Vector control, also called field-oriented control (FOC), is a
variable-frequency drive
A variable-frequency drive (VFD) is a type of motor drive used in electro-mechanical drive systems to control AC motor speed and torque by varying motor input frequency and, depending on topology, to control associated voltage or current va ...
(VFD) control method in which the
stator currents of a
three-phase AC or brushless DC electric motor are identified as two orthogonal components that can be visualized with a vector. One component defines the
magnetic flux of the motor, the other the torque. The control system of the drive calculates the corresponding current component references from the flux and torque references given by the drive's speed control. Typically
proportional-integral (PI) controllers are used to keep the measured current components at their reference values. The
pulse-width modulation of the variable-frequency drive defines the
transistor
upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink).
A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch ...
switching according to the stator voltage references that are the output of the PI current controllers.
FOC is used to control
AC synchronous
Synchronization is the coordination of events to operate a system in unison. For example, the conductor of an orchestra keeps the orchestra synchronized or ''in time''. Systems that operate with all parts in synchrony are said to be synchronou ...
and
induction motors.
It was originally developed for high-performance motor applications that are required to operate smoothly over the full
speed
In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (ma ...
range, generate full
torque
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
at zero speed, and have high dynamic performance including fast
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
and
deceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
. However, it is becoming increasingly attractive for lower performance applications as well due to FOC's motor size, cost and
power consumption
Electric energy consumption is the form of energy consumption that uses electrical energy. Electric energy consumption is the actual energy demand made on existing electricity supply for transportation, residential, industrial, commercial, and ot ...
reduction superiority.
It is expected that with increasing computational power of the microprocessors it will eventually nearly universally displace single-variable
scalar volt
The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827).
Defin ...
s-per-
Hertz
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that o ...
(V/f) control.
Development history
![US3824437](https://upload.wikimedia.org/wikipedia/commons/8/8b/US3824437.jpg)
Technische Universität Darmstadt's K. Hasse and Siemens' F. Blaschke pioneered
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
control of AC motors starting in 1968 and in the early 1970s. Hasse in terms of proposing indirect vector control, Blaschke in terms of proposing direct vector control.
Technical University Braunschweig's Werner Leonhard further developed FOC techniques and was instrumental in opening up opportunities for
AC drives to be a competitive alternative to
DC drives.
Yet it was not until after the commercialization of
microprocessor
A microprocessor is a computer processor where the data processing logic and control is included on a single integrated circuit, or a small number of integrated circuits. The microprocessor contains the arithmetic, logic, and control circ ...
s, that is in the early 1980s, that general purpose AC drives became available.
Barriers to use FOC for AC drive applications included higher cost and complexity and lower maintainability compared to DC drives, FOC having until then required many electronic components in terms of sensors, amplifiers and
so on.
The
Park transform
The direct-quadrature-zero (DQZ or DQ0 or DQO, sometimes lowercase) transformation or zero-direct-quadrature (0DQ or ODQ, sometimes lowercase) transformation is a tensor that rotates the reference frame of a three-element vector or a three-by-three ...
ation has long been widely used in the analysis and study of
synchronous
Synchronization is the coordination of events to operate a system in unison. For example, the conductor of an orchestra keeps the orchestra synchronized or ''in time''. Systems that operate with all parts in synchrony are said to be synchronou ...
and induction machines. The transformation is by far the single most important concept needed for an understanding of how FOC works, the concept having been first conceptualized in a 1929 paper authored by
Robert H. Park.
Park's paper was ranked second most important in terms of impact from among all power engineering related papers ever published in the twentieth century. The novelty of Park's work involves his ability to transform any related machine's linear
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
set from one with time varying coefficients to another with time ''invariant'' coefficients
resulting in a
linear time-invariant system or LTI system.
Technical overview
Overview of key competing VFD control platforms:
While the analysis of AC drive controls can be technically quite involved ("See also" section), such analysis invariably starts with modeling of the drive-motor circuit involved along the lines of accompanying
signal flow graph
A signal-flow graph or signal-flowgraph (SFG), invented by Claude Shannon, but often called a Mason graph after Samuel Jefferson Mason who coined the term, is a specialized flow graph, a directed graph in which nodes represent system variables, ...
and equations.
:Induction motor model equations
::
:where
::
::
:
:
![IFOC](https://upload.wikimedia.org/wikipedia/commons/a/a2/IFOC.jpg)
![VectorSensorlesssBD](https://upload.wikimedia.org/wikipedia/commons/5/50/VectorSensorlesssBD.jpg)
In vector control, an AC induction or synchronous motor is controlled under all operating conditions like a separately
excited DC motor.
[Bose (2006), p. 429] That is, the AC motor behaves like a DC motor in which the
field flux linkage and
armature flux linkage created by the respective field and armature (or torque component) currents are
orthogonally aligned such that, when torque is controlled, the field flux linkage is not affected, hence enabling dynamic torque response.
Vector control accordingly generates a three-phase
PWM motor voltage output derived from a
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
voltage vector to control a complex current vector derived from motor's three-phase stator current input through
projections or
rotations back and forth between the three-phase speed and time dependent system and these vectors' rotating reference-frame two-
coordinate
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
time invariant system.
Such complex
stator current space vector can be defined in a (d,q) coordinate system with orthogonal components along d (direct) and q (quadrature) axes such that field flux linkage component of current is aligned along the d axis and torque component of current is aligned along the q axis.
The induction motor's (d,q) coordinate system can be superimposed to the motor's instantaneous (a,b,c) three-phase
sinusoidal system as shown in accompanying image (phases b & c not shown for clarity). Components of the (d,q) system current vector allow conventional control such as proportional and integral, or
PI, control, as with a DC motor.
Projections associated with the (d,q) coordinate system typically involve:
* Forward projection from instantaneous currents to (a,b,c) complex
stator current space vector representation of the three-phase
sinusoidal system.
* Forward three-to-two phase, (a,b,c)-to-(
,
) projection using the
Clarke
Clarke is a surname which means "clerk". The surname is of English and Irish origin and comes from the Latin . Variants include Clerk and Clark. Clarke is also uncommonly chosen as a given name.
Irish surname origin
Clarke is a popular surname i ...
transformation. Vector control implementations usually assume ungrounded motor with balanced three-phase currents such that only two motor current phases need to be sensed. Also, backward two-to-three phase, (
,
)-to-(a,b,c) projection uses space vector PWM modulator or inverse Clarke transformation and one of the other PWM modulators.
* Forward and backward two-to-two phase,(
,
)-to-(d,q) and (d,q)-to-(
,
) projections using the Park and inverse Park transformations, respectively.
The idea of using the park transform is to convert the system of three phase currents and voltages into a two coordinate linear time-invariant system. By making the system LTI is what enables the use of simple and easy to implement PI controllers, and also simplifies the control of flux and torque producing currents.
However, it is not uncommon for sources to use combined transform three-to-two, (a,b,c)-to-(d,q) and inverse projections.
While (d,q) coordinate system rotation can arbitrarily be set to any speed, there are three preferred speeds or reference frames:
* Stationary reference frame where (d,q) coordinate system does not rotate;
* Synchronously rotating reference frame where (d,q) coordinate system rotates at synchronous speed;
* Rotor reference frame where (d,q) coordinate system rotates at rotor speed.
Decoupled
'' Decoupled'' is an Indian English-language comedy web series for Netflix written by Manu Joseph and directed by Hardik Mehta. The series stars R. Madhavan and Surveen Chawla in the lead roles.
Plot
A misanthropic writer and his startup-foun ...
torque and field currents can thus be derived from raw stator current inputs for control algorithm development.
Whereas magnetic field and torque components in DC motors can be operated relatively simply by separately controlling the respective field and armature currents, economical control of AC motors in variable speed application has required development of microprocessor-based controls
with all AC drives now using powerful DSP (
digital signal processing) technology.
Inverters can be implemented as either
open-loop sensorless or closed-loop FOC, the key limitation of open-loop operation being minimum speed possible at 100% torque, namely, about 0.8 Hz compared to standstill for closed-loop operation.
There are two vector control methods, direct or
feedback vector control (DFOC) and indirect or
feedforward vector control (IFOC), IFOC being more commonly used because in closed-loop mode such drives more easily operate throughout the speed range from zero speed to high-speed field-weakening. In DFOC, flux magnitude and angle feedback signals are directly calculated using so-called voltage or current models. In IFOC, flux space angle feedforward and flux magnitude signals first measure stator currents and
rotor
Rotor may refer to:
Science and technology
Engineering
* Rotor (electric), the non-stationary part of an alternator or electric motor, operating with a stationary element so called the stator
*Helicopter rotor, the rotary wing(s) of a rotorcraft ...
speed for then deriving flux space angle proper by summing the rotor angle corresponding to the rotor speed and the calculated reference value of
slip
Slip or SLIP may refer to:
Science and technology Biology
* Slip (fish), also known as Black Sole
* Slip (horticulture), a small cutting of a plant as a specimen or for grafting
* Muscle slip, a branching of a muscle, in anatomy
Computing and ...
angle corresponding to the slip frequency.
[Bose (2006), p. 423-425]
Sensorless control (see Sensorless FOC Block Diagram) of AC drives is attractive for cost and reliability considerations. Sensorless control requires derivation of rotor speed information from measured stator voltage and currents in combination with open-loop estimators or closed-loop observers.
Application
# Stator phase currents are measured, converted to complex space vector in (a,b,c) coordinate system.
# Current is converted to (
,
) coordinate system.
Transformed to a coordinate system rotating in
rotor
Rotor may refer to:
Science and technology
Engineering
* Rotor (electric), the non-stationary part of an alternator or electric motor, operating with a stationary element so called the stator
*Helicopter rotor, the rotary wing(s) of a rotorcraft ...
reference frame, rotor position is derived by
integrating the speed by means of
speed measurement sensor.
# Rotor
flux linkage
In circuit theory, flux linkage is a property of a two-terminal element. It is an extension rather than an equivalent of magnetic flux and is defined as a time integral
:\lambda = \int \mathcal \,dt,
where \mathcal is the voltage across the de ...
vector is estimated by multiplying the stator current vector with magnetizing inductance L
m and
low-pass filter
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filt ...
ing the result with the rotor no-load
time constant In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a s ...
L
r/R
r, namely, the rotor inductance to rotor resistance ratio.
# Current vector is converted to (d,q) coordinate system.
# d-axis component of the stator current vector is used to control the rotor flux linkage and the imaginary q-axis component is used to control the motor torque. While PI controllers can be used to control these currents,
bang-bang type current control provides better dynamic performance.
# PI controllers provide (d,q) coordinate voltage components. A decoupling term is sometimes added to the controller output to improve control performance to mitigate cross coupling or big and rapid changes in speed, current and flux linkage. PI-controller also sometimes need
low-pass filter
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filt ...
ing at the input or output to prevent the current ripple due to transistor switching from being amplified excessively and destabilizing the control. However, such filtering also limits the dynamic control system performance. High switching frequency (typically more than 10 kHz) is typically required to minimize filtering requirements for high-performance drives such as servo drives.
# Voltage components are transformed from (d,q) coordinate system to (
,
) coordinate system.
# Voltage components are transformed from (
,
) coordinate system to (a,b,c) coordinate system or fed in
Pulse-Width Modulation (PWM) modulator, or both, for signaling to the power inverter section.
Significant aspects of vector control application:
* Speed or position measurement or some sort of estimation is needed.
* Torque and flux can be changed reasonably fast, in less than 5-10 milliseconds, by changing the references.
* The
step response
The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the out ...
has some
overshoot if PI control is used.
* The switching frequency of the transistors is usually constant and set by the modulator.
* The accuracy of the torque depends on the accuracy of the motor parameters used in the control. Thus large errors due to for example rotor temperature changes often are encountered.
* Reasonable processor performance is required; typically the control algorithm is calculated every PWM cycle.
Although the vector control algorithm is more complicated than the
Direct Torque Control Direct torque control (DTC) is one method used in variable-frequency drives to control the torque (and thus finally the speed) of three-phase electric motor, AC electric motors. This involves Calculation, calculating an estimate of the motor's magn ...
(DTC), the algorithm need not be calculated as frequently as the DTC algorithm. Also the current sensors need not be the best in the market. Thus the cost of the processor and other control hardware is lower making it suitable for applications where the ultimate performance of DTC is not required.
See also
*
transform
*
Adaptive control Adaptive control is the control method used by a controller which must adapt to a controlled system with parameters which vary, or are initially uncertain. For example, as an aircraft flies, its mass will slowly decrease as a result of fuel consumpt ...
*
Control engineering
Control engineering or control systems engineering is an engineering discipline that deals with control systems, applying control theory to design equipment and systems with desired behaviors in control environments. The discipline of controls o ...
*
Control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
*
Dqo transformation
*
Eigenvalues and eigenvectors
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
*
Extended Kalman filter
In estimation theory, the extended Kalman filter (EKF) is the nonlinear version of the Kalman filter which linearizes about an estimate of the current mean and covariance. In the case of well defined transition models, the EKF has been considered t ...
*
Filter (signal processing)
*
Frequency response
In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of s ...
*
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the functi ...
*
Impulse response
*
Linear time-invariant system
*
Kalman filter
For statistics and control theory, Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, and produces estima ...
*
Robust control In control theory, robust control is an approach to controller design that explicitly deals with uncertainty. Robust control methods are designed to function properly provided that uncertain parameters or disturbances are found within some (typicall ...
*
Root locus
In control theory and stability theory, root locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly a gain within a feedback system. This is a technique used as a ...
*
Perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...
*
Signal-flow graph
A signal-flow graph or signal-flowgraph (SFG), invented by Claude Shannon, but often called a Mason graph after Samuel Jefferson Mason who coined the term, is a specialized flow graph, a directed graph in which nodes represent system variables, ...
*
Small-signal model
Small-signal modeling is a common analysis technique in electronics engineering used to approximate the behavior of electronic circuits containing nonlinear devices with linear equations. It is applicable to electronic circuits in which the AC si ...
*
Sliding mode control
In control systems, sliding mode control (SMC) is a nonlinear control method that alters the dynamics of a nonlinear system by applying a discontinuous control signal (or more rigorously, a set-valued control signal) that forces the system to ...
*
State observer
*
State space representation
In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations or difference equations. State variables are variables wh ...
*
Symmetrical components
*
Systems analysis
Systems analysis is "the process of studying a procedure or business to identify its goal and purposes and create systems and procedures that will efficiently achieve them". Another view sees system analysis as a problem-solving technique that ...
*
Transient response
In electrical engineering and mechanical engineering, a transient response is the response of a system to a change from an equilibrium or a steady state. The transient response is not necessarily tied to abrupt events but to any event that affec ...
*
Transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that theoretically models the system's output for each possible input. They are widely used ...
References
{{Authority control
Electric motors