Designing Solenoids: From Field to Form

1. From a Single Loop to a Thick Solenoid

The magnetic field of any coil, no matter how complex, can be understood by starting with the field of a single current-carrying loop and integrating.

The Field of a Single Loop

Recall the Biot-Savart law, which gives the magnetic field from an infinitesimal current element. For a circular loop of radius $r$ carrying current $I$, the magnetic field at a point $z$ along its central axis is:

$B_z(z)=\frac{{\mu }_{0}I{r}^2}{2(r^2+z^2{)}^{3/2}}$

The Field of a Thin Solenoid

A thin solenoid is a collection of such loops stacked side-by-side. To find the field, we can integrate the contribution from each loop along the length of the solenoid. Let the solenoid have $n$ turns per unit length, a length $L$, and radius $R$.

As the image shows, this integration leads to the well-known formula for the on-axis field of a finite, thin solenoid. At the geometric center of the solenoid, the formula simplifies to:

$B_z(\text{center})=\frac{{\mu }_{0}NI}{\sqrt{L^2+(2R{)}^2}}=\frac{{\mu }_0(nL)I}{\sqrt{L^2+4R^2}}$
where $N=nL$ is the total number of turns.

The Field of a Thick Solenoid

A real magnet coil has a significant thickness. We can model a thick solenoid (a multi-layer coil with a rectangular cross-section) by treating it as a nested set of thin solenoids. We find the total field by integrating the thin solenoid formula over the radius, from the inner radius $a_1$ to the outer radius $a_2$.

Let's assume a uniform current density $j$ throughout the conductor cross-section. The term $nI$ in the thin solenoid formula represents the total current per unit length of the solenoid. For an elemental thin solenoid of thickness $dr$ at radius $r$, this is equivalent to $j\cdot dr$.

So, the contribution to the central field from a thin shell at radius $r$ is:
$d{B}_z=\frac{{\mu }_0(j\cdot dr)L}{\sqrt{L^2+4r^2}}$

Integrating this from $a_1$ to $a_2$ gives the total central field $B_0$:
$B_0={\int }_{a_1}^{a_2}\frac{{\mu }_{0}jL}{\sqrt{L^2+4r^2}}dr=\frac{{\mu }_{0}jL}{2}{\int }_{a_1}^{a_2}\frac{dr}{\sqrt{(L/2{)}^2+r^2}}$
This is a standard integral, yielding:
$B_0=\frac{{\mu }_{0}jL}{2}{\left[\ln \left(r+\sqrt{r^2+(L/2{)}^2}\right)\right]}_{a_1}^{a_2}$
Evaluating at the limits gives the final expression for the central field of a thick solenoid:
$B_0=\frac{{\mu }_{0}jL}{2}\ln \left(\frac{a_2+\sqrt{a_2^2+(L/2{)}^2}}{a_1+\sqrt{a_1^2+(L/2{)}^2}}\right)$

This is a cornerstone formula for magnet design.

2. The Engineer's Design Equation: The Fabry Factor

The formula we just derived is correct, but it's often more practical to express it in terms of dimensionless geometric ratios. This separates the problem into a "scale" part (set by the bore size and current density) and a "shape" part (set by the coil's proportions).

Let's define two dimensionless parameters:

• Winding Ratio ($\alpha$): The ratio of the outer to the inner radius.
  $\alpha =\frac{a_2}{a_1}$
• Aspect Ratio ($\beta$): The ratio of the coil's half-length to its inner radius.
  $\beta =\frac{L/2}{a_1}$

Substituting $a_2=\alpha a_1$ and $L/2=\beta a_1$ into our formula for $B_0$ and simplifying, we get:
$B_0={\mu }_{0}j{a}_1\left[\beta \ln \left(\frac{\alpha +\sqrt{{\alpha }^2+{\beta }^2}}{1+\sqrt{1+{\beta }^2}}\right)\right]$

This is the form we'll use for design. We can group the terms:

$B_0={\mu }_{0}j{a}_{1}F(\alpha ,\beta )$

where $F(\alpha ,\beta )$ is the dimensionless Fabry factor (sometimes called the geometry factor), which depends only on the shape of the coil:

$F(\alpha ,\beta )=\beta \ln \left(\frac{\alpha +\sqrt{{\alpha }^2+{\beta }^2}}{1+\sqrt{1+{\beta }^2}}\right)$

This elegant equation tells us that for a given bore radius ($a_1$), the central field is directly proportional to the current density ($j$) and this geometric efficiency factor $F(\alpha ,\beta )$.

3. Applying the Formula for Design

Now we can address our learning outcome: designing a coil for a target field and bore size. Let's say we need to generate a peak field $B_{target}$ in a bore of radius $a_1$. Our design equation is:

$B_{target}={\mu }_{0}j{a}_{1}F(\alpha ,\beta )$

We are given $B_{target}$ and $a_1$. This leaves us with three variables to choose:

1. Current Density ($j$): How hard we drive the conductor. This is limited by thermal constraints (Joule heating, which we will study in Module 7) and mechanical strength (Lorentz forces, which we will address later in this module).
2. Winding Ratio ($\alpha$): How thick the winding is. A larger $\alpha$ means more conductor, which generates more field but also increases cost, size, and mechanical stress.
3. Aspect Ratio ($\beta$): How long and skinny versus short and fat the coil is.

The design process is an exercise in managing these trade-offs. To help, it's useful to understand how the Fabry factor $F(\alpha ,\beta )$ behaves. A larger $F$ means you get more field for your current and size—it represents a more efficient geometry.

The reading below explores this concept, though it uses a related factor G which is tied to power dissipation in DC magnets. The underlying principle of a geometric factor is the same.

The key insight from the paper is that there is an optimal geometry. For the power-based factor $G$, the maximum is around $\alpha \approx 3$ and $\beta \approx 2$. A similar optimization exists for our current-density-based factor $F(\alpha ,\beta )$. For a fixed volume of conductor, there is a shape that gives the maximum central field. In practice, other constraints (like field homogeneity or mechanical stress) often lead designers to choose geometries that are slightly different from this "Fabry optimum".

4. A Practical Design Example

Let's walk through a basic design. Your goal is to design a multi-turn pulsed solenoid that meets the following specifications:

• Target Central Field ($B_0$): 25 T
• Bore Diameter: 50 mm (so, inner radius $a_1=25$ mm = 0.025 m)

We will use the design parameters from a conceptual magnet design for this exercise. This will help you see how real-world numbers fit into the analytical formula.

Let's use our formula to check their result.

1. Calculate Dimensionless Ratios:

• Inner radius $a_1=0.025$ m
• Outer radius $a_2=0.036$ m
• Length $L=0.600$ m

$\alpha =\frac{a_2}{a_1}=\frac{0.036}{0.025}=1.44$
$\beta =\frac{L/2}{a_1}=\frac{0.300}{0.025}=12$
This is a very long, thin coil ($\beta$ is large, $\alpha$ is small).

2. Calculate the Fabry Factor $F(\alpha ,\beta )$:
$F(1.44,12)=12\times \ln \left(\frac{1.44+\sqrt{1.44^2+12^2}}{1+\sqrt{1^2+12^2}}\right)$
$F(1.44,12)=12\times \ln \left(\frac{1.44+\sqrt{2.07+144}}{1+\sqrt{145}}\right)=12\times \ln \left(\frac{1.44+12.086}{1+12.042}\right)$
$F(1.44,12)=12\times \ln \left(\frac{13.526}{13.042}\right)=12\times \ln (1.037)\approx 12\times 0.0363=0.436$

3. Calculate the Central Field $B_0$:
Let's use the average current density $j\approx 5.0\times 10^9{\text{A/m}}^2$.
$B_0={\mu }_{0}j{a}_{1}F(\alpha ,\beta )$
$B_0=(4\pi \times 10^{-7}\text{T}\cdot \text{m/A})\times (5.0\times 10^9{\text{A/m}}^2)\times (0.025\text{m})\times 0.436$
$B_0\approx 27.4\text{T}$

Our simplified calculation gives a result in the same ballpark as the 25 T target from the thesis. The discrepancy arises because the actual magnet in the thesis is not a simple solenoid with uniform current density; it's a more complex, force-optimized structure with varying pitch angles, which our simple model doesn't capture. Nonetheless, this demonstrates that the analytical formula provides a solid first approximation for the design.

Conclusion

In this lesson, we established the fundamental analytical tool for solenoid design. By starting from the Biot-Savart law and integrating, we derived the formula for the central magnetic field of a thick solenoid and recast it into a practical design equation using dimensionless geometric factors.

Key Takeaways:

• The central magnetic field $B_0$ of a thick solenoid is given by $B_0={\mu }_{0}j{a}_{1}F(\alpha ,\beta )$.
• The design process involves choosing the current density $j$ and the coil's shape, defined by the winding ratio $\alpha =a_2/a_1$ and the aspect ratio $\beta =(L/2)/a_1$.
• The Fabry factor $F(\alpha ,\beta )$ quantifies the geometric efficiency of the coil shape in producing a central field.
• This analytical formula provides an excellent first-order approximation for designing a coil to meet a specified field and bore size, forming the basis for more detailed numerical optimization.

Next Lesson Preview:
Knowing the field our coil will produce is only half the story. The next crucial question is: will the coil survive? In the next lesson, we will begin our study of the immense forces at play. We will calculate the axial and radial magnetic field profiles throughout the coil, which is the first step toward understanding the Lorentz force distribution that threatens to tear the magnet apart.