Inductance Calculation for Coils

1. The Fundamental Definition of Inductance

From Faraday's Law of Induction, we know that a changing magnetic flux ${\Phi }_B$ through a coil with $N$ turns induces an electromotive force (EMF):
$\mathcal{E}=-N\frac{d{\Phi }_B}{dt}$
For a single coil, the magnetic flux it produces is directly proportional to the current $I$ flowing through it. We can therefore define a constant of proportionality, the self-inductance $L$, such that the total flux linkage $N{\Phi }_B$ is given by:
$N{\Phi }_B=LI$
Substituting this into Faraday's law gives the familiar circuit equation for an inductor:
$\mathcal{E}=-L\frac{dI}{dt}$
From this, we arrive at the primary definition we will use for calculating inductance:
$L=\frac{N{\Phi }_B}{I}$
Inductance is the total magnetic flux linkage per unit of current. It is a purely geometric property (and depends on the material's permeability, $\mu$).

Similarly, if the flux from a coil 1 (with $N_1$ turns carrying current $I_1$) links a second coil 2 (with $N_2$ turns), we define the mutual inductance $M_{21}$ as:
$M_{21}=\frac{N_2{\Phi }_{21}}{I_1}$
where ${\Phi }_{21}$ is the flux through a single turn of coil 2 due to the current in coil 1. A remarkable result from electromagnetic theory is the reciprocity theorem, which states that $M_{21}=M_{12}$.

To start with a clear conceptual picture and a simple derivation, watch the following video.

2. Self-Inductance of a Long Solenoid

As you just saw in the video, the long solenoid is the archetypal example for calculating self-inductance. Let's formalize the steps, as this pattern of calculation is one we will reuse.

1. Find the Magnetic Field $B$: For a long solenoid (length $l$, total turns $N$, current $I$), the magnetic field inside is uniform and given by Ampere's Law:
   $B={\mu }_{0}nI={\mu }_0\frac{N}{l}I$

2. Calculate Flux Through a Single Turn ${\Phi }_B$: The flux is the field multiplied by the cross-sectional area $A$.
   ${\Phi }_B=B\cdot A=\left({\mu }_0\frac{N}{l}I\right)A$

3. Calculate Total Flux Linkage $N{\Phi }_B$: Multiply the single-turn flux by the total number of turns $N$.
   $N{\Phi }_B=N\left({\mu }_0\frac{N}{l}AI\right)=\frac{{\mu }_0{N}^{2}A}{l}I$

4. Calculate Inductance $L$: Divide the total flux linkage by the current $I$.
   $L=\frac{N{\Phi }_B}{I}=\frac{{\mu }_0{N}^{2}A}{l}$
   This formula is a cornerstone, but it is an approximation that assumes an infinitely long solenoid. For the precise work of magnet design, we need more accurate formulae for realistic, finite geometries.

3. Inductance of Practical Coil Geometries

The calculation of inductance for finite coils is a classic and complex problem in electromagnetism. Fortunately, this work was extensively catalogued in the early 20th century. We will refer to a landmark publication from the National Bureau of Standards (now NIST) that collects these invaluable formulae.

a) Single-Layer Solenoid of Finite Length

For a real-world single-layer solenoid of finite length $b$ and radius $a$, the simple formula is no longer sufficient.

• Rayleigh and Niven's Formula (Short Coils): For coils where the length is small compared to the radius ($b\ll a$), a highly accurate series expansion is given by Formula 54 in the NIST bulletin (page 39).
  $L_s=4\pi a{n}^2\left[{\log }_e\frac{8a}{b}-\frac{1}{2}+\frac{b^2}{8a^2}\left({\log }_e\frac{8a}{b}+\frac{1}{4}\right)-\dots \right]$
  Here, $n$ is the total number of turns. Note the subscript $s$, which denotes a "current sheet" approximation, where the discrete wires are modeled as a continuous sheet of current.

• Lorenz's Formula (Any Length): For a solenoid of any length, Lorenz provided an exact formula (Formula 58 on page 40 of the bulletin) in terms of complete elliptic integrals, which you encountered in our first lesson on off-axis fields. This is computationally more intensive but exact for the current sheet model.

• Correction for Wires: Real coils are wound with discrete wires. The bulletin provides a crucial correction (Formula 59, page 42) to adjust the current-sheet value $L_s$ to the actual inductance $L$:
  $L=L_s-4\pi an(A+B)$
  The terms $A$ and $B$ are correction factors that depend on the wire's diameter, the pitch of the winding, and the number of turns. Tables for these are provided in the bulletin's appendix. This level of detail is essential for accurate magnet design.

b) Circular Coil of Rectangular Cross-Section

Most high-field pulsed magnets are not single-layer solenoids but are thick, multi-layer coils with a rectangular cross-section of breadth $b$ and depth $c$.

• Stefan's Formula: A very useful and accurate formula for such coils is given by Stefan (Formula 69, page 47 of the bulletin):
  $L_u=4\pi a{n}^2\left({\log }_e\frac{8a}{d}-y_1+y_2\frac{c^2}{a^2}\right)$
  Here, $d=\sqrt{b^2+c^2}$ is the diagonal of the coil's cross-section, and $y_1,y_2$ are coefficients that depend on the aspect ratio $b/c$ (given in Table VI of the bulletin). The subscript $u$ denotes that this formula assumes a uniform current distribution over the rectangular cross-section.

• Maxwell's Correction: Again, to account for the use of round, insulated wires instead of a uniform current block, a correction must be applied (Formula 72, page 49).

4. A General Approach: Inductance from Energy

While the collection of formulae in the NIST bulletin is powerful, it's useful to understand a more modern and general framework for deriving inductance from first principles. This method connects inductance to magnetic energy and the magnetic vector potential, $\vec{A}$.

The magnetic energy $U$ stored in a system is:
$U=\frac{1}{2}{\int }_V\vec{J}\cdot \vec{A}dV=\frac{1}{2}L{I}^2$
By expressing $\vec{A}$ in terms of $\vec{J}$ and solving this integral, we can find $L$ for very general geometries.

The key result from this derivation is a general formula (Eq. A.19) for the inductance of a coil formed by extruding a planar curve $C$ to a height $h$. This powerful formula can describe simple solenoids, rectangular coils, and even more complex shapes by defining the appropriate curve $C$.

5. Mutual Inductance of Coupled Systems

When two or more coils are in proximity, the magnetic field from one coil links the others, creating mutual inductance. This is crucial for designing systems with primary and secondary coils (like transformers) or for understanding parasitic coupling between different parts of a magnet system.

The calculation of mutual inductance follows a similar pattern to self-inductance.

1. Calculate the magnetic field ${\vec{B}}_1$ produced by current $I_1$ in coil 1.
2. Integrate ${\vec{B}}_1$ over the area of a single turn of coil 2 to find the flux ${\Phi }_{21}$.
3. Multiply by the number of turns in coil 2 to get the total flux linkage $N_2{\Phi }_{21}$.
4. Divide by the current $I_1$ to get $M_{21}$.

The NIST bulletin (LINK) provides an extensive set of formulae for mutual inductance in its initial sections.

• Two Coaxial Circles: The foundational case is finding the mutual inductance between two circular filaments. Maxwell's formula (Formula 1, page 6) gives an exact solution in terms of elliptic integrals. Series expansions (Formulas 7, 9, 10, 11) provide excellent approximations for circles that are close together.

• Two Coaxial Coils: For thick coils, one typically calculates the mutual inductance $M_0$ between the central turns of the coils and then adds correction terms. Rowland's formula (Formula 14, page 12) is a classic example that uses derivatives of $M_0$ with respect to the coil dimensions to form a Taylor-series-like correction.

Partial Inductance

The concept of "partial inductance," which you encountered in the MIT thesis (LINK, Sec. A.3), is a powerful tool for complex systems. The idea is to break down a conductor or a system of conductors into segments. The total inductance is then the sum of all the partial self-inductances of the segments and all the partial mutual-inductances between every pair of segments.
$L_{\text{total}}={\sum }_n{L}_n+{\sum }_{n\ne m}{M}_{nm}$
This is the fundamental principle behind many numerical inductance calculation tools, including the Finite Element Method (FEM) software you are interested in.

Conclusion

In this lesson, we have defined self and mutual inductance and explored how to calculate them for geometries relevant to magnet design.

Key Takeaways:

• Self-inductance $L=N{\Phi }_B/I$ is the flux linkage per unit current in a coil. Mutual inductance $M_{21}=N_2{\Phi }_{21}/I_1$ is the flux linkage in one coil due to current in another.
• The calculation for a long solenoid ($L={\mu }_0{N}^{2}A/l$) provides a simple, foundational formula.
• For realistic, finite coils, we rely on more complex solutions, such as series expansions (e.g., Stefan's formula) or exact formulas involving elliptic integrals (e.g., Lorenz's formula).
• Corrections must be applied to these idealized "current sheet" formulas to account for the geometry of discrete wire windings.
• Inductance can be derived from first principles using magnetic energy and the vector potential, providing a general framework for any geometry.
• The concept of partial inductance is fundamental to numerical methods for calculating inductance in complex systems.

In our next lesson, we will explore the consequences of placing a bulk conductor inside a time-varying magnetic field. This will lead us to the concepts of eddy currents and the magnetic diffusion equation, which are absolutely essential for understanding the behavior of pulsed magnets and their sample environments.