Designing Nulling Coils for Field Gradient Correction

1. Describing Inhomogeneity: The Harmonic Expansion

To correct a field, we first need a precise mathematical language to describe its imperfections. In the previous lesson, we used a one-dimensional Taylor series. A more powerful and general approach, standard in magnet design, is the harmonic expansion.

In a source-free region, the magnetic field $\vec{B}(\vec{r})$ can be expressed as a sum over a basis of polynomial functions, ${\Pi }_{lm}(\vec{r})$, which are solutions to Maxwell's equations.

Here:

• ${\Pi }_{lm}(\vec{r})$ are the harmonic polynomials. They describe the spatial shape of the field components. For example, ${\Pi }_{00}$ represents a uniform vertical field, while ${\Pi }_{10}$ represents a linear vertical gradient along the z-axis ($\partial B_z/\partial z$).
• $G_{lm}$ are the generalized magnetic gradients. These coefficients represent the strength of each harmonic component in the total field. Our goal is to make the unwanted $G_{lm}$ coefficients as close to zero as possible.

The core idea of field correction is to use auxiliary coils to generate a field with the same harmonic shape (${\Pi }_{lm}$) as the unwanted gradient but with the opposite sign, such that the sum of the main field's gradient and the correction field's gradient is zero.

To get a concrete feel for these harmonic polynomials, we'll consult a paper describing the coil system for the n2EDM experiment. This resource provides the rigorous foundation we need.

This framework gives us our targets. If we can measure or simulate the field of our main magnet and decompose it into its harmonic components, we will get a list of unwanted gradient coefficients $G_{lm}$ that we need to cancel.

2. From Geometry to Gradients: The Coil Constant

Now, how do we design a coil that produces a specific harmonic, for instance, a linear gradient ${\Pi }_{10}$? The geometry of the coil windings determines the spectrum of harmonics it generates.

Consider a simple coil to generate a linear gradient along the z-axis, $\partial B_z/\partial z$. This is often called a $G_{10}$ coil.

From the n2EDM paper (resource LINK, Appendix D), these are geometries for coils designed to produce specific gradients. Left: A G10 coil for a linear z-gradient. It's an anti-Helmholtz configuration with two pairs of loops carrying opposite currents. Middle: A G20 coil for a quadratic z-gradient. Right: A G30 coil for a cubic z-gradient.

Let's focus on the G10 coil on the left. It consists of two pairs of square loops. The inner pair and outer pair have currents flowing in opposite directions. At the center ($z=0$), the vertical fields from the top and bottom halves cancel out, so $B_z(0)=0$. However, since the field from the top loops decreases with distance and the field from the bottom loops increases (becomes less negative) with distance, their derivatives add up, creating a strong, nearly linear gradient across the center.

For any given correction coil geometry, we can define a coil constant, $C_{lm}$, which quantifies how efficiently it generates a specific harmonic gradient per unit of current.

This constant depends only on the coil's geometry. It can be calculated by:

1. Using the Biot-Savart law to find the magnetic field $\vec{B}(\vec{r})$ produced by 1 Ampere of current in the coil.
2. Projecting this field onto the harmonic polynomials ${\Pi }_{lm}(\vec{r})$ to find the resulting gradient coefficients $G_{lm}$.

While the full calculation is a complex numerical integration, the principle is straightforward. For simple on-axis gradients ($m=0$), this is equivalent to calculating the field's spatial derivatives at the center.

To see the kind of formulas involved in step 1, we can refer to the thesis on Helmholtz coil design.

3. The Calculation: Nulling the Gradients

With the concepts of harmonic gradients and coil constants, we can now set up the calculation.

Imagine our main pulsed coil, when fired, produces the desired high field $B_0$ (the $G_{00}$ term) but also has a significant, unwanted linear z-gradient, $G_{10}^{\text{main}}$.

Problem: We have a $G_{10}$ correction coil (like the one pictured above) with a known coil constant $C_{10}$. What current, $I_{10}$, do we need to pass through it to null the main coil's gradient?

Solution:
The total gradient in the system is the sum of the gradient from the main coil and the gradient from the correction coil:

We know that $G_{10}^{\text{corr}}=C_{10}\cdot I_{10}$. To null the total gradient, we set $G_{10}^{\text{total}}=0$:

Solving for the required current gives our answer:

This is the fundamental calculation. The negative sign simply means the correction coil's current must generate a gradient that opposes the unwanted one.

4. The General Solution: The Coil Matrix

In a real system, the situation is more complex.

1. A coil designed to be a "pure" $G_{10}$ generator might also produce small, higher-order gradients ($G_{30}$, $G_{50}$, etc.).
2. The main magnet will have many unwanted gradients ($G_{10}$, $G_{20}$, $G_{11}$, etc.), not just one.

Therefore, a real compensation system uses an array of different correction coils. The effect of this array is captured by a coil constant matrix, often called a shimming matrix.

Let's say we want to correct $N$ different harmonic gradients using $M$ different correction coils. We can write the problem in matrix form:

Where:

• ${\vec{G}}_{\text{main}}$ is a vector of the $N$ unwanted gradients from the main magnet.
• $\vec{I}$ is a vector of the $M$ currents we want to find for our correction coils.
• $\mathbf{C}$ is the $N\times M$ coil constant matrix. The element $C_{ij}$ is the strength of the $i$-th harmonic gradient produced by a unit current in the $j$-th coil.

To null the gradients, we set ${\vec{G}}_{\text{total}}=0$ and solve for the currents $\vec{I}$. If the number of coils equals the number of gradients we want to control ($M=N$), the matrix $\mathbf{C}$ is square and we can solve by inverting it:

This matrix equation is the heart of a modern, automated field correction system. Before an experiment, the system performs a field map to determine ${\vec{G}}_{\text{main}}$. Since the coil matrix $\mathbf{C}$ is a fixed property of the hardware (determined once by simulation and measurement), the control computer can instantly calculate the required correction currents $\vec{I}$ and apply them.

The n2EDM paper provides a perfect description of this exact process in a real-world application.

Conclusion

In this lesson, we transitioned from the principles of field shaping to a concrete, calculable methodology. We have established a complete framework for designing and operating a compensation coil system.

Key Takeaways:

• Magnetic field inhomogeneity is best described by a harmonic expansion, where $G_{lm}$ coefficients quantify the strength of each unwanted gradient.
• Correction coils are designed with specific geometries to target specific harmonics. The relationship between the coil's current and the gradient it produces is its coil constant, $C_{lm}$.
• For a single unwanted gradient, the required correction current is simple to calculate: $I=-G^{\text{main}}/C$.
• For a realistic system with multiple gradients and multiple correction coils, the problem is solved using the coil matrix: $\vec{I}=-{\mathbf{C}}^{-1}\cdot {\vec{G}}_{\text{main}}$. This is the basis for automated shimming systems in modern magnet facilities.

Next Lesson Preview:
We have now covered the design of both the main field coil and the correction coils needed to ensure field quality. Our next lesson, "Explain the physical principles of single-turn coil field generation and the extreme mechanical stresses involved," will shift gears to an entirely different and more extreme method of generating pulsed fields, opening up the topic of ultra-high field generation.