Special algorithms make ac induction motors respond like variablespeed dc motors for fast PID control.



Steve Bowling
Application Segments Engineer
Microchip Technology Inc.
Chandler, Ariz.
Synchronous dc motors often top the field in highperformance motor controls. This group of motors includes brushed, brushless, woundfield, and permanentmagnet varieties. They are widely used simply because it's easier to regulate the operation of dc motors than that of ac motors. This is especially true if the application demands tight control of motor torque, velocity, or position.
Within specific limits, the torque of a dc motor is a linear function of input current. So it is relatively easy to get solid performance out of a dc motor with standard proportionalintegralderivative (PID) controllers.
In reality, the type of motor that works best in any given application depends on many other system variables. Ease of motor maintenance, failure modes, operating environments, motor temperature rise and cooling, and costs are just a few of the factorsto consider.
Many times a motor that operates only at a constant speed needs only on/off control. Typically, acinduction motors (ACIMs) are the choice for fixedspeed applications. ACIMs are rugged, inexpensive, and have a long life. However, ACIMs normally cannot match the performance of a dc motor in delivering variable speeds. Fortunately, microcontrollers (MCUs) and highpower electronic devices have evolved to make inexpensive speed control of ACIMs possible. The application of digitalsignal controllers (DSCs), that combine a digitalsignal processor with an MCU, now opened the door to fieldoriented control (FOC) of ACIMs. In a nutshell, FOC lets dc control techniques work with variablespeed ac motors.
ACIMs work well in highpower applications because their windings are fixed to the iron frame of the motor. The frame acts as a heat sink for additional cooling. The simple steel cage of the rotor is more durable then an armature and can withstand high temperatures. And the ACIM has no commutator to wear or brushes to replace.
With ac power widely available, ACIMs are typically designed with a specific line voltage and frequency in mind. For example, a typical nameplate on an ACIM might list these parameters: Voltage, 230 Vac; Frequency, 60 Hz; FullLoad Amperage, 1.4 A; Horsepower, 1/3 hp; and an rpm of 3,450.
Among other things, the nameplate specifies a motor's rated power, operating voltage, frequency, and rpm. The stator windings of the motor are arranged such that a rotating magnetic field is created when energized with threephase ac currents. The rpms at which the magnetic field rotates is called the synchronous speed.
The rotor of an ACIM must turn at a slightly slower rpm than the synchronous speed. The difference between the field's synchronous speed and the rotor's true speed is called slip. Slip is expressed as either a ratio of true speed divided by synchronous speed or as a frequency. In this case it's more helpful to consider slip frequency. The synchronous speed of the example motor is 3,600 rpm, or 60 rev/sec. But the nameplate rpm under full load is only 3,450 rpm, or 57.5 rev/sec. So the slip frequency is 60 57.5 Hz, or 2.5 Hz.
The 2.5Hz slip frequency is a source of ac power that supplies energy to the rotor via transformer coupling. The rotor becomes energized with ac currents that produce the rotor's magnetic field, creating motor torque. ACIM slip also gives the motor the ability to selfregulate its own speed — to a certain extent. As motor load goes up, rotor speed drops. The slip frequency rises, boosting rotor currents and thus motor torque.
An ACIM can operate at different speeds and torque levels by varying the frequency and voltage supplied to the motor. Suppose it's necessary that the example motor operate at only half its rated speed. To accomplish this, cut the input frequency to the motor in half, to 30 Hz. If the motor were to operate at quarter speed, then the frequency must drop to 15 Hz.
The stator magnetic field should remain relatively constant by keeping stator currents constant. Because the ACIM motor is inductive, stator currents will rise as input frequency drops. Therefore, stator voltage must also go down by a proportional amount when frequency decreases. This relationship is known as the V/Hz profile.
A constant V/Hz profile is often used to vary the speed of ACIMs. The V/Hz constant for a motor is calculated by dividing the operating frequency into the operating voltage. The K factor for the example motor becomes 230 V/60 Hz, or 3.83 V/Hz.
The constant is used to calculate the stator drive voltage for any given input frequency: V = K × Hz. To operate the example motor at 20 Hz, the stator input voltage should be 3.83 V/Hz × 20 Hz, or 76.6 V.
There is no fixed rule that says the drive voltage must maintain a fixed linear relationship to frequency. In fact, the shape of the V/Hz profile is often altered in specific frequency ranges to optimize the drive performance in that particular range. For example, some profiles offer higher voltages in the lowfrequency range that boosts motor torque to help overcome load friction and inertia on startup.
Within the mechanical limits of the motor, drive frequency can also rise beyond nameplate values to get higher speeds. However, higher frequencies translate into less stator current because of higher inductive reactance. This means the motor will have lower torque because of the reduced current.
For applications that don't need frequent speed or load variations, the V/Hz method for controlling an ACIM works well. This is especially true when control loops are used to regulate speed or motor current. Many MCUs used in motor controls contain specialized pulsewidthmodulation (PWM) circuits that drive sixtransistor power inverters. The MCU measures the output of the motor tachometer, calculates the speed error, and generates a drive demand using a PIDlike control loop. The drive demand translates into a required voltage and frequency using the V/Hz profile. The PWM software varies the duty cycle over time to generate sinusoidal drive signals with the proper amplitude and frequency.
However, this won't work for applications that need fast dynamic response. The V/Hzprofile control response is quite sluggish. Furthermore, motor currents will be high during load or speed changes. Response is poor because the current for motor torque and the current for the rotor field are the same current. A change in either drive voltage or frequency changes both torque and rotor currents.
Ideally, we would like to use an algorithm that controls motor torque independently of other motor variables. The FOC algorithm accomplishes this goal. FOC controls the voltage, frequency, and instantaneous phase of the motor voltage to produce the desired stator currents. The V/Hz control method does not control the phase. FOC maintains both motor efficiency and dynamic response for a given application.
If the motor is observed electrically from the perspective of the input terminals, all signals inside the motor appear sinusoidal. Sinusoidal signals are difficult to process in software, especially if PID controllers are used to regulate motor currents. However, changing the point of reference used in the calculations makes the signals inside the ac motor look mathematically like dc values under steadystate conditions.
Specifically, FOC measures ac motor currents. The threephase stator currents combine to form a single rotating current vector in time when a stationary reference plane is used. Instead of using a stationary reference, the reference plane rotates synchronously with the motor. With the rotating reference plane, steadystate ac quantities look stationary.
Here is a helpful analogy: Imagine you are standing on the side of a circular racecar track. From your stationary perspective, all of the cars seem to be speeding around the track. It is hard to tell which car is winning the race with the entire pack of cars going by so quickly.
Instead of watching the cars from the side of the track, hop into the pace car and drive next to the lead car. From your perspective, the pack of cars becomes more or less stationary. The only thing that changes over time is the relative position of all the other cars to the lead position, the moving reference point. The actual speed of the cars moving around the track becomes irrelevant — assuming you're not afraid of high speeds!
In this analogy, the speed of the cars moving around the track is analogous to the motordrive frequency.The relative position of the cars is analogous to the phase of the stator current vector.
FOC uses a pair of conversions called the Clarke Transform and the Park Transform to get from the stationary reference plane to the rotating reference plane. First, two of the three phase currents are measured for their instantaneous value. The third phase current can just be computed because the sum of the three instantaneous currents should equal zero. The measured currents represent the vector components of the current in a threeaxis coordinate system with each axis separated by 120°. It's easier to represent the rotating current vector in a twoaxis orthogonal coordinate system, so the Clarke Transform converts the measured currents to represent the current vector with only two vector components. The two vector components calculated using the Clarke Transform still vary with time.
Clarke Transform Equations:
Iα = Ia
Iβ = 0.577(Ia + 2Ib)
The Park Transform is now used to rotate the twoaxis coordinate system so that it is aligned with the rotating motor. The rotation angle is represented by θ. When using FOC for a synchronous threephase motor, the rotating reference plane would always align with the rotor and θ is obtained directly from the rotor position using a sensor.
Park Transform Equations:
Id = Iα cos θ + Iβ sin θ
Iq = Iα cos θ + Iβ sin θ
However, an ACIM is an asynchronous machine that needs slip to operate.
One method of calculating θ is to use equations that model the rotor currents. The rotorcurrent model calculates the required slip frequency from the measured stator currents. The rotor current model also demands knowledge of the rotor resistance and inductance. These values form a time constant that adjusts the motor slip to the correct value during transient current events. After a slip frequency is determined, a value of θ is calculated using the rotor velocity. This will align the reference plane ahead of the rotor to give the requisite slip. So, the rotating reference plane aligns with the applied statorcurrent vector that spins faster than the rotor.
The key to FOC is that the Clarke and Park transformations provide dc representations of the statorphase currents under steadystate conditions. But motor current is really an ac signal represented as a rotating current vector. It is only because the coordinate system is synchronously rotating with the current vector that the transformed current components appear as dc values. If the value of either current component changes over time, then the amplitude and phase of the motorcurrent vector has changed.
Most importantly, one component of the transformed statorcurrent vector determines the amount of motor torque. The other component determines the rotor field. With FOC, the component of current responsible for motor torque is isolated and controlled separately. This is why FOC lets ac motors be controlled like dc motors.
In practice, the two transformed current components are separately regulated using PID controllers in software. The outputs of the two PID controllers provide two voltage vector components that determine how to energize the motor phases to produce the desired stator currents. The reference input for one PID controller is set to a constant value so the rotor will generate a constant field. The reference input to the other PID controller determines the amount of motor torque. The reference torque level is usually supplied from a third PID control loop that regulates motor speed.
The last step in the FOC process is to "unwind" the voltagevector components that were generated in the rotating reference plane. The value of θ that was calculated in the rotor current model equations is used, along with inverse Clarke and Park transformations. The equations are similar to the forward transforms.
FOC controls the amplitude, frequency, and phase of the voltage vector to produce the desired amplitude, frequency, and phase of motor currents. While the equations for FOC are not complex, they must be executed relatively frequently to get good performance. One reason is that the old value of θ from the prior iteration of the FOC equations must be used to transform the measured current values. Then, a new value is calculated using the rotor current model equations. The FOC equations are typically executed every 50 μsec to minimize the amount of angular error between iterations. Frequent calculation of the FOC equations demands use of a fast 16bit MCU or DSC.
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