and I 0xy-the existing value of the current in the time point t n . The expression (8) is valid in the specified time interval between the successive switching of the inverter states V k(n) and V k(n+1) . The time interval is assumed to be extremely short in comparison to the time constant of the load circuit and the supposed period of the counter RMF.
where the time constant is τ = L xy R xy and I 0xy initial current in the time point t n . The expressions (7) and (8) define the dependence between phase-to-phase voltage and the current load using the suitable symbols of the vector index k binary expansion.  Figure 4 presents the five-phase two-level inverter model with a load arrangement that is connected in a star. For the considered class of inverters, it can be assumed more than one mathematical definition of the five-phase two-level inverter state vector V k . For instance, the following definition:

Star Connected Load of the Five-Phase Two-Level Inverter
is effective in view of the fact that u ab k + u bc k + u cd k + u de k + u ea k = 0.
Energies 2020, 13, x FOR PEER REVIEW 7 of 22 Figure 4 presents the five-phase two-level inverter model with a load arrangement that is connected in a star. For the considered class of inverters, it can be assumed more than one mathematical definition of the five-phase two-level inverter state vector . For instance, the following definition:

Star Connected Load of the Five-Phase Two-Level Inverter
is effective in view of the fact that For any vector switched on, in the time t = tn, the inverter model is reduced to an equivalent circuit consisting of four closed loops and presented in Figure 5.  For any vector V k switched on, in the time t = t n , the inverter model is reduced to an equivalent circuit consisting of four closed loops and presented in Figure 5.  Figure 4 presents the five-phase two-level inverter model with a load arrangement that is connected in a star. For the considered class of inverters, it can be assumed more than one mathematical definition of the five-phase two-level inverter state vector . For instance, the following definition:

Star Connected Load of the Five-Phase Two-Level Inverter
is effective in view of the fact that For any vector switched on, in the time t = tn, the inverter model is reduced to an equivalent circuit consisting of four closed loops and presented in Figure 5.  If at a time point t = t n the control system switched on the vector V k at that time, a quadruple of phase-to-phase voltages was connected to the circuit. The selected vector V k acts in a time interval t tt n , t n+1 and it is assumed-as it was aforesaid-that this interval is very short in relation to a time constant of the circuit and a period of the interphase voltage. Then, the following assumptions might be acceptable: It means that in the time interval t tt n , t n+1 all above voltages are constant. Additional assumptions result in the fact that the system considered in Figure 4 is not equipped with a neutral conductor, so the following dependencies are obligatory: If the phase load is fully symmetrical it is profitable to assume the equal opportunity of resistances and inductances as well: As a matter of fact, the mathematical model of the circuit is described by the system of four equations: Solving the system, it is possible to acquire all analytical phase voltage and current waveforms. The Laplace transform is a good tool to accomplish that task. For any vector V k the relation between the Laplace transform of phase and loop currents is evident: In order to obtain a final solution, it is necessary to introduce expressions describing the relations between phase and phase-to-phase voltages. The applicable interdependence between phase voltage U x (s) and phase-to-phase voltage U xy (s) does not depend on selected vectors and remains the following: For instance, the dependence defining the phase voltage U a can be proved by the use of the appropriate voltage vectors. This situation is illustrated in Figure 6.
Knowledge of phase voltages allows the solving of the system of equations and finding the successive phase currents. For instance, the expression describing the Laplace transform of the phase a current is the following: Energies 2020, 13, 4385 9 of 21 Energies 2020, 13, x FOR PEER REVIEW 9 of 22 Knowledge of phase voltages allows the solving of the system of equations and finding the successive phase currents. For instance, the expression describing the Laplace transform of the phase a current is the following: The inverse Laplace transform leads to analytical expressions describing phase currents. They are given as: The output phase-to-phase voltage of the two-level inverter, independently of the phase number, may assume only three values: 0, ± . Solving the four-loop circuit in the way presented in [18] and converting (14) it is possible to write the generalized expression describing the phase voltages: , , , , , of the five-phase two-level five-phase VSI as well as phase load currents , , , , (16). They all can be described using binary digits of the vector index = ( ) binary expansion.
where = is a time constant of the phase load, = ( + ) and − phase x current in the time point .
The inverse Laplace transform leads to analytical expressions describing phase currents. They are given as: The output phase-to-phase voltage of the two-level inverter, independently of the phase number, may assume only three values: 0, ±U D . Solving the four-loop circuit in the way presented in [18] and converting (14) it is possible to write the generalized expression describing the phase voltages: u ak , u bk , u ck , u dk , u ek , of the five-phase two-level five-phase VSI as well as phase load currents i a , i b , i c , i d , i e (16). They all can be described using binary digits of the vector index k = (a k b k c k d k e k ) 2 binary expansion.
where τ = L R is a time constant of the phase load, E x = Esin(ωt n + ϕ x ) and I 0x − phase x current in the time point t n .
These expressions describe phase voltages and currents of the two-level five-phase voltage source inverter. Converting similarly Equations (4) and (14) it makes it possible to write expressions describing phase-to-phase voltages and the resulting currents. The five-phase two-level VSI has 32 states of operation described by vectors representing the consecutive sets of phase-to-phase voltages. Every single set of 32 vectors has its own connection diagram to the supply voltage U D . Some possible arrangements are presented in Figure 7.
Energies 2020, 13, x FOR PEER REVIEW 10 of 22 These expressions describe phase voltages and currents of the two-level five-phase voltage source inverter. Converting similarly Equations (4) and (14) it makes it possible to write expressions describing phase-to-phase voltages and the resulting currents.
The five-phase two-level VSI has 32 states of operation described by vectors representing the consecutive sets of phase-to-phase voltages. Every single set of 32 vectors has its own connection diagram to the supply voltage . Some possible arrangements are presented in Figure 7.    vectors are V 31(11111) , V 0(00000) . According to the model presented in Figure 4 these two vectors cause that all phase-to-phase voltages u xy(0 or 31) are equal to zero, so they do not influence a passage of loop currents. These vectors are usually called "zero vectors". These expressions together with Equations (6) and (12) describe the voltages and currents of the two-level five-phase voltage source inverter.

Space Vectors of the Five-Phase Two-Level Voltage Source Inverter
The polar voltage space vector of the two-level five-phase VSI is determined analogically to the way presented in previous sections. The definition was established as: where q = e j2π/5 = cos α + j sin α α = 72 • .
Applying the Euler's formula and introducing interdependencies among the angles as well as symbols a k , b k , c k , d k , e k , the space vector is given as follows: The Equation (20) expresses the space vector which is defined in symbols of the index k binary expansion. The index k is a decimal number discriminating the state vector k = (a k b k c k d k e k ) 2 , After transformation the space vector is a complex number which may be represented by the modulus M k and the argument ϕ k . But in the case of the five-phase inverter the situation is more complicated compared to the three-phase inverters. The expression allowing calculation of the space vector modulus is the following: where coefficients γ 1 , γ 2 are: The position of the successive vector is determined by its modulus M k and the shift angle ϕ k . The angle is determined as: In order to determine correctly the shift angle, it might be necessary to use additionally the arctg function (24). It results in periodicity of trigonometrical functions and a different period of sine and tangent functions.
It was assumed at this point that the denominator of the expressions (23) and (24) did not equal zero. The result of these two formulae is indeterminate for 0 = a k = b k = c k = d k = e k or 1 = a k = b k = c k = d k = e k . The circumstances, where digits are equal to 0 or 1, denote that the state vector modulus reaches 0 and its shift angle is indeterminate. The relevant two vectors V 0(00000) and V 31(11111) are usually called "zero vectors" and they do not influence output current.
According to (21) and after calculations, the modulus M k assumes three different results: In such a case among the 32 vectors there are three groups of 10 active vectors each beside two nonactive zero vectors. The parameters: moduli M k , coefficients γ 1 , γ 2 and shift angles ϕ k of all 30 active vectors are collected in Table 1. The moduli values are approximated. Table 1. The active vectors' parameters of the two-level five-phase inverter. Figure 8 presents the active polar space vectors of the five-phase inverter. They are presented on the complex plane (α − jβ), where α denotes the real axis and β-the imaginary one. The diagram presents their position. For every successive ϕ k = nα/2 (n·36 • ), where n = 0, 1, 2 . . . 9, the position of the corresponding three vectors is marked as one vector because they only differ in moduli.
The modulus values of each triple are given in order: M k = 0.8U D (the upper one), M k = 0.494U D , M k = 1.294U D respectively.
The voltage space vectors prove a convenient mathematical tool. They are a basic tool that allow the generation of the vector control method, the most widespread contemporary control method used to control AC induction or synchronous motors. In addition, they are very useful when studying possible control algorithms and strategy. The controller of the AC drive uses voltage space vectors in order to generate a desirable five-phase PWM (Pulse-Width Modulation) inverter output voltage supplying the motor.
The next Figures present a few examples of phase and phase-to-phase voltage waveforms obtained by using the simplest method of control. For illustration purposes, it was assumed that every leg of the inverter was switched on only once during one period of the referenced voltage and every phase was connected to the +U D pole during the first half of a period while to the −U D pole during the other half. Figure 9 depicts a set of polar U a0 , U b0 , U c0 , U d0 , U e0 and phase-to-phase U abk , U bck , U cdk , U dek , U eak voltage waves. The phase-to-phase voltage relates to the pentagon connected load. The control algorithm consists in successive (every 36 • ) activation of the following sequence of vectors: V 19(10011) , V 17(10001) , V 25(11001) , V 24(11000) , V 28(11100) , V 12(01100) , V 14(01110) , V 6(00110) , V 7(00111) , V 3(00011) . The parameters of the vectors as well as their index binary expansion are collected in Table 2. The operation time of every vector corresponds to the angle ωt = 36 • where ω = 2π f and f denotes frequency of the reference voltage.
Using the star connection load, the same sequence of successively switched vectors results in the phase voltage waveforms presented in Figure 10. The voltage space vectors prove a convenient mathematical tool. They are a basic tool that allow the generation of the vector control method, the most widespread contemporary control method used to control AC induction or synchronous motors. In addition, they are very useful when studying possible control algorithms and strategy. The controller of the AC drive uses voltage space vectors in order to generate a desirable five-phase PWM (Pulse-Width Modulation) inverter output voltage supplying the motor.
The next Figures present a few examples of phase and phase-to-phase voltage waveforms obtained by using the simplest method of control. For illustration purposes, it was assumed that every leg of the inverter was switched on only once during one period of the referenced voltage and every phase was connected to the + pole during the first half of a period while to the − pole during the other half.  Table 2. The operation time of every vector corresponds to the angle ωt = 36° where ω = 2 and f denotes frequency of the reference voltage. Using the star connection load, the same sequence of successively switched vectors results in the phase voltage waveforms presented in Figure  10.
Energies 2020, 13, x FOR PEER REVIEW 15 of 22 Another example is presented in Figure 11. It demonstrates a set of polar and phase-to-phase voltage waveforms when the inverter is controlled completely dissimilarly. Phase-to-phase voltage waveforms relate to the pentagon connected load. The sequence of vectors executed by the control algorithm is the following: (  Table 3. Table 3. Collected parameters of the vectors. Red color denotes potential + , green colorpotential 0. Another example is presented in Figure 11. It demonstrates a set of polar and phase-to-phase voltage waveforms when the inverter is controlled completely dissimilarly. Phase-to-phase voltage waveforms relate to the pentagon connected load. The sequence of vectors executed by the control algorithm is the following:  Table 3. Energies 2020, 13, x FOR PEER REVIEW 16 of 22 Figure 11. Polar (a) and phase-to-phase (b) voltage waveforms: the pentagon connected load. Vector sequence: , , , It is worth recognizing that this way of control guarantees higher phase-to-phase voltage than was obtained in the previous example. The fundamental harmonic of the waveform presented in Figure 9 did not go beyond 0.44 while the fundamental of the phase-to-phase voltage presented above reaches 1.15 . The RMS values of these voltage waveforms reach approximately 0.63 and 0.89 , respectively.

Experimental Results
This mathematical tool was verified during simulation tests using the PLECS program. The following parameters were assumed for the purposes of simulation tests: voltage = 600 , frequency = 50 and load: = 10 Ω and = 5 , = 10 , = 20 . Figure 12 presents the main part of the model of the five-phase two-level inverter made in the PLECS program. The control signals are presented in Figure 13.
It is worth recognizing that this way of control guarantees higher phase-to-phase voltage than was obtained in the previous example. The fundamental harmonic of the waveform presented in Figure 9 did not go beyond 0.44U D while the fundamental of the phase-to-phase voltage presented above reaches 1.15U D . The RMS values of these voltage waveforms reach approximately 0.63U D and 0.89U D , respectively.

Experimental Results
This mathematical tool was verified during simulation tests using the PLECS program. The following parameters were assumed for the purposes of simulation tests: voltage U D = 600 V, frequency f = 50 Hz and load: R = 10 Ω and L = 5 mH, L = 10 mH, L = 20 mH. Figure 12 presents the main part of the model of the five-phase two-level inverter made in the PLECS program. The control signals are presented in Figure 13.                 During the simulation, important coefficients characterizing electrical signals such as the THD (Total Harmonic Distortion) and the RMS (Root Mean Square) were calculated. With selected load parameters, these coefficients are presented in Table 4. During the simulation, important coefficients characterizing electrical signals such as the THD (Total Harmonic Distortion) and the RMS (Root Mean Square) were calculated. With selected load parameters, these coefficients are presented in Table 4. Summarizing the results of the simulation tests described in this section, it can be stated that they confirm the correctness of the mathematical tool presented in the paper. In the spectral analysis of voltage and current waveforms it can be seen that the first harmonic and the third harmonic dominate. A significant advantage of the described control method is that the current third harmonics are significantly lower even using the simplest way of control, in particular, without the use of PWM modulation. In order to assess the content of higher harmonics, the THD coefficients for currents and voltages were also calculated.

Conclusions
In multiphase inverters the ascending number of active vectors causes difficulties in designing sophisticated control methods and algorithms. It requires accessibility to proper mathematical tools and inverter models which are handy for this purpose and provide easy implementation as well as fast performance of the converter. The main contribution of this paper was to present a very simple mathematical system of notation and formulas. The inverter may be defined by the use of definite state vectors or standard space vectors obtained with the use of the polar voltage transform. The paper presents a notation system of all five-phase two-level 32 converter states. Every state is defined by two vectors: state vector and polar voltage space vector. Both vectors are specified by the use of the same digits resulting in binary expansion of the decimal vector index. The method could be advanced to other multiphase and multilevel inverters. This construction of the notation system provides an easy-to-use mathematical tool. It enables selection of a suitable vector sequence assuring the desirable voltage or current waveform. The discussed vector sequences were based on the imperative of only one switching in one phase during the whole voltage period. Three adequate examples were presented in Figures 9-11. This mathematical tool makes it easier to define state and space vectors as well as to calculate the available phase and phase-to-phase voltages and the resulting load currents. This mathematical tool was verified during simulation tests using the PLECS program. After applying the control signals calculated in accordance with the proposed mathematical algorithm in the simulation model, similar waveforms of phase voltages and inter-phase voltages were obtained. Additionally, in experimental studies, the operation of the model with the RL type load was tested. The most important electrical factors such as the THD and the RMS were also calculated. In particular, the analysis of the THD coefficient shows that the harmonic content in voltage and current waveforms is satisfactory, especially taking into account the fact that PWM modulation was not used. In the spectral analysis of voltage and current waveforms it can be seen that the first harmonic and the third harmonic dominate. A significant advantage of the described control method is that the current third harmonics are significantly lower even using the simplest way of control, in particular, without the use of PWM modulation.