39.3.1 Pulse Width Modulation Generation

In typical ac motor-controller design, both hardware and software considerations are involved in the process of generating the PWM signals that are ultimately used to turn on or off the power devices in the three-phase inverter. In typical digital control environments, the controller generates a regularly timed interrupt at the PWM switching frequency (nominally 10–20 kHz). In the interrupt service routine, the controller software computes new duty-cycle values for the PWM signals used to drive each of the three legs of the inverter. The computed duty cycles depend on both the measured state of the motor (torque and speed) and the desired operating state. The duty cycles are adjusted on a cycle-by-cycle basis in order to make the actual operating state of the motor follow the desired trajectory.

Once the desired duty-cycle values have been computed by the processor, a dedicated hardware PWM generator is needed to ensure that the PWM signals are produced over the next PWM and controller cycle. The PWM generation unit typically consists of an appropriate number of timers and comparators that are capable of producing very accurately timed signals. Typically, 10- to 12-bit performance in the generation of the PWM timing waveforms is desirable.

Fig. 39.4 shows a typical PWM waveform for a single-leg inverter. In general, there is a small delay required between turning off one power device like A-phase lower device and turning on the complementary power device A-phase upper device. This dead time is required to ensure that the device being turned off has sufficient time to regain its blocking capability before the other device is turned on. Otherwise, a short circuit of the dc voltage could result.

In LF2407, the compare units have been used to generate the PWM signals. The PWM output signal is high when the output of current PI regulation matches the value of T1CNT and is set to low when the timer underflow occurs. The switch states are controlled by the ACTR register. In order to minimize the switching losses, the lower switches are always kept on, and the upper switches are chopped on/off to regulate the phase current.

39.3.2 Analog-to-Digital Conversion Requirements

For the control of high-performance motor drives, fast, high-accuracy, simultaneous-sampling A/D conversion of the measured current values is required. The drives have a rated operation range—a certain power level that they can sustain continuously, with an acceptable temperature rise in the motor and power converter. They also have a peak rating—the ability to handle a current far in excess of the rated current for short periods of time. This allows a large torque to be applied transiently, to accelerate or decelerate the drive very quickly, and then to revert to the continuous range for normal operation. This also means that in the normal operating mode of the drive, only a small percentage of the total input range is being used.

At the other end of the scale, in order to achieve the smooth and accurate rotations desired in these machines, it is wise to compensate for small offsets and nonlinearities such as core saturation and parameter detuning. In any current-sensor electronics, the analog signal processing is often subject to gain and offset errors. Gain mismatches, for example, can exist between the current-measuring systems for different windings. These effects combine to produce undesirable oscillations in the torque. To meet both of these conflicting resolution requirements, modern motor drives use 10-bit A/D converters, depending on the cost/performance trade-off required by the application.

The bandwidth of the system is essentially limited by the amount of time it takes to input information and then perform the calculations. The A/D converters that take many microseconds to convert can produce intolerable delays in the system. A delay in a closed-loop system will degrade the achievable bandwidth of the system, and bandwidth is one of the most important figures of merit in these high-performance drives. Therefore, fast analog-to-digital conversion is a necessity for these applications.

A third important characteristic of the A/D converter used in these applications is timing. In addition to high resolution and fast conversion, simultaneous sampling is needed. In any three-phase motor, it is necessary to measure the currents in the three windings of the motor at exactly the same time in order to get an instantaneous “snapshot” of the torque in the machine. Any time skew (time delay between the measurements of the different currents) is an error factor that is artificially inserted by the means of measurement. Such a nonideality translates directly into a ripple of the torque—a very undesirable characteristic.

The analog-to-digital converter (ADC) on the LF2407 allows the DSP to sample analog or “real-world” voltage signals. The output of the ADC is an integer number that represents the voltage level sampled. The integer number may be used for calculations in an algorithm. The resolution of the ADC is 10 bits, meaning that the ADC will generate a 10-bit number for every conversion it performs. However, the ADC stores the conversion results in registers that are 16-bit wide. The 10 most significant bits are the ADC result, while the least significant bits (LSBs) are filled with “0”s. There are a total of 16 input channels to the single input ADC. The control logic of the ADC consists of autosequencers, which control the sampling of the 16 input channels to the ADC. The autosequencers control not only which channels (input channels) will be sampled by the ADC but also the order of the channels that the ADC performs conversions on. The two 8-conversion autosequencers can operate independently or cascade together as a “virtual” 16-conversion ADC.

39.3.3 Position Sensing and Encoder Interface Units

Usually, the motor position is measured through the use of an encoder mounted on the rotor shaft. The incremental encoder produces a pair of quadrature outputs (A and B), each with a large number of pulses per revolution of the motor shaft. For a typical encoder with 1024 lines, both signals produce 1024 pulses per revolution. Using a dedicated quadrature counter, it is possible to count both the rising and falling edges of both the A and B signals so that one revolution of the rotor shaft may be divided into 4096 different values. In other words, a 1024-line encoder allows the measurement of rotor position to 12-bit resolution. The direction of rotation may also be inferred from the relative phasing of quadrature signals A and B.

Fig. 39.5 shows the structure of an encoder. It consists of a light source, a radially slotted disk, and the photoelectric sensors. The disk rotates with the rotor. The two photosensors detect the light passing through the slots in the disk. When the light is hidden, a logic “0” is generated by the sensors. When the light passes through the slots of the disk, a logic “1” is produced. These logic signals are shown in Fig. 39.5. By counting the number of pulses, the motor speed can be calculated. The direction of rotation can be determined by detecting the leading signal between signals A and B.

This is all very well, but there is an increasing class of cost-sensitive motor drive applications with lower performance demands that can afford neither the cost nor the space requirements of the rotor position transducer. In these cases, the same motor-control algorithms can be implemented with estimated rather than measured rotor position.

The DSP core is quite capable of computing rotor position using sophisticated rotor-position estimation algorithms, such as extended Kalman estimators that extract estimates of the rotor position from measurements of the motor voltages and currents. These estimators rely on the real-time computation of a sufficiently accurate model of the motor in the DSP. In general, these sensorless algorithms can be made to work and the sensored algorithms at medium to high speeds of rotation. But as the speed of the motor decreases, the extraction of reliable speed-dependent information from voltage and current measurements becomes more difficult. In general, sensorless motor control is applicable principally to applications such as compressors, fans, and pumps, where continuous operation at zero or low speeds is not required.

39.3.4 The PI Regulator

An electric drive based on the field-orientated control (FOC) needs two constants as control parameters, the torque component reference i_qs^e* and the flux component reference i_ds^e*. The classical PI regulator is well suited to regulate the torque and flux feedback to the desired values. This is because it is able to reach constant references by correctly setting both the proportional term (K_p) and the integral term (K_i), which are, respectively, responsible for the error sensibility and for the steady-state error. The numerical expression of the PI regulatoris as follows:

$Y_{\left(k\right)}=K_p{e}_{\left(k\right)}+K_i{e}_{\left(k\right)}+{\displaystyle \sum _{n=0}^{k-1}{e}_{\left(n\right)}}$

which is represented in Fig. 39.6.

During normal operation, large reference value variations or disturbances may occur that result in the saturation and overflow of the regulator variables and output. To solve this problem, one solution is to add a correction of the integral component as depicted in Fig. 39.7.

The constants K_p, K_i, and K_c, proportional, integral, and integral correction components, respectively, are selected based on the sampling period and on the motor parameters. After defining the DSP-controlled motor drive requirements, in the following, we describe the digital control algorithms for permanent-magnet motors and induction motors.

39.4.1 Mathematical Model of the BLDC Motor

The phase variables are used to model the BLDC motor due to its nonsinusoidal back EMF and phase current. The terminal voltage equation of the BLDC motor can be written as

$\left[\begin{array}{l}{\nu }_a\hfill \\ {\nu }_b\hfill \\ {\nu }_c\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill R+p{L}_s\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill R+p{L}_s\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill R+p{L}_s\hfill \end{array}\right]\cdot \left[\begin{array}{l}{i}_a\hfill \\ i_b\hfill \\ i_c\hfill \end{array}\right]+\left[\begin{array}{l}{e}_a\hfill \\ e_b\hfill \\ e_c\hfill \end{array}\right]$

where v_a, v_b, and v_c are the phase voltages; i_a, i_b, and i_c are the phase currents; e_a, e_b, and e_c are the phase back-EMF voltages; R is the phase resistance; L_s is the synchronous inductance per phase and includes both leakage and armature reaction inductances; and p represents d/dt. The electromagnetic torque is given by

$T_e=\frac{\left(e_a{i}_a+e_b{i}_b+e_c{i}_c\right)}{{\omega }_m}$

where ω_m is the mechanical speed of the rotor. The equation of motion is

$\frac{d}{dt}{\omega }_m=\frac{\left(T_e-T_L-B{\omega }_m\right)}{J}$

where T_L is the load torque, B is the damping constant, and J is the moment of inertia of the rotor shaft and the load.

39.4.2 Torque Generation

From Eq. (39.3), the electromagnetic torque of the BLDC motor is related to the product of the phase back EMF and current. The back EMFs in each phase are trapezoidal in shape and are displaced by 120 electrical degrees with respect to each other in a three-phase machine. A rectangular current pulse is injected into each phase so that current coincides with the crest of the back-EMF waveform; hence, the motor develops an almost constant torque. This strategy, commonly called six-step current control, is shown Fig. 39.9. The amplitude of each phase's back EMF is proportional to the rotor speed and is given by

$E=k\varphi {\omega }_m$

where k is a constant and depends on the number of turns in each phase, φ is the permanent-magnet flux, and ω_m is the mechanical speed. In Fig. 39.9, during any 120 degrees interval, the instantaneous power converted from electric to mechanical, P_o, is the sum of the contributions from two phases in series and is given by

$P_0={\omega }_m{T}_e=2EI$

where T_e is the output torque and I is the amplitude of the phase current. From Eqs. (39.4) and (39.6), the expression for output torque can be written as

$T_e=2k\varphi I=k_{t}I$

where K_t is the torque constant. Since the electromagnetic torque is only proportional to the amplitude of the phase current in Eq. (39.7), torque control of the BLDC motor is essentially accomplished by phase current control.

39.4.3 BLDC Motor Control Topology

Based on the previously discussed concept, a BLDC motor drive system is shown in Fig. 39.10. It can be seen that the total drive system consists of the BLDC motor, power electronic converter, sensor, and controller.

The BLDC motors are predominantly surface-magnet machines with wide magnet pole arcs. The stator windings are usually concentrated windings, which produce a square waveform distribution of flux density around the air gap. The design of the BLDC motor is based on the crest of each half cycle of the back-EMF waveform. In order to obtain a smooth output torque, the back-EMF waveform should be wider than 120 degrees electrical degrees. A typical BLDC motor with 12 stator slots and 4 poles on the rotor is shown in Fig. 39.11. The inverter is usually responsible for the electronic commutation and current regulation. For the six-step current control, if the motor windings are Y-connected without the neutral connection, only two of the three-phase currents flow through the inverter in series. This results in the amplitude of the DC-link current always being equal to that of the phase currents. The PWM current controllers are typically used to regulate the actual machine currents in order to match the rectangular current reference waveforms shown in Fig. 39.9. For example, during one 60 degrees interval, when switches T₁ and T₆ are active, phases A and B conduct. The lower switch T₆ is always turned on, and the upper switch T₁ is chopped on/off using either a hysteresis current controller with variable switch frequency or a PI controller with fixed switch frequency.

When T₁ and T₆ are conducting, current builds up in the path as shown in Fig. 39.12A with dashed line. When switch T₁ is turned off, the current decays through diode D₄ and switch T₆ as depicted in Fig. 39.12B. In the next interval, switch T₂ is on, and T₁ is chopped so that phase A and phase C conduct. During the commutation interval, the phase-B current rapidly decreases through the freewheeling diode D₃ until it becomes zero and the phase C current builds up.

From the above analysis, each of the upper switches is always chopped for one 120 degrees interval, and the corresponding lower switch is always turned on per interval. The freewheeling diodes provide the necessary paths for the currents to circulate when the switches are turned off and during the commutation intervals. There are two types of sensors for the BLDC drive system: a current sensor and a position sensor. Since the amplitude of the dc-link current is always equal to the motor phase current in six-step current control, the dc-link current is measured instead of the phase current. Thus, a shunt resistor, which is in series with the inverter, is usually used as the current sensor. Hall-effect position sensors typically provide the position information needed to synchronize the stator excitation with rotor position in order to produce constant torque. Hall-effect sensors detect the change in magnetic field. The rotor magnets are used as triggers for the Hall sensors. A signal-conditioning circuit is needed for noise cancellation in Hall-effect sensor circuits. In six-step current control algorithm, rotor position needs to be detected at only six discrete points in each electric cycle. The controller tracks these six points so that the proper switches are turned on or off for the correct intervals. Three Hall-effect sensors, spaced 120 electrical degrees apart, are mounted on the stator frame. The digital signals from the Hall sensors are then used to determine the rotor position and switch gating signals for the inverter switches.

39.4.4 DSP Controller Requirements

The controller of BLDC drive systems reads the current and position feedback, implements the speed or torque control algorithm, and finally generates the gate signals. The connectivity of the LF2407 in this application is illustrated in Fig. 39.13. Three capture units in the LF2407 are used to detect both the rising and falling edges of Hall-effect signals. Hence, in every 60 electrical degrees of motor rotation, one capture unit interrupt is generated that ultimately causes a change in the gating signals and the motor to move to the next position. One input channel of the 10-bit A/D converter reads the dc-link current. The output pins PWM1–PWM6 are used to supply the gating signals to the inverter.

39.4.5 Implementation of the BLDC Motor Control Algorithm Using LF2407

A block diagram of the BLDC motor-control system is shown in Fig. 39.14. The dashed line separates the software from the hardware components introduced in the previous section. It is necessary to choose hardware components carefully in order to ensure high processing speed and precision in the overall control system. The overall control algorithm of the BLDC motor consists of nine modules:

• Initialization procedure

• Detection of Hall-effect signals

• Speed control subroutine

• Measurement of current

• Speed profiling

• Calculation of actual speed

• PID regulation

• PWM generation

• DAC output

The flowchart of the overall control algorithm is illustrated in Fig. 39.15.

39.5.1 Mathematical Model of PMSM

Fig. 39.16 depicts the simplified three-phase surface-mounted PMSM motor for our discussion. The stator windings, as-as′, bs-bs′, and cs-cs′, are shown as lumped windings for simplicity but are actually distributed around the stator. The rotor has two poles. Mechanical rotor speed and position are denoted as ω_rm and θ_rm, respectively. Electric rotor speed and position, ω_r and θ_r, are defined as P/2 times the corresponding mechanical quantities, where P is the number of poles.

Based on the above motor definition, the voltage equation in the abc stationary reference frame is given by

$V_{\mathit{abcs}}=R_s{i}_{\mathit{abcs}}+\frac{d}{dt}{\lambda }_{\mathit{abcs}}$

where

$f_{\mathit{abcs}}={\left[f_{as}{f}_{bs}{f}_{cs}\right]}^T$

and the stator resistance matrix is given by

$R_s=\mathrm{diag}\left[\begin{array}{ccc}\hfill r_s\hfill & \hfill r_s\hfill & \hfill r_s\hfill \end{array}\right]$

The flux linkages equation can be expressed by

${\lambda }_{\mathit{abcs}}=L_s{i}_{\mathit{abcs}}+{\lambda }_m^{\prime }\left[\begin{array}{c}\hfill sin{\vartheta }_r\hfill \\ \hfill sin\left({\vartheta }_r-\frac{2\pi }{3}\right)\hfill \\ \hfill sin\left({\vartheta }_r-\frac{4\pi }{3}\right)\hfill \end{array}\right]$

where λ′_m denotes the amplitude of the flux linkages established by the permanent magnet as viewed from the stator phase windings. Note that in Eq. (39.11), the back EMFs are sinusoidal waveforms that are 120 degrees apart from each other.

The stator self-inductance matrix, L_s, is given as

$L_s=\left\{\begin{array}{ccc}{L}_{ls}+L_A-L_{B}cos2{\theta }_r& -\frac{1}{2}{L}_A-L_{B}cos2\left({\theta }_r-\mathrm{\pi }/3\right)& -\frac{1}{2}{L}_A-L_{B}cos2\left({\theta }_r+\mathrm{\pi }/3\right)\\ -\frac{1}{2}{L}_A-L_{B}cos2\left({\theta }_r-\mathrm{\pi }/3\right)& L_{ls}+L_A-L_{B}cos2\left({\theta }_r-2\mathrm{\pi }/3\right)& -\frac{1}{2}{L}_A-L_{B}cos2\left({\theta }_r+\mathrm{\pi }\right)\\ -\frac{1}{2}{L}_A-L_{B}cos2\left({\theta }_r+\mathrm{\pi }/3\right)& -\frac{1}{2}{L}_A-L_{B}cos2\left({\theta }_r+\mathrm{\pi }\right)& L_{ls}+L_A-L_{B}cos2\left({\theta }_r+2\mathrm{\pi }/3\right)\end{array}\right.$

The electromagnetic torque may be written as

$\begin{array}{c}{T}_e=\frac{\mathrm{P}}{2}\left\{{\lambda }_m^{\prime }\left[\left(i_{as}-\frac{1}{2}{i}_{bs}-\frac{1}{2}{i}_{cs}\right)cos{\theta }_r-\frac{\sqrt{3}}{2}\left(i_{bs}-i_{cs}\right)sin{\vartheta }_r\right]\right.\\ +\frac{L_{md}-L_{mq}}{3}\left[\left(i_{as}^2-\frac{1}{2}{i}_{bs}^2-\frac{1}{2}{i}_{cs}^2-i_{as}{i}_{bs}-i_{as}{i}_{cs}+2i_{bs}{i}_{cs}\right)\right.\\ sin2{\theta }_r+\frac{\sqrt{3}}{2}\left.\left.\left(i_{bs}^2{i}_{cs}^2-2i_{as}{i}_{bs}+2i_{as}{i}_{cs}\right)cos2{\theta }_r\right]\right\}+T_{\mathit{cog}}\left({\theta }_r\right)\end{array}$

In Eq. (39.13), T_cog (θ_r) represents the cogging torque, and the d- and q-axes magnetizing inductances are defined by

$L_{md}=\frac{3}{2}\left(L_A-L_B\right)$

and

$L_{md}=\frac{3}{2}\left(L_A+L_B\right)$

The torque and speed are related by the electromechanical motion equation

$J\frac{d}{dt}{\omega }_{rm}=\frac{P}{2}\left(T_e-T_L\right)-B_m{\omega }_{rm}$

where J is the rotational inertia, B_m is the approximated mechanical damping due to friction, and T_L is the load torque.

39.5.2 Mathematical Model of PMSM in Rotor Reference Frame

The voltage and torque equations can be expressed in the rotor reference frame in order to transform the time-varying variables into steady-state constants. The transformation of the three-phase variables in the stationary reference frame to the rotor reference frame is defined as

$f_{qd0r}=K_r{f}_{\mathit{abcs}}$

where

$k_r=\frac{2}{3}\left[\begin{array}{ccc}\hfill cos{\theta }_r\hfill & \hfill cos\left({\theta }_r-\frac{2\pi }{3}\right)\hfill & \hfill cos\left({\theta }_r+\frac{2\pi }{3}\right)\hfill \\ \hfill sin{\theta }_r\hfill & \hfill sin\left({\theta }_r-\frac{2\pi }{3}\right)\hfill & \hfill sin\left({\theta }_r+\frac{2\pi }{3}\right)\hfill \\ \hfill \frac{1}{2}\hfill & \hfill \frac{1}{2}\hfill & \hfill \frac{1}{2}\hfill \end{array}\right]$

If the applied stator voltages are given by

$\left\{\begin{array}{l}{V}_{as}=\sqrt{2}{V}_{s}cos{\theta }_{ev}\\ V_{bs}=\sqrt{2}{V}_{s}cos\left({\theta }_{ev}=\frac{2\pi }{3}\right)\\ V_{cs}=\sqrt{2}{V}_{s}cos\left({\theta }_{ev}+\frac{2\pi }{3}\right)\end{array}\right.$

then applying Eq. (39.16) to Eqs. (39.8), (39.11), and (39.17) yields

${\nu }_{qs}^r=r_s{i}_{qs}^r+{\omega }_r{\lambda }_{ds}^r+\frac{d}{dt}{\lambda }_{qs}^r$

${\nu }_{ds}^r=r_s{i}_{ds}^r-{\omega }_r{\lambda }_{qs}^r+\frac{d}{dt}{\lambda }_{ds}^r$

${\lambda }_{qs}^r=L_{qs}{i}_{qs}^r$

${\lambda }_{ds}^r=L_{ds}{i}_{ds}^r+{\lambda }_m^{{\prime }_r}$

where the q- and d-axes self-inductances are given by L_qs=L_ls+L_mq and L_ds=L_ls+L_md, respectively.

The electromagnetic torque can be written as

$T_e=\frac{3}{2}\frac{P}{2}\left[{\lambda }_m^{\text{'}r}{i}_{qs}^r+\left(L_{ds}-L_{qs}\right)i_{qs}{i}_{ds}\right]$

From Eq. (39.22), it can be seen that torque is related only to the d- and q-axes currents. Since L_q≥L_d (for surface-mounted PMSM, both inductances are equal), the second item contributes a negative torque if the flux-weakening control has been used. In order to achieve the maximum torque/current ratio, the d-axis current is set to zero during the constant torque control so that the torque is proportional only to the q-axis current. Hence, this results in the control of the q-axis current for regulating the torque in rotor reference frame.

39.5.3 PMSM Control Topology

Based on the above analysis, a PMSM drive system is developed as shown in Fig. 39.17. The total drive system looks similar to that of the BLDC motor and consists of a PMSM, power electronic converter, sensors, and controller. These components are discussed in detail in the following sections.

The design consideration of the PMSM is to first generate the sinusoidal back EMF. Unlike the BLDC, which needs concentrated windings to produce the trapezoidal back EMF, the stator windings of PMSM are distributed in as many slots per pole as deemed practical to approximate a sinusoidal distribution. To reduce the torque ripple, standard techniques such as skewing and chorded windings are applied to the PMSM. With the sinusoidally excited stator, the rotor design of the PMSM becomes more flexible than the BLDC motor where the surface-mounted permanent magnet is a favorite choice. Besides the common surface-mounted nonsalient pole PM rotor, the salient pole rotors, like inset and buried magnet rotors, are often used because they offer appealing performance characteristics during the flux-weakening region. A typical PMSM with 36 stator slots in stator and 4 poles on the rotor is shown in Fig. 39.18.

Due to the sinusoidal nature of the PMSM, control algorithms such as V/f and vector control, developed for other AC motors, can be directly applied to the PMSM control system. If the motor windings are Y-connected without a neutral connection, three-phase currents can flow through the inverter at any moment. With respect to the inverter switches, three switches, one upper and two lower in three different legs, conduct at any moment as shown in Fig. 39.19. The PWM current control is still used to regulate the actual machine current. Either a hysteresis current controller, a PI controller with sine triangle, or an SVPWM strategy is employed for this purpose. Unlike the BLDC motor, the three switches are switched at any time.

39.5.4 DSP Controller Requirements

The LF2407 is used as the controller to implement speed control of the PMSM system. The interface of the LF2407 is illustrated in Fig. 39.20. Similar to the BLDC motor-control system, three input channels are selected to read the two-phase currents and resolver signal. Because a resolver is used in one case, the quadrature-encoder-pulse (QEP) inputs are not used. The QEP inputs work only with a QEP signal that a rotary encoder supplies. The DSP output pins PWM1–PWM6 are used to supply the gating signals to the switches and form the output of the control part of the system.

39.5.5 Implementation of the PMSM Algorithm Using the LF2407

The block diagram of the PMSM drive system is displayed in Fig. 39.21.

The flowchart of the developed software is shown in Fig. 39.22. The control program of the PMSM has one main routine and includes four modules:

• Initialization procedure

• DAC module

• ADC module

• Speed control module

In the following section, speed control module is discussed.

39.6.1 Induction Motor Field-oriented Control

The term “vector” control refers to the control technique that controls both the amplitude and the phase of ac excitation voltage. Vector control controls the spatial orientation of the EMF in the machine. This has led to the coining of the term FOC, which is used for controllers that maintain a 90 degrees spatial orientation between the critical field components.

The required 90 degrees of spatial orientation between key field components can be compared with the dc motor, where the armature winding magnetic field and the field winding magnetic field are always in quadrature. The objective is to force the control of the induction machine to be similar to the control of a dc motor, that is, torque control. In dc machines, the field and the armature winding axes are orthogonal to one another, making the magnetomotive forces (MMFs) established orthogonal. If the iron saturation is ignored, then the orthogonal fields can be considered to be completely decoupled.

It is important to maintain a constant field flux for proper torque control. It is also important to maintain an independently controlled armature current in order to overcome the effects of the detuning of resistance of the armature winding and leakage inductance. A spatial angle of 90 degrees between the flux and MMF axes has to be maintained in order to limit the interaction between the MMF and the flux. If these conditions are met at every instant of time, the torque will always follow the current.

With vector control, the mechanically robust induction motors can be used in high-performance applications where dc motors were previously used. The key feature of the control scheme is the orientation of the synchronously rotating q-d-0 frame to the rotor flux vector. The d-axis component is aligned with the rotor flux vector and regarded as the flux-producing current component. On the other hand, the q-axis current, which is perpendicular to the d-axis, is solely responsible for torque production.

In order to apply a rotor flux field-orientation condition, the rotor flux linkage is aligned with the d-axis, so the q-axis rotor flux in excitation reference frame λ_qr^e will be zero, and the d-axis rotor flux in the excitation reference frame will be the rotor flux, ${\lambda }_{dr}^e={\hat{\lambda }}_r$. Therefore, we have

$i_{ds}^e=\frac{{\mathrm{\lambda }}_{dr}^e}{L_m}$

${\omega }_{\mathit{slip}}=\frac{r_r}{{\hat{\mathrm{\lambda }}}_r}\left(\frac{L_m}{L_r}\right)i_{qs}^e=\frac{L_m{i}_{qs}^e}{{\tau }_r{\mathrm{\lambda }}_r}$

$T_e=\frac{3}{2}\frac{P}{2}\frac{L_m}{L_r}{\hat{\mathrm{\lambda }}}_d{i}_{qs}^e$

where τ_r is rotor time constant, L_m magnetizing inductance, L_r, rotor leakage inductance, r_r, rotor resistance, i_qs^e q-axis stator current in excitation frame, i_ds^e d-axis stator current in excitation frame, and ω_slip angular frequency of slip. We can find out that in this case, i_ds^e controls the rotor flux linkage and i_qs^e controls the electromagnetic torque. The reference currents of the q-d-0 axis (i_qs^e*,i_ds^e*) are converted to the reference phase voltages (v_ds^e*,v_qs^e*) as the commanded voltages for the control loop. Given the position of the rotor flux and two-phase currents, this generic algorithm implements the instantaneous direct torque and flux control by means of coordinate transformations and PI regulators, thereby achieving accurate and efficient motor control.

It is clear that for implementing vector control, we have to determine the rotor flux position. This usually is performed by measuring the rotor position and utilizing the slip relation to compute the angle of the rotor flux relative to the rotor axis.

Equations (39.23) and (39.24) show that we can control torque and field by i_ds and i_qs in the excitation frame. However, in the implementation of FOC, we need to know i_ds and i_qs in the stationary reference frame. So, we have to know the angular position of the rotor flux to transform i_ds and i_qs from the excitation frame to the stationary frame. By using ω_slip, which is shown in Eq. (39.24), and using actual rotor speed, the rotor flux position is obtained:

${\displaystyle \underset{0}{\overset{t}{\int }}{\omega }_{\mathit{slip}}}dt+{\theta }_{re}\left(t\right)={\theta }_r\left(t\right)$

where θ_re(t) is electric angular rotor position and θ_r(t) angular rotor flux position.

The current model takes i_ds and i_qs as inputs and the rotor mechanical speed and gives the rotor flux position as an output. Fig. 39.23 shows the block diagram of the vector-control strategy in which speed regulation is possible using a control loop.

As shown in Fig. 39.23, two-phase currents are measured and fed to the Clarke transformation block. These projection outputs are indicated as i_ds^s and i_qs^s. These two components of the current provide the inputs to Park's transformation, which gives the currents in the qds^e excitation reference frame. The i_ds^e and i_qs^e components, which are outputs of the Park transformation block, are compared with their reference values i_ds^e*, the flux reference, and i_qs^e*, the torque reference. The torque command, i_qs^e*, comes from the output of the speed controller. The flux command, i_ds^e*, is the output of the flux controller that indicates the right rotor flux command for every speed reference. For i_ds^e*, we can use the fact that the magnetizing current is usually between 40% and 60% of the nominal current. For operating in speeds above the nominal speed, a field-weakening section should be used in the flux controller section. The current regulator outputs, v_ds^e* and v_qs^e*, are applied to the inverse Park transformation. The outputs of this projection are v_ds^e and v_qs^s, which are the components of the stator voltage vector in the dqs^s orthogonal reference frame. They form the inputs of the space-vector PWM block. The outputs of this block are the signals that drive the inverter.

Note that both the Park and the inverse Park transformations require the exact rotor flux position, which is given by the current model block. This block needs the rotor resistance or rotor time constant as a parameter. Accurate knowledge of the rotor resistance is essential to achieve the highest possible efficiency from the control structure. The lack of this knowledge results in the detuning of the FOC. In Fig. 39.23, a space-vector PWM has been used to emulate v_ds^s and v_qs^s in order to implement current regulation.

39.6.2 DSP Controller Requirements

The controller of the induction motor-control system is used to read the feedback current and position signals, to implement the speed or torque control algorithm, and to generate the gate signals based on the control signal. Analog controllers or digital signal processors, such as LF2407, can perform these tasks.

The interface of the LF2407 is illustrated in Fig. 39.24. Two quadrature counters detect the rising and falling edges of the encoder signals. Two input channels related to the 10-bit ADC are selected to read the two-phase currents. The pins PWM1–PWM6 output the gating signals to the gate drive circuitry.

39.6.3 Implementation of Field-Oriented Speed Control of Induction Motor

Some practical aspects of implementing the block diagram of Fig. 39.24 are discussed in this section and subsections. The software organization, the utilization of different variables, and the handling of the DSP controller resources are described. In addition, the control structure for the per-unit model is presented. Next, some numerical considerations have been made in order to address the problems inherent within the fixed-point calculation. As described, current model is one of the most important blocks in the block diagram depicted in Fig. 39.23. The inputs of this block are the currents and mechanical speed of rotor. In the next sections, technical points that should be considered during current and speed measurement and their scaling are discussed.

39.6.3.1 Software Organization

The body of the software consists of two main modules: the initialization module and the PWM interrupt service routine (ISR) module. The initialization model is executed only once at start-up. The PWM ISR module interrupts the waiting infinite loop when the timer underflows. When the underflow interrupt flag is set, the corresponding ISR is served. Fig. 39.25 shows the general structure of the software. The complete FOC algorithm is executed within the PWM ISR so that it runs at the same frequency as the switching frequency or at a fraction of it. The wait loop could be easily replaced with a user interface.

39.6.3.3 Speed Estimation During High-Speed Region

As previously mentioned, this method is based on counting the number of encoder pulses in a specified time interval. The QEP assigned timer counts the number of pulses and records it in the timer counter register (TxCNT). As the mechanical time constant is much slower than the electric one, the speed regulation loop frequency might be lower than the current loop frequency. The speed regulation loop frequency is obtained in this algorithm by means of a software counter. This counter accepts the PWM interrupt as input clock, and its period is the software variable called SPEEDSTEP. The counter variable is named speedstep. When speedstep is equal to SPEEDSTEP, the number of pulses counted is stored in another variable called n_p, and thus, the speed can be calculated. The scheme depicted in Fig. 39.26 shows the structure of the speed feedback generator.

Assuming that n_p is the number of encoder pulses in one SPEEDSTEP period when the rotor turns at the nominal speed, a software constant K_speed should be chosen as follows:

$01000h=K_{\mathit{speed}}.n_p$

The speed feedback can then be transformed into a Q4.12 format, which can be used in the control software. In the proposed control system, the nominal speed is 1800 rpm, and SPEEDSTEP is set to 125. The n_p can be calculated as follows:

$n_p=\frac{1800\times 64\times 4}{60}\times \mathrm{SPEEDSTEP}\times T_p=288$

  (39.28)

where $T_p=\frac{3}{f_{pwm}}=3\times {10}^{-4}$ (PWM frequency is 10 kHz, but the program is running at 3333 Hz) and hence K_speed is given by

$K_{\mathit{speed}}=\frac{4096}{288}=14.22\iff 0E38h$

  Q8.8

Note that K_speed is out of the Q4.12 format range. The most appropriate format to handle this constant is the Q8.8 format. The speed feedback in Q4.12 format is then obtained from the encoder by multiplying n_p by K_speed. The flowchart of speed measurement is presented in Fig. 39.27.

39.6.3.4 Speed Measurement During Low-Speed Region

To detect the edges of two successive encoder pulses, the developed program can use either the QEP counter or the capture unit input pins. The program has to measure the time between two successive pulses; therefore, it must utilize another GP timer. In this program, timer 3 has been dedicated to the time measurement. During the interrupt service routine of the capture unit or counter QEP, speed can be calculated. To obtain the actual speed of the motor, the appropriate number is divided by the value in the count register of timer 3.

As it can be inferred, at very low speeds, an overflow may occur in timer 3. The counter would then reset itself to zero and start counting up again. This event results in a large error in speed measurement. To avoid this event, timer 3 will be disabled in the overflow interrupt service routine. However, this timer is enabled in the capture unit (counter QEP) interrupt.

The flowchart of this implementation is presented in Fig. 39.28.

39.6.3.5 The Current Model

The current model is used to find the rotor flux position. This module takes i_ds and i_qs as inputs plus the rotor electric speed and then calculates the rotor flux position. The current model is based on Eqs. (39.23) and (39.24). Eq. (39.23) in transient form can be written as

$\frac{L_r}{r_r{L}_m}\frac{d{\lambda }_{dr}}{dt}+\frac{{\lambda }_{dr}}{L_m}=i_{ds}$

  (39.29)

Assume λ_dr/L_m=i_m where i_m is the magnetizing current; therefore, Eq. (39.29) can be written as follows:

$T_r\frac{d}{dt}{i}_m+i_m=i_{ds}$

  (39.30)

Rotor flux speed in a per-unit system can be shown by

$f_s=\frac{1}{{\omega }_b}\frac{d\theta }{dt}={\omega }_{re}+\frac{i_{qs}}{T_r{i}_m{\omega }_b}$

  (39.31)

where θ is the rotor flux position and T_r=L_r/r_r and ω_re are the rotor time constant and rotor electric speed, respectively. The rotor time constant is critical to the correct functionality of the current model. This system outputs the rotor flux speed, which in turn will be integrated to get the rotor flux position. Assuming that ${i_{qs}}_{{}_{\left(k+1\right)}}\approx {i_{qs}}_{{}_{\left(k\right)}}$, Eqs. (39.30) and (39.31) can be discretized as follows:

$\begin{array}{c}{i}_{m{r}_{\left(k+1\right)}}=i_{m{r}_{\left(k\right)}}+\frac{T_p}{T_r}\left(i_{d{s}_{\left(k\right)}}-i_{m{r}_{\left(k\right)}}\right)\\ f_{S_{\left(k+1\right)}}=n_{\left(K+1\right)}+\frac{1}{T_r{\omega }_b}\frac{{i_{qs}}_{{}_{\left(k\right)}}}{i_{m{r}_{\left(k+1\right)}}}\end{array}$

  (39.32)

For example, let the constants T_p/T_r and 1/T_rω_b be renamed to K_t and K_t, respectively. Here, L_r=73.8 mH, r_r=0.73 Ω, and f_n=60 Hz. So for K_t and K_r, we have

$\begin{array}{c}{K}_r=\frac{T_p}{T_r}=\frac{{\left(10000/3\right)}^{-1}}{101.09\times {10}^{-3}}\\ =2.967\times {10}^{-3}\iff 000Ch\end{array}$

  Q4.12

$\begin{array}{c}{K}_t=\frac{1}{T_r{\omega }_b}=\frac{1}{30.232\times {10}^{-3}\times 377}\\ =26.237\times {10}^{-3}\iff 006Bh\end{array}$

  Q4.12

By knowing the rotor flux speed (f_s), the rotor flux position (θ_cm) is computed by the integration formula in the per-unit system:

${{\theta }_{\mathit{cm}}}_{{}_{\left(k+1\right)}}={{\theta }_{\mathit{cm}}}_{{}_{\left(k\right)}}+{\omega }_b\cdot f_{s_{\left(k\right)}}\cdot T$

  (39.33)

In Eq. (39.33), let ω_bf_sT be called θ_inc. This variable is the rotor angle variation within one sampling period. Thus, the current model module has three input variables i_ds, i_ds, and ω_re and one output, which is the rotor flux position θ_cm represented as a 16-bit integer value. The flowchart of the field-oriented speed control of induction motor is presented in Fig. 39.29. This routine is placed inside the PWM interrupt service routine.