This section discusses the principle and implementation steps of Space Vector Pulse Width Modulation (SVPWM), and builds a SVPWM simulation model in the Matlab/Simulink environment .

• Field-oriented control (FOC), also known as vector control, is a commonly used PMSM control method that can achieve independent control of torque and magnetic field. Space vector pulse width modulation (SVPWM) is used to generate the PWM signal of the inverter drive motor.

• Space vector modulation (SVPWM) is a common method for field-oriented control of PMSMs to generate pulse width modulation signals to control the switches of the inverter, thereby generating the required modulation voltage to drive the motor at the desired speed or torque.

1. Basic principles of SVPWM

Space Vector Modulation (SVPWM) is responsible for generating the pulse width modulation signals to control the switches of the inverter, thereby producing the required modulation voltage to drive the motor at the desired speed or torque.

1.1 Advantages of SVPWM

The goal of SPWM control is to make the output current as close to a sine wave as possible. The three-phase AC load also requires the balance and phase difference of the three-phase current to be maintained. Control according to a circular rotating magnetic field can achieve better performance .
Sinusoidal PWM modulation has advantages in its simplicity, but it cannot fully utilize the DC bus voltage. Since the line voltage is defined as the phase-to-phase voltage difference, it is easy to see that the line voltage is lower than the DC bus voltage.

SVPWM is based on the theory of space vector coordinate transformation. It controls the space vector trajectory to approach the ideal magnetic flux through the changes in different switching states of the inverter, obtains a quasi-circular rotating magnetic field, and has higher control performance for PMSM motors than SPWM.

The advantages of SVPWM are:

• Good dynamic performance, small output voltage and current harmonics, can reduce the torque pulsation of the motor and improve the running performance of the motor;
• The voltage utilization efficiency is high, and the maximum output voltage is$U^d/\sqrt{3}$The maximum line voltage is$U^d$, DC voltage utilization can reach 100%;
• The switching loss is small. Only one switch is switched at each state change, so the switching loss is small.

1.2 SVPWM circuit topology

Consider the space vector modulation method for three-phase inverter motor control. The three-phase bridge PWM inverter has three groups of bridge arms and six switching devices. The switch states of the upper and lower bridge arms are complementary, which can produce eight switch combinations. Each switch combination will generate a specific voltage applied to the motor terminals.

The three-phase inverter circuit connected to the motor stator winding is shown in the figure below. Switches S2, S4, and S6 are complementary to S1, S3, and S5 respectively.

AC phase voltage measurement$V^{AN}$,$V^{BN}$,$V^{CN}$The relationship with the switch function is:

{ V A N = U d c ( 2 s a − s b − s c ) / 3 V B N = U d c ( 2 s b − s a − s c ) / 3 V C N = U d c ( 2 s c − s a − s b ) / 3

⎩ ⎨ ⎧​VA N​VBN​VCN​​=Ud c​( 2 sa​−sb​−sc​) /3=Ud c​( 2 sb​−sa​−sc​) /3=Ud c​( 2 sc​−sa​−sb​) /3​​

Voltage is a basic space vector, and its magnitude and direction are represented by a space vector hexagon. The 8 switching combinations correspond to 8 basic voltage space vectors V_0 (0,0,0), V_1 (0,0,1), V_2 (0,1,0), V_3 (0,1,1), V_4 (1,0,0), V_5 (1,0,1), V_6 (1,1,0) and V_7 (1,1,1).
2 zero vectors$V^0$,$V^7$Located at the origin, with a magnitude of zero, it is called the zero vector.$V^1{V}^6$The amplitudes are the same (the phase voltage amplitudes are all U_dc*2/3), and the spatial positions differ by 60°, and are located at the six vertices of a regular hexagon.

Substituting the 8 switch state functions into the above formula, the relationship between the 8 switch combinations and voltage is shown in the following table.

These eight voltage space vectors can form a closed regular hexagonal flux vector trajectory, but cannot produce a continuous voltage space vector operation trajectory with a large harmonic component, which will cause torque pulsation.

1.3 Continuously rotating space vector

By dividing a working cycle into 6 sectors and allocating the working time of 8 vectors, the desired voltage space vector required to generate a regular 6N-gon rotating magnetic field can be synthesized, which can approximate a circular rotating magnetic field.

In each sector, two adjacent voltage vectors and the zero vector are selected, and any voltage vector in each sector is synthesized according to the principle of volt-second balance:

$${\int }^0{U}^{ref}dt={\int }^0{U}^{x}dt+{\int }^{T^x}{U}^{y}dt+{\int }^{T^y}{U}^{0}dt$$

$$U^{ref}T=U^x{T}^x+U^y{T}^y+U^0{T}^0$$

In the formula, Uref is the expected voltage vector, T is the sampling period, Tx, Ty, T0 are the action time of two non-zero voltage vectors Ux, Uy and zero voltage vector U0 in one sampling period, respectively. U0 includes two zero vectors U0 and U7.

The meaning of the above formula is that the integral value generated by the vector Uref in time T is the same as the sum of the integral values ​​generated by Ux, Uy, and U0 in time Tx, Ty, and T0 respectively.

Since the three-phase sinusoidal voltage synthesizes an equivalent rotating voltage in the voltage space vector, its rotation speed is the input power angular frequency, and the trajectory of the equivalent rotating voltage will be a circle as shown in Figure 2-9. To generate a three-phase sinusoidal voltage, the above voltage vector synthesis method can be used. On the voltage space vector, the set voltage vector starts from the U4 (100) position and increases a small increment each time. Each small increment set voltage vector can be synthesized with two adjacent basic non-zero vectors and a zero voltage vector in the area. The set voltage vector obtained in this way is equivalent to a voltage space vector that rotates smoothly on the voltage space vector plane, thereby achieving the purpose of voltage space vector pulse width modulation.

By adjusting the action time of the basic space vector (direction) and the zero vector (amplitude) in the switching interval, the voltage vector at any position and any amplitude in the space vector hexagon can be approximated. For example, in the figure, in a pulse width modulation (PWM) cycle, two adjacent space vectors (U3 and U4 in the figure) are selected to act for a period of time, and the zero vector (U7 or U8) acts during the rest of the cycle, thereby obtaining an approximate average reference vector Uref.

By controlling the switching sequence, i.e. the on-time of the pulses, any voltage vector with varying magnitude and direction can be obtained in each PWM cycle. The goal of the space vector modulation method is to generate a switching sequence that matches the reference voltage vector in each PWM cycle to achieve a continuously rotating space vector.

2. SVPWM algorithm implementation

2.1 Voltage vector combination scheme

By adjusting the action time of the basic space vector (direction) and the zero vector (amplitude) within the switching interval, the voltage vector at any position and with any amplitude within the space vector hexagon can be approximately obtained.

Use zero vector$V^0$and the two nearest adjacent non-zero vectors$V^k$,$V^{k+1}$Synthetic reference vector, zero vector action time$T^0$and a non-zero vector$V^k$,$V^{k+1}$Time of action$T^k$,$T^{k+1}$They are:

{ T 0 = ( T k + T k + 1 ) T s / 2 [ T k T k + 1 ] = 3 2 T s U d [ s i n ( k π / 3 ) − c o s ( k π / 3 ) − s i n ( ( k − 1 ) π / 3 ) c o s ( ( k − 1 ) π / 3 ) ] [ u r α u r β ]

⎩ ⎨ ⎧​T0​[Tk​Tk + 1​​]​=( Tk​+Tk + 1​) Ts​/2=23 ​​Ud​Ts​​[s in ( kπ /3 )− s in (( k−1 ) π /3 )​− cos ( kπ /3 )cos (( k−1 ) π /3 )​][ur α​ur β​​]​​

There are many combinations of voltage vectors, the most commonly used is the 7-segment combination shown in the following table.

Table 2-2: 7-segment combination scheme of SVPWM output voltage vector

2.2 Implementation steps of SVPWM

SVPWM operates based on a reference voltage vector and generates appropriate turn-on signals for the inverter in each PWM cycle, with the goal of achieving a continuously rotating space vector.

SVPWM mainly includes the following steps:
(1) Determine the sector where the reference space voltage vector is located;
(2) Calculate the action time of each vector according to the signal timing of the 7-segment combination;
(3) Determine the time point of sector vector switching.
(4) Use the triangular carrier model to compare with the switching points of each sector vector to generate the required PWM switching signal.

(1) Determine the sector

To reference voltage$U^{r}ef$
To control, we must first determine the sector where the reference voltage is located .$U^{r}ef$The components in the α-β rectangular coordinate system are$u^{\alpha }$,$u^{\beta }$, define the variable$U^{ref1}$,$U^{ref2}$,$U^{ref3}$:

{ U r e f 1 = u β U r e f 2 = 3 2 u α − 1 2 u β U r e f 3 = − 3 2 u α − 1 2 u β

⎩ ⎨ ⎧​Ure f 1​Ure f 2​Ure f 3​​=uβ​=23 ​​uα​−21​uβ​=−23 ​​uα​−21​uβ​​​

Then judge:
(1) If U1>0, then A=1, otherwise A=0;
(2) If U2>0, then B=1, otherwise B=0;
(3) If U3>0, then C=1, otherwise C=0.

make$N=4C+2B+A$, we can get the relationship between N and sector, and get the reference voltage$U^{r}ef$The sector is:

(2) Calculation of vector action time

(3) Calculate the vector switching time point

when$U^{ref}$After the sector and the corresponding effective voltage vector action time are determined, the value of each corresponding comparator is calculated according to the PWM modulation principle:

{ t a o n = ( T s − T x − T y ) / 4 t b o n = t a o n + T x / 2 t c o n = t b o n + T y / 2

⎩ ⎨ ⎧​ta o n​tb o n​tkon​​​=( Ts​−Tx​−Ty​) /4=ta o n​+Tx​/2=tb o n​+Ty​/2​​

In the formula$t^{aon}$,$t^{bon}$,$t^{kon}$are the values ​​of the corresponding comparators respectively.

The values ​​of the comparators for different sectors are assigned as follows: