## Abstract

In this chapter Maxwell’s equations are described and common ways to solve them analytically are discussed. The equations imply certain properties of matter with which it interacts and full solutions that describes this behavior analytically are provided from first principles. The chapter shows specifically that one can derive very fundamental properties with simple calculations. Furthermore, the concept of inductance and capacitance are highlighted by reference to their duality. Various high-speed phenomena are studied in some detail with particular attention to the current distributions induced by the magnetic field.

Maxwell’s equations in source form

Using estimation analysis in connection with Maxwell’s equations

Duality of capacitance vs inductance

One to three-dimensional solutions to Maxwell’s equations relevant to integrated circuit designers

Current distributions in various situations in order to estimate inductance

### 4.1 Introduction

This chapter discusses the basis of electromagnetism in terms of Maxwell’s equations. This topic that has been studied extensively and there are many great books that discuss its various aspects: see [1–14] for a small selection. Items [2, 5] focus on microwave aspects of the theory. Items [4, 8, 12] are standard physics graduate student texts. A more recent treatment is, for example [13, 14], which showcases engineering aspects of electromagnetism. We will follow the presentation in [1, 2] fairly closely. The intention in this chapter is to be self-consistent and in that spirit we will present common solution techniques found in the literature. The estimation techniques we discuss in this book will be heavily applied toward the end of the chapter where the concepts of capacitance and inductance are introduced. It is assumed that the reader has encountered electromagnetism before in elementary classes and we will not discuss the basic discoveries and the history that led to the remarkable formulation of the fundamental equations by Maxwell in a series of papers around 1865. The history of this development is a fascinating read and a great example of how science evolves [3].

We will start with a brief discussion of Maxwell’s equations and show how to reformulate them to be suitable in various situations encountered in integrated circuit design. Common solution techniques and handling of boundary conditions are presented next. Thereafter we will discuss the important concept of energy and power relating to the electromagnetic fields. These concepts will naturally lead to the definitions of capacitance and inductance both in general and for circuit theory. We will show that the concepts are naturally very similar, or dual, and we attempt to dispel some of the mystery that sometimes surrounds these phenomena. The chapter wraps up with a handful of examples where the estimation analysis technique is applied to calculate capacitance, inductance, skin effect, and other such effects. Most of these examples will start directly from Maxwell’s equations.

### 4.2 Maxwell’s Equations

This section presents Maxwell’s equations and we will follow the general outline presented in [1, 2]. Maxwell’s work was based on a large body of empirical and theoretical knowledge developed by Gauss, Ampere, Faraday, and others.

It is assumed that the reader has some familiarity with Maxwell’s equations and the history leading to their discovery. Here we will simply state them and highlight some of the historical events that surrounds them. The equations will be presented in their differential form. We believe most readers are familiar with the MKS or SI system of units and we will use them throughout the book.

With this we have for the equations:

**=**

*D**ρ*.

**= 0.**

*B*

We have the quantities defined as:

H is the magnetic field in amperes per meters [A/m]*H*

D is the electric flux density, in coulombs per meter squared [coul/m*D*^{2}]

J is the electric current density in amperes per meter squared [A/m*J*^{2}]

*ρ*ρ is the electric charge density in coulombs per meter cubed [coul/m^{3}]

E is the electric field in volts per meter [V/m]*E*

B is the magnetic flux density in webers per meter squared [Wb/m*B*^{2}]

The fields and their corresponding fluxes are related by the constitutional equations:

**=**

*D**ϵ*

**.**

*E***=**

*B**μ*

*H*The factors *ϵ*, *μ*ϵ,μ are matrices in general and dependent on position. Throughout the book we will assume them to be scalar functions that are occasionally dependent on position.

#### Vector Potential and Elementary Gauge Theory

Having established Maxwell’s equations we can now take note of some interesting properties. For one, there are no magnetic charges, hence ∇ ⋅ ** B** = 0∇⋅B=0. This means that instead of using

**B we can define another entity**

*B***A by using the fact that for functions that are smooth, all derivatives exists and are continuous, we have the vector identity ∇ ⋅ (∇ ×**

*A***) = 0∇⋅∇×A=0. This has important implications. If**

*A***= ∇ ×**

*B*

*A*then equation (4.4) is automatically fulfilled. Substituting this into (4.3) we get

We now use the vector identity: ∇ × ∇ *φ* ≡ 0∇×∇φ≡0, where *φ*(** x**)φxt is any smooth function of coordinates

**x and time**

*x**t*, to integrate

**E and we find**

*E*The vector field ** A**A is commonly referred to as the vector potential field and scalar

*φ*φ is known as the potential field, or voltage field. Together they are called gauge potentials in physics literature.

The equation for the *E*E-field should be familiar to most readers with the possible exception of the last term. Equation (4.8) is simply the normal elementary textbook definition of the electric field as a gradient of a voltage but with an additional time-derivative (dynamic) term. It means we can have an electric field without a voltage drop in a dynamic situation. With the potential fields we can write Maxwell’s equations to be a set of equations for *φ* and ** A**φandA. Let us rewrite equation (4.1):

or

and from equations (4.2) and (4.5) we find

We now have another version of Maxwell’s equations through equations (4.7)–(4.10). As the reader will have noticed, the potentials, ** A**,

*φ*A,φ are not uniquely defined. One can, for instance, add a term ~ ∇

*f*∼∇f where

*f*f is some function to

**A and the**

*A***B field is not affected (∇ × ∇**

*B**f*≡ 0)∇×∇f≡0. This freedom in the choice of the potentials is known as gauge invariance. Let us look at this in some more detail (compare with [15] for a similar argument).

Let *γ*(** x**,

*t*)γxt be an arbitrary scalar field. Then let us change the gauge potentials according to the following transformation

Now

**→ ∇ × (**

*B***+ ∇**

*A**γ*) = ∇ ×

*A*and

Both fields are unchanged! The transformation in equation (4.11) is known as a gauge transformation, and since the fields are unchanged under this transformation we speak of gauge symmetry. We see that a physical system is described by a whole family of gauge potentials that differ by a gauge transformation. By picking a particular set of gauge potentials we are making a gauge choice. The above might seem a trivial matter but it has profound importance in theoretical physics. The interested reader is highly encouraged to refer to the literature on this matter.

We will now show that we can always find a solution that satisfies

by making an appropriate gauge choice. Assume we have a particular solution *A*^{′}, *φ*^{′}A′,φ′. We look for a particular gauge transformation that will satisfy (4.12). We have using (4.11)

or

The right-hand side is the known solution which acts as a source term for a wave equation which we can solve for *γ*γ. This way we can always find gauge potentials that satisfy (4.12). We do not have to find *γ*γ explicitly, but we can use this calculation as a motivation to use (4.12) as an additional requirement on ** A**,

*φ*A,φ. Equation (4.12) is known as the Lorenz gauge. Another typical choice is the Coulomb gauge

**= 0.**

*A*The Lorenz gauge is typically used for situations where the wavelength is comparable to the physical sizes. It is standard in microwave theory and antenna theory for obvious reasons. For integrated circuits one can often get by with the Coulomb gauge which corresponds to the Lorenz gauge in the long-wavelength approximation.

#### Maxwell’s Equations in Terms of External Sources

In electrical engineering it is natural to think of currents and charges as being impressed on an electric system through voltage or current sources. These impressed entities will then give rise to electromagnetic fields. We will now write Maxwell’s equation in terms of such impressed currents and charges.

The current can be divided into two parts. A conduction component,

*J*_{c}=

*σ*

**(ohm ‘ s law)**

*E*and an impressed current, *J*_{i}Ji. The charge can be divided the same way. By taking the divergence of equation (4.1) we recover the continuity equation:

which relates the charge to the current. The addition of the time derivative of the ** D**D-field to Ampere’s law (∇ ×

**=**

*H***)∇×H=J was Maxwell’s famous generalization that created a self-consistent description of the field equations. For the current we have**

*J*Putting this together we get

These equations show the fields as a result of external source currents and charges.

##### Full-Wave Approximation – Single Frequency Tone Formulation

In this book we will generally look at these equations not as a function of time but as a function of frequency. We get there by simply assuming the time dependence scales as *e*^{jωt}ejωt where we follow the convention in most engineering books. By doing this we get

From the continuity equation for the conduction component we get

This together with the charge equation (4.17) gives

We can now define an effective permittivity

and we have

**=**

*H**jω ϵ*

^{′}

**+**

*E*

*J*_{i}

*ϵ*

^{′}

**) =**

*E**ρ*

_{i}.

Using equations (4.6)–(4.8) we get

We use the Lorenz gauge which in frequency domain looks like

**+**

*A**jϵ*

^{′}

*μω φ*= 0

and get

**) − ∇**

*A*^{2}

**= ∇ (−**

*A**jϵ*

^{′}

*μω φ*) − ∇

^{2}

**=**

*A**jωμ ϵ*

^{′}(− ∇

*φ*−

*jω*

**) +**

*A**μ*

*J*_{i}

After rewriting we find

^{2}

**+**

*A**ω*

^{2}

*μ ϵ*

^{′}

**= −**

*A**μ*

*J*_{i}

We also have from equation (4.23)

After rewriting we find

Equations (4.25) and (4.26) are Maxwell’s equation in yet another form. Knowing ** A**,

*φ*A,φ will give us

**and**

*B***BandE through equations (4.7) and (4.8). We will use these in numerous examples in this and the following chapters.**

*E*##### Long Wavelength Approximation

In the long wavelength approximation, *λ* ≫ *l*λ≫l, where *λ* = 2*πc*/*ω*λ=2πc/ω is the wavelength and *l*l is a length scale of the model, we find the second term on the left-hand side of (4.25), (4.26) disappears and we are left with:

^{2}

**= −**

*A**μ*

*J*_{i}.

We can also write this directly from the fields as

**=**

*H**μ*

*J*_{i}(Ampere ‘ s law),

where (4.27) follows if we use gauge ∇ ⋅ ** A** = 0∇⋅A=0 and

For integrated circuits the long wavelength approximation is often appropriate since the dimensions are much smaller than any wavelength.

#### Solutions to Maxwell’s Equations

Note the equations are of the same form where the only difference is the vector form for the vector potential and scalar from for the voltage field equation. The general solutions are known for certain sources. Here we will almost exclusively look at cases where the sources are Dirac delta functions of one sort or other. The specific solution will depend on the boundary conditions and most of the time will be spent establishing those.

For completeness, in this section we will discuss a common approach to solve wave style equations such as Maxwell’s. The first subsection covers the general solutions and we will discuss how to handle the all-important boundary conditions in the following subsection. We follow the presentation given in [1].

##### General Solution

We will start discussing the general solution of the one-dimensional case and follow with the two- and three-dimensional versions in the following subsections. There is a rich literature describing these methods and also a list of references at the end of the chapter.

###### 1D – Solution

Let us consider an equation similar to (4.26) in free space

This is known as Helmholz’s equation in one-dimensional free space subject to the boundary condition at infinity, *φ*(±∞) = 0φ±∞=0. The response at *x*x is due to the delta source at *x*_{0}x0. Let us consider the homogeneous equation

This is the same as (4.30) when *x* ≠ *x*_{0}x≠x0. The solution to this equation that satisfies the boundary conditions at infinity is

The unknown constants *A*A, *B*B can be determined by the boundary condition at *x* = *x*_{0} ± Δx=x0±Δ where ΔΔ denotes an infinitesimally small interval. Integrating (4.30) from *x* = *x*_{0} − Δx=x0−Δ to *x* = *x*_{0} + Δx=x0+Δ we find

Since *φ*(*x*)φx is continuous the last term on the left-hand side disappears when Δ → 0Δ→0.

We get

Solving for *A*A and *B*B gives

###### Long Wavelength Approximation

When *kx*_{0} ≪ 1kx0≪1 (long wavelength approximation) we find

Since in the long wavelength approximation Helmholz equation reduces to Poisson equation where *φ*_{lw}φlw is defined with an arbitrary constant factor, we can simply relabel the constant term for *φ*_{lw}φlw and end up with

Let us verify by examining the Poisson equation

When *x* ≠ *x*_{0}x≠x0 we see ∂2φlwx∂x2≡0. Let us integrate around the singularity as we did earlier

Indeed, in one dimension and the long wavelength approximation (4.32) solves the Poisson equation.

###### 2D – Solution

In two dimensions equation (4.30) becomes

We can now use the Fourier transform

This gives

The solution to this equation is the one-dimensional free-space Green’s function

where κ=k2−β2. We find

where the last equality relates the Hankel function of order 0, H01 to the integral. We see the solution is simply a composition of a continuous spectrum of plane waves.

###### Long Wavelength Approximation

The two-dimensional solution in the long wavelength approximation can be found using similar techniques as in the one-dimensional case we noted earlier. Here we will show the solution for the special case of cylindrical symmetry which we will take advantage of later in this chapter when we discuss inductance and current elements.

Helmholz equation in cylindrical symmetry becomes, with a delta function at *x* = 0x=0,

*φ*=

*Cδ*(

**).**

*x*Outside of *x* = 0x=0 we have

To further simplify we will assume there is no *θ*θ-dependence and we find

This has the general solution

*φ*(

*r*) =

*D*ln

*r*+

*B*.

To find out the constants we need to integrate (4.34) around ** x** = 0x=0. Let us choose a sphere centered at

**= 0x=0 with radius ΔΔ as an integration volume. We find for the left-hand side using the divergence theorem (see Appendix B)**

*x*The right-hand side of (4.35) becomes, as before, *C*C. Putting all this together we have

We have for the long wavelength solution to the Helmholz equation in two dimensions:

###### 3D – Solution

We finally present the 3D solution. We will use it in Chapter 6. Equation (4.30) becomes with a source at ** r** =

*r*_{0}r=r0

Let us first do a change of variables ** ρ** =

**−**

*r*

*r*_{0}ρ=r−r0. We find we have created a spherically symmetric model. By realizing

*δ*(

**−**

*r*

*r*_{0})

*dV*= ∫

*δ*(

**)4**

*ρ**πρ*

^{2}

*dρ*,

we find

By substituting

we find for the Helmholz equation

The solution for *u*(*ρ*) = *Ae*^{jkρ}uρ=Aejkρ and we find

The boundary condition at *ρ* = 0ρ=0 must be used to determine *A*A. The Helmholz equation is

After integrating over a small volume we find, as with the one-dimensional case,

After applying the divergence theorem we get

So

The general solution is

Similarly, for the vector potential we get

###### Long Wavelength Approximation

The solution to the long wavelength equations (4.27) and (4.29) in three dimensions are well known. We see the equations are essentially identical and the general solution that vanishes at infinity is

The solution for *φ*φ is similarly

For a point charge at the origin, *ρ*_{i}(*r*^{′}) = *qδ*(*r*^{′})ρir′=qδr′ we find

which is the familiar electrostatic potential from a point charge.

###### Boundary Conditions

Identifying the relevant equations and how to solve them is a helpful exercise. It is often the easiest part of any investigation. The real problem comes when taking into account what happens at the boundaries, either in time and/or in space. To the novice, the opposite often appears true. To help us there are many great examples in the literature on how to handle the boundaries, and here we will go through the basic methods and leave it to the reader to explore more if needed.

Fundamentally, what one does is to put a “pill box” at the boundary that extends a little into each material with a large surface area. In Figure 4.1 it extends *ε*/2ε/2 into each region. The volume is thus infinitesimal, while the area is macroscopic. They key thing to note here is that the equations are still valid in this volume and to find out what the boundary conditions are one simply integrates the equations over the small volume. For some entities, portions of the equation will be proportional to the volume and thus small while other entities will be proportional to the area and thus large. We will go through Maxwell’s equations with this method to show explicitly how the conditions work out.

Figure 4.1 Boundary conditions pill box.

Let us look at Ampere’s law (4.28)

**=**

*H***.**

*J*At the boundary between two media we put a small pill box of height *ε*ε which is much smaller than any other dimension in the problem. We integrate Ampere’s law over this volume

*H**dV*= ∫

*J**dV*.

We can here use Stoke’s theorem for the left-hand side and denoting by *A*_{rea} = *A*_{rea}** n**Area=Arean where

*A*

_{rea}Area is the area and

**n is the outward normal to the area segment**

*n*

*H**dV*= ∮

**×**

*H*

*A*_{rea}

*dA*

_{rea}=

*A*

_{rea}(

*H*

_{t, +}−

*H*

_{t, −}) +

*εH*

_{n}→

*A*

_{rea}(

*H*

_{t, +}−

*H*

_{t, −}),

*ε*→ 0.

###### Volume Current

For the right-hand side without delta functions, which we call volume current, we get

*J**dV*~

*ε*→ 0 when

*ε*→ 0.

Putting it all together we get at the boundary

*H**dV*=

*A*

_{rea}(

*H*

_{t, +}−

*H*

_{t, −}) = ∫

*J**dV*= 0.

or

*H*

_{t, +}=

*H*

_{t, −}.

###### Surface Current

We also see when we have a delta function on the right-hand side, a surface current,

*J**dV*= ∫

*J*

_{s}

*δ*(

*y*)

*dV*=

*J*

_{s}

*A*

_{rea}when

*ε*→ 0.

Putting this together with the left-hand side we get

*A*

_{rea}(

*H*

_{t, +}−

*H*

_{t, −}) =

*J*

_{s}

*A*

_{rea}

or

*H*

_{t, +}−

*H*

_{t, −}) =

*J*

_{s}

Similarly, we can use the charge equation (4.23)

*ϵ*

^{′}

**) =**

*E**ρ*

_{i}.

We integrate over volume

*ϵ*

^{′}

**)**

*E**dV*= ∫

*ρ*

_{i}

*dV*

For the left-hand side we use Gauss’ law

###### Volume Charge

For the case of volume charge, the right-hand side is now treated the same way as for the volume current earlier so we end up with

Finally, for the case of surface charges we have

#### Field Energy Definitions

Let us look at the concept of energy in these fields. It is outside the scope of this book to fully derive this here but we will make some plausible arguments as to their validity: see [4] for details. Let us start with a static electric field where there are no currents.

###### Electric Field Energy

We assume for simplicity there is one conductor at a constant voltage *φ*φ. If we have an infinitesimal charge *δρ*δρ moving from infinity to the conductor we will need to apply an energy

*δW*=

*φ δρ*

Physicists like to call this entity “work,” but we will stick to using a less stringent definition of energy. We know from the boundary conditions, the charge on the conductor is

*ρ*= − ∮

*D*

_{n}

*dS*= − ∮

**⋅**

*D**d*

*S*Here *dS*dS is a surface element and *d*** S**dS is surface element in the direction normal to the conductor surface. Since the potential is constant on the surface of the conductor we have

*δW*=

*φ δρ*= − ∫

*φ δ*

**⋅**

*D**d*

**= − ∫ ∇ ⋅ (**

*S**φ δ*

**)**

*D**dV*