Name: dc motor modeling in electromechanical systems
Uploaded: 2024-09-26T13:17:19.000Z
Duration: 1 min 19 s
Channel: JoVE

Electromechanical systems represent a unique blend of electrical and mechanical components that integrate seamlessly.
A DC motor is a unit that transforms voltage into angular displacement, producing a mechanical outcome from an electrical input.
As current flows in the armature circuit placed inside the magnetic field, it experiences a force, leading to torque that rotates the rotor.
Since the armature carries current while rotating in a magnetic field, its voltage becomes proportional to speed.
Applying a loop equation to the Laplace transformed armature circuit reveals a correlation between the armature current, the applied armature voltage, and the back EMF.
The motor&#39;s torque is proportional to the armature current, with the proportionality constant depending on the motor and magnetic field characteristics.
The torque is related to the inertia and viscous damping at the armature.
The torque equation is first rewritten to express it in terms of the angular position of the motor shaft.
Following this, it is assumed that the armature inductance is considerably smaller than the armature resistance.
Upon simplification, the transfer function is obtained.

electro-mechanical systems

Electromechanical systems are intricate configurations that effectively combine electrical and mechanical elements to achieve a desired outcome. Central to many of these systems is the DC motor, a device that converts electrical energy into mechanical motion, enabling various applications ranging from simple fans to complex robotic mechanisms.
A key component of the DC motor is the armature, a rotating circuit positioned within a magnetic field. As an electric current passes through the armature, it encounters a force due to the interaction with the magnetic field, producing torque. This torque initiates the rotation of the rotor, thus converting electrical energy into mechanical motion. The voltage induced in the armature is directly proportional to its speed, a phenomenon known as back electromotive force (EMF).
To analyze the behavior of a DC motor, we apply electrical principles to the armature circuit. By employing a loop equation and transforming it via the Laplace method, we can elucidate the relationship between the armature current (ia), the applied armature voltage (Va), and the back EMF (Eb). The equation is given by:
<img alt="Electrical circuit equation Va=iaRa+Eb; key concept in circuit analysis and Ohm's Law." src="/files/ftp_upload/16098/16098_PT%20EQUATION_1.svg" style="text-align: center; height: 32px;" data-altupdatedat="1762173717000" data-alt="Equation1" />
Where Ra represents the armature resistance and Eb represents the back EMF.
In the s domain, the torque (T) produced by the motor is directly proportional to the armature current, described by:
<img alt="Motor torque formula \(T_m(s) = K_tI_a(s)\); control systems equation." src="/files/ftp_upload/16098/16098_PT%20EQUATION_2.svg" style="text-align: center; height: 20px;" data-altupdatedat="1762173717000" data-alt="Equation2" />
&#8203;Here, kt is the torque constant. This torque can also be written in terms of the inertia (J) of the rotor:
<img alt="Control system transfer function equation, diagram for dynamic system analysis." src="/files/ftp_upload/16098/16098_PT%20EQUATION_3.svg" style="text-align: center; height: 20px;" data-altupdatedat="1762173717000" data-alt="Equation3" />
By expressing the torque in terms of the angular position (&#952;) of the motor shaft and simplifying it, one can derive the transfer function. Assuming the armature inductance to be negligible compared to the armature resistance, the simplified transfer function of the DC motor becomes:
<img alt="Mechanical feedback control equation, Laplace transform, control theory formula." src="/files/ftp_upload/16098/16098_PT%20EQUATION_4.svg" style="text-align: center; height: 64px;" data-altupdatedat="1762173717000" data-alt="Equation4" />
This transfer function provides a comprehensive understanding of the motor's dynamic response, linking the electrical input to the mechanical output and facilitating the design and control of electromechanical systems.

Electro-mechanical Systems

Electromechanical systems represent a unique blend of electrical and mechanical components that integrate seamlessly.
A DC motor is a unit that transforms ...

Education

JoVE Core

Electrical Engineering

Unit 1

Modeling in Time and Frequency Domain

KEY TERMS AND CONCEPTS

dc motor modeling in electromechanical systems

DC motor modeling is a key part of electromechanical systems. These systems combine electrical parts and mechanical parts to produce a useful output. A DC motor is one common example because it turns electrical energy into mechanical motion. It is used in many places, from simple fans to complex robotic mechanisms.

Inside the DC motor, the armature is a rotating circuit placed in a magnetic field. When current flows through the armature, it experiences a force from the magnetic field. That force creates torque, which starts the rotor spinning. As the motor turns, it also produces a voltage in the armature that rises with speed. This voltage is called back electromotive force, or back EMF.

To study how the motor behaves, electrical principles are applied to the armature circuit. A loop equation is written for the armature current, the applied armature voltage, and the back EMF. The equation can then be transformed with the Laplace method to help show the motor response in the s domain. This makes it easier to relate the input voltage to the motor’s electrical and mechanical behavior.

The torque produced by the motor is directly proportional to the armature current in the s domain. The torque constant, k_t, sets that relationship. Torque can also be written in terms of the rotor inertia, J, and the motion of the motor shaft. By expressing the torque through the angular position, θ, and simplifying the result, a transfer function can be derived. When armature inductance is assumed to be negligible compared with armature resistance, the DC motor transfer function becomes simpler and shows how electrical input leads to mechanical output.

DC Motor Modeling in Electromechanical Systems

dc-motor-modeling-electromechanical-systems

transfer function in control systems

Transfer Function from Input to Output

The transfer function describes how a linear time-invariant, or LTI, system maps an input to an output in the frequency domain. It gives a compact way to study system behavior without working only in the time domain. It also connects the differential equations of the system to a form that is easier to handle mathematically. For an nth-order LTI system, the behavior is written as a differential equation with constant coefficients. In that equation, r(t) is the input and c(t) is the output. The...

electrical systems

RLC Circuit Transfer Functions

RLC circuits use resistors, capacitors, and inductors to show how electrical signals change over time. In circuit analysis, these passive linear components are arranged so the link between an input and an output can be described with a transfer function. For this topic, the output is the voltage across the capacitor, compared with the input voltage. A series RLC circuit can be analyzed with Kirchhoff's Voltage Law, or KVL. KVL says that the sum of all voltages around a closed loop is zero.

mechanical systems

Transfer Functions in Mechanical Motion

Mechanical motion can be modeled with springs, masses, and viscous dampers. In these models, springs behave like inductors and masses act like capacitors in an electrical network. A viscous damper works like a resistor because it removes energy from the system. A force acts on the mass in the direction of motion. It is balanced by the spring force, the damping force, and the force from the mass’s acceleration. Newton’s second law is used to describe this balance, and the sum of the forces on...

DC motor modeling is a key part of electromechanical systems. These systems combine electrical parts and mechanical parts to produce a useful output. A DC motor is one common example because it turns electrical energy into mechanical motion. It is used in many places, from simple fans to complex robotic mechanisms. Inside the DC motor, the armature is a rotating circuit placed in a magnetic field. When current flows through the armature, it experiences a force from the magnetic field. That...

linear approximation in frequency domain

Linearizing Nonlinear RL Circuits

Linearizing nonlinear RL circuits helps turn a hard circuit problem into a simpler linear one. Linear systems follow superposition, which means the total response equals the sum of the responses to separate inputs. They also follow homogeneity, which means scaling the input scales the output by the same amount. Nonlinear systems do not automatically have those properties. For small changes around an operating point, though, a nonlinear system can often be treated as linear. This is done with a...

state space representation

RLC Circuit State Equations

RLC circuit state equations show how a circuit changes over time using state variables. This time-domain method is useful for linear, time-invariant systems, but it also helps when frequency-domain analysis is not enough. It is especially useful for nonlinear systems, time-varying systems, and multiple-input multiple-output systems. The state-space approach turns an nth-order system into a set of simultaneous first-order differential equations called state equations. For a common second-order...

transfer function to state space

Phase-Variable State Equations from Transfer Functions

Phase-variable state equations show how a transfer function can be rewritten for computer simulation. In state-space form, the system is described with state variables, which are the output and its first several derivatives. This makes it easier to model a physical system on a digital computer. The starting point is an nth-order linear differential equation with constant coefficients. Systems like an RLC circuit can be written this way. To build the state equations, the transfer function is...

state space to transfer function

Finding a Transfer Function from State Space

Finding a transfer function from state-space form is a key step in system analysis. It changes a time-domain model into a frequency-domain form. That makes it easier to study and design control systems. The process starts with the state-space equations. These include the state equation and the output equation. In these equations, x(t) is the state vector, u(t) is the input vector, y(t) is the output vector, and A, B, C, and D are the matrices that define the system dynamics. Next, the Laplace...

linear approximation in time domain

Linearizing Nonlinear State-Space Models

Linearizing nonlinear state-space models helps describe systems that change with time or operating conditions. State-space representation is often effective for this kind of modeling. It is especially useful for systems such as a simple pendulum or a translational mechanical system with nonlinear springs. A simple pendulum can be modeled by summing torques around the pivot point. In this case, the mass is evenly distributed along the pendulum, and the center of mass is located at half its...

key terms and concepts

Learn how a DC motor turns electrical input into motion, how back EMF affects armature current, and how Laplace methods help derive the transfer function.

DC Motor Modeling in Electromechanical Systems | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

DC Motor Modeling in Electromechanical Systems

Description

Transcript

Tags