Electrical Transformers and Rotating Machines
4th Edition
ISBN: 9781305494817
Author: Stephen L. Herman
Publisher: Cengage Learning
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Textbook Question
Chapter 3, Problem 10RQ
An inductor has an inductive reactance of 250 Ω when connected to a 60-Hz line. What will be the inductive reactance if the inductor is connected to a 400-Hz line?
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QUESTION 1
Consider the equations of motion of an armature controlled DC motor given by
Jw(t)+bw (t) Ki (t)
=
di (l)
L + Ri (t) + Kw (t)
=
-Tfriction - Tload (t)
Vin (t),
dt
where vin(t) is the input voltage and Tload(t) is the input load torque. The outputs are the armature current i(t) and the motor speed w(t).
When the input voltage and torque are constant, the transient response of this system has the form of the homogeneous solution.
What is the time constant of the transient response? Find the characteristic roots of this system first. Keep 3 significant figures, and do not include units. Use scientific
notation.
The constants are given below:
Nominal voltage, Vin(t) = 17 V
Moment of inertia of the rotor, J = 0.02 kg-m²
Motor viscous friction constant, b = 0.009 N-m-s
Friction torque, Tfric = 0.003 N-m
Armature inductance, L = 0.1 H
Armature resistance, R = 0.35
Back emf constant and motor torque constant, K = 0.21 N-m/A.
**Recall that the time constant (or relaxation time) is…
The following EOMS describe the behavior of an electric motor attached to a torsional spring and damper:
[Li + Rz + vb = Vin(t)
IÖ = k0b0 + T
Importantly, the equations are coupled by the fact that 7 = k₁z and v₂ = k₂0.
Here, L, R, kv, I, k, b, and kt are constants. z is the current in the motor, and is the angular displacement of the
motor (clearly then, is the motor's angular velocity). Finally, the input to the system is vin (t).
Your tasks:
A Substituting in the coupling terms (7 = k₁² and v = k„0) into the EOMs, write the state space form of the
system x = Ax + Bu, assuming your states are 2₁=2, 22 = 0, and x3 = 0. Clearly identify your matrices.
B Assume that the desired outputs of the system (to be measured) are the current z, the angle and the input
Vin all as separate elements in the vector y. Write the output equation y = Cx + Du. Hint: D will NOT be
zero in this case. Clearly identify your matrices
Assume
represen
angular velocity and 9(s) to
The following EOMS describe the behavior of an electric motor attached to a torsional spring and damper:
[Li + Rz + vb = Vin(t)
IÖ= k0b0 + T
Importantly, the equations are coupled by the fact that 7 = k₁z and v₂ = k₂0.
Here, L, R, kv, I, k, b, and kt are constants. z is the current in the motor, and is the angular displacement of the
motor (clearly then, is the motor's angular velocity). Finally, the input to the system is vin (t).
Your tasks:
A Substituting in the coupling terms (7 = ktz and v k0) into the EOMs, write the state space form of the
system x = Ax + Bu, assuming your states are x₁ = z, x₂ = 0, and x3 = 0. Clearly identify your matrices.
B Assume that the desired outputs of the system (to be measured) are the current z, the angle and the input
Vin all as separate elements in the vector y. Write the output equation y = Cx + Du. Hint: D will NOT be
zero in this case. Clearly identify your matrices
C Assume there is no stiffness to the spring (k = 0). Using w(t) to represent…
Chapter 3 Solutions
Electrical Transformers and Rotating Machines
Ch. 3 - How many degrees are the current and voltage out...Ch. 3 - How many degrees are the current and voltage out...Ch. 3 - To what is inductive reactance proportional?Ch. 3 - Four inductors, each having an inductance of 0.6...Ch. 3 - Three inductors are connected in parallel....Ch. 3 - Prob. 6RQCh. 3 - An inductor is connected to a 240-V, 1000-Hz line....Ch. 3 - Prob. 8RQCh. 3 - Prob. 9RQCh. 3 - An inductor has an inductive reactance of 250 Ω...
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