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The transfer function of the motor with the PID controller is given by: K G(s) JR Sp+ K₂K (K₂ +. T+JR² K₁ S²+ +Kas) In this system, there are various parameters of the motor denoted by Kt, Ke, R, L, J. and b. The controller has three gains: proportional gain (Kp), integral gain (Ki), and derivative gain (Kd). The input voltage V(t) has a Laplace transform V(s), which is a function of the complex frequency variable s. The transfer function of the motor, when used with the PID controller, relates V(s) to the Laplace transform of the output angular velocity omega(s). To express this transfer function as a rational function of s, we apply the Laplace transform to the differential equation of the motor with the PID controller. This transfer function can then be analyzed to determine system stability and performance, and to design the PID controller gains to meet specific performance requirements.

The transfer function of the motor with the PID controller is given by: K G(s) JR Sp+ K₂K (K₂ +. T+JR² K₁ S²+ +Kas) In this system, there are various parameters of the motor denoted by Kt, Ke, R, L, J. and b. The controller has three gains: proportional gain (Kp), integral gain (Ki), and derivative gain (Kd). The input voltage V(t) has a Laplace transform V(s), which is a function of the complex frequency variable s. The transfer function of the motor, when used with the PID controller, relates V(s) to the Laplace transform of the output angular velocity omega(s). To express this transfer function as a rational function of s, we apply the Laplace transform to the differential equation of the motor with the PID controller. This transfer function can then be analyzed to determine system stability and performance, and to design the PID controller gains to meet specific performance requirements.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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Q8 The dynamic equation of a DC motor can be expressed as: dw(t) + bw(t) = K₁i(t) dt where omega(t) is the angular velocity of the motor, J is the moment of inertia of the rotor, b is the damping coefficient of the motor, Kt is the torque constant of the motor, and i(t) is the current flowing through the motor. Now using Kirchhoff's voltage law to the circuit of the motor, we can obtain the follow the equation below: di(t) V(t) = Ri(t) + L + K₂w(t) dt where V(t) is the voltage applied to the motor, R is the resistance of the motor, L is the inductance of the motor, and Ke is the back-EMF constant of the motor. By substituting Kt i(t) from the mechanical equation into the electrical equation, we can obtain a single equation that relates the input voltage V(t) to the output angular velocity omega(t): V(t) - Kew (t) R dw (t) + bw(t) = K₂( dt dw (t) dt Simplifying this equation, we get: - + bw(t) = K / V (t) - R KtKe -w (t) R This equation can be written in the standard form of a second-order differential equation: d² w(t) bdw(t), K+Ke + w (t) + dt² J dt JR Kt -V(t) JR Explanation: The equation governing the behavior of a DC motor establishes a connection between the input voltage and the output angular velocity, and is a second-order differential equation. The equation is obtained by applying Kirchhoff's voltage law to the motor's electrical circuit and taking into consideration its mechanical properties. Writing the equation in the standard form of a second-order differential equation facilitates the analysis and control of the motor. Applying the Laplace transform to the differential equation and simplifying the expression leads to the transfer function of the motor with the PID controller. This transfer function mathematically links the Laplace transform of the input voltage to the Laplace transform of the output angular velocity, making it a valuable tool for designing and evaluating control systems for the motor. To regulate the motor's output, we can employ a PID controller. The transfer function of the PID controller is given by: K₁ G(s) = K₂ + + Kas S where Kp, Ki, and Kd are the proportional, integral, and derivative gains of the controller, respectively. The transfer function of the motor with the PID controller can be obtained by substituting s with frac{d}{dt} in the above equation and multiplying both sides by s: KtKe s²w (t) + sbw (t) + -w(t) = K₂V (t) R sb ⇒ (s² + b + Keke) w(t) = KV (t) JR The transfer function of the motor with the PID controller is then given by: G(s) K₂K₂ + S² + We can simplify this expression by multiplying the numerator and denominator by JRs²: G(s) = Kt Kt + sbJR JRs² (KJRS + KJR + KaJRs³) G(s) K₁ JR sb, KK₂ + J JR -(K₂ + + Kas) K₁ S Taking the Laplace transform of the input voltage V Explanation: The transfer function of the motor with the PID controller is given by: Kt JR S²+ K₁ sb KK K₂ + S + J JR + Kas) In this system, there are various parameters of the motor denoted by Kt, Ke, R, L, J, and b. The controller has three gains: proportional gain (Kp), integral gain (Ki), and derivative gain (Kd). The input voltage V(t) has a Laplace transform V(s), which is a function of the complex frequency variable s. The transfer function of the motor, when used with the PID controller, relates V(s) to the Laplace transform of the output angular velocity omega(s). To express this transfer function as a rational function of s, we apply the Laplace transform to the differential equation of the motor with the PID controller. This transfer function can then be analyzed to determine system stability and performance, and to design the PID controller gains to meet specific performance requirements.
Transcribed Image Text:Q8 The dynamic equation of a DC motor can be expressed as: dw(t) + bw(t) = K₁i(t) dt where omega(t) is the angular velocity of the motor, J is the moment of inertia of the rotor, b is the damping coefficient of the motor, Kt is the torque constant of the motor, and i(t) is the current flowing through the motor. Now using Kirchhoff's voltage law to the circuit of the motor, we can obtain the follow the equation below: di(t) V(t) = Ri(t) + L + K₂w(t) dt where V(t) is the voltage applied to the motor, R is the resistance of the motor, L is the inductance of the motor, and Ke is the back-EMF constant of the motor. By substituting Kt i(t) from the mechanical equation into the electrical equation, we can obtain a single equation that relates the input voltage V(t) to the output angular velocity omega(t): V(t) - Kew (t) R dw (t) + bw(t) = K₂( dt dw (t) dt Simplifying this equation, we get: - + bw(t) = K / V (t) - R KtKe -w (t) R This equation can be written in the standard form of a second-order differential equation: d² w(t) bdw(t), K+Ke + w (t) + dt² J dt JR Kt -V(t) JR Explanation: The equation governing the behavior of a DC motor establishes a connection between the input voltage and the output angular velocity, and is a second-order differential equation. The equation is obtained by applying Kirchhoff's voltage law to the motor's electrical circuit and taking into consideration its mechanical properties. Writing the equation in the standard form of a second-order differential equation facilitates the analysis and control of the motor. Applying the Laplace transform to the differential equation and simplifying the expression leads to the transfer function of the motor with the PID controller. This transfer function mathematically links the Laplace transform of the input voltage to the Laplace transform of the output angular velocity, making it a valuable tool for designing and evaluating control systems for the motor. To regulate the motor's output, we can employ a PID controller. The transfer function of the PID controller is given by: K₁ G(s) = K₂ + + Kas S where Kp, Ki, and Kd are the proportional, integral, and derivative gains of the controller, respectively. The transfer function of the motor with the PID controller can be obtained by substituting s with frac{d}{dt} in the above equation and multiplying both sides by s: KtKe s²w (t) + sbw (t) + -w(t) = K₂V (t) R sb ⇒ (s² + b + Keke) w(t) = KV (t) JR The transfer function of the motor with the PID controller is then given by: G(s) K₂K₂ + S² + We can simplify this expression by multiplying the numerator and denominator by JRs²: G(s) = Kt Kt + sbJR JRs² (KJRS + KJR + KaJRs³) G(s) K₁ JR sb, KK₂ + J JR -(K₂ + + Kas) K₁ S Taking the Laplace transform of the input voltage V Explanation: The transfer function of the motor with the PID controller is given by: Kt JR S²+ K₁ sb KK K₂ + S + J JR + Kas) In this system, there are various parameters of the motor denoted by Kt, Ke, R, L, J, and b. The controller has three gains: proportional gain (Kp), integral gain (Ki), and derivative gain (Kd). The input voltage V(t) has a Laplace transform V(s), which is a function of the complex frequency variable s. The transfer function of the motor, when used with the PID controller, relates V(s) to the Laplace transform of the output angular velocity omega(s). To express this transfer function as a rational function of s, we apply the Laplace transform to the differential equation of the motor with the PID controller. This transfer function can then be analyzed to determine system stability and performance, and to design the PID controller gains to meet specific performance requirements.
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