Internal Inductance

Inductance is a fundamental property in electromagnetism, often introduced as a measure of a circuit's opposition to changes in current. While typically associated with coils and the magnetic flux they enclose, this view overlooks a crucial component: the magnetic field that exists inside the current-carrying conductors themselves. This internal field stores energy, giving rise to what is known as internal inductance. This article delves into this often-neglected concept, revealing it to be not just a minor correction but a key principle with far-reaching consequences. We will build a complete understanding by first exploring the core Principles and Mechanisms, dissecting how internal inductance arises from stored energy, its dependence on current distribution, and its behavior in AC circuits due to the skin effect. Following this, the Applications and Interdisciplinary Connections section will demonstrate its practical importance, from designing high-frequency electronics to controlling the stability of star-hot plasmas in fusion reactors.

Now that we have been introduced to the idea of internal inductance, let us embark on a journey to truly understand it. We will not be satisfied with mere definitions; we want to build an intuition, to see the world through the lens of this concept. Like any good journey of discovery, we will start with the simplest of things—a piece of wire—and find that it leads us to the very heart of a star confined in a magnetic bottle.

You have likely been taught that when a current $I$ flows through a circuit, it creates a magnetic field $\vec{B}$, and this field generates a magnetic flux $\Phi$. The self-inductance $L$ is then simply the constant of proportionality between the two: $\Phi = L I$. This is a fine and useful definition, but it hides a beautiful subtlety. Where, exactly, is this magnetic field?

A current running through a wire creates a magnetic field that loops around it. Some of this field exists in the space outside the wire, stretching out to infinity. But some of it must also exist inside the physical volume of the wire itself. After all, the current is what creates the field, so why wouldn't the field be present where the current is?

This simple observation allows us to split the total inductance into two distinct parts. We call the part arising from the magnetic field outside the wire the ​​external inductance​​, $L_{ext}$. And we call the part that comes from the magnetic field inside the conductor the ​​internal inductance​​, $L_{int}$. The total inductance, the one you measure with an instrument, is simply the sum of the two: $L = L_{ext} + L_{int}$.

Think of it like this: inductance is a measure of electrical inertia. It’s the resistance to a change in current. The external inductance is like the inertia you feel from pushing the air around you as you start to run. The internal inductance is like the inertia of your own limbs; you have to get your own arms and legs moving. To accelerate, you must overcome both. In most circuits, especially those with coils, the external part dominates because the coil is designed to maximize the flux in the space it encloses. But the internal part is never zero, and as we will see, it holds some fascinating secrets.

Defining inductance through flux is convenient, but a more fundamental and powerful way to think about it is through energy. Creating a magnetic field costs energy, and this energy is stored in the field itself. The total energy $W$ stored in an inductor is given by the famous relation $W = \frac{1}{2} L I^2$.

This gives us a brilliant way to isolate our quarry, the internal inductance. If we can calculate the magnetic energy stored only inside the volume of the conductor, $W_{int}$, then we can define the internal inductance as:

$W_{int} = \frac{1}{2} L_{int} I^2$

Let's try this. Imagine a long, straight, cylindrical wire of radius $R$ carrying a steady DC current $I$, distributed uniformly across its cross-section. What is the magnetic field inside the wire? Using Ampere's law, we find that the field at a distance $r$ from the center is not constant. It starts at zero at the very center, and grows linearly with radius: $B(r) \propto r$. This makes perfect sense: the farther out you go, the more current you have enclosed within your Amperian loop.

The energy density of a magnetic field is $u_m = \frac{B^2}{2\mu_0}$. Since $B \propto r$ inside the wire, the energy density must go like $u_m \propto r^2$. The energy is most densely packed near the surface of the wire and is zero at its center. To find the total internal energy per unit length, we must integrate this energy density over the volume of a slice of the wire. When we perform this calculation, a truly remarkable result appears. The internal inductance per unit length, which we'll call $L'_{int}$, is:

$L'_{int} = \frac{\mu_0}{8\pi}$

Look at this expression carefully. The internal inductance per unit length for a wire with uniform current depends on nothing but a fundamental constant of nature, the permeability of free space $\mu_0$! It does not depend on the wire's radius, its material (as long as it's non-magnetic), or the amount of current it carries. There is a universal, inherent "inductiveness" to a uniform current flowing in a cylinder. This is the kind of profound simplicity that physics often reveals.

We found a beautiful, simple result by assuming the current was uniformly distributed. But nature is rarely so neat. What if the current density is not uniform? What if it's more concentrated at the center, or perhaps pushed out towards the surface?

Let's imagine the current density $J$ follows a power law, $J(r) = C r^\alpha$, for some constant $\alpha \ge 0$.

• If $\alpha=0$, we have $J(r) = C$, which is our old friend, the uniform current distribution.
• If $\alpha > 0$, the current density is zero at the center and increases as we move outwards, becoming most concentrated near the surface. This describes a "hollow" current profile.

If we repeat our energy calculation for this generalized current profile, we find a new expression for the internal inductance per unit length:

$L'_{int} = \frac{\mu_0}{4\pi(\alpha+2)}$

Let's check this. If we plug in $\alpha=0$ for a uniform current, we get $L'_{int} = \frac{\mu_0}{8\pi}$, exactly what we had before. But now we can see what happens when the current profile changes. As we increase $\alpha$, making the current more hollow and pushing it toward the surface, the denominator $(\alpha+2)$ gets larger, and the internal inductance decreases.

This is a key insight! ​​By changing the shape of the current distribution, we change the internal inductance.​​ Pushing the current towards the outer edge of the conductor reduces the magnetic field in the bulk of the material. Less field inside means less stored energy, which means less internal inductance. This simple idea is the gateway to understanding how inductance behaves in the real world of alternating currents.

So far, we have only talked about steady, direct currents (DC). What happens when we switch to alternating currents (AC)? Something wonderful, and deeply intuitive, occurs.

An AC current is constantly changing. According to Faraday's law of induction, a changing magnetic flux induces an electric field. Inside the wire, the changing magnetic field of the AC current induces swirling eddy currents. Lenz's law tells us these eddy currents will flow in a direction that opposes the change that created them. In the center of the wire, these eddy currents flow against the main current, effectively canceling it out. Near the surface, they reinforce it.

The net result is that the current is no longer uniform. It is pushed away from the center and forced to flow in a thin layer near the conductor's surface. This phenomenon is called the ​​skin effect​​, and the characteristic thickness of this current-carrying layer is the ​​skin depth​​, $\delta$. The skin depth depends on the frequency $\omega$ of the current, the conductivity $\sigma$, and permeability $\mu$ of the material: $\delta = \sqrt{\frac{2}{\omega \mu \sigma}}$. As the frequency increases, the skin depth becomes smaller and smaller.

But wait! We just learned that pushing the current towards the surface reduces internal inductance. The skin effect does exactly this. This means that ​​the internal inductance of a wire is not a constant; it is frequency-dependent​​. As you increase the frequency, the current is squeezed into a thinner and thinner skin, and the internal inductance continuously decreases.

What happens in the extreme high-frequency limit, as $\omega \to \infty$? The skin depth $\delta$ shrinks to zero. The current flows only on the mathematical surface of the wire. In this case, the magnetic field inside the conductor material becomes zero everywhere. If there's no magnetic field, there's no stored magnetic energy. And if there's no stored internal energy, the internal inductance must be zero!

In this high-frequency limit, a fascinatingly simple relationship emerges. The impedance of the wire has a resistive part, $R$, due to ohmic heating, and a reactive part, $X = \omega L_{int}$, due to the inductance. It turns out that for the skin effect, these two are exactly equal: $R = \omega L_{int}$. This means that the energy dissipated as heat in one AC cycle is directly proportional to the peak magnetic energy stored inside the wire during that cycle. It is a beautiful balance, a dance between energy stored and energy lost, governed by the physics of magnetic diffusion.

From a simple wire, our journey now takes an audacious leap: to the inside of a tokamak, a device designed to harness nuclear fusion, the power source of the sun. A tokamak confines a donut-shaped cloud of plasma, hotter than the sun's core, using immensely powerful magnetic fields. A key part of this confinement is driving a massive electrical current—millions of amperes—through the plasma itself.

This plasma is a fluid conductor, and the distribution of its current is of paramount importance. Physicists characterize the shape of this current distribution using a dimensionless quantity called the ​​normalized internal inductance​​, denoted $l_i$. Just like our factor $\alpha$ before, $l_i$ is a "shape factor." It's a single number that tells physicists whether the plasma current is sharply peaked at the hot core or spread out more broadly.

• A high value of $l_i$ corresponds to a current profile that is sharply peaked at the center. A parabolic profile, for instance, gives an $l_i$ of about $0.92$.
• A low value of $l_i$ means the profile is broad or even hollow. A profile that is zero at the center and peaked halfway out might have an $l_i$ around $0.73$.

Why does this single number matter so much? Because the stability of the entire multi-billion dollar experiment depends on it. A plasma with a very peaked current profile (high $l_i$) is prone to violent instabilities, particularly the "kink" instability, which is exactly what it sounds like: the entire plasma column can rapidly buckle and twist like a firehose, crashing into the reactor walls in milliseconds and extinguishing the fusion reaction.

The value of $l_i$ is a direct input into the complex equations of magnetohydrodynamics (MHD) that govern the plasma's behavior. In fact, $l_i$ is deeply connected to other crucial stability parameters, like the magnetic shear—a measure of how the twist of the magnetic field lines changes with radius. By carefully controlling the heating and current drive systems, physicists can sculpt the current profile, changing $l_i$ in real-time to navigate a narrow path to stable operation.

And so, we have come full circle. The same fundamental principle—that the distribution of current determines the stored magnetic energy—that governs the behavior of a simple copper wire also holds the key to controlling a miniature star on Earth. The internal inductance is not just a minor correction to a circuit diagram; it is a profound expression of the geometry of electromagnetism, a concept whose consequences reach from our everyday electronics to the frontiers of clean energy.

Now that we have grappled with the principles of internal inductance—this idea that a conductor’s own magnetic lifeblood stores energy—we might be tempted to ask, “So what?” Is this just a subtle correction, a footnote for the meticulous engineer, or does it have a deeper significance? The answer, as is so often the case in physics, is that what begins as a detail in one corner of the world blossoms into a central principle in another. The journey of internal inductance takes us from the design of everyday electronic circuits to the monumental challenge of confining a star in a laboratory. It is a beautiful illustration of the unity and reach of physical law.

Let's begin in the tangible world of electrical engineering. When we first learn about inductance, we often imagine current flowing through infinitely thin wires, where all the magnetic field is outside the conductor. But real wires have thickness, and if current flows through them, a magnetic field must exist inside them. This internal field carries energy, and this energy gives rise to internal inductance.

Consider a simple twin-lead cable, the kind that used to bring television signals to our homes. It’s just two parallel wires. Its total inductance per unit length is the sum of two parts: an "external" part that depends on how far apart the wires are, and an "internal" part that depends only on the nature of the wires themselves. For a standard wire, this internal contribution is a fixed constant, a sort of "entry fee" for using a thick conductor instead of an idealized filament.

But the story gets far more interesting when the currents are not steady. Suppose we send a high-frequency alternating current down the wire. The changing magnetic fields induce eddy currents within the conductor that oppose the main flow in the center and reinforce it at the edges. The result is the famous ​​skin effect​​: the current is squeezed out of the conductor's bulk and forced to flow in a thin layer near its surface.

What does this do to the internal inductance? Well, if the current is no longer flowing through the interior of the wire, then the magnetic field inside the wire must dramatically decrease! Less internal field means less stored internal magnetic energy, and therefore, a lower internal inductance. At very high frequencies, the current flows in an infinitesimally thin skin, and the internal magnetic field—and thus the internal inductance—vanishes completely.

This is not a mere academic curiosity. The total inductance of a component like a solenoid, which we might naively treat as a constant, actually changes with frequency. The inductance we measure at DC, $L_{DC}$, includes both the external inductance from the coil's geometry and the internal inductance of the wire itself. At high frequencies, the internal part disappears, and the inductance becomes $L_{HF} = L_{ext}$. This frequency dependence is a critical factor in designing circuits for radio, radar, and modern high-speed computers. In fact, one can calculate the precise frequency at which the AC internal inductance of a wire drops to, say, one-tenth of its DC value. This frequency depends directly on the wire's radius and its conductivity, providing a clear, quantitative link between a material property and its high-frequency electrical behavior.

Now, let us take a giant leap from a solid copper wire to one of the most exotic conductors known: a plasma. Specifically, let's look inside a ​​tokamak​​, a doughnut-shaped magnetic bottle designed to confine a plasma at over 100 million degrees Celsius to achieve controlled nuclear fusion. This plasma is, in essence, a fluid conductor carrying an immense electrical current—millions of amperes—that generates a magnetic field to confine itself.

In this extreme environment, the concept of internal inductance is not a minor correction; it is a central character in the drama of fusion. Physicists use a normalized, dimensionless version of internal inductance, denoted by the symbol $l_i$. This parameter is no longer just about the energy in the field, but a crucial measure of the shape of the current density profile. A low $l_i$ corresponds to a broad, flat current profile, while a high $l_i$ signifies a current that is sharply peaked in the center of the plasma.

Why does this matter? Because in a fusion plasma, everything depends on the precise shape of the magnetic field.

​​1. Equilibrium and Position:​​ A hot plasma has immense pressure and naturally wants to expand. It is held in place by magnetic forces. The outward force, which leads to a shift in the plasma's position known as the ​​Shafranov shift​​, depends on both the plasma pressure and the internal magnetic forces. The internal inductance, $l_i$, directly quantifies this internal magnetic pressure. A more peaked current profile (higher $l_i$) alters the confining forces and changes the equilibrium position of the plasma inside the tokamak. To keep a star-hot plasma from touching the machine's walls, one must understand and control its internal inductance.

​​2. Stability and Performance:​​ Holding the plasma in place is only half the battle. It must also be stable against a zoo of violent instabilities, like kinks and balloons, that can cause it to lose confinement in an instant. The stability of the plasma is profoundly sensitive to the shape of the current profile. A key goal in fusion research is to maximize the plasma pressure that can be stably confined, a metric captured by the normalized beta, $\beta_N$. Theoretical and experimental work shows that this stability limit, $\beta_N^{\text{max}}$, is directly related to the internal inductance $l_i$. By carefully tailoring the current profile—that is, by "tuning" $l_i$—scientists can navigate the treacherous waters of MHD stability to operate the tokamak at higher performance, bringing us closer to a viable fusion power plant.

​​3. Dynamics and Diagnostics:​​ The internal inductance of a plasma is not a static quantity. It evolves as the current profile changes due to resistive diffusion. Sometimes, it changes with astonishing speed. In a phenomenon known as a "sawtooth crash," the magnetic field lines in the core of the plasma rapidly reconnect, causing the peaked current profile to suddenly flatten. This corresponds to a sharp drop in the internal inductance, $\Delta l_i$. According to Faraday's Law, a rapid change in inductance at a constant current induces a voltage. This is exactly what is observed: a sawtooth crash inside the plasma broadcasts a sharp, negative spike on the loop voltage measured at the edge of the machine. It is a spectacular demonstration of fundamental physics: a violent convulsion in the magnetic skeleton of the plasma is felt externally as an electrical signal, all because its internal inductance changed.

But how can we possibly know the value of $l_i$ inside a fiery plasma hotter than the sun's core? We cannot simply insert a probe. The answer is a beautiful piece of scientific detective work. The current flowing within the plasma generates a magnetic field that extends outside the plasma. By placing an array of magnetic sensors around the vacuum vessel, scientists can meticulously map the shape of this external field. From the details of this shape—specifically, its dipole and quadrupole moments—one can mathematically reconstruct the key parameters of the distribution inside, including the all-important internal inductance $l_i$.

From a small correction in a wire to a master parameter governing the life and death of a fusion plasma, the story of internal inductance is a testament to the power of a single physical idea. It reminds us that to truly understand the world, we must not only look at the grand structures but also appreciate the physics at work deep within them.