Question

Prove that Z sin(θ + ϕ) = 0 is an equation of a straight line in the R–X plane. What is the slope of this line? Recall that this equation is obtained by dividing the torque equation by I . Thus, the straight-line characteristic is not valid for I = 0. If the spring constant is not negligible, show that the performance equation describes a circle through the origin, with the diameter of the circle depending upon the magnitude of the voltage [3]. Even for rather small values of the voltage, the diameter of the circle is so large that a straight-line approximation is still valid.

Answer 1

Z sin(θ + φ) = 0 is a straight line in the R-X plane with either a slope of 0 or undefined. The exact slope depends on specific angle values. When considering a non-negligible spring constant and voltage source, the system follows a circular performance equation.

Step-by-step explanation:

The equation Z sin(θ + φ) = 0 represents an equation of a straight line in the R–X plane when Z represents a fixed nonzero constant, and θ and φ represent angles. This equation suggests that for the line to exist, either Z = 0, which is not the case since Z is constant and nonzero, or sin(θ + φ) = 0, which occurs when θ + φ is an integer multiple of π, implying that the line is horizontal or vertical depending on the values of θ and φ. Thus, this line has a slope of 0 if θ + φ = nπ where n is an even integer, and undefined if n is an odd integer.

When considering a system with a non-negligible spring constant, the equation describing the system's behavior in the presence of a voltage source is shaped by the interplay between magnetic and elastic forces, leading to a circular path rather than a straight line. The performance equation, therefore, describes a circle through the origin, with the diameter influenced by the voltage's magnitude. However, for small voltage values, the circle's diameter is so large that a straight-line approximation near the origin is still valid.