To achieve better dynamic performance, a more complex control scheme needs to be applied, to control the PM motor. With the mathematical processing power offered by the microcontrollers, we can implement advanced control strategies, which use mathematical transformations to decouple the torque generation and the magnetization functions in PM motors. Such de-coupled torque and magnetization control is commonly called rotor flux oriented control, or simply Field Oriented Control (FOC).

In a direct current (DC) Motor, the excitation for the stator and rotor is independently controlled, the produced torque and the flux can be independently tuned as shown in Figure 2-4. The strength of the field excitation (for example, the magnitude of the field excitation current) sets the value of the flux. The current through the rotor windings determines how much torque is produced. The commutator on the rotor plays an interesting part in the torque production. The commutator is in contact with the brushes, and the mechanical construction is designed to switch into the circuit the windings that are mechanically aligned to produce the maximum torque. This arrangement then means that the torque production of the machine is fairly near optimal all the time. The key point here is that the windings are managed to keep the flux produced by the rotor windings orthogonal to the stator field.

To achieve better dynamic performance, a more complex control scheme needs to be applied, to control the PM motor. With the mathematical processing power offered by the microcontrollers, we can implement advanced control strategies, which use mathematical transformations to decouple the torque generation and the magnetization functions in PM motors. Such de-coupled torque and magnetization control is commonly called rotor flux oriented control, or simply Field Oriented Control (FOC).

The goal of the FOC (also called vector control) on synchronous and asynchronous machine is to be able to separately control the torque producing and magnetizing flux components. FOC control will allow us to decouple the torque and the magnetizing flux components of stator current. With decoupled control of the magnetization, the torque producing component of the stator flux can now be thought of as independent torque control. To decouple the torque and flux, it is necessary to engage several mathematical transforms, and this is where the microcontrollers add the most value. The processing capability provided by the microcontrollers enables these mathematical transformations to be carried out very quickly. This in turn implies that the entire algorithm controlling the motor can be executed at a fast rate, enabling higher dynamic performance. In addition to the decoupling, a dynamic model of the motor is now used for the computation of many quantities such as rotor flux angle and rotor speed. This means that their effect is accounted for, and the overall quality of control is better.

According to the electromagnetic laws, the torque produced in the synchronous machine is equal to vector cross product of the two existing magnetic fields as Equation 6.

This expression shows that the torque is maximum if stator and rotor magnetic fields are orthogonal meaning if we are to maintain the load at 90°. If we are able to ensure this condition all the time, if we are able to orient the flux correctly, we reduce the torque ripple and we ensure a better dynamic response. However, the constraint is to know the rotor position: this can be achieved with a position sensor such as incremental encoder. For low-cost application where the rotor is not accessible, different rotor position observer strategies are applied to get rid of position sensor.

In brief, the goal is to maintain the rotor and stator flux in quadrature: the goal is to align the stator flux with the q axis of the rotor flux, for example, orthogonal to the rotor flux. To do this, the stator current component in quadrature with the rotor flux is controlled to generate the commanded torque, and the direct component is set to zero. The direct component of the stator current can be used in some cases for field weakening, which has the effect of opposing the rotor flux, and reducing the back-emf, which allows for operation at higher speeds.

The Field Orientated Control consists of controlling the stator currents represented by a vector. This control is based on projections which transform a three phase time and speed dependent system into a two co-ordinate (d and q co-ordinates) time invariant system. These projections lead to a structure similar to that of a DC machine control. Field orientated controlled machines need two constants as input references: the torque component (aligned with the q co-ordinate) and the flux component (aligned with d co-ordinate). As Field Orientated Control is simply based on projections, the control structure handles instantaneous electrical quantities. This makes the control accurate in every working operation (steady state and transient) and independent of the limited bandwidth mathematical model. The FOC thus solves the classic scheme problems, in the following ways:

• The ease of reaching constant reference (torque component and flux component of the stator current)
• The ease of applying direct torque control because in the (d, q) reference frame the expression of the torque is defined in Equation 7.
  Equation 7. ${\tau }_{em}\propto {\psi }_R\times i_{sq}$

By maintaining the amplitude of the rotor flux (ψ_R) at a fixed value we have a linear relationship between torque and torque component (i_Sq). We can then control the torque by controlling the torque component of stator current vector.

Space Vector Definition and Projection

The 3-phase voltages, currents and fluxes of AC-motors can be analyzed in terms of complex space vectors. With regard to the currents, the space vector can be defined as follows. Assuming that i_a, i_b, i_c are the instantaneous currents in the stator phases, then the complex stator current vector is defined in Equation 8.

where $\alpha =e^{j\frac{2}{3}\pi }$ and ${\alpha }^2=e^{j\frac{4}{3}\pi }$ represent the spatial operators.

Figure 2-5 shows the stator current complex space vector.

Where (a,b,c) are the three phase system axes. This current space vector depicts the three phase sinusoidal system. It still needs to be transformed into a two time invariant co-ordinate system. This transformation can be split into two steps:

• $(a, b)\Rightarrow (\alpha ,\beta )$ (Clarke transformation) which outputs a 2-coordinate time-variant system
• $\left(\alpha ,\beta \right)\Rightarrow (d, q)$ (Park transformation) which outputs a 2-coordinate time-invariant system

The $(a, b)\Rightarrow (\alpha ,\beta )$ Clarke Transformation

The space vector can be reported in another reference frame with only two orthogonal axis called (α, β). Assuming that the axis a and the axis αlpha are in the same direction we have the following vector diagram as shown in Figure 2-6.

The projection that modifies the 3-phase system into the (α, β) 2-dimension orthogonal system is presented in Equation 9.

The two phase (α, β) currents are still depends on time and speed.

The $\left(\alpha ,\beta \right)\Rightarrow (d, q)$ Park Transformation

This is the most important transformation in the FOC. In fact, this projection modifies a 2-phase orthogonal system (α, β) in the (d, q) rotating reference frame. If we consider the d axis aligned with the rotor flux, Figure 2-7 shows the relationship for the current vector from the two reference frame.

The flux and torque components of the current vector are determined by Equation 10.

where θ is the rotor flux position

These components depend on the current vector (α, β) components and on the rotor flux position; if we know the right rotor flux position then, by this projection, the d,q component becomes a constant. Two phase currents now turn into dc quantity (time-invariant). At this point the torque control becomes easier where constant i_sd (flux component) and i_sq (torque component) current components controlled independently.

The Basic Scheme of FOC for AC Motor

Figure 2-8 summarizes the basic scheme of torque control with FOC:

Two motor phase currents are measured. These measurements feed the Clarke transformation module. The outputs of this projection are designated i_sα and i_sβ. These two components of the current are the inputs of the Park transformation that gives the current in the d,q rotating reference frame. The i_sd and i_sq components are compared to the references i_sdref (the flux reference component) and i_sqref (the torque reference component). At this point, this control structure shows an interesting advantage: it can be used to control either synchronous or induction machines by simply changing the flux reference and obtaining rotor flux position. As in synchronous permanent magnet a motor, the rotor flux is fixed determined by the magnets; there is no need to create one. Hence, when controlling a PMSM, i_sdref should be set to zero. As an AC induction motor needs a rotor flux creation to operate, the flux reference must not be zero. This conveniently solves one of the major drawbacks of the classic control structures: the portability from asynchronous to synchronous drives. The torque command i_sqref could be the output of the speed regulator when we use a speed FOC. The outputs of the current regulators are Vsdref and V_sqref; they are applied to the inverse Park transformation.

The outputs of this projection are V_sαref and V_sβref which are the components of the stator vector voltage in the (α, β) stationary orthogonal reference frame. These are the inputs of the Space Vector PWM. The outputs of this block are the signals that drive the inverter. Note that both Park and inverse Park transformations need the rotor flux position. Obtaining this rotor flux position depends on the AC machine type (synchronous or asynchronous machine).

Rotor Flux Position

Knowledge of the rotor flux position is the core of the FOC. In fact if there is an error in this variable the rotor flux is not aligned with d-axis and i_sd and i_sq are incorrect flux and torque components of the stator current. Figure 2-9 shows the (a, b, c), (α, β) and (d, q) reference frames, and the correct position of the rotor flux, the stator current and stator voltage space vector that rotates with d,q reference at synchronous speed.

The measure of the rotor flux position is different if we consider synchronous or asynchronous motor:

• In the synchronous machine the rotor speed is equal to the rotor flux speed. Then θ (rotor flux position) is directly measured by position sensor or by integration of rotor speed.
• In the asynchronous machine the rotor speed is not equal to the rotor flux speed (there is a slip speed), then it needs a particular method to calculate θ. The basic method is the use of the current model which needs two equations of the motor model in d, q reference frame.

Theoretically, the field oriented control for the PMSM drive allows the motor torque be controlled independently with the flux like DC motor operation. In other words, the torque and flux are decoupled from each other. The rotor position is required for variable transformation from stationary reference frame to synchronously rotating reference frame. As a result of this transformation (so called Park transformation), q-axis current will be controlling torque while d-axis current is forced to zero. Therefore, the key module of this system is the estimation of rotor position using enhance Sliding-Mode Observer (eSMO) or FAST estimator.

Figure 2-10 shows the overall block diagram of sensorless FOC of a PMSM motor using eSMO with flying start in this document.

Figure 2-11 shows the overall block diagram of sensorless FOC of a PMSM motor using eSMO with field weakening control (FWC) and maximum torque per ampere (MTPA) in this document.

Figure 2-13 shows the overall block diagram of sensorless FOC of a PMSM motor using FAST with flying start in this document.

Figure 2-13 shows the overall block diagram of sensorless FOC of a PMSM motor using FAST with field weakening control (FWC) and maximum torque per ampere (MTPA) in this document.