Skin Depth and Current Distribution in Pulsed Operation

1. The Physics of the Skin Effect

The skin effect is another consequence of Faraday's Law of Induction, the same physics that drives eddy currents. When an alternating current flows through a conductor, it generates an alternating magnetic field that permeates the conductor itself.

1. This changing magnetic flux induces eddy currents that circulate within the conductor.
2. According to Lenz's Law, these eddy currents flow in a direction that opposes the change in the main current's flux.
3. In the center of the conductor, the eddy currents flow against the main current, reducing the net current flow there.
4. Near the surface of the conductor, the eddy currents loop back around and flow with the main current, reinforcing it.

The net result is a redistribution of the total current, with a higher current density near the surface and a lower current density in the core.

The strength of this effect is frequency-dependent. Higher frequencies mean a faster rate of change of magnetic flux ($d{\Phi }_B/dt$), which induces stronger eddy currents and pushes the main current more forcefully to the surface.

The following video provides an excellent visualization of this process and introduces the concept of skin depth.

2. Deriving the Skin Depth Formula

We can make this concept quantitative by solving the magnetic diffusion equation for a sinusoidal field. Let's consider a simple case: a semi-infinite conductor occupying the space $z>0$, with a sinusoidal magnetic field $\vec{H}(z,t)=\text{Re}[{\vec{H}}_0{e}^{i(kz-\omega t)}]$ propagating into it.

To find the relationship between the wave number $k$ and the frequency $\omega$, we will follow the rigorous derivation provided in David Tong's lecture notes on electromagnetism, which should be familiar from your advanced E&M coursework.

As you read, the key steps are:

1. Start with the magnetic diffusion equation we saw last lesson, but written in the frequency domain: ${\nabla }^2\vec{H}=i\omega \mu \sigma \vec{H}$.
2. For a plane wave varying only in $z$, this becomes a simple ordinary differential equation: $\frac{d^{2}H}{d{z}^2}=i\omega \mu \sigma H$.
3. The solution that decays as $z\to \infty$ is of the form $H(z)=H_{surface}{e}^{-kz}$, where $k^2=i\omega \mu \sigma$.
4. Solving for $k$ gives $k=\sqrt{i\omega \mu \sigma }=\sqrt{\frac{\omega \mu \sigma }{2}}(1+i)$.
5. The field inside the conductor is therefore $H(z,t)=H_{surface}{e}^{-z/\delta }{e}^{i(z/\delta -\omega t)}$, where the real part gives the physical field. The amplitude decays exponentially as $e^{-z/\delta }$.

The skin depth, $\delta$, is defined as the distance over which the field (and current) amplitude decays to $1/e$ (about 37%) of its surface value. From the derivation, we arrive at the formula:

$\delta =\sqrt{\frac{2}{\omega \mu \sigma }}=\sqrt{\frac{2\rho }{\omega \mu }}$

where:

• $\omega =2\pi f$ is the angular frequency of the current.
• $\mu$ is the magnetic permeability of the conductor (for copper, $\mu \approx {\mu }_0$).
• $\sigma$ is the electrical conductivity (and $\rho =1/\sigma$ is the resistivity).

This formula confirms our intuition: higher frequency $\omega$ or higher conductivity $\sigma$ leads to a smaller skin depth $\delta$.

3. Calculating Skin Depth for Pulsed Magnets

A current pulse is not a perfect sine wave, but we can approximate its "effective" frequency to estimate the skin depth. For a sinusoidal pulse of duration $T$ (e.g., 10 ms), the fundamental angular frequency is $\omega \approx \pi /T$.

Let's calculate the skin depth for a copper conductor, which is a common material for pulsed coils.

• Resistivity of copper at room temp ($\sim 300$ K): ${\rho }_{300K}\approx 1.72\times 10^{-8}\Omega \cdot \text{m}$
• Permeability: $\mu \approx {\mu }_0=4\pi \times 10^{-7}\text{H/m}$

Case 1: 10 ms pulse (typical "short pulse")

• Effective frequency: $f\approx 1/(2T)=1/(0.02s)=50$ Hz, so $\omega =2\pi f\approx 314$ rad/s.
• Skin depth:
  ${\delta }_{10ms,300K}=\sqrt{\frac{2\times (1.72\times 10^{-8}\Omega \cdot \text{m})}{(314\text{rad/s})(4\pi \times 10^{-7}\text{H/m})}}\approx 9.3\mathbf{mm}$

Case 2: 100 ms pulse (typical "long pulse")

• Effective frequency: $f\approx 1/(2T)=1/(0.2s)=5$ Hz, so $\omega \approx 31.4$ rad/s.
• Skin depth:
  ${\delta }_{100ms,300K}=\sqrt{\frac{2\times (1.72\times 10^{-8}\Omega \cdot \text{m})}{(31.4\text{rad/s})(4\pi \times 10^{-7}\text{H/m})}}\approx 29.4\mathbf{mm}$

These numbers are significant. The conductors used in pulsed magnets often have a thickness on the order of several millimeters to a centimeter. For a 10 ms pulse, the skin depth is comparable to the conductor size, meaning the current distribution will be noticeably non-uniform.

The Critical Effect of Temperature

Pulsed magnets are often pre-cooled with liquid nitrogen to 77 K to reduce their initial resistance. The resistivity of high-purity copper drops dramatically at this temperature. For OFHC (Oxygen-Free High-Conductivity) copper, the resistivity at 77 K can be roughly ${\rho }_{77K}\approx 0.2\times 10^{-8}\Omega \cdot \text{m}$.

Let's re-calculate for the 10 ms pulse at 77 K:
${\delta }_{10ms,77K}=\sqrt{\frac{2\times (0.2\times 10^{-8}\Omega \cdot \text{m})}{(314\text{rad/s})(4\pi \times 10^{-7}\text{H/m})}}\approx 3.2\mathbf{mm}$

Cooling the coil significantly reduces the skin depth, concentrating the current even more tightly at the surface.

4. Impact on Magnet Design and a More Rigorous View

The skin effect has three profound consequences for pulsed magnet design:

1. Increased Effective Resistance: By forcing the current into a smaller cross-sectional area ($\approx \text{perimeter}\times \delta$), the AC resistance of the conductor becomes much higher than its DC resistance. This leads to significantly more Joule heating ($P=I^{2}R$) than a simple DC calculation would predict.
2. Non-uniform Heating: Since the current density is highest at the surface, heating is also concentrated there. This creates temperature gradients within the conductor, which can lead to thermal stresses.
3. Non-uniform Forces: The Lorentz force density ($\vec{f}=\vec{J}\times \vec{B}$) is highest where the current density is highest. This means the mechanical load is not borne uniformly by the conductor but is concentrated on its surfaces, which must be strong enough to withstand it.

Our sinusoidal approximation is a powerful tool for estimation, but a real current pulse is a transient event, not a steady-state sine wave. For a more rigorous analysis, one must solve the time-dependent magnetic diffusion equation for the specific pulse shape.

This more advanced treatment confirms that field and current penetration is a diffusive process. A useful rule of thumb for the penetration depth $d$ of a field after a time $t$ has elapsed is $d(t)\sim \sqrt{t/(\mu \sigma )}$. For a pulse of duration $T$, setting $t=T$ gives a characteristic length scale that is functionally equivalent to the skin depth $\delta$ we derived, reinforcing the validity of our estimation.

Conclusion

Key Takeaways:

• The skin effect is the tendency of AC current to concentrate near the surface of a conductor, caused by opposing eddy currents induced in the conductor's core.
• The skin depth, $\delta$, is the characteristic distance over which the current density drops to $1/e$ of its surface value. It is given by $\delta =\sqrt{2\rho /(\omega \mu )}$.
• Skin depth depends inversely on the square root of frequency and conductivity. For pulsed magnets, this means shorter pulses and lower temperatures (higher conductivity) lead to a more pronounced skin effect.
• This effect is not a minor correction; it fundamentally alters the magnet's thermal and mechanical behavior by increasing effective resistance and concentrating heating and forces at the conductor's surface.

Next Lesson Preview:
Now that we have a better understanding of how current distributes itself within a magnet's windings, our next step is to calculate the immense forces this current generates. In the next lesson, we will derive the Maxwell stress tensor, a powerful mathematical tool that allows us to compute electromagnetic forces, such as the immense outward pressure that a high-field magnet exerts on itself.