Transmission line fundamentals

The chapter introduces the electromagnetics and presents the origin physics of the Maxwell’s equations. Electromagnetic wave propagation equations in both free space and conductive media are derived. Transmission line theory laying the foundation for signal integrity analysis and interconnect design is discussed. Commonly used transmission lines in today’s high-speed systems and new development trends for them are presented in the last section of the chapter.

Keywords

Maxwell’s equations; transmission line theory; plane wave; electromagnetics

The chapter introduces electromagnetics and presents the origin physics of Maxwell’s equations. Electromagnetic wave propagation equations in both free space and conductive media are derived. Transmission line theory laying the foundation for signal integrity analysis and interconnect design is discussed. Commonly used transmission lines in today’s high-speed systems and new development trends for them are presented in the last section of the chapter.

Basic Electromagnetics

Starting with the introduction of integral and derivative forms of Maxwell’s equations, four physical laws composing Maxwell’s equations are explained. Four fundamental electromagnetic field vectors, building blocks of electromagnetics, are also presented. Lastly the propagation of electromagnetic waves is covered.

Electromagnetics Field Theory

Maxwell’s equations

By introducing the concept of displacement current, Maxwell’s summarized the famous equation set describing the electromagnetic phenomenon that electric field can induce magnetic field and vice versa. By combining the equations representing four electromagnetic physics laws, the integral form of Maxwell’s equations are written as below:

∮1H⋅dl=∫s(Jc+∂D∂t)⋅dS∮1E⋅dl=−∫s∂B∂t⋅dS∮SD⋅dS=∫vρdv∮SB⋅dS=0 (1.1)

Symbols used are defined as follows: electric field intensity E (V/m), electric flux density D (C/m2), magnetic field intensity H (A/m), and magnetic flux intensity B (T). Jc is the conducting current density (A/m2), σ is the media conductivity (S/m). E and D and B and H are dependent. Relationships between D and E, Jc and E, and B and H in isotropic media are:

=εE,B=μH,andJc=σE (1.2)

Electric field intensity E is defined as the electric force experienced by a unit positive charge in an electric field:

=Fq (1.3)

Symbol q in the equation is the quantity of charge on the test charge experiencing the force. F is the force experienced by the test charge.

Electric flux density D is used to define the electric field in dielectric materials where the dielectric can be polarized by an applied electric field. The induced dielectric polarization density is defined as P:

=ε0E+P (1.4)

where 0 is the electric permittivity of free space, 0=8.85×10−12F/m.

In a linear and isotropic media, P is defined as:

=χeε0E (1.5)

where e is the electric susceptibility of the dielectric material. It is a measure of how easily it polarizes in response to an electric field. The electric flux density D is further written by:

=ε0εrE=εE (1.6)

where =ε0(1+χe), the electric permittivity of the dielectric material, and r=1+χe is the relative permittivity of the dielectric.

Magnetic field strength or magnetic flux density B is a vector used to describe magnetic field. It relates the magnetic force experienced by a particle carrying a charge of coulomb to the magnetic field the charge is passing through at a speed of ,

=qv×B (1.7)

As dielectric being polarized by an applied electric field, magnetic media can be magnetized by an applied magnetic field. The magnetization is defined as M. In a linear and isotropic media, M is defined as:

=χmH (1.8)

where m is the magnetic susceptibility.

The magnetic flux density can be written as:

=μ0H+μ0M (1.9)

where 0 is the permeability of free space, 0=4π×10−7H/m. And,

=μ0μrH=μH (1.10)

where =μ0(1+χm) is the permeability of the magnetic media. r=1+χm is the relative permeability of the media.

Magnetic field is given by:

=Bμ0−M (1.11)

Maxwell’s equations formulate the interactions of vector fields D, E, B, and H. They are the cornerstones for the study of electromagnetic field and electromagnetic waves. The integral form of Maxwell’s equation describes the relationship between different vector fields in certain regions—for example, a symmetric distribution of charges and currents. However, in practical applications less symmetric situations and/or vector fields at a certain location in the region are usually desired. In these cases the differential form of Maxwell’s equations are more often used. The differential forms of Maxwell’s equations are summarized below:

∇×H=Jc+∂D∂t∇×E=−∂B∂t∇⋅D=ρ∇⋅B=0 (1.12)

Each equation of Maxwell’s equations represents a physics law observed in experiments. Their details are discussed in following sections.

Ampere’s law

Starting with Ampere’s law with displacement current correction, the origin of Ampere’s law states that magnetic fields can be generated by electrical current:

lH⋅dl=I (1.13)

The equation shows that the circulation or the line integral of the magnetic field is equal to the sum of the current inside the curl. However, this form of Ampere’s law does not apply to a non-continuous conducting current. Consider the situation in Figure 1.1: when a capacitor is being charged there will be a continuous conducting current outside of the capacitor. However, inside the capacitor there will be no conducting current. The magnetic circulation has different values depending on which surface surrounded is selected. When the current inside the surface S1 is I, and in contrast the current inside the surface S2 is 0, both the surfaces are bounded by path l. Maxwell’s correction by introducing displacement current filled this gap in the original Ampere’s law. The correction shows that not only does a continuous conducting current induce a magnetic field, but also a changing electric field induces a magnetic field. Ampere’s law can be rewritten as:

lH⋅dl=Ic+Id

where Ic is the conducting current and Id is the displacement current.

Since,

=∫sJ⋅dS

1H⋅dl=∫s(Jc+Jd)⋅dS=∫s(Jc+∂D∂t)⋅dS (1.14)

where Jc and Jd are conducting current density and displacement current density, respectively.

Faraday’s law

Faraday’s law summarizes that a voltage or electromotive force (EMF) can be produced by the altering magnetic flux in an electric circuit. The induced EMF (V) is equal to the negative change rate of magnetic flux B:

=−dΦBdt (1.15)

where is EMF. B is the magnetic flux (Wb). ΦB is defined as:

B=∫sB⋅dS (1.16)

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