ECE 375: Electromagnetic Fields and Waves

These notes cover the full scope of ECE 375 as taught at the University of Waterloo (Winter 2025, instructor: Hamed Majedi). The course spans transmission line theory, microwave network analysis, scattering parameters, Maxwell’s equations in differential and integral form, electromagnetic plane waves, polarization, power flow, and wave reflection and transmission at planar boundaries. The treatment is derivation-heavy and self-contained. Primary reference: F. T. Ulaby and U. Ravaioli, Fundamentals of Applied Electromagnetics, 7th ed., Pearson, 2014.

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Sources and References

The following publicly available references support the content of these notes:

• F. T. Ulaby and U. Ravaioli, Fundamentals of Applied Electromagnetics, 7th ed., Pearson/Prentice Hall, 2014. (Primary course textbook.)
• D. M. Pozar, Microwave Engineering, 3rd ed., John Wiley, 2004. (Transmission lines, S-parameters, waveguides, and cavity resonators.)
• D. J. Griffiths, Introduction to Electrodynamics, 4th ed., Cambridge University Press, 2017. (Electrostatics, magnetostatics, Faraday’s law, and Maxwell’s equations with exceptional physical clarity.)
• W. H. Hayt and J. A. Buck, Engineering Electromagnetics, 8th ed., McGraw-Hill, 2011. (Boundary conditions, plane waves, transmission lines.)
• A. S. Inan, U. S. Inan, and R. A. Said, Engineering Electromagnetics and Waves, 2nd ed., Pearson, 2015. (Alternate reference used in parallel ECE 375 offerings.)
• S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, 3rd ed., John Wiley, 1994.
• MIT OpenCourseWare 6.013 — Electromagnetics and Applications, freely available at ocw.mit.edu. (Open lectures and problem sets on Maxwell’s equations, waves, and guided structures.)

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Chapter 1: Mathematical Foundations

1.1 Vector Algebra and Coordinate Systems

Electromagnetic theory is formulated in terms of vector fields defined on three-dimensional space, varying in time. Facility with coordinate systems is therefore prerequisite to everything that follows.

1.1.1 Cartesian Coordinates

In Cartesian coordinates \((x, y, z)\), the position vector is

\[ \mathbf{r} = x\,\hat{x} + y\,\hat{y} + z\,\hat{z}. \]

The unit vectors \(\hat{x}, \hat{y}, \hat{z}\) are mutually orthogonal and constant throughout space. A general vector field \(\mathbf{A}\) is written \(\mathbf{A} = A_x\hat{x} + A_y\hat{y} + A_z\hat{z}\). The dot and cross products satisfy \(\hat{x}\cdot\hat{y} = 0\), \(\hat{x}\times\hat{y} = \hat{z}\), and cyclic permutations thereof.

The differential arc length element is

\[ d\boldsymbol{\ell} = dx\,\hat{x} + dy\,\hat{y} + dz\,\hat{z}, \]

the differential surface element on a surface of constant \(z\) is \(d\mathbf{S} = dx\,dy\,\hat{z}\), and the differential volume element is \(dV = dx\,dy\,dz\).

1.1.2 Cylindrical Coordinates

Cylindrical coordinates \((\rho, \phi, z)\) are related to Cartesian by \(x = \rho\cos\phi\), \(y = \rho\sin\phi\), \(z = z\). The unit vectors \(\hat{\rho}\) and \(\hat{\phi}\) are position-dependent:

\[ \hat{\rho} = \cos\phi\,\hat{x} + \sin\phi\,\hat{y}, \qquad \hat{\phi} = -\sin\phi\,\hat{x} + \cos\phi\,\hat{y}. \]

The differential elements are \(d\boldsymbol{\ell} = d\rho\,\hat{\rho} + \rho\,d\phi\,\hat{\phi} + dz\,\hat{z}\) and \(dV = \rho\,d\rho\,d\phi\,dz\).

1.1.3 Spherical Coordinates

Spherical coordinates \((r, \theta, \phi)\) satisfy \(x = r\sin\theta\cos\phi\), \(y = r\sin\theta\sin\phi\), \(z = r\cos\theta\), with \(r \geq 0\), \(0 \leq \theta \leq \pi\), \(0 \leq \phi < 2\pi\). The differential volume is \(dV = r^2\sin\theta\,dr\,d\theta\,d\phi\).

1.2 Differential Operators

1.2.1 The Gradient

For a scalar field \(V\), the gradient \(\nabla V\) points in the direction of maximum rate of increase and has magnitude equal to that rate. In Cartesian coordinates,

\[ \nabla V = \frac{\partial V}{\partial x}\hat{x} + \frac{\partial V}{\partial y}\hat{y} + \frac{\partial V}{\partial z}\hat{z}. \]

In cylindrical coordinates,

\[ \nabla V = \frac{\partial V}{\partial\rho}\hat{\rho} + \frac{1}{\rho}\frac{\partial V}{\partial\phi}\hat{\phi} + \frac{\partial V}{\partial z}\hat{z}. \]

The gradient is related to the directional derivative: for unit vector \(\hat{u}\),

\[ \frac{dV}{d\ell}\bigg|_{\hat{u}} = \nabla V \cdot \hat{u}. \]

1.2.2 The Divergence

The divergence of a vector field \(\mathbf{A}\) measures the net outward flux per unit volume at a point. In Cartesian coordinates,

\[ \nabla \cdot \mathbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}. \]

In cylindrical coordinates,

\[ \nabla \cdot \mathbf{A} = \frac{1}{\rho}\frac{\partial(\rho A_\rho)}{\partial\rho} + \frac{1}{\rho}\frac{\partial A_\phi}{\partial\phi} + \frac{\partial A_z}{\partial z}. \]

1.2.3 The Curl

The curl of \(\mathbf{A}\) measures the circulation (rotation) per unit area about a point. In Cartesian coordinates,

[ \nabla \times \mathbf{A} = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \ \partial_x & \partial_y & \partial_z \ A_x & A_y & A_z \end{vmatrix} = \left(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\right)\hat{x}

• \left(\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}\right)\hat{y}
• \left(\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right)\hat{z}. ]

1.2.4 The Laplacian

The Laplacian of a scalar field is the divergence of its gradient:

\[ \nabla^2 V = \nabla \cdot (\nabla V) = \frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2} \quad\text{(Cartesian)}. \]

The vector Laplacian satisfies \(\nabla^2 \mathbf{A} = \nabla(\nabla\cdot\mathbf{A}) - \nabla\times(\nabla\times\mathbf{A})\), which is the vector identity most frequently used in deriving wave equations.

Two identities will recur constantly in what follows:

\[ \nabla \times (\nabla V) = \mathbf{0} \quad\text{(curl of a gradient is zero)}, \]\[ \nabla \cdot (\nabla \times \mathbf{A}) = 0 \quad\text{(divergence of a curl is zero)}. \]

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Chapter 2: Transmission Line Theory

2.1 When Circuit Theory Fails

At low frequencies, the wavelength \(\lambda = c/f\) is much larger than the circuit dimensions, so signals propagate instantaneously across the circuit — phase variations along conductors are negligible. At microwave frequencies (GHz range), wavelengths shrink to centimeters or millimeters, comparable to interconnect lengths. A signal at \(f = 1\,\text{GHz}\) has \(\lambda \approx 30\,\text{cm}\) in free space. If the interconnect is \(15\,\text{cm}\) long, the voltage and current vary significantly along its length, and lumped-element circuit theory is no longer adequate. Transmission line theory extends circuit analysis to the distributed-parameter regime.

2.2 The Transmission Line Circuit Model

A transmission line is characterized by four distributed (per-unit-length) parameters:

• \(R'\) — resistance per unit length \([\Omega/\text{m}]\), representing conductor losses;
• \(L'\) — inductance per unit length \([\text{H/m}]\), representing magnetic energy storage;
• \(G'\) — conductance per unit length \([\text{S/m}]\), representing dielectric losses;
• \(C'\) — capacitance per unit length \([\text{F/m}]\), representing electric energy storage.

Consider an infinitesimal segment of length \(\Delta z\). Applying KVL and KCL to the equivalent circuit of this segment and taking \(\Delta z \to 0\) yields the telegrapher’s equations:

\[ \frac{\partial v(z,t)}{\partial z} = -R'\,i(z,t) - L'\frac{\partial i(z,t)}{\partial t}, \]\[ \frac{\partial i(z,t)}{\partial z} = -G'\,v(z,t) - C'\frac{\partial v(z,t)}{\partial t}. \]

These are coupled first-order PDEs. For time-harmonic (sinusoidal steady-state) excitation at angular frequency \(\omega\), we introduce phasors \(V(z)\) and \(I(z)\) so that \(v(z,t) = \text{Re}[V(z)e^{j\omega t}]\). The telegrapher’s equations become

\[ \frac{dV}{dz} = -(R' + j\omega L')\,I(z), \]\[ \frac{dI}{dz} = -(G' + j\omega C')\,V(z). \]

2.3 Wave Solutions and the Propagation Constant

Differentiating the first phasor equation with respect to \(z\) and substituting the second:

\[ \frac{d^2 V}{dz^2} = \gamma^2\,V(z), \]

where the complex propagation constant is

\[ \gamma = \sqrt{(R' + j\omega L')(G' + j\omega C')} = \alpha + j\beta. \]

Here \(\alpha \geq 0\) is the attenuation constant in \(\text{Np/m}\) and \(\beta > 0\) is the phase constant in \(\text{rad/m}\). The general solution is

\[ V(z) = V_0^+\,e^{-\gamma z} + V_0^-\,e^{+\gamma z}. \]

The first term represents a wave travelling in the \(+z\) direction; the second represents a wave travelling in the \(-z\) direction. The corresponding current is

\[ I(z) = \frac{1}{Z_0}\left(V_0^+\,e^{-\gamma z} - V_0^-\,e^{+\gamma z}\right), \]

where the characteristic impedance is

\[ Z_0 = \sqrt{\frac{R' + j\omega L'}{G' + j\omega C'}}. \]

2.4 Voltage Reflection Coefficient and Input Impedance

Suppose the line of length \(\ell\) is terminated at \(z = 0\) by a load impedance \(Z_L\). The voltage reflection coefficient at the load is

\[ \Gamma_L = \frac{V_0^-}{V_0^+} = \frac{Z_L - Z_0}{Z_L + Z_0}. \]

Note \(\Gamma_L = 0\) when \(Z_L = Z_0\) (matched load — no reflection) and \(|\Gamma_L| = 1\) when \(Z_L\) is purely imaginary (open or short circuit — total reflection).

The input impedance looking into the line from \(z = -\ell\) toward the load is

\[ Z_\text{in} = Z_0\,\frac{Z_L + j Z_0\tan(\beta\ell)}{Z_0 + j Z_L\tan(\beta\ell)}. \]

2.5 Standing Waves and the Voltage Standing Wave Ratio

When the load is mismatched, forward and backward waves superpose to form a standing wave pattern. The voltage magnitude along the line is

\[ |V(z)| = |V_0^+|\left|1 + \Gamma_L e^{j2\beta z}\right| \cdot e^{-\alpha(z+\ell)}, \]

evaluated with \(z \in [-\ell, 0]\). For a lossless line, the ratio of maximum to minimum voltage magnitude is the voltage standing wave ratio:

\[ S = \text{VSWR} = \frac{1 + |\Gamma_L|}{1 - |\Gamma_L|}. \]

\(S = 1\) for a matched load; \(S \to \infty\) for an open or short circuit.

2.6 Power on a Transmission Line

The time-averaged power carried in the \(+z\) direction on a lossless line is

\[ P_\text{av} = \frac{|V_0^+|^2}{2Z_0}(1 - |\Gamma_L|^2). \]

The factor \((1 - |\Gamma_L|^2)\) represents the fraction of incident power delivered to the load; \(|\Gamma_L|^2\) is reflected.

2.7 Impedance Matching Techniques

2.7.1 Single-Stub Matching

A shunt stub (short- or open-circuited section of line) placed at a carefully chosen distance from the load can cancel the susceptance presented by the load, achieving a matched condition. The design proceeds in two steps: (1) find the distance \(d\) from the load to the stub location where the real part of the line admittance equals \(Y_0 = 1/Z_0\); (2) choose the stub length \(\ell_s\) so that its susceptance cancels the imaginary part.

2.7.2 The Smith Chart

The Smith Chart is a graphical representation of the complex reflection coefficient \(\Gamma\) plotted on a unit disk in the \(\Gamma\)-plane. Constant-resistance circles and constant-reactance arcs (from the normalized impedance \(z = Z/Z_0\)) are superimposed. Moving along the transmission line rotates \(\Gamma\) clockwise on a circle of constant radius \(|\Gamma|\) at a rate of \(\pi\) radians per half-wavelength. The Smith Chart makes impedance transformation, stub design, and matching immediately visual and was historically indispensable before computer-aided design tools.

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Chapter 3: Microwave Network Analysis

3.1 Impedance and Admittance Matrices

At sufficiently high frequencies a multi-port circuit is conveniently described by relating port voltages and currents. For an \(N\)-port network with port voltages \(V_i\) and port currents \(I_i\) (flowing into port \(i\)), the impedance matrix \([Z]\) is defined by

\[ \begin{pmatrix} V_1 \\ V_2 \\ \vdots \\ V_N \end{pmatrix} = [Z] \begin{pmatrix} I_1 \\ I_2 \\ \vdots \\ I_N \end{pmatrix}, \quad Z_{ij} = \frac{V_i}{I_j}\bigg|_{I_k=0,\,k\neq j}. \]

The element \(Z_{ij}\) is the open-circuit transfer impedance from port \(j\) to port \(i\), measured with all other ports open. The admittance matrix \([Y] = [Z]^{-1}\) is defined analogously with short-circuit conditions.

For a reciprocal network (no active elements or ferrites), \([Z]\) and \([Y]\) are symmetric. For a lossless network, \([Z]\) and \([Y]\) are purely imaginary.

3.2 Scattering Parameters

At microwave frequencies, open- and short-circuit terminations are difficult to realize precisely and cause instability in active devices. Scattering parameters (\(S\)-parameters) use incident and reflected wave amplitudes normalized to the port characteristic impedance \(Z_0\), and are measured with all non-driven ports terminated in matched loads — a far more practical condition.

Define the normalized wave amplitudes at port \(i\):

\[ a_i = \frac{V_i + Z_0 I_i}{2\sqrt{Z_0}}, \qquad b_i = \frac{V_i - Z_0 I_i}{2\sqrt{Z_0}}. \]

Here \(|a_i|^2/2\) and \(|b_i|^2/2\) are the incident and reflected powers at port \(i\). The \(S\)-matrix relates \(\mathbf{b}\) to \(\mathbf{a}\):

\[ \mathbf{b} = [S]\,\mathbf{a}, \qquad S_{ij} = \frac{b_i}{a_j}\bigg|_{a_k=0,\,k\neq j}. \]

The element \(S_{ij}\) is the ratio of outgoing wave amplitude at port \(i\) to incoming wave amplitude at port \(j\), with all other ports matched. In particular, \(S_{11}\) is the input reflection coefficient and \(S_{21}\) is the forward transmission coefficient.

3.3 Vector Network Analyzers

A Vector Network Analyzer (VNA) measures the complex \(S\)-parameters of a device under test over a range of frequencies. It generates a stimulus at one port, measures the amplitude and phase of incident and transmitted/reflected signals, and displays \(S\)-parameters on a Smith Chart or in Cartesian format. Calibration (e.g., SOLT: Short, Open, Load, Through) removes systematic errors due to cables, connectors, and the instrument itself.

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Chapter 4: Time-Varying Electromagnetics and Maxwell’s Equations

4.1 Faraday’s Law of Electromagnetic Induction

Faraday’s experimental discovery (1831) is that a changing magnetic flux through a circuit induces an electromotive force (EMF). The integral form is

\[ \mathcal{E} = \oint_C \mathbf{E} \cdot d\boldsymbol{\ell} = -\frac{d\Phi_B}{dt} = -\frac{d}{dt}\int_S \mathbf{B} \cdot d\mathbf{S}, \]

where \(S\) is any surface bounded by the closed contour \(C\). Using Stokes’ theorem and assuming a stationary surface,

\[ \int_S \left(\nabla \times \mathbf{E} + \frac{\partial \mathbf{B}}{\partial t}\right) \cdot d\mathbf{S} = 0. \]

Since this holds for any surface, we obtain the differential form of Faraday’s law:

\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}. \]

This equation, together with Ampere’s law (modified by Maxwell), forms the heart of electromagnetic wave theory.

4.2 Ampere’s Law with Maxwell’s Displacement Current

The static Ampere’s law, \(\nabla \times \mathbf{H} = \mathbf{J}\), is inconsistent with charge conservation in time-varying situations. Taking the divergence of both sides gives \(\nabla \cdot \mathbf{J} = 0\), but the continuity equation requires \(\nabla \cdot \mathbf{J} = -\partial\rho_v/\partial t\), which need not vanish. Maxwell (1865) resolved this by adding the displacement current density \(\mathbf{J}_d = \partial\mathbf{D}/\partial t\):

\[ \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}. \]

The displacement current is not a physical current of charges; it is the time rate of change of the electric flux density \(\mathbf{D} = \varepsilon\mathbf{E}\). Its existence is verified by electromagnetic radiation — the very phenomenon it predicts.

4.3 Maxwell’s Equations in Full

The four Maxwell equations in differential form for a linear, isotropic medium with constitutive relations \(\mathbf{D} = \varepsilon\mathbf{E}\), \(\mathbf{B} = \mu\mathbf{H}\), and Ohm’s law \(\mathbf{J} = \sigma\mathbf{E}\) are:

\[ \nabla \cdot \mathbf{D} = \rho_v \quad\text{(Gauss's law for electric fields)}, \]\[ \nabla \cdot \mathbf{B} = 0 \quad\text{(Gauss's law for magnetic fields — no magnetic monopoles)}, \]\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \quad\text{(Faraday's law)}, \]\[ \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} \quad\text{(Ampere's law with displacement current)}. \]

The integral forms follow by applying the divergence and Stokes theorems:

\[ \oint_S \mathbf{D}\cdot d\mathbf{S} = Q_\text{enc}, \qquad \oint_S \mathbf{B}\cdot d\mathbf{S} = 0, \]\[ \oint_C \mathbf{E}\cdot d\boldsymbol{\ell} = -\frac{d}{dt}\int_S \mathbf{B}\cdot d\mathbf{S}, \qquad \oint_C \mathbf{H}\cdot d\boldsymbol{\ell} = I_\text{enc} + \frac{d}{dt}\int_S \mathbf{D}\cdot d\mathbf{S}. \]

4.4 Boundary Conditions

At an interface between two media (1 and 2), the boundary conditions derived from Maxwell’s equations in integral form are:

These four conditions are not independent — any two of the tangential conditions imply the other two for time-varying fields. They are indispensable for matching wave solutions at interfaces.

4.5 Time-Harmonic Fields and Phasor Notation

For sinusoidal steady-state analysis at angular frequency \(\omega\), every field quantity is written as

\[ \mathbf{E}(\mathbf{r}, t) = \text{Re}\left[\widetilde{\mathbf{E}}(\mathbf{r})\,e^{j\omega t}\right], \]

where \(\widetilde{\mathbf{E}}(\mathbf{r})\) is the complex phasor amplitude. The operation \(\partial/\partial t \to j\omega\) converts PDEs into algebraic or ordinary differential equations. Maxwell’s equations in phasor form (source-free region, \(\rho_v = 0\), \(\mathbf{J} = \sigma\mathbf{E}\)) become:

\[ \nabla \cdot \widetilde{\mathbf{E}} = 0, \quad \nabla \cdot \widetilde{\mathbf{H}} = 0, \]\[ \nabla \times \widetilde{\mathbf{E}} = -j\omega\mu\,\widetilde{\mathbf{H}}, \]\[ \nabla \times \widetilde{\mathbf{H}} = (\sigma + j\omega\varepsilon)\,\widetilde{\mathbf{E}} = j\omega\varepsilon_c\,\widetilde{\mathbf{E}}, \]

where the complex permittivity is \(\varepsilon_c = \varepsilon' - j\varepsilon'' = \varepsilon - j\sigma/\omega\).

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Chapter 5: Plane Wave Propagation

5.1 The Wave Equation

Starting from the phasor-form Maxwell equations in a homogeneous, source-free medium, take the curl of Faraday’s law:

\[ \nabla \times (\nabla \times \widetilde{\mathbf{E}}) = -j\omega\mu\,\nabla \times \widetilde{\mathbf{H}} = -j\omega\mu\,(\sigma + j\omega\varepsilon)\,\widetilde{\mathbf{E}}. \]

Using the vector identity \(\nabla \times (\nabla \times \widetilde{\mathbf{E}}) = \nabla(\nabla\cdot\widetilde{\mathbf{E}}) - \nabla^2\widetilde{\mathbf{E}}\) and \(\nabla\cdot\widetilde{\mathbf{E}} = 0\):

\[ \nabla^2\widetilde{\mathbf{E}} - \gamma^2\widetilde{\mathbf{E}} = \mathbf{0}, \]

where

\[ \gamma^2 = j\omega\mu(\sigma + j\omega\varepsilon) = j\omega\mu\cdot j\omega\varepsilon_c. \]

This is the vector Helmholtz equation. An identical equation holds for \(\widetilde{\mathbf{H}}\). The propagation constant \(\gamma = \alpha + j\beta\) generalizes what we found for transmission lines — not by coincidence, since both systems describe wave propagation governed by Maxwell’s equations.

5.2 Uniform Plane Waves in Lossless Media

Consider a plane wave propagating in the \(+z\) direction in a lossless medium (\(\sigma = 0\)), polarized in \(\hat{x}\). The Helmholtz equation reduces to

\[ \frac{d^2\widetilde{E}_x}{dz^2} + k^2\widetilde{E}_x = 0, \qquad k = \omega\sqrt{\mu\varepsilon} \equiv \beta. \]

The forward-propagating solution is

\[ \widetilde{E}_x(z) = E_0\,e^{-jkz}, \]

with corresponding time-domain field

\[ E_x(z,t) = E_0\cos(\omega t - kz). \]

From Faraday’s law, the associated magnetic field is

\[ \widetilde{H}_y(z) = \frac{E_0}{\eta}\,e^{-jkz}, \qquad \eta = \sqrt{\frac{\mu}{\varepsilon}}. \]

Here \(\eta\) is the intrinsic impedance of the medium, the analogue of the characteristic impedance \(Z_0\) for transmission lines. For free space, \(\eta_0 = \sqrt{\mu_0/\varepsilon_0} \approx 377\,\Omega\).

Several features of plane waves in lossless media:

• \(\mathbf{E}\) and \(\mathbf{H}\) are mutually perpendicular and both perpendicular to the direction of propagation \(\hat{k}\). This is a transverse electromagnetic (TEM) wave.
• The ratio \(|\mathbf{E}|/|\mathbf{H}| = \eta\) at every point.
• The fields are in phase: \(\mathbf{E}\) and \(\mathbf{H}\) reach their peaks and zeros simultaneously.

5.3 Plane Waves in Lossy Media

For a lossy medium with conductivity \(\sigma \neq 0\), the complex propagation constant is

\[ \gamma = \alpha + j\beta = j\omega\sqrt{\mu\varepsilon_c} = j\omega\sqrt{\mu\varepsilon\left(1 - j\frac{\sigma}{\omega\varepsilon}\right)}, \]

yielding the explicit forms

\[ \alpha = \omega\sqrt{\frac{\mu\varepsilon}{2}\left[\sqrt{1 + \left(\frac{\sigma}{\omega\varepsilon}\right)^2} - 1\right]}, \]\[ \beta = \omega\sqrt{\frac{\mu\varepsilon}{2}\left[\sqrt{1 + \left(\frac{\sigma}{\omega\varepsilon}\right)^2} + 1\right]}. \]

The time-domain electric field decays as

\[ E_x(z,t) = E_0\,e^{-\alpha z}\cos(\omega t - \beta z). \]

The amplitude decays exponentially with distance. The intrinsic impedance becomes complex:

\[ \eta_c = \sqrt{\frac{j\omega\mu}{\sigma + j\omega\varepsilon}} = |\eta_c|\,e^{j\theta_\eta}, \]

and the magnetic field lags the electric field by phase angle \(\theta_\eta > 0\).

5.3.1 Limiting Cases

Good dielectric (low loss, \(\sigma/\omega\varepsilon \ll 1\)):

\[ \alpha \approx \frac{\sigma}{2}\sqrt{\frac{\mu}{\varepsilon}}, \qquad \beta \approx \omega\sqrt{\mu\varepsilon}, \qquad \eta_c \approx \sqrt{\frac{\mu}{\varepsilon}}. \]

Good conductor (\(\sigma/\omega\varepsilon \gg 1\)):

\[ \alpha \approx \beta \approx \sqrt{\frac{\omega\mu\sigma}{2}}, \qquad \eta_c \approx (1+j)\sqrt{\frac{\omega\mu}{2\sigma}}. \]

5.3.2 Skin Depth

The skin depth \(\delta_s\) is the distance over which the field amplitude decays to \(e^{-1}\) of its surface value:

\[ \delta_s = \frac{1}{\alpha}. \]

For a good conductor, \(\delta_s = \sqrt{2/(\omega\mu\sigma)}\). For copper (\(\sigma = 5.8\times10^7\,\text{S/m}\)) at \(1\,\text{GHz}\), \(\delta_s \approx 2.1\,\mu\text{m}\). This is why high-frequency currents flow only in a thin layer near the conductor surface — the skin effect. The resistance per square of a conductor surface is the surface resistance \(R_s = 1/(\sigma\delta_s) = \sqrt{\omega\mu/(2\sigma)}\).

5.4 Group Velocity and Dispersion

In a dispersive medium, \(\beta\) is a nonlinear function of \(\omega\). Different frequency components of a wavepacket travel at different speeds, causing spreading. The group velocity, the speed of energy transport for a narrowband signal, is

\[ u_g = \left(\frac{d\beta}{d\omega}\right)^{-1}. \]

For a lossless nondispersive medium, \(u_g = u_p\). In a plasma (ionosphere), \(\beta = \sqrt{(\omega^2 - \omega_p^2)/c^2}\), giving \(u_p u_g = c^2\) — the phase velocity exceeds \(c\) while the group velocity (carrying information) is less than \(c\).

5.5 Wave Polarization

The polarization of a plane wave describes the trajectory traced by the electric field vector at a fixed point in space as time progresses.

Polarization is crucial in antenna design, optical fiber communications, and radar systems. Circular polarization is used in satellite links to reduce Faraday rotation effects in the ionosphere.

5.6 The Poynting Vector and Power Density

The electromagnetic power flow is described by the Poynting vector. The instantaneous power density is

\[ \mathbf{S}(t) = \mathbf{E}(t) \times \mathbf{H}(t) \quad [\text{W/m}^2]. \]

For time-harmonic fields, the time-averaged Poynting vector is

\[ \mathbf{S}_\text{av} = \frac{1}{2}\text{Re}[\widetilde{\mathbf{E}} \times \widetilde{\mathbf{H}}^*]. \]

For a plane wave in a lossless medium with \(\widetilde{\mathbf{E}} = E_0 e^{-jkz}\hat{x}\) and \(\widetilde{\mathbf{H}} = (E_0/\eta)e^{-jkz}\hat{y}\):

\[ \mathbf{S}_\text{av} = \frac{1}{2}\text{Re}\left[E_0\hat{x} \times \frac{E_0^*}{\eta}\hat{y}\right] = \frac{|E_0|^2}{2\eta}\hat{z}. \]

The power density is proportional to the square of the field amplitude and inversely proportional to the medium impedance.

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Chapter 6: Reflection and Transmission at Planar Boundaries

6.1 Normal Incidence

Consider a plane wave in medium 1 (\(\mu_1, \varepsilon_1, \eta_1\)) incident normally (along \(\hat{z}\)) on a planar interface at \(z = 0\) with medium 2 (\(\mu_2, \varepsilon_2, \eta_2\)).

The incident, reflected, and transmitted fields are:

\[ \widetilde{\mathbf{E}}^i = E_0^i\,e^{-j k_1 z}\hat{x}, \qquad \widetilde{\mathbf{H}}^i = \frac{E_0^i}{\eta_1}\,e^{-j k_1 z}\hat{y}, \]\[ \widetilde{\mathbf{E}}^r = E_0^r\,e^{+j k_1 z}\hat{x}, \qquad \widetilde{\mathbf{H}}^r = -\frac{E_0^r}{\eta_1}\,e^{+j k_1 z}\hat{y}, \]\[ \widetilde{\mathbf{E}}^t = E_0^t\,e^{-j k_2 z}\hat{x}, \qquad \widetilde{\mathbf{H}}^t = \frac{E_0^t}{\eta_2}\,e^{-j k_2 z}\hat{y}. \]

Applying the boundary conditions (tangential \(\mathbf{E}\) and \(\mathbf{H}\) continuous at \(z = 0\)):

\[ E_0^i + E_0^r = E_0^t, \]\[ \frac{E_0^i}{\eta_1} - \frac{E_0^r}{\eta_1} = \frac{E_0^t}{\eta_2}. \]

Solving these simultaneously:

\[ \Gamma = \frac{E_0^r}{E_0^i} = \frac{\eta_2 - \eta_1}{\eta_2 + \eta_1}, \qquad \tau = \frac{E_0^t}{E_0^i} = \frac{2\eta_2}{\eta_2 + \eta_1}. \]

These are the reflection and transmission coefficients for normal incidence. Note the consistency relation \(1 + \Gamma = \tau\), which follows directly from the boundary condition on \(\mathbf{E}\). Power conservation requires \(1 - |\Gamma|^2 = (\eta_1/\eta_2)|\tau|^2\) (for lossless media).

6.2 Oblique Incidence: Geometry and Snell’s Law

For a plane wave incident at angle \(\theta_i\) (measured from the normal to the interface), the phase matching condition at the boundary requires that all phase velocities along the interface be equal. This gives Snell’s law of refraction:

\[ k_1\sin\theta_i = k_2\sin\theta_t \quad\Longrightarrow\quad n_1\sin\theta_i = n_2\sin\theta_t, \]

where \(n = \sqrt{\mu_r\varepsilon_r}\) is the refractive index, and \(\theta_t\) is the transmission (refraction) angle. The law of reflection is \(\theta_r = \theta_i\). These two laws are necessary conditions; the amplitude coefficients (Fresnel equations) follow from the full boundary conditions.

6.3 Fresnel Equations for TE Polarization

For TE (s-) polarization, the electric field is perpendicular to the plane of incidence (the plane containing the incident wave vector and the surface normal). Taking the plane of incidence as the \(xz\)-plane with the interface at \(z = 0\):

\[ \widetilde{\mathbf{E}}^i = E_0^i\,e^{-j(k_{1x}x + k_{1z}z)}\hat{y}, \]

where \(k_{1x} = k_1\sin\theta_i\) and \(k_{1z} = k_1\cos\theta_i\). Matching tangential \(\mathbf{E}\) and tangential \(\mathbf{H}\) at \(z = 0\):

\[ \Gamma_\text{TE} = \frac{\eta_2\cos\theta_i - \eta_1\cos\theta_t}{\eta_2\cos\theta_i + \eta_1\cos\theta_t}, \]\[ \tau_\text{TE} = \frac{2\eta_2\cos\theta_i}{\eta_2\cos\theta_i + \eta_1\cos\theta_t}. \]

6.4 Fresnel Equations for TM Polarization

For TM (p-) polarization, the magnetic field is perpendicular to the plane of incidence (equivalently, the electric field lies in the plane of incidence):

\[ \Gamma_\text{TM} = \frac{\eta_2\cos\theta_t - \eta_1\cos\theta_i}{\eta_2\cos\theta_t + \eta_1\cos\theta_i}, \]\[ \tau_\text{TM} = \frac{2\eta_2\cos\theta_i}{\eta_2\cos\theta_t + \eta_1\cos\theta_i}. \]

These reduce to the normal incidence results when \(\theta_i = \theta_t = 0\).

6.5 Brewster’s Angle

For TM polarization, \(\Gamma_\text{TM} = 0\) when the numerator vanishes:

\[ \eta_2\cos\theta_t = \eta_1\cos\theta_i. \]

For non-magnetic media (\(\mu_1 = \mu_2 = \mu_0\)), this combined with Snell’s law yields the Brewster angle:

\[ \theta_B = \arctan\left(\frac{n_2}{n_1}\right). \]

At \(\theta_i = \theta_B\), TM-polarized light is completely transmitted — the reflected wave vanishes. TE polarization has no such Brewster angle for non-magnetic materials. This effect is exploited in polarizing filters and laser Brewster windows.

6.6 Total Internal Reflection

When a wave travels from a denser to a less dense medium (\(n_1 > n_2\)), Snell’s law predicts

\[ \sin\theta_t = \frac{n_1}{n_2}\sin\theta_i. \]

Since \(n_1/n_2 > 1\), \(\sin\theta_t > \sin\theta_i\), and as \(\theta_i\) increases, \(\theta_t\) reaches \(90^\circ\) at the critical angle:

\[ \theta_c = \arcsin\left(\frac{n_2}{n_1}\right). \]

For \(\theta_i > \theta_c\), there is no real solution for \(\theta_t\), and \(|\Gamma| = 1\) — total internal reflection (TIR). No power is transmitted into medium 2, but an evanescent wave propagates along the interface and decays exponentially perpendicular to it.

The evanescent wave has the form \(e^{-\alpha_2 z}e^{-jk_x x}\) where \(\alpha_2 = k_1\sqrt{\sin^2\theta_i - (n_2/n_1)^2} > 0\). It carries no time-average power in the \(z\)-direction.

6.7 Reflection from Multiple Layers

For a slab of thickness \(d\) between two half-spaces, the analysis generalizes via the transmission line analogy. The input impedance looking into the slab from medium 1 is

\[ Z_\text{in} = \eta_2\frac{\eta_3 + j\eta_2\tan(k_{2z}d)}{\eta_2 + j\eta_3\tan(k_{2z}d)}, \]

where \(k_{2z} = k_2\cos\theta_t\) and \(\eta_3\) is the impedance of medium 3. The overall reflection coefficient is then \(\Gamma = (Z_\text{in} - \eta_1)/(Z_\text{in} + \eta_1)\).

This framework underpins the design of anti-reflection (AR) coatings (used on lenses and solar cells), high-reflectance (HR) coatings, and Bragg reflectors (used in vertical-cavity surface-emitting lasers, VCSELs, and distributed Bragg reflector filters in fiber optics).

Anti-reflection coating: A quarter-wave layer (\(k_{2z}d = \pi/2\)) with \(\eta_2 = \sqrt{\eta_1\eta_3}\) (equivalently \(n_2 = \sqrt{n_1 n_3}\) for normal incidence on non-magnetic media) gives \(Z_\text{in} = \eta_3\), so \(\Gamma = 0\) at the design wavelength.

Bragg reflector: Alternating quarter-wave layers of high and low refractive index materials create constructive interference of reflections. With \(N\) periods, the peak reflectivity approaches unity exponentially in \(N\), with a bandwidth (stop-band) determined by the index contrast.

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Chapter 7: Transmission Lines — Advanced Topics

7.1 Transients on Transmission Lines

The transmission line equations, treated as PDEs in the time domain for a lossless line, admit wave solutions propagating at \(u_p = 1/\sqrt{L'C'}\). When a step voltage \(V_s\) is switched onto a line of characteristic impedance \(Z_0\) at \(t = 0\), an incident wave of amplitude \(V^+ = V_s Z_0/(Z_s + Z_0)\) launches from the source end and propagates toward the load. Upon reaching the load, a reflected wave of amplitude \(\Gamma_L V^+\) returns, and upon reaching the source, a reflection \(\Gamma_s \Gamma_L V^+\) re-launches. The steady-state voltage is reached after multiple reflections; the solution is most easily tracked via a bounce diagram.

The source reflection coefficient is \(\Gamma_s = (Z_s - Z_0)/(Z_s + Z_0)\). For a matched source (\(Z_s = Z_0\)), \(\Gamma_s = 0\) and steady state is reached after one round trip.

7.2 Microstrip Lines

Microstrip is the most widely used planar transmission line in microwave integrated circuits. A conductor strip of width \(w\) is separated from a ground plane by a dielectric substrate of thickness \(h\) and relative permittivity \(\varepsilon_r\). The electromagnetic mode supported is quasi-TEM: the fields are approximately TEM but not exactly so, because part of the field exists in air above the strip. An effective permittivity

\[ \varepsilon_\text{eff} = \frac{\varepsilon_r + 1}{2} + \frac{\varepsilon_r - 1}{2}\left(1 + \frac{12h}{w}\right)^{-1/2} \]

captures the mixed-media effect, with \(1 < \varepsilon_\text{eff} < \varepsilon_r\). The characteristic impedance and effective permittivity determine the phase velocity \(u_p = c/\sqrt{\varepsilon_\text{eff}}\) and thus the guided wavelength \(\lambda_g = \lambda_0/\sqrt{\varepsilon_\text{eff}}\).

7.3 Source and Load Mismatch

In a system with source impedance \(Z_s\) connected to a transmission line of characteristic impedance \(Z_0\) terminated in load \(Z_L\), the power delivered to the load is

\[ P_L = P_\text{avs}(1 - |\Gamma_s|^2)(1 - |\Gamma_\text{in}|^2)\cdot\frac{1}{|1 - \Gamma_s\Gamma_\text{in}|^2}, \]

where \(P_\text{avs}\) is the available power from the source and \(\Gamma_\text{in}\) is the input reflection coefficient of the loaded line. Maximum power transfer requires conjugate matching: \(Z_\text{in} = Z_s^*\).

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Chapter 8: Electromagnetic Wave Polarization and Power — Extended Treatment

8.1 Stokes Parameters

The polarization state of a wave can be described by four real quantities called the Stokes parameters \((I, Q, U, V)\):

\[ I = |E_{x0}|^2 + |E_{y0}|^2 \quad\text{(total intensity)}, \]\[ Q = |E_{x0}|^2 - |E_{y0}|^2 \quad\text{(preference for }x\text{ vs. }y\text{ linear)}, \]\[ U = 2|E_{x0}||E_{y0}|\cos\delta \quad\text{(preference for 45° linear)}, \]\[ V = 2|E_{x0}||E_{y0}|\sin\delta \quad\text{(circularity)}. \]

For a fully polarized wave, \(I^2 = Q^2 + U^2 + V^2\). For unpolarized (natural) light, \(Q = U = V = 0\). Stokes parameters are directly measurable with intensity detectors and polarizers, making them central to remote sensing and optical communications.

8.2 Power Budget: Reflection and Absorption

For a plane wave impinging on a planar interface between media 1 and 2, define the reflectance \(R\) and transmittance \(T\) as fractions of incident power:

\[ R = |\Gamma|^2, \qquad T = 1 - R \quad\text{(for lossless media)}. \]

For TE and TM polarization at oblique incidence, the transmittances are

\[ T_\text{TE} = \frac{\eta_1}{\eta_2}\frac{\cos\theta_t}{\cos\theta_i}|\tau_\text{TE}|^2, \qquad T_\text{TM} = \frac{\eta_1}{\eta_2}\frac{\cos\theta_t}{\cos\theta_i}|\tau_\text{TM}|^2. \]

The factor \(\cos\theta_t/\cos\theta_i\) accounts for the change in beam cross-sectional area upon refraction.

8.3 Radiation Pressure

The electromagnetic field exerts a radiation pressure on matter. The time-average radiation pressure on a perfectly absorbing surface is

\[ P_\text{rad} = \frac{S_\text{av}}{c} = \frac{|E_0|^2}{2\eta_0 c}. \]

On a perfectly reflecting mirror, the radiation pressure doubles. Although tiny at ordinary intensities, radiation pressure is significant in solar sailing, laser trapping of atoms, and optomechanics.

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Chapter 9: Maxwell’s Equations in Matter

9.1 Electric Fields in Matter

9.1.1 Dielectrics and Polarization

In a dielectric, applied electric fields displace bound positive and negative charges to create electric dipoles. The macroscopic effect is described by the polarization \(\mathbf{P}\) (dipole moment per unit volume). The electric flux density is

\[ \mathbf{D} = \varepsilon_0\mathbf{E} + \mathbf{P} = \varepsilon_0(1 + \chi_e)\mathbf{E} = \varepsilon_0\varepsilon_r\mathbf{E} = \varepsilon\mathbf{E}, \]

where \(\chi_e\) is the electric susceptibility and \(\varepsilon_r = 1 + \chi_e\) is the relative permittivity. Bound surface and volume charge densities from the polarization are \(\rho_{bs} = \mathbf{P}\cdot\hat{n}\) and \(\rho_b = -\nabla\cdot\mathbf{P}\).

9.1.2 Conductors

In a conductor, free charges redistribute to oppose internal electric fields. In electrostatic equilibrium, the interior field is zero, all excess charge resides on the surface, and the surface is an equipotential. At microwave frequencies, the field penetrates only to depth \(\delta_s\) — the skin effect governs the current distribution and loss.

9.2 Magnetic Fields in Matter

9.2.1 Magnetic Materials

Materials respond to magnetic fields via three mechanisms: diamagnetism, paramagnetism, and ferromagnetism. For linear magnetic materials, the magnetization \(\mathbf{M}\) is proportional to \(\mathbf{H}\):

\[ \mathbf{M} = \chi_m\mathbf{H}, \qquad \mathbf{B} = \mu_0(\mathbf{H} + \mathbf{M}) = \mu_0(1 + \chi_m)\mathbf{H} = \mu_0\mu_r\mathbf{H} = \mu\mathbf{H}. \]

For most engineering materials at microwave frequencies, \(\mu_r \approx 1\). Ferrites are magnetically lossy but non-conducting, making them useful for circulators and isolators in microwave circuits.

9.3 Energy Stored in Electromagnetic Fields

The electric and magnetic energy densities are:

\[ w_e = \frac{1}{2}\mathbf{D}\cdot\mathbf{E} = \frac{1}{2}\varepsilon|E|^2, \qquad w_m = \frac{1}{2}\mathbf{B}\cdot\mathbf{H} = \frac{1}{2}\mu|H|^2. \]

The total electromagnetic energy density in a lossless medium is \(w = w_e + w_m\). For a plane wave in a lossless medium, \(w_e = w_m\) — the electric and magnetic energy densities are equal. The time-averaged energy transport velocity is \(u_\text{energy} = S_\text{av}/\langle w\rangle = u_p\) for a lossless, non-dispersive medium.

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Chapter 10: Guided Waves and Waveguides

10.1 From Plane Waves to Guided Waves

In free space, plane waves propagate in any direction. When conducting boundaries confine the wave, only certain discrete field distributions — called modes — can exist. Each mode has a characteristic field pattern, and modes with insufficient frequency cannot propagate — they are cut off.

The simplest guiding structure beyond the transmission line is the rectangular metallic waveguide, which supports TE and TM modes. Unlike transmission lines, single-conductor waveguides cannot support a TEM mode (which would require both a center conductor and a return path), explaining why a hollow metallic pipe does not guide DC or low-frequency signals.

10.2 Rectangular Waveguide Modes

Consider a rectangular waveguide with perfectly conducting walls, internal dimensions \(a \times b\) (with \(a > b\)), filled with a lossless medium. The waveguide axis is along \(z\). All fields vary as \(e^{-j\beta_z z}\) in the propagation direction.

10.2.1 TE Modes

For TE\(_{mn}\) modes, \(E_z = 0\) everywhere. The longitudinal magnetic field satisfies

\[ \nabla_t^2 H_z + k_c^2 H_z = 0, \]

where \(k_c^2 = k^2 - \beta_z^2\) and \(\nabla_t^2\) is the transverse Laplacian. With the boundary condition that the normal derivative of \(H_z\) vanishes at perfectly conducting walls, the solution is

\[ H_z = H_0\cos\left(\frac{m\pi x}{a}\right)\cos\left(\frac{n\pi y}{b}\right)e^{-j\beta_z z}, \]

for integers \(m, n \geq 0\) (not both zero). The transverse fields are derived from \(H_z\) via

\[ \mathbf{E}_t = \frac{j\omega\mu}{k_c^2}(\hat{z}\times\nabla_t H_z), \qquad \mathbf{H}_t = \frac{-j\beta_z}{k_c^2}\nabla_t H_z. \]

10.2.2 TM Modes

For TM\(_{mn}\) modes, \(H_z = 0\). The longitudinal electric field satisfies the same equation:

\[ \nabla_t^2 E_z + k_c^2 E_z = 0, \]

but now \(E_z = 0\) on all walls, giving

\[ E_z = E_0\sin\left(\frac{m\pi x}{a}\right)\sin\left(\frac{n\pi y}{b}\right)e^{-j\beta_z z}, \]

with \(m \geq 1\), \(n \geq 1\) (both must be nonzero, otherwise \(E_z\) vanishes identically). The transverse fields follow analogously.

10.2.3 Cutoff Frequency and Propagation

The cutoff wavenumber for mode \((m, n)\) is

\[ k_{c,mn} = \pi\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2}. \]

The cutoff frequency is

\[ f_{c,mn} = \frac{k_{c,mn}}{2\pi\sqrt{\mu\varepsilon}} = \frac{c}{2\sqrt{\mu_r\varepsilon_r}}\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2}. \]

The phase constant is

\[ \beta_z = \sqrt{k^2 - k_{c,mn}^2}, \]

which is real (propagating mode) when \(f > f_{c,mn}\) and imaginary (evanescent mode) when \(f < f_{c,mn}\).

10.2.4 Phase and Group Velocity in Waveguide

Unlike a transmission line, the waveguide is dispersive even with a lossless, non-dispersive filling:

\[ u_p = \frac{\omega}{\beta_z} = \frac{u}{\sqrt{1 - (f_c/f)^2}} > u, \]\[ u_g = \frac{d\omega}{d\beta_z} = u\sqrt{1 - (f_c/f)^2} < u, \]

where \(u = 1/\sqrt{\mu\varepsilon}\) is the speed of light in the filling medium. At cutoff, \(u_p \to \infty\) and \(u_g \to 0\). The product \(u_p u_g = u^2\) holds generally.

10.2.5 Wave Impedance

The wave impedance for TE modes is

\[ Z_\text{TE} = \frac{E_t}{H_t} = \frac{\eta}{\sqrt{1 - (f_c/f)^2}} > \eta, \]

and for TM modes,

\[ Z_\text{TM} = \eta\sqrt{1 - (f_c/f)^2} < \eta. \]

At frequencies well above cutoff, both approach the plane-wave intrinsic impedance \(\eta\). The frequency dependence of the wave impedance means that waveguide transitions and connections must be carefully engineered.

10.3 Field Patterns of the TE\(_{10}\) Mode

For the TE\(_{10}\) mode in a waveguide with \(a\)-dimension along \(x\) and \(b\)-dimension along \(y\):

\[ H_z = H_0\cos\left(\frac{\pi x}{a}\right)e^{-j\beta z}, \qquad E_y = -\frac{j\omega\mu a}{\pi}H_0\sin\left(\frac{\pi x}{a}\right)e^{-j\beta z}, \]\[ H_x = \frac{j\beta a}{\pi}H_0\sin\left(\frac{\pi x}{a}\right)e^{-j\beta z}, \qquad E_x = H_y = 0. \]

The electric field has only a \(y\)-component that is maximum at the center (\(x = a/2\)) and zero at the side walls — satisfying the boundary condition that tangential \(\mathbf{E}\) vanishes at a perfect conductor. The TE\(_{10}\) mode is linearly polarized.

10.4 Cavity Resonators

A cavity resonator is a closed metallic enclosure that supports standing electromagnetic waves at discrete resonant frequencies. For a rectangular cavity of dimensions \(a \times b \times d\), the resonant frequencies of the TE\(_{mnp}\) and TM\(_{mnp}\) modes are

\[ f_{mnp} = \frac{c}{2\sqrt{\mu_r\varepsilon_r}}\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2 + \left(\frac{p}{d}\right)^2}. \]

The quality factor \(Q\) of a resonator is

\[ Q = \omega_0\frac{W_\text{stored}}{P_\text{loss}}, \]

where \(W_\text{stored}\) is the total stored electromagnetic energy and \(P_\text{loss}\) is the power dissipated (in conductor walls and dielectric). For a metallic cavity with walls of conductivity \(\sigma\), the \(Q\) due to conductor loss is \(Q_c = V_\text{eff}/(R_s \cdot A_\text{eff})\), where \(R_s\) is the surface resistance. Cavity resonators achieve \(Q\) values of \(10^3\) to \(10^5\) — far higher than lumped-element resonators — making them useful as frequency references and filters.

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Chapter 11: Electrostatics Review

11.1 Coulomb’s Law and the Electric Field

The electric field \(\mathbf{E}\) at a point \(\mathbf{r}\) is the force per unit positive test charge. For a continuous charge distribution with volume charge density \(\rho_v(\mathbf{r}')\):

\[ \mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0}\int_V \rho_v(\mathbf{r}')\frac{\hat{\mathscr{R}}}{\mathscr{R}^2}\,dV', \]

where \(\boldsymbol{\mathscr{R}} = \mathbf{r} - \mathbf{r}'\) is the vector from source to field point.

11.2 Gauss’s Law

Gauss’s law is most useful when the charge distribution has enough symmetry (spherical, cylindrical, planar) that the magnitude of \(\mathbf{E}\) is constant over an appropriate Gaussian surface.

11.3 Electric Potential

The electric field is conservative (in the static case): \(\nabla \times \mathbf{E} = 0\). Therefore \(\mathbf{E} = -\nabla V\) where the scalar potential \(V\) satisfies Poisson’s equation:

\[ \nabla^2 V = -\frac{\rho_v}{\varepsilon}. \]

In charge-free regions this reduces to Laplace’s equation \(\nabla^2 V = 0\). Solutions to Laplace’s equation satisfying the boundary conditions on conductors give the electrostatic field configuration.

11.4 Capacitance

For two conductors carrying charges \(\pm Q\) with potential difference \(V_{12}\), the capacitance is \(C = Q/V_{12}\). The energy stored is \(W_e = \frac{1}{2}CV^2 = \frac{1}{2}Q^2/C\). Capacitance depends only on geometry and material properties.

11.5 Laplace’s Equation and Boundary Value Problems

Uniqueness: in a bounded region with specified boundary conditions on \(V\) (Dirichlet) or \(\partial V/\partial n\) (Neumann), the solution to Laplace’s equation is unique. Practical solution methods include:

• Separation of variables: assume \(V = X(x)Y(y)Z(z)\); the PDE separates into three ODEs. In Cartesian coordinates, this yields sinusoidal or hyperbolic solutions in each direction.
• Method of images: replace a conductor by image charges that reproduce the same boundary conditions on the conductor surface.

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Chapter 12: Magnetostatics Review

12.1 Biot-Savart Law

The magnetic field due to a steady current element \(Id\boldsymbol{\ell}'\) is

\[ d\mathbf{H} = \frac{I\,d\boldsymbol{\ell}'\times\hat{\mathscr{R}}}{4\pi\mathscr{R}^2}. \]

Integrating over a complete circuit gives the full field. For an infinite straight wire carrying current \(I\) along \(\hat{z}\), Biot-Savart (or Ampere’s law) yields

\[ \mathbf{H} = \frac{I}{2\pi\rho}\hat{\phi}, \]

a fundamental result for coaxial line inductance and two-wire line calculations.

12.2 Ampere’s Law (Static)

For magnetostatics,

\[ \oint_C \mathbf{H}\cdot d\boldsymbol{\ell} = I_\text{enc} \quad\Longleftrightarrow\quad \nabla\times\mathbf{H} = \mathbf{J}. \]

This is the static form — the displacement current term \(\partial\mathbf{D}/\partial t\) is absent, as discussed in Chapter 4.

12.3 Inductance and Magnetic Energy

The inductance of a circuit is defined by \(L = \Phi_\text{total}/I\), where \(\Phi_\text{total} = N\Phi_B\) is the total flux linkage for an \(N\)-turn coil. The energy stored in the magnetic field is

\[ W_m = \frac{1}{2}LI^2 = \frac{1}{2}\int_V \mu|\mathbf{H}|^2\,dV. \]

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Chapter 13: Synthesis — From Circuits to Fields to Waves

13.1 The Electromagnetic Spectrum

Maxwell’s equations predict electromagnetic waves spanning an enormous range of frequencies. The free-space wavelength \(\lambda = c/f\) ranges from kilometers (AM radio) to nanometers (X-rays). The microwave band (\(300\,\text{MHz}\) to \(300\,\text{GHz}\), wavelengths \(1\,\text{m}\) to \(1\,\text{mm}\)) is where the content of this course is most directly applied: radar, wireless communications, satellite links, and microwave imaging.

13.2 The Unification of Electricity, Magnetism, and Optics

One of the profound achievements of 19th-century physics was Maxwell’s demonstration that Faraday’s law and the modified Ampere’s law together imply the existence of self-sustaining electromagnetic waves propagating at speed \(c = 1/\sqrt{\mu_0\varepsilon_0}\). When the numerical values of \(\mu_0\) and \(\varepsilon_0\) were inserted, the result matched the measured speed of light, unifying electromagnetism and optics. Light is an electromagnetic wave; so are radio waves, microwaves, infrared radiation, ultraviolet, X-rays, and gamma rays — differing only in frequency.

13.3 The Telegrapher’s Equations as Maxwell’s Equations in 1D

The transmission line telegrapher’s equations are Maxwell’s equations restricted to a quasi-TEM mode on a two-conductor system. The inductance \(L'\) per unit length captures the energy in the magnetic field between conductors, \(C'\) captures the energy in the electric field, \(R'\) models conductor loss (skin effect), and \(G'\) models dielectric loss. The characteristic impedance \(Z_0 = \sqrt{L'/C'}\) parallels the intrinsic impedance \(\eta = \sqrt{\mu/\varepsilon}\) of a plane wave in the surrounding medium; in fact, for a coaxial line, \(Z_0\) and \(\eta\) are related by the geometry of the cross-section.

13.4 Smith Chart as a Reflection-Plane Map

The Smith Chart provides a conformal map from the normalized impedance half-plane \(\text{Re}(z) \geq 0\) to the unit disk in the \(\Gamma\)-plane. Moving away from the load toward the generator on a lossless line rotates the reflection coefficient at constant radius. Stub matching, quarter-wave transformers, and multi-section matching networks can all be visualized and designed graphically on the Smith Chart, complementing the numerical methods available in modern RF design software.

13.5 Power Engineering vs. Microwave Engineering

At power frequencies (50–60 Hz), the wavelength is \(\approx 5000\,\text{km}\) — far larger than any power grid component. Circuit theory with lumped elements applies. At microwave frequencies, interconnect lengths comparable to \(\lambda/10\) require transmission line treatment, and lengths comparable to \(\lambda\) require full-wave electromagnetic simulation. ECE 375 marks the transition from the lumped-element paradigm of circuits courses to the distributed-parameter and full-wave paradigm of microwave and photonics engineering.

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Appendix A: Key Formulas Reference Sheet

A.1 Transmission Lines

Quantity	Formula
Propagation constant	\(\gamma = \sqrt{(R' + j\omega L')(G' + j\omega C')}\)
Characteristic impedance	\(Z_0 = \sqrt{(R' + j\omega L')/(G' + j\omega C')}\)
Reflection coefficient	\(\Gamma_L = (Z_L - Z_0)/(Z_L + Z_0)\)
Input impedance	\(Z_\text{in} = Z_0(Z_L + jZ_0\tan\beta\ell)/(Z_0 + jZ_L\tan\beta\ell)\)
VSWR	\(S = (1 + \lvert\Gamma\rvert)/(1 - \lvert\Gamma\rvert)\)
Average power	\(P_\text{av} = \lvert V^+\rvert^2(1 - \lvert\Gamma\rvert^2)/(2Z_0)\)

A.2 Plane Waves

Quantity	Lossless	Lossy
Phase constant	\(\beta = \omega\sqrt{\mu\varepsilon}\)	\(\beta = \omega\sqrt{\mu\varepsilon/2}\left[\sqrt{1+({\sigma}/{\omega\varepsilon})^2} + 1\right]^{1/2}\)
Attenuation	\(\alpha = 0\)	\(\alpha = \omega\sqrt{\mu\varepsilon/2}\left[\sqrt{1+({\sigma}/{\omega\varepsilon})^2} - 1\right]^{1/2}\)
Intrinsic impedance	\(\eta = \sqrt{\mu/\varepsilon}\)	\(\eta_c = \sqrt{j\omega\mu/(\sigma + j\omega\varepsilon)}\)
Skin depth (good cond.)	—	\(\delta_s = \sqrt{2/(\omega\mu\sigma)}\)
Phase velocity	\(u_p = 1/\sqrt{\mu\varepsilon}\)	\(u_p = \omega/\beta\)

A.3 Boundary Conditions Summary

\[ \hat{n}\times(\mathbf{E}_1 - \mathbf{E}_2) = 0, \qquad \hat{n}\times(\mathbf{H}_1 - \mathbf{H}_2) = \mathbf{J}_s, \]\[ \hat{n}\cdot(\mathbf{D}_1 - \mathbf{D}_2) = \rho_s, \qquad \hat{n}\cdot(\mathbf{B}_1 - \mathbf{B}_2) = 0. \]

A.4 Fresnel Equations (TE and TM)

\[ \Gamma_\text{TE} = \frac{\eta_2\cos\theta_i - \eta_1\cos\theta_t}{\eta_2\cos\theta_i + \eta_1\cos\theta_t}, \qquad \tau_\text{TE} = \frac{2\eta_2\cos\theta_i}{\eta_2\cos\theta_i + \eta_1\cos\theta_t}, \]\[ \Gamma_\text{TM} = \frac{\eta_2\cos\theta_t - \eta_1\cos\theta_i}{\eta_2\cos\theta_t + \eta_1\cos\theta_i}, \qquad \tau_\text{TM} = \frac{2\eta_2\cos\theta_i}{\eta_2\cos\theta_t + \eta_1\cos\theta_i}. \]

A.5 Rectangular Waveguide TE\(_{mn}\) and TM\(_{mn}\)

\[ k_{c,mn} = \pi\sqrt{(m/a)^2 + (n/b)^2}, \qquad f_{c,mn} = \frac{u}{2}\sqrt{(m/a)^2 + (n/b)^2}, \]\[ \beta_z = \sqrt{k^2 - k_{c,mn}^2}, \qquad Z_\text{TE} = \frac{\eta}{\sqrt{1-(f_c/f)^2}}, \qquad Z_\text{TM} = \eta\sqrt{1-(f_c/f)^2}. \]