Question

L(x)= 1−x/d/L 0 For a particular solenoid coil, the constant d=7.8 mm and the inductance at zero stroke is L 0 =0.006H. Note that inductance L(x) increases with stroke x as the armature moves toward the center of the coil.

Develop a linearized approximation for inductance L(x) about a nominal stroke x =1 mm.

Answer 1

To develop a linearized approximation for the inductance L(x) about a nominal stroke x =1 mm in the solenoid coil, we can use the concept of Taylor series expansion. The linearized approximation formula is L(x) = L0 + ((dL)/(dx))|(x=1mm) * (x - 1mm).

Step-by-step explanation:

To develop a linearized approximation for the inductance L(x) about a nominal stroke x =1 mm, we can use the concept of Taylor series expansion. The linearized approximation for L(x) can be expressed as:

L(x) = L0 + \frac{(dL)}{(dx)}|x=1mm * (x - 1mm)

In this case, d = 7.8 mm is the constant and L0 = 0.006 H is the inductance at zero stroke.

Answer 2

A linearized approximation for the inductance L(x) about a nominal stroke x = 1mm can be found by using the provided formula, substituting the known values for L0 and d, and calculating the derivative to determine the slope at x = 1 mm.

Step-by-step explanation:

To develop a linearized approximation for the inductance L(x) of the given solenoid coil about the nominal stroke x = 1 mm, we can start by considering the provided inductance formula L(x) = 1 - (x / d) * L0. Here, L0 represents the inductance at zero stroke, x is the stroke, and d is a constant.

Using the given values, L0 = 0.006 H and d = 7.8 mm, we can substitute these into the formula to get the inductance at x = 1 mm. Then, to linearize it, we take the derivative of L(x) with respect to x and evaluate it at x = 1 mm, which will give us the slope of L(x) at this point. Multiplying the derivative (slope) by (x - 1) and adding it to L(1) provides us with the linear approximation around x = 1 mm.

Calculation Steps

1. Substitute L0 and d into the formula to get L(1 mm).
2. Calculate the derivative dL/dx at x = 1 mm.
3. Use the derivative to find the linear approximation around x = 1 mm.

The linearized equation will approximate the behavior of the solenoid's inductance very close to x = 1 mm.