Induction Motor: Methods / Devices for Forming Characteristics

By its very nature resulting from solid rotor windings and lack of power supply to its windings, an induction motor is most suitable for operation under steady conditions and with a small slip. In such a case the angular speed results from the frequency of the supply to the stator windings, number of pole pairs and value of the slip. Traditionally, it was applied in drives in which neither frequent changes of speed nor variable control were required (examples of such devices include pumps, blowers, compressors, belt conveyors, cranes, industrial hoists). There was virtually no possibility of controlling induction motors within wide range of speeds while concurrently preserving high energetic efficiency until 1970s. Drives in which the control of speed was necessary most frequently applied slip ring induction motors, in which it’s possible to control rotational speed as a result of use of external elements.

However, such systems are either complex, costly and problematic in control due to the use of cascaded systems. Alternatively, they have lower energetic efficiency due to additional resistance in the rotor's circuit. In addition, the start-up properties of an induction motor under direct connection to the network are adverse due to the initial period of oscillations of electromagnetic torque with a high amplitude and high value of the start-up current. Despite these drawbacks the induction motor has become the most common machine in electric drive systems due to the fundamental advantages including long service life and reliability as well as low price and accessible supply source. Following the development of power electronics and control elements enabling arbitrary shaping of voltages and currents, induction motors became widely applied in complex drives due to a new angular speed control potential and general robustness at heavy duty.

This section will be devoted to the presentation of the methods of forming characteristics of induction motors and will cover the devices that make it possible to realize the required characteristics. The possibility of modeling characteristics results directly from the relation defining angular speed where: - synchronous angular speed of a rotating field. Each of the p values in relation offers the possibility of modeling mechanical characteristics: number of pole pairs p, slip s as well as the frequency fs of the supply volt ages. The control of slip s is possible to a large extent as a result of the external interference in the rotor circuit and also voltage changes but within a small range of rotational speeds. The presentation of methods used for modeling characteristics associated with rotor slip changes will follow in the subsequent sections. Concurrently, a separate section will be devoted to an extensive presentation of control as a result of modifying the frequencies of the supply voltages. The application of the various number of pole pairs p for changing motor speed appears to be most straightforward to explain. A series of synchronous speeds ?f for a given supply frequency consists of a discreet values. For the successive number of pole pairs p = 1,2,3,4,5,6,... and, e.g., for the frequency of the supply fs = 50 [Hz] they are, approximately:

This finds application in multi-pole motors, in which the windings can be switched to two or three synchronous speeds, which leads to a stepwise change of rotor speed. This type of drive is applied in cranes and industrial hoists mainly with two speeds - transit speed with a higher value and a slower approach speed.

Control of Supply Voltage

The control of the supply voltage can offer only limited possibility of adjusting rotational speed of an induction motor. This results from the basic mechanical characteristic of the motor which indicates that the slip under a given load can be increased up to the limit of s < sb, which means it has to keep below the break-torque slip beyond which a loss of the stability occurs and the motor stops.

In addition, this type of control is achieved at the expense of efficiency loss since under a constant load the losses in the motor are ?P > Pf s. This comes as a con sequence of the increase of the current and losses in the motor windings. At the same time, the control of the supply voltage is currently used in order to reduce the start-up current and perform a soft-start. This is realized with the use of an electronic device called a soft-starter. A diagram of such a device is found.

+-+-+- Basic diagram of a soft-starter for an induction motor.

The introduction of semiconductor elements (SCRs, IGBTs, GTOs, MOSFETs etc.) for the two directions of current flow for each line supplying the motor windings makes it possible to employ current flow with a selected delay angle a in relation to the zero crossing of the supply voltage curve. As a result, at some expense of altering the current and voltage from sinusoidal shape, it’s possible to control the value of voltage and synchronize the motor with the network at the instant of connecting the particular motor phase windings during start-up. Soft-starters may, accordingly, realize the following functions related to the start-up and stopping of an induction motor:

- synchronization of the connection of particular phase winding to the network and thus enabling the reduction of the variable component of the torque

- limitation of the start-up current in a selected range,…

- braking with the use of direct current and conduct con trolled stop of a drive.

Not all of the above functions have to be realized by a single type of soft starter.

In the most economic versions designed for smaller drives, a soft starter some times contains switches in the two supply lines, which only leads to limitation of the start-up currents and does not provide symmetry of the supply voltages. The following are examples of application of a soft-starter for an medium power induction motor with a delay angle a = 40º and the basic value of the moment of inertia J = Js. The figures present a comparison between start up versions without synchronization during the connection of phases to the net work and the one with synchronization involving the connection of line L1, L2 for phase angle d1,2 = 0.48p [rad] and a later connection of the third supply line L3 for angle: d3 = a + d1,2 -0.1 [rad]. As a result, we obtain a very soft starting curve during the initial stage of the start-up of the motor accompanied by a very favorable torque waveform. The synchronized connection for such a large delay angle a = 40º also results in the reduction of the duration of the start-up since the value of the constant component of the motor torque increases during the initial stage of the start-up. The current wave form in the phase winding of the motor for such a supply. The delay angle in the range of around 40° is virtually the sharpest one for which it’s possible to conduct start-up of the motor during idle run within a sensible time, due to the considerable reduction of the value of electromagnetic torque of the motor. The approximate illustration of the effect of delay angle a on characteristics of the motor. Soft-starters find application in drives with an easy start-up due to the considerable reduction of the torque following the fall of the value of the supply voltage.

+-+-+-Line current of the medium power induction motor during free acceleration with a soft starter (a = 40°): a) without synchronization b) with synchronization: d=0.48p; d3 = a + d1,2 - 0.1

+-+-+- Relative velocity curve for the medium power motor during the soft-start free acceleration, under conditions.

+-+-+- Electromagnetic torque curve for the medium power motor during the soft-start free acceleration, under conditions

+-+-+- Steady-state line current for the medium power motor during the soft-start free acceleration, under conditions.

+-+-+- Induction motor starting characteristics (relative values) for the medium power motor during the soft-start free acceleration in relation to delay angle a: a) starting current b) break torque c) starting torque d) idle run free-acceleration time

Slip Control

It’s possible to control slip in an induction motor when electric power is delivered through the rotor windings to the external devices. This comes as a consequence of the fact that for a constant electromagnetic torque Te and constant supply frequency f_s the power P? delivered by the rotating field from the stator to the rotor has to remain constant. This is so since in the steady state ...

After some power is extracted from the rotor windings, the electric power Pel = P? s increases and the mechanical power Pm = P? (1-s) decreases, which is possible as a result of an increase of slip s, i.e. the reduction of the rotational speed of the rotor. As we can see, the control of the slip is only possible in slip-ring motors, due to access to the rotor windings from outside. The other possibility associated with power supply to the rotor is hardly ever practically encountered. It’s possible for instance in a motor with power supply from two sides and this case won’t be discussed. The process of power extraction from windings is conducted in two ways. An inclusion of an additional resistance Rd in the rotor circuit is the oldest method of performing soft-start and possibly speed control; however, it’s accompanied by huge losses associated with the produced heat. Another method involves power output to external devices whose role is to transform the power to useful forms, for instance its return to the supply net work. Such devices, which used to be electromechanical, now predominantly are power electronic ones are called cascades. One of them is the Bi-directional drive, and is a subject in the latter part of this section.

Additional Resistance in the Rotor Circuit

This method of control results in changes of static characteristics of the torque. The break-torque slip sb increases proportionally to the increase of the resistance of rotor windings Rr, while the break-torque Tb does not change. This comes as a consequence of the maintenance of the constant relation ar /s, which means that slip s rises proportionally to the increase of ar. This in a way results in the change of the scale of the slip which extends the characteristics of the torque in the direction of higher values of the slip. This leads to an improvement of motor start-up since the start-up current is reduced for s = 1 and the static start-up torque increases. Unfortunately, the operation of the motor in the steady state with an additional resistance in order to reduce the rotational speed is not applied since it results in the reduction of the energetic efficiency of the drive. This is so because for an induction motor the following relation is maintained:

... which means that for instance the reduction of rotor speed theta_r to reach the half of the synchronic speed ?f with the use of resistance based control leads to the reduction of the efficiency to 5 . 0 < ? , which is unacceptable.

+-+-+- Torque-slip characteristics of a wound induction motor with additional external resistance in the rotor for ard = ar·(1…20); ard includes additional resistance Rd connected to the slip-rings

Bi-directional Drive

As a result of the application of Bi-directional drive it’s possible to control the speed of a slip-ring induction motor as a consequence of electric power extraction from rotor windings and its return after the desired transformation into the network. In its historical model the Bi-directional drive contains an electromechanical frequency converter connected on one side to the rotor of an induction motor and the other one to the supply network into which the power returns. In a modern solution of the Bi-directional drive, the currents of the rotor windings are rectified in a 3-phase rectifying bridge and subsequently supply a converter which returns the energy into the network via an adapting transformer. Between the two bridges there is an inductor that smoothens the flow of the current and whose role is to se cure the continuity of current flow across the rotor even for small mechanical load of the motor shaft. The control parameter is delay angle a of the thyristor bridge, which for the desired inverter mode is contained in the range:

... where µ is the emergency angle which prevents the inverter back-feed and has to be bigger then the maximum calculated commutation angle. Such control corresponds to the feeding of voltage Ud (a) into the rotor, which offers a possibility of controlling the slip of an idle run s0. The slip during idle run s0 corresponds to the theoretical idle run of an induction motor in which there is an equilibrium between the mean values of electromotive forces E theta_r and voltage at the output of the inverter Ud (a).

For the case of an adequately selected transformer rate supplying the inverter the following is fulfilled: U'L = Ur0' and, as a consequence: ... where ß = 180-a is the inverter advancing angle. An increase of the control angle a of the thyristor, leads to an increase of the slip s0 of the idle run thus shifting the mechanical characteristics of the motor in the direction of lower rotational speeds.

+-+-+- Diagram of a semiconductor Bi-directional drive

+-+-+- Diagram of the simplified Bi-directional drive for mathematical modeling

+-+-+-Starting of the 450 [kW] Bi-directional drive with s0 ˜ 0.3, Tl = 0.15Tn, J = 3Js: a) stator current b) rotor current c) bridge current d) DC link current e) angular speed f) electromagnetic torque.

The control of the semiconductor cascaded drive has been modeled in a simplified form, where an inverter is reduced to lumped elements Ld, Rd, Ud(a). An induction motor is modeled so that the electric variables of the stator are transformed into orthogonal axes u,v while we have to do with natural variables ir1, ir2, ir3 in the rotor windings. This model is discussed and is described by the system of equations. The combination of the model of a slip-ring motor with a bridge on the side of the rotor and a circuit of direct current for an inverter is described by a system of equations with se = 7 electrical degrees of freedom and a single sm = 1 degree of freedom for the rotational motion of the rotor. The generalized coordinates for the electric component of the model are the transformed currents of the stator windings isu, isv, phase currents of the rotor windings ir1, ir3 currents of the rectifier bridge i1, i3 and the current id in the inverter circuit.

This model for electric circuits can take the form of a matrix equation:

Subsequently, +-+-+- presents the waveforms of electric and mechanical variables of the drive in steady state for U”_d = 1200 [V], which corresponds to s0 = 0.148 and ?0 = 133.8 [rad/s], respectively for the load of Tl = 0.5 Tn.

+-+-+-Steady state time-curves of the 450 [kW] Bi-directional drive with s0 ˜ 0.15, Tl = 0.5Tn, J = 3Js: a) stator current b) rotor current c) bridge current d) DC link current e) relative speed f) electromagnetic torque.

One can note that start-up of this drive does not normally occur in a cascaded system, but in system with an additional resistance Rd in the rotor circuit, which ensures a faster start-up with a larger motor torque.

After this initial stage of start-up rotor is reconnected to the Bi-directional drive sys tem. As indicated by the comparison of static characteristics of the torque with the start-up characteristics in the Bi-directional drive system, in the latter case the motor develops much smaller torque due to the deformations of the rotor currents from the sine curve accompanied by a considerable decrease of voltages associated with the components of semiconductor bridges in the rotor.

To give an illustration of a transient operation of the Bi-directional drive a stepwise change in control of inverter voltage was introduced. The output voltage of the inverter changed abruptly from U”_d = 1200 [V] to U”_d = 2400 [V], without a change of the load being Tl = 0.5 Tn. The drive has to slow down due to the change of the idle slip value from 15 . 0 0 ˜ s to 25 . 0 0 ˜ s . The resulting transients are presented.

+-+-+- Transient characteristics of the Bi-directional drive during inverter control change from s0 ˜ 0.15 to s0 ˜ 0.25: a) stator phase current b) rotor phase current c) bridge current d) DC link current e) relative rotor speed f) relative electromagnetic torque

Static mechanical characteristics of the semiconductor

Bi-directional drive were determined on the basis of a series of calculations of the steady state for various inverter bridge control angles.

Torque-speed characteristics the Bi-directional drive for U”_d = 0, 600, 1200, 1800, 2400, 3000, 4200, 4800 [V] or equivalent idle run slip values: s0 = 0.0, 0.074, 0.148, 0.296, 0.370, 0.444, 0.518, 0.594

Supply Frequency fs Control

One of the fundamental methods applied for control of angular speed of an induction motor in accordance with is based on changing the frequency fs of the voltage supply to the stator's windings. Although it was difficult to execute in the past, this method has become widespread as a result of the application of various power electronic converters. It finds application in induction motor drives in the range of power ratings from a fraction of a [kW] to powerful machines exceeding 10 MW [10]. Depending on the power output and design it’s possible to apply various solutions of inverters and various frequencies of energy conversion in the range from several hundred to 30 [kHz] for small and medium power devices. In this section an emphasis will be on the basic solutions applied in induction drives with power inverters, including:

- direct frequency converters -- cycloconverters,

- two-level voltage source inverters,

- three-level voltage source inverters,

- PWM current source inverters.

This list of converter drives does not form the complete record of the applied drives - in particular with regard to large power drives but contains the most common ones. Moreover, resonance based current inverters and load commuted inverters are applied in addition to the listed ones. For each one of the systems it’s possible to apply several methods of control realizing the various voltages wave forms and output currents. The issues thereof are very extensive and are widely discussed in the references.

Direct Frequency Converter-Cycloconverter

The role of a cycloconverter is the conversion of 3-phase alternating voltage and current of the supply network with the frequency fL into single-phase voltage and current of a load with the frequency of fs without conversion into direct current. In order to obtain a 3-phase system of voltages and currents supplying the motor on the output of a converter it’s necessary to apply three separate conversion unit or two units in the economical versions of a converter. Each of such units consists of 2 antiparallel groups of controlled rectifiers most common of which include 6-pulse rectifiers (q = 6). The necessity of using two antiparallel rectifying groups results from the need of symmetric conducting currents in two opposite directions. Thyristors (SCRs) are applied in the rectifying groups of the converter and hence in a typical frequency converter we have to do with current commutation. This comes as a consequence of the fact that the typical application of a converter is the controlled large power induction or synchronous motor drive with a capacity of up to a dozen MW. A SCR-Silicon Controlled Rectifier is a power electronic component with the highest operating voltages and high conduction cur rents; hence, it’s used in large power converters. One of the standard applications of a frequency converter in a 3-phase load (ac motor).

+-+-+- A 3-phase cycloconverter with a 6-pulse rectifier bridges and separated outputs.

The system with inductors limiting the circulating current.

This is a converter with bridge rectifying units and separate phases of the load, which, however, is supplied from a transformer with a single secondary winding.

Between the rectifiers supplying the windings of the motor there are inductors limiting the impact of the equalizing current in the circuit of the antiparallel rectifier units since in this solution both of them are controlled by the delay angle over the entire range of the operating conditions of the converter. This is a solution that does not require detection of the instant of the load current crossing zero and a subsequent separate control of the two rectifier units. Another solution of the cycloconverter system. In this case the 3-phase load is connected in a star, which leads to the supply of particular systems of the anti-parallel converters from separate secondary windings of the transformer in order to avoid shorts.

+-+-+- A 3-phase cycloconverter with a 6-pulse rectifier bridges and the Y - connected output, which requires 3 separate secondary windings of the supply transformer. The system without circulating currents and consequently without inductors.

Inductors are not applied between antiparallel systems, which means that such units don’t involve simultaneous control of both rectifying bridges. Each of them feeds the current into one direction of conducted current that is singular for it. This is associated with a need to apply more advanced control of the converter, which involves the detection of the instant of a current flow direction change, and a short break during the conduction of both bridges in this period to restore blocking ability. The basic distinction in the applied control system involves selection the control method of a cycloconverter with equalizing currents or without them and it’s possible to use one or the other in every type of converter. However, the limitation of the equalizing current is associated with the need to use massive and ex pensive inductors designed for conducting currents with large values. This requirement leads to a tendency to apply a system without equalizing currents.

The principle for the control of cyclo-converters results from the adequate control of a 3-phase rectifier in such a way that ensures an output voltage whose basic harmonic is a sine waveform with the frequency of fs. Since the output voltage and current originate from co-operation of the two rectifying units (bridges) forming the input of a single phase of a load, it’s necessary to control both of them, simultaneously or in succession, in order to ensure that they supply uniform output volt age within the range of the basic harmonic:

- amplitude modulation factor

- frequency modulation factor

- maximum value of the basic harmonic of the output voltage of a converter

- mean voltage of the rectifying unit (q = 6) for delay

- RMS voltage of the secondary side of a transformer.

The delay angle waveform of the rectifier group 1 in the function of the phase angle of the output voltage.

The figures that follow present how the output voltage is formed by the units 1 and 2 of antiparallel rectifiers. Rectifier unit 1 performs the descending section of the modulated voltage, while unit 2 is responsible for the ascending section of the voltage by application of 3-phase voltages of the supply network. Both voltages for these units of the converter generate an identical harmonic of the output volt age with the frequency of fs, while in the range of the higher harmonics the wave forms are different. Hence, the equalizing current occurs in the antiparallel system of the rectifying units for the case of controlling both groups over the entire period of the output voltage.

+-+-+- a1 control angle of a cycloconverter anode group as a function of the output volt age phase angle, for amplitude modulation factor m_a = 0, 0.1,…0.9, 1.0

+-+-+- Cycloconverter's output voltages for ma = 0.55, mf = 0.1667, fs = 10 [Hz] and for fL = 60 [Hz]: a) unit 1 voltages b) unit 2 voltages c) output voltage for separate control of rectifying units - control without circulating current.

+-+-+- Cycloconverter's output voltages for ma = 0.75, mf = 0.3333, fs = 20 [Hz] and for fL = 60 [Hz]: a) unit 1 voltages b) unit 2 voltages c) output voltage for separate control of rectifying units - control without circulating current.

From the illustration of the transformed voltage curves, whose basic frequency is fs it results that the content of higher harmonics in the output voltage is high and increases along with the decrease of the amplitude modulation factor ma. The analysis of the output frequency indicates that the dominant part is occupied by harmonics with the frequencies of ... ... which for a low output frequency fs means that the major higher harmonics have a frequency around qfL. This means around 300 [Hz] for a converter with 6-pulse rectifying units and the frequency of the supply network of fL = 50 [Hz]. Practical considerations lead to the limitation of the upper boundary of the out put frequency to around 0.4 fL and the adaptation of voltage U2 of the transformer supplying the converter to this frequency. This comes as a consequence of the principle, which defines the adaptation of the value of the supply volt age to the frequency, while preserving an adequate surplus of voltage. The aim of this is to apply a potentially high amplitude modulation factor ma and, thus, the limitation of the amplitudes of higher harmonics of the output voltage.

+-+-+- Cycloconverter's output voltages for ma = 0.45, mf = 0.1, fs = 5 [Hz] and for fL = 50 [Hz]: a) unit 1 voltages b) unit 2 voltages c) output voltage for separate control of rectifying units - control without circulating current

+-+-+-Diagram of a 2-level 3-phase voltage source inverter (VSI)

Two-Level Voltage Source Inverter

Voltage Source Inverter - VSI is a power electronic device capable of transforming the DC voltage and direct current into voltage and alternating current with the desired characteristics. It’s possible to design single- and multiphase inverters. 3 phase inverters are commonly applied for the supply of induction motors. The basic diagram of a 2-level voltage inverter is presented.

This inverter is commonly referred to as VSI inverter since it forms the source with voltage characteristics and the voltage curve on the output (load) is not relative to the value of the load current in a wide range of operating conditions. This is made possible due to the powerful voltage source with small internal impedance additionally boosted by a capacitor with adequately large capacity Cs on the input of the inverter. A 3-phase inverter has 3 branches with two semiconductor switches and free-wheeling diodes presented. The control switches apply IGBTs (Isolated Gate Bipolar Transistors) or GCTs (Gate Commuted Thyristors) or MOSFET (Metal-Oxide Semiconductor Field Effect Transistor) de pending on required working conditions, firstly supply voltage, load current and switching frequency. An output voltage filter with capacitors with the capacity of Cf is connected to the load. The name for a two-level converter comes as a consequence of the fact that voltages U1G, U2G, U3G can only assume two values, i.e. Ud or 0. Regardless of the specific manner of control of the inverter's gates at any instant only one of the semiconductor switches can conduct current in a specific branch. The commutation involves the process in which in a single branch one of the switches terminates the conduction while the other switch starts the conduction process only after an adequate break to prevent shorts. At any instant with the only exception of the commutation break in the inverter we have to do with conduction of three semiconductor switches, i.e. one in each branch. The desired waveform of the output voltage is gained as a result of an adequate control of inverter semiconductor switches. For the supply of an induction motor drive it’s desirable to have 3-phase voltage with a sine waveform having controllable frequency and amplitude. The basic method applied for the modeling of the output voltage consists in Pulse Width Modulation-(PWM). Under standard conditions it involves adequate switching of the potential U and potential 0 at the output by semiconductor switches from the inverter branches in a short intervals corresponding to a small fraction of the period of the output voltage. Concurrently, there is a large number of modulation methods, some of which will be discussed in this section. Every modulation method should result in output voltages close to 3-phase symmetrical sinusoidal system with small content of higher harmonics. The other postulate regards the application of possibly small number of switchings between the control semiconductor elements corresponding to a single cycle of the output voltage. Every commutation in an inverter branch is associated with resistive losses in the power electronic switches, hence the effort to make transition period short. Also the commutation is associated with losses in the dielectric in the windings' insulation leading to wear of the insulation layer. Hence, the postulate of the limitation of the number of connections follows. The discussion here will focus on sinusoidal PWM (SPWM) modulation in which the triangular carrier signal is modulated with the sinusoidal waveform as well as several varieties of the space vector modulation (SVM). The first of the listed methods has its origin in the analogue technique of control and was thus undertaken in this area, while the other one corresponds to the digital technique of control.

In the description of the inverter's mode of operation we apply amplitude and frequency modulation factors.

Sinusoidal Pulse Width Modulation (SPWM): Sinusoidal pulse width modulation (SPWM-Sinusoidal PWM) involves appropriate employment of the crossing points between saw carrier signal and sinusoidal modulation wave. When modulation voltages um1, um2, um3 are higher then the voltages of the carrier wave.

... the potential of the output semiconductor switch i = 1,2,3 assumes the value of the supply Ud. In the opposite case this potential has the value of 0 since the ground semiconductor switch of the adequate branch of the inverter is in the ON state.

The formation of the carrier wave is presented. The output voltage results from the difference of the potential between the appropriate pairs of output points between inverter's branches i = 1,2,3 like:

+-+-+-Formation of an output voltage by the Sinusoidal PWM.

When the carrier frequency f_saw is an integer multiple of the output frequency f_s we have to do with synchronic modulation. The case when ma<1 is called proper modulation and in that case the frequency of switching is ... while for ma>1 we have to do with overmodulation, the switching frequency is smaller than it results from, and the voltage waveform is distorted. The highest attainable RMS value of the output voltage basic harmonic with the frequency of f is equal to …

...which indicates a relatively small application of the supply voltage. Concurrently, higher harmonic orders in this curve for the synchronic modulation amount to ... and don’t contain low order harmonics being the multiple of f_s. In order to in crease the inverter's scope of application of the supply voltage a method with the injection of the third harmonic of the modulating voltage has been developed. As a result, the saw carrier wave is modulated with the voltage of ... in a manner that ensures the wave um of the modulating voltage does not exceed the voltage of the carrier wave U . This condition is fulfilled when ...

The introduction of the third harmonic into the modulating voltage results in the distortion of the voltage waveforms in relation to the reference potential U1G, U2G, U3G. However, it does not result in the distortion of the output voltages in the load U12, U23, U31 since the compensation of the effect of the third harmonic.

+-+-+- Forming an output voltage by the SPWM method with a 3rd harmonic injection In the result we have to do with an increase of the output voltage to which means that it’s over 15% more comparing with.

Space Vector Modulation (SVM):

Two-level voltage inverter has three branches, each of which is in one of conduction states. The conduction states of an inverter can be determined as: For example the state S1 = 1 means that in the first branch the upper semiconductor switch is in the conduction state and the output potential is equal to U1G = Ud.

Concurrently, S2 = 0 means that the ground semiconductor switch in the second branch is 'ON' and then U2G = 0. In this method the inverter's output short time (Tp) averaged voltage vector Vs could be defined and effectively constructed by use of a concept of the space vector (complexor) Vk, which is determined by the basic harmonic U1ph of the required output voltage and the states of the particular branches S1, S2, S3:

Since there are 3 branches, each of which can be in either of two states, the instantaneous outputs Vk from the inverter can assume any of 8 states illustrated graphically.

+-+-+- Space vector plane with the switching states <S1,S2,S3> defined for each vector

Along with the given vector Vk, the states of the inverter's branches for which either one occurs. There are also two zero states of the inverter outputs V0, V8 for which semiconductor switches in all branches connect the load either to the ground (G) or positive (P) supply bar. In result the output voltages are equal to zero. Directly from the states of the branches it’s possible to determine output voltages ... which corresponds to the length of vector Vk of the voltage star. The amplitude of this value is achievable only for phase angles theta = kp/3. The instantaneous position of vector Vs is determined by a phase angle ?. The method of modulation using space vectors SVM will be presented on the example of the synthesis of vector Vs situated in the first sector of the voltage star of the inverter.

In the other sectors the situation is similar, as a value of phase angle theta could be reduced to the range of the first sector. This vector is synthesized by adequately selected switching times of the states that determine vectors V1 and V2 as well as zero vectors V0 and V8.

+-+-+- Phase voltage of the two-level inverter

+-+-+- A method of synthesis of the Vs vector in the first segment of the space vector plane

The construction of short time averaged vector Vs involves the fact that within sufficiently short pulsation time Tp, which corresponds to a fraction of the total cycle, voltages V1, V2 and V0 or V8 are switched on for the selected duration tx, ty, tn. These intervals are obviously relative to the instant position of vector Vs deter mined by angle ?. As a matter of simplification it’s assumed that within a single pulsation time the angle theta is invariable. The determination of time intervals tx, ty, tn is performed using the relation.