Electromagnetics

Theory

Scattering-parameters (S-parameters) are a fundamental concept in electromagnetics and RF/microwave engineering used to describe how electromagnetic waves behave in high-frequency networks. They quantify the relationship between incident (input) and reflected/transmitted (output) power waves at different ports of a system.

S-parameters are essential for characterizing high-frequency circuits because voltages and currents are difficult to measure directly at microwave frequencies.
They are measured using a Vector Network Analyzer (VNA).
All S-parameters are complex numbers, possessing both magnitude and phase.

For a 2-port network , there are four S-parameters, represented in a matrix equation relating the reflected (b) waves to the incident (a) waves:

[\begin{matrix} b_{1} \\ b_{2} \end{matrix}] = [\begin{matrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{matrix}] [\begin{matrix} a_{1} \\ a_{2} \end{matrix}]

Where,

a₁a₂: Incident waves at ports 1 and 2, respectively
b₁b₂: Reflected (scattered) waves at ports 1 and 2, respectively

Table 1.
S-Parameter	Meaning	Description
S₁₁	Input reflection coefficient	Reflected wave at port 1 when input is at port 1
S₂₁	Forward transmission	Transmitted wave at port 2 when input is at port 1
S₁₂	Reverse transmission	Transmitted wave at port 1 when input is at port 2
S₂₂	Output reflection coefficient	Reflected wave at port 2 when input is at port 2

Perfect electric conductor (PEC) is a theoretical material defined by an infinite electrical conductivity (σ→∞). In theory, it offers zero resistance to the flow of electric current.

Key Properties and Boundary Conditions of a PEC:

Electric Field Inside is Zero ( $E_{i n s i d e} = 0$ ): Charges rearrange instantly to cancel any internal electric field.
Surface Current Flows with No Resistance: Any current flows only on the surface of the PEC.
Tangential Electric Field at the Surface is Zero: This is a critical boundary condition for solving Maxwell's equations.
Normal Component of Magnetic Field at the Surface is Zero: A time-varying magnetic field induces surface currents that cancel any magnetic flux into the conductor.
Reflects All Incident Electromagnetic Waves: It acts as a perfect mirror for EM waves, with no transmission or absorption.

A rectangular waveguide is a hollow metallic pipe, often modeled with PEC walls to simplify calculations.

Key Idea:

The PEC boundaries force the tangential electric field to zero at the walls, which in turn creates discrete field patterns called modes inside the waveguide.

The dominant TE₁₀ mode (Transverse electric, m=1,n=0) is the lowest-frequency mode that can propagate.

The electric field is zero at the PEC side walls (for example, at x=0 and x=a).
This mode is considered the fundamental (first) mode.
For a rectangular waveguide with dimensions a>b, the cutoff frequency (fc) for the TE_mn mode is:
$f_{c} = \frac{1}{2 π \sqrt{με}} \sqrt{{(\frac{m π}{a})}^{2} + {(\frac{n π}{b})}^{2}}$

The lowest cutoff frequency occurs for m=1,n=0→TE₁₀.

Material Properties

Relative permittivity ( $\in_{r}$ ) is the ratio of a material's permittivity (

\in

) to the permittivity of free space (

\in_{0}

). It indicates a material's capacity to store an electric field per unit voltage compared to a vacuum.

Higher $\in_{r}$ results in a slower wave speed and shorter wavelength in the medium.

Electrical conductivity ( $δ$ ) measures how easily a material allows the flow of electric current. Units are Siemens per meter (S/m).

Table 2.
Region	$δ$ = 0 (Perfect insulator)	$δ$ = ∞ (Perfect conductor / PEC)
Waveguide Fill (Interior)	✔ No conduction loss (ideal dielectric)	✖ Not physically meaningful Filling with a perfect conductor would block all EM waves
	✔ EM wave can propagate freely
	✖ No energy dissipation
Waveguide Walls	✖ Cannot confine wave	✔ Ideal case for waveguiding
	✖ Wave leaks or is not guided	✔ No Ohmic loss
	✖ No proper reflection of fields	✔ Total reflection of fields
	✖ No proper reflection of fields	✔ Used in most EM simulations (PEC)

Dielectric loss tangent ( $\tan δ_{\in}$ ) describes how much energy from an electric field is dissipated as heat within the material.

A higher value means more loss.

Table 3.
Region	$\tan δ$ =0 (Lossless dielectric)	$\tan δ$ →∞ (Extremely lossy material)
Waveguide Fill (Interior)	✔ Ideal case	✖ Very high loss
	✔ Wave propagates with no dielectric loss	✖ Wave is heavily attenuated
	✔ Used for air, vacuum, PTFE	✖ Used only for absorbers or lossy coatings
Waveguide Walls	✖ Not applicable — waveguide walls are metals or PEC, not dielectrics	✖ Not applicable — walls are not modeled with dielectric loss tangent

Relative permeability ( $μ_{r}$ ) is the ratio of a material's magnetic permeability (

μ

) to the permeability of free space (

μ_{0}

). It indicates how much the material supports the formation of a magnetic field inside it.

$μ_{r}$ =1 for non-magnetic materials (like air, most dielectrics)
High $μ_{r}$ means the material concentrates magnetic flux.

Magnetic loss tangent ( $\tan δ_{μ}$ ) describes magnetic losses in a material. It is defined as

δ_{μ} = \frac{μ "}{μ'}

, where

μ'

is the real part (energy storage) and

μ "

is the imaginary part (energy loss) of the permeability.

Table 4.
Region	$\tan δ_{μ}$ =0 (Lossless magnetic)	$\tan δ_{μ}$ →∞ (Highly magnetic lossy)
Waveguide Fill (Interior)	✔ No magnetic loss	✖ Strong attenuation
	✔ Wave propagates ideally if μ′>1	✖ EM energy absorbed as magnetic loss
	✔ Used in low-loss ferrites	✔ Used in magnetic absorbers
Waveguide Walls	✖ Not applicable — waveguide walls are metal, not magnetic materials	✖ Not applicable — metals don't use magnetic loss tangent

Modal Electromagnetics Analysis

Modal electromagnetics analysis evaluates the frequencies and corresponding modes of electromagnetic fields within a cavity in a model.

The first natural frequency is also called the fundamental frequency of the cavity. Modes are listed in ascending order so the fundamental frequency is the first mode.

The number of modes value indicates how many total modes you want to investigate.

In the case of damped oscillations, the output contains eigenvalues and damping factors, as well as frequencies. The real part of the eigenvalue represents the damping rate (in [1/sec]), and the imaginary part represents the angular frequency (in [rad/sec]).