2.2 Electrical systems

Electrical systems are the backbone of modern control theory. They enable precise measurement, signal processing, and actuation in control systems. From sensors to actuators, these components work together to monitor and manipulate physical systems.

Understanding electrical components is crucial for implementing effective control strategies. This section covers sensors, actuators, signal conditioning, power electronics, and system modeling. We'll explore how these elements interact to create robust control systems.

Electrical components in control systems

• Electrical components are essential for implementing control systems by enabling the measurement of system variables, processing of control signals, and actuation of system inputs
• Control systems rely on a variety of electrical components to interface with the physical system being controlled and to perform the necessary computations and signal conditioning
• Key electrical components in control systems include sensors, actuators, signal conditioning circuits, power electronics, and microcontrollers or PLCs (programmable logic controllers)

Sensors for measuring system variables

• Sensors are used to measure various physical quantities of interest in a control system, providing feedback to the controller about the current state of the system
• The choice of sensor depends on the specific variable being measured, the required accuracy and resolution, and the operating conditions of the system
• Common types of sensors used in control systems include position sensors, velocity sensors, acceleration sensors, temperature sensors, pressure sensors, and flow sensors

Position sensors

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• Measure linear or angular displacement of an object relative to a reference point
• Examples include potentiometers, encoders (optical or magnetic), and LVDTs (linear variable differential transformers)
• Potentiometers provide a voltage output proportional to the position of a sliding contact along a resistive element
• Encoders generate digital pulses as a shaft rotates, allowing for high-resolution position measurement
• LVDTs use the principle of magnetic induction to measure linear displacement without physical contact

Velocity sensors

• Measure the speed of an object or the rate of change of position
• Can be derived from position sensors by differentiating the position signal over time
• Tachometers are commonly used velocity sensors, generating a voltage proportional to the rotational speed of a shaft
• Doppler effect sensors use the frequency shift of reflected waves (ultrasonic or laser) to measure velocity

Acceleration sensors

• Measure the rate of change of velocity or the second derivative of position
• Accelerometers are the most common type of acceleration sensor, using the deflection of a small mass to measure acceleration
• Piezoelectric accelerometers generate an electrical charge proportional to the applied acceleration
• MEMS (microelectromechanical systems) accelerometers use the deflection of a microscale cantilever beam to measure acceleration

Temperature sensors

• Measure the temperature of a system or its environment
• Common types include thermocouples, RTDs (resistance temperature detectors), and thermistors
• Thermocouples generate a voltage proportional to the temperature difference between two dissimilar metals
• RTDs change resistance with temperature, typically using a platinum element
• Thermistors are semiconductor devices that exhibit a large change in resistance with temperature

Pressure sensors

• Measure the force per unit area exerted by a fluid (liquid or gas) on a surface
• Can be used to monitor system pressures, detect leaks, or control fluid flow
• Pressure sensors often use a diaphragm or bellows to convert pressure into a mechanical displacement, which is then measured using a strain gauge or capacitive sensor
• Differential pressure sensors measure the difference in pressure between two points, commonly used for flow measurement

Flow sensors

• Measure the rate of fluid flow through a pipe or channel
• Can be based on various principles, such as differential pressure, turbine rotation, or thermal mass flow
• Orifice plate and Venturi tube flow sensors use the pressure drop across a restriction to infer flow rate
• Turbine flow sensors measure the rotation speed of a turbine driven by the fluid flow
• Thermal mass flow sensors use the heat transfer between a heated element and the fluid to measure flow rate

Actuators for system control

• Actuators are devices that convert electrical signals from the controller into physical actions to manipulate the system being controlled
• They provide the means to apply control inputs to the system, such as force, torque, or displacement
• The choice of actuator depends on the type of system, the required force or torque, the speed of response, and the operating conditions
• Common types of actuators used in control systems include electric motors, solenoids, relays, servomotors, and stepper motors

Electric motors

• Convert electrical energy into mechanical energy, providing rotational motion
• DC motors are widely used in control systems due to their simplicity and ease of control
• AC motors, such as induction motors and synchronous motors, are used for high-power applications
• The speed and torque of electric motors can be controlled by varying the voltage or frequency of the input signal

Solenoids

• Electromagnetic devices that convert electrical energy into linear motion
• Consist of a coil of wire wound around a movable iron core (armature)
• When current flows through the coil, a magnetic field is generated, pulling the armature into the coil
• Used for applications requiring quick, short-stroke linear actuation, such as valves or locks

Relays

• Electrically operated switches that use an electromagnet to open or close one or more sets of contacts
• Allow a low-power control signal to switch a high-power load
• Commonly used for isolation between control circuits and power circuits
• Can also be used for logic operations in control systems

Servomotors

• Rotary actuators that allow for precise control of angular position
• Consist of a DC motor, a position sensor (potentiometer or encoder), and a control circuit
• The control circuit compares the desired position (set by the input signal) with the actual position (measured by the sensor) and adjusts the motor input to minimize the error
• Widely used in robotics, CNC machines, and other applications requiring accurate position control

Stepper motors

• Brushless DC motors that divide a full rotation into a number of equal steps
• Can be controlled to rotate a specific number of steps, allowing for precise position control without the need for a separate position sensor
• Commonly used in printers, scanners, and other devices requiring precise incremental motion
• Require a specific sequence of pulses to be applied to the motor windings to achieve rotation

Signal conditioning of sensor outputs

• Signal conditioning is the process of modifying sensor output signals to make them suitable for input to a controller or data acquisition system
• It involves amplifying low-level signals, filtering out noise, and converting signals from analog to digital form
• Proper signal conditioning ensures that the measured data accurately represents the physical quantity being monitored and is compatible with the controller's input requirements

Amplification of low-level signals

• Many sensors produce low-voltage or low-current output signals that require amplification before they can be processed by the controller
• Operational amplifiers (op-amps) are commonly used for signal amplification
• The gain of the amplifier is set by the ratio of feedback resistors, allowing for adjustable amplification
• Instrumentation amplifiers are a special type of op-amp circuit designed for accurate, low-noise amplification of small differential signals

Filtering of noise

• Sensor outputs often contain unwanted noise components that can interfere with the accurate measurement of the desired signal
• Filters are used to remove or attenuate specific frequency components of the signal
• Low-pass filters remove high-frequency noise, such as electromagnetic interference (EMI) or power line noise
• High-pass filters remove low-frequency noise, such as drift or offset voltages
• Band-pass filters allow a specific range of frequencies to pass while attenuating others
• Active filters use op-amps to achieve high-order filter characteristics with adjustable cut-off frequencies

Analog-to-digital conversion

• Most modern controllers and data acquisition systems operate in the digital domain, requiring sensor outputs to be converted from analog to digital form
• Analog-to-digital converters (ADCs) sample the continuous analog signal at discrete time intervals and quantize the amplitude into a finite number of digital values
• The resolution of the ADC (number of bits) determines the smallest detectable change in the analog signal
• The sampling rate of the ADC must be at least twice the highest frequency component of the analog signal to avoid aliasing (Nyquist theorem)
• Multiplexers allow multiple analog signals to be sequentially connected to a single ADC, reducing the number of required converters

Power electronics for actuator control

• Power electronics are used to efficiently control the flow of electrical power to actuators, such as motors and solenoids
• They allow for the precise regulation of voltage, current, or frequency to achieve the desired actuator performance
• Power electronic circuits use semiconductor devices, such as transistors and thyristors, to switch and modulate electrical power

Transistors vs thyristors

• Transistors, such as MOSFETs (metal-oxide-semiconductor field-effect transistors) and IGBTs (insulated-gate bipolar transistors), are commonly used in low to medium power applications
• Transistors can be fully turned on or off by applying a control signal to their gate terminal, allowing for efficient switching and pulse-width modulation (PWM) control
• Thyristors, such as SCRs (silicon-controlled rectifiers) and triacs, are used in high-power applications
• Thyristors are latching devices that can be turned on by a control signal but require the current to drop below a certain threshold to turn off

Pulse-width modulation (PWM)

• PWM is a technique used to control the average voltage or current delivered to an actuator by rapidly switching the power on and off
• The duty cycle (ratio of on-time to total period) of the PWM signal determines the effective voltage or current applied to the actuator
• PWM allows for efficient power control and reduces losses compared to linear regulation techniques
• The frequency of the PWM signal must be chosen to minimize ripple in the actuator current while avoiding excessive switching losses

H-bridges for motor control

• H-bridges are power electronic circuits used to control the direction and speed of DC motors
• They consist of four switches (transistors or thyristors) arranged in an H configuration, allowing current to flow through the motor in either direction
• By controlling the switches, the H-bridge can apply positive, negative, or zero voltage across the motor, enabling forward, reverse, and braking operation
• The switches are typically controlled using PWM signals to regulate the motor speed

Protection circuits

• Power electronic circuits must include protection features to prevent damage to the components and the actuator in case of faults or overloads
• Overcurrent protection, such as fuses or electronic current limiting, prevents excessive current from flowing through the switches or the actuator
• Overvoltage protection, such as transient voltage suppressors or snubber circuits, limits voltage spikes caused by inductive loads or switching transients
• Thermal protection, such as temperature sensors and shutdown circuits, prevents overheating of the power electronic components
• Isolation circuits, such as optocouplers or transformers, provide electrical isolation between the control and power stages to protect the controller from high voltages or noise

Electrical system modeling

• Electrical system modeling involves creating mathematical representations of electrical components and their interactions to predict system behavior and design control strategies
• Models can be developed using fundamental laws of electrical circuits, such as Kirchhoff's laws, and can be represented in various forms, such as transfer functions or state-space equations
• Accurate modeling is essential for simulating system response, optimizing control parameters, and ensuring stability and performance

Kirchhoff's laws

• Kirchhoff's current law (KCL) states that the sum of currents entering a node in a circuit must equal the sum of currents leaving the node
• Kirchhoff's voltage law (KVL) states that the sum of voltages around any closed loop in a circuit must equal zero
• These laws form the basis for analyzing electrical circuits and developing system equations
• KCL and KVL can be used to derive the relationships between currents, voltages, and impedances in a circuit

Impedance and admittance

• Impedance is a measure of the opposition to current flow in an electrical circuit, considering both resistance and reactance
• Reactance is the opposition to current flow caused by inductors (inductive reactance) and capacitors (capacitive reactance)
• Impedance is a complex quantity, with the real part representing resistance and the imaginary part representing reactance
• Admittance is the reciprocal of impedance and represents the ease with which current flows in a circuit
• Impedance and admittance are used to characterize the frequency-dependent behavior of electrical components and systems

Transfer functions of electrical components

• Transfer functions describe the input-output relationship of electrical components or systems in the frequency domain
• They are obtained by applying Laplace transforms to the differential equations governing the system behavior
• Transfer functions are expressed as the ratio of the output variable to the input variable, with the Laplace variable s as the complex frequency
• Common transfer functions for electrical components include:
  • Resistor: $V(s) = R \cdot I(s)$
  • Inductor: $V(s) = s \cdot L \cdot I(s)$
  • Capacitor: $I(s) = s \cdot C \cdot V(s)$
• Transfer functions can be combined using series and parallel connection rules to obtain the overall system transfer function

State-space representation

• State-space representation is an alternative method for modeling electrical systems, particularly for multi-input, multi-output (MIMO) systems
• The state-space model consists of a set of first-order differential equations that describe the evolution of the system state variables over time
• The state variables are a minimal set of variables that fully characterize the system at any given time
• The state-space model includes an input matrix (B) that relates the system inputs to the state variables and an output matrix (C) that relates the state variables to the system outputs
• The general form of a state-space model is:
  • $\dot{x} = A \cdot x + B \cdot u$
  • $y = C \cdot x + D \cdot u$ where $x$ is the state vector, $u$ is the input vector, $y$ is the output vector, and $A$, $B$, $C$, and $D$ are the state-space matrices
• State-space representation allows for the analysis of system stability, controllability, and observability using linear algebra techniques

Electrical system analysis

• Electrical system analysis involves evaluating the performance and stability of electrical systems using various techniques and tools
• The goal is to understand the system's response to different inputs, identify potential issues, and design appropriate control strategies
• Key aspects of electrical system analysis include frequency response, stability analysis, pole-zero analysis, and transient and steady-state response

Frequency response

• Frequency response describes how a system responds to sinusoidal inputs of different frequencies
• It is typically represented using Bode plots, which show the magnitude and phase of the system's transfer function as a function of frequency
• The magnitude plot illustrates the system's gain (ratio of output to input) at each frequency, while the phase plot shows the phase shift between the input and output signals
• Frequency response analysis helps identify resonant frequencies, bandwidth, and stability margins of the system
• It is also used to design frequency-domain control techniques, such as lead-lag compensation or PID tuning

Stability analysis using Nyquist and Bode plots

• Stability analysis determines whether a system will remain bounded and converge to a steady-state value for a given input
• The Nyquist stability criterion uses the Nyquist plot (a polar plot of the system's open-loop transfer function) to assess stability
• If the Nyquist plot encircles the point -1+j0 counterclockwise as many times as there are unstable poles in the open-loop transfer function, the closed-loop system is stable
• The Bode plot can also be used for stability analysis by examining the gain and phase margins
• The gain margin is the amount of additional gain that can be added to the system before it becomes unstable, while the phase margin is the amount of additional phase lag that can be introduced before instability occurs
• A stable system should have positive gain and phase margins, with typical values being at least 6 dB for gain margin and 45 degrees for phase margin

Poles and zeros

• Poles and zeros are the roots of the denominator and numerator polynomials, respectively, of a system's transfer function
• Poles represent the natural frequencies and damping of the system, while zeros represent the frequencies at which the system's response is nullified
• The location of poles and zeros in the complex plane determines the stability and transient response of the system
• Poles in the left-half plane (LHP) indicate a stable system, while poles in the right-half plane (RHP) indicate an unstable system
• Poles on the imaginary axis result in sustained oscillations, while poles near the imaginary axis lead to lightly damped or resonant behavior
• Zeros in the LHP can introduce undershoot or non-minimum phase behavior in the system response

Transient response vs steady-state response

• Transient response refers to the system's behavior during the initial period after an input is applied or a disturbance occurs
• It is characterized by the rise time (time required to reach a specified percentage of the final value), overshoot (maximum deviation from the final value), settling time (time required to stay within a specified tolerance of the final value), and peak time (time at which the overshoot occurs)
• Steady-state response refers to the system's behavior long after the transient period has ended
• It is characterized by the steady-state error (difference between the input and output values) and the type of input (step, ramp, or parabolic) that the system can follow with zero error
• The transient and steady-state responses are influenced by the system's poles, zeros, and gain, as well as the type and magnitude of the input signal
• Control systems are often designed to achieve a desired transient response (e.g., fast rise time, minimal overshoot) while maintaining acceptable steady-state performance (e.g., zero or small steady-state error)

Electrical system design considerations

• Designing electrical systems for control applications involves considering various factors to ensure reliable, efficient, and safe operation
• Key design considerations include power supply requirements