Microwave Engineering Foundations

Introduction and Context

Microwave engineering concerns the generation, propagation, control, measurement, and application of electromagnetic energy at frequencies conventionally from approximately 300 MHz to 300 GHz, corresponding to wavelengths from 1 m to 1 mm in free space. The term is historical rather than absolute: modern systems extend below and above this range, and engineering practice often includes millimetre-wave and sub-terahertz technologies.

At microwave frequencies, electromagnetic wavelength becomes comparable with the physical dimensions of circuits, packages, interconnects, antennas, and measurement fixtures. Consequently, voltage and current are no longer uniform throughout a component. Transmission-line and field concepts must supplement, and sometimes replace, ordinary low-frequency circuit models.

Microwave engineering provides the foundation for radar, satellite communication, cellular and wireless networks, navigation, remote sensing, radio astronomy, industrial heating, medical systems, material characterization, and emerging 5G/6G and automotive sensing technologies. A sound foundation requires both field theory and practical judgment concerning losses, dispersion, impedance, power flow, manufacturability, and measurement uncertainty.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Describe the microwave spectrum and identify major frequency bands and applications.

2. State and interpret Maxwell’s equations in differential and integral form.

3. Apply electromagnetic boundary conditions at material interfaces.

4. Explain plane-wave propagation, phase, group velocity, and polarization.

5. Calculate average power flow using the Poynting vector.

6. Explain skin effect, conductor loss, dielectric loss, and dispersion.

7. Determine why lumped-circuit models become inadequate at high frequency.

8. Relate field behavior to practical microwave components and systems.

9. Analyze a basic high-frequency interconnect or waveguide transition.

1. The Microwave Spectrum

Frequency f, wavelength λ, and wave velocity v are related by

v = fλ.

In free space, v = c ≈ 3.00 × 10^8 m/s, so

λ0 = c/f,

where λ0 is free-space wavelength. In a homogeneous, nondispersive dielectric with relative permittivity εr and relative permeability μr,

v = c/√(μrεr),

and λ = λ0/√(μrεr).

These relations assume a uniform medium and sinusoidal steady-state operation. In a dispersive material, phase velocity varies with frequency and the simple interpretation of a single wavelength must be used carefully.

Common microwave designations include:

• UHF: 0.3–1 GHz
• L band: 1–2 GHz
• S band: 2–4 GHz
• C band: 4–8 GHz
• X band: 8–12 GHz
• Ku band: 12–18 GHz
• K band: 18–27 GHz
• Ka band: 27–40 GHz
• Millimetre-wave bands: approximately 30–300 GHz

Band boundaries can differ slightly among standards and industries. Therefore, a design should state the actual operating frequency rather than relying only on a band name.

Applications are strongly influenced by propagation. Lower microwave frequencies generally penetrate vegetation, walls, or the ionosphere more effectively and support longer wavelengths. Higher frequencies provide larger available bandwidth and permit physically smaller antennas, but atmospheric absorption, surface roughness, fabrication tolerances, and conductor loss become more significant. Oxygen absorption is prominent near 60 GHz, while water-vapour absorption has important features near 22 GHz and higher frequencies.

For an aperture antenna, approximate directivity increases as the aperture becomes electrically larger. Thus, at a fixed physical aperture, increasing frequency can produce a narrower beam. This is central to radar imaging, satellite links, and automotive radar.

2. Electromagnetic Foundations: Maxwell’s Equations

Maxwell’s equations describe how electric and magnetic fields are generated and coupled. In differential form, using SI units,

∇ × E = −∂B/∂t,

∇ × H = J + ∂D/∂t,

∇ · D = ρv,

∇ · B = 0.

Here E is electric field intensity (V/m), H is magnetic field intensity (A/m), D is electric flux density (C/m²), B is magnetic flux density (T), J is conduction-current density (A/m²), and ρv is volume charge density (C/m³).

The constitutive relations for a linear, isotropic medium are

D = εE,

B = μH,

J = σE,

where ε is permittivity (F/m), μ is permeability (H/m), and σ is conductivity (S/m). More complex materials may be anisotropic, nonlinear, magnetic, or dispersive; in such cases ε, μ, and σ can be tensors or frequency-dependent quantities.

The first equation is Faraday’s law: a time-varying magnetic field produces a circulating electric field. The second is the Ampère–Maxwell law: magnetic circulation is produced by conduction current and by displacement current ∂D/∂t. The third is Gauss’s electric law, and the fourth expresses the absence of isolated magnetic charge.

Integral Interpretation

The corresponding integral equations are

∮C E · dl = − d/dt ∫S B · dS,

∮C H · dl = ∫S J · dS + d/dt ∫S D · dS,

∮S D · dS = ∫V ρv dv,

∮S B · dS = 0.

The contour C bounds surface S, and V is the volume enclosed by a closed surface. The integral form is especially useful for deriving boundary conditions and analyzing symmetry.

Source-Free Wave Equation

In a homogeneous, source-free, linear medium, ρv = 0 and J = 0. Taking the curl of Faraday’s law and using the vector identity

∇ × (∇ × E) = ∇(∇ · E) − ∇²E,

with ∇ · E = 0, gives

∇²E − με ∂²E/∂t² = 0.

Similarly,

∇²H − με ∂²H/∂t² = 0.

For a sinusoidal field represented by phasors with time dependence e^{jωt}, the electric-field equation becomes

∇²E + k²E = 0,

where k = ω√(με) is the phase constant in a lossless medium and ω = 2πf. In a conducting or lossy dielectric, k is complex and describes both phase progression and amplitude attenuation.

3. Boundary Conditions at Material Interfaces

Boundary conditions determine how fields change across a surface separating two media. Let n̂ point from medium 1 to medium 2, and let ρs be free surface charge density.

The tangential electric field is continuous when no ideal impulsive magnetic surface current is present:

n̂ × (E2 − E1) = 0.

The tangential magnetic field changes according to free surface current density Js:

n̂ × (H2 − H1) = Js.

The normal electric flux density satisfies

n̂ · (D2 − D1) = ρs.

The normal magnetic flux density is continuous:

n̂ · (B2 − B1) = 0.

These conditions follow directly from applying the integral Maxwell equations to an infinitesimally small loop or pillbox crossing the interface. They are essential in analyzing reflection, transmission, waveguides, coaxial connectors, dielectric interfaces, and conductor surfaces.

For a perfect electric conductor (PEC), the interior electric field is zero. At its surface, the tangential electric field must vanish:

E_t = 0.

The magnetic field may be tangential and is related to surface current by

Js = n̂ × H.

A PEC is an idealization. Real conductors have finite conductivity, so fields penetrate a small distance and cause dissipation.

4. Plane-Wave Propagation

Consider a uniform plane wave propagating in the +z direction in a lossless, homogeneous medium. A possible electric field is

E(z,t) = E0 cos(ωt − βz) x̂,

where E0 is field amplitude, β = ω√(με) is phase constant, and x̂ indicates polarization. The associated magnetic field is

H(z,t) = (E0/η) cos(ωt − βz) ŷ,

where

η = √(μ/ε)

is the intrinsic impedance of the medium. In free space, η0 ≈ 376.7 Ω, commonly rounded to 377 Ω.

The vectors E, H, and the direction of propagation are mutually perpendicular. Their orientation follows the right-hand rule. The phase velocity is

vp = ω/β = 1/√(με).

In free space, vp = c. In a nonmagnetic dielectric, vp ≈ c/√εr.

Reflection at a Normal Interface

For a normally incident plane wave moving from medium 1 with impedance η1 to medium 2 with impedance η2, the electric-field reflection coefficient is

Γ = (η2 − η1)/(η2 + η1).

The transmitted electric-field amplitude relative to the incident field is

τE = 2η2/(η2 + η1).

These expressions assume planar, steady-state waves and normal incidence. A nonzero Γ indicates impedance mismatch. If η1 = η2, there is no reflection. For a wave entering a higher-permittivity nonmagnetic dielectric from air, η2 < η1 and Γ is negative, indicating a 180-degree phase reversal of the reflected electric field.

At oblique incidence, reflection depends on polarization, producing TE and TM coefficients and phenomena such as Brewster-angle transmission. Those effects become important in radomes, remote sensing, multilayer coatings, and dielectric windows.

5. Polarization

Polarization describes the time-varying direction and magnitude of the electric-field vector at a fixed point. It is determined by two orthogonal field components and their relative phase.

For

E = x̂ Ex cos(ωt) + ŷ Ey cos(ωt + δ),

linear polarization occurs when δ = 0 or π, because the field direction remains fixed. Circular polarization occurs when Ex = Ey and δ = ±π/2. The sign determines right-hand or left-hand rotation according to the adopted convention; engineering documents should state that convention explicitly. All other combinations generally produce elliptical polarization.

Polarization is important because antennas receive maximum power when their polarization states match. For two linearly polarized antennas with relative angle ψ, the polarization power factor is cos²ψ. A 90-degree mismatch ideally produces zero received power. In practical systems, cross-polarization discrimination is finite because of scattering, asymmetry, multipath, and fabrication imperfections.

Circular polarization is useful in satellite communication because the orientation of a linearly polarized antenna can rotate during propagation or vehicle motion. It is also used in radar and navigation, while dual-polarized systems support polarization diversity and multiple-input multiple-output operation.

6. Power Flow and Energy Conservation

The instantaneous electromagnetic power-flow density is represented by the Poynting vector

S = E × H,

with units W/m². Its direction indicates the local direction of energy transport. For sinusoidal phasor fields, the time-average Poynting vector is

⟨S⟩ = 1/2 Re(E × H*),

where the asterisk denotes complex conjugation. The average power through a surface A is

Pav = ∫A ⟨S⟩ · dA.

The factor 1/2 applies when E and H are peak-amplitude phasors. If root-mean-square phasors are used, the factor is omitted.

Poynting’s theorem expresses conservation of electromagnetic energy:

∇ · S + ∂u/∂t + J · E = 0,

where u is electromagnetic energy density. In a linear, nondispersive medium,

u = 1/2(E · D + B · H).

The term J · E represents power converted into heat or other material processes. In microwave devices, Poynting’s theorem provides a field-based interpretation of insertion loss, conductor heating, dielectric heating, radiation, and stored energy.

Worked Example 1: Power Density of a Plane Wave

A uniform plane wave in free space has a peak electric-field amplitude of 10 V/m. Find its average power density.

For a lossless plane wave, H0 = E0/η0. Therefore,

⟨S⟩ = E0²/(2η0)

= 10²/(2 × 376.7)

≈ 0.133 W/m².

The result assumes a uniform plane wave, free-space propagation, and peak rather than rms field amplitude.

7. Conductors, Skin Effect, and Loss

At low frequency, current may be treated as approximately uniform across a conductor cross-section. At microwave frequencies, electromagnetic fields induce currents concentrated near the conductor surface. This phenomenon is the skin effect.

For a good conductor, the skin depth is

δs = √(2/(ωμσ)) = 1/√(πfμσ),

where f is frequency, μ is conductor permeability, and σ is conductivity. Skin depth is the distance over which field amplitude decreases to 1/e of its surface value. The expression assumes a good conductor, meaning σ is sufficiently large that conduction current dominates displacement current.

For copper at 10 GHz, using σ ≈ 5.8 × 10^7 S/m and μ ≈ μ0 = 4π × 10^−7 H/m,

δs ≈ 0.66 μm.

Thus, bulk conductor dimensions do not eliminate high-frequency loss; surface quality, plating, roughness, current crowding, and geometry become decisive.

The surface resistance of a good conductor is

Rs = 1/(σδs) = √(πfμ/σ),

in ohms per square. It increases as the square root of frequency. Conductor loss is reduced by using high-conductivity materials, smooth surfaces, appropriate plating, wide current paths, and geometries that avoid excessive current concentration. However, plating choices require care: a low-loss surface layer must be sufficiently thick relative to skin depth and compatible with mechanical, thermal, and environmental requirements.

Dielectric loss is often characterized by loss tangent,

tan δ = ε''/ε',

where ε' is the energy-storing component and ε'' represents dielectric dissipation under a common phasor convention. For a low-loss dielectric, attenuation due to dielectric loss approximately increases with frequency and loss tangent, although exact values depend on field distribution and material dispersion.

Total attenuation may include conductor loss, dielectric loss, radiation loss, leakage, connector loss, and mismatch-related effects. These mechanisms should not be conflated: mismatch changes delivered power through reflection, whereas dissipation converts electromagnetic energy into heat.

8. Dispersion and Group Velocity

A medium or structure is dispersive when its phase constant β depends nonlinearly on frequency. The phase velocity is

vp = ω/β,

while group velocity is

vg = dω/dβ.

In a nondispersive medium, vp and vg are equal. In a dispersive medium they generally differ. Group velocity describes the motion of a narrowband wave packet under suitable conditions; it is not universally identical to signal velocity or energy velocity, particularly near strong absorption or anomalous dispersion.

Waveguides are inherently dispersive. For a rectangular waveguide with broad dimension a and operating in the dominant TE10 mode, the cutoff frequency is

fc = 1/(2a√(με)).

For a lossless guide, the propagation constant is

β = ω√(με)√(1 − (fc/f)²), f > fc.

The phase velocity is

vp = 1/√(με) · 1/√(1 − (fc/f)²),

and the group velocity is

vg = 1/√(με) · √(1 − (fc/f)²).

For an air-filled guide, vp exceeds c while vg is less than c; this does not violate relativity because no information is transmitted faster than the appropriate signal velocity. Near cutoff, group velocity approaches zero and attenuation, dispersion, and sensitivity to imperfections become significant.

Single-mode operation is normally preferred. For a rectangular waveguide, the next-mode cutoff must be above the operating frequency, with adequate margin for manufacturing tolerances and perturbations. Waveguide dimensions are therefore selected by balancing bandwidth, power handling, loss, and mechanical constraints.

9. Why Lumped-Circuit Models Fail at Microwave Frequencies

Kirchhoff’s voltage and current laws remain consequences of Maxwell’s equations, but the ideal lumped-element approximation requires that the physical size of a circuit be electrically small. A common engineering criterion is

l ≪ λ,

often interpreted conservatively as l < λ/10 for a simple interconnect. This is a guideline, not a universal boundary. Whether a model is adequate depends on required phase accuracy, impedance variation, bandwidth, and electromagnetic coupling.

Several effects cause failure of lumped models:

1. Distributed phase: voltage and current vary along an interconnect. A transmission line must be described by per-unit-length inductance L′ and capacitance C′, with characteristic impedance approximately Z0 = √(L′/C′) for a lossless line.

2. Parasitic reactance: pads, vias, package leads, and component bodies contribute inductance and capacitance.

3. Radiation and coupling: nearby conductors exchange energy through electric and magnetic fields, creating crosstalk and electromagnetic interference.

4. Skin and proximity effects: current distribution changes resistance and inductance with frequency.

5. Resonance: physical structures may behave as resonators rather than simple capacitors or inductors.

6. Discontinuities: bends, connectors, launches, and changes in cross-section generate reflections.

7. Material dispersion: dielectric constant and loss vary with frequency.

For a transmission line with propagation constant γ = α + jβ, a forward wave varies as e^−γz. Here α is attenuation constant in nepers per metre and β is phase constant in radians per metre. A lumped circuit cannot represent this spatial dependence without being replaced by a distributed model or a sufficiently fine ladder approximation.

At microwave frequencies, designers use scattering parameters rather than relying exclusively on open- and short-circuit impedance measurements. S-parameters describe incident and reflected traveling waves referenced to defined ports and impedances, usually 50 Ω. They are practical because perfect open and short standards are difficult to realize at high frequency, and because power flow and matching are naturally expressed through traveling waves.

10. Practical Microwave Structures

Coaxial Lines

A coaxial line supports a TEM mode over its usable frequency range. Its inner and outer conductors confine fields, reducing radiation and electromagnetic interference. For ideal concentric conductors filled with a homogeneous dielectric,

Z0 = (1/(2π))√(μ/ε) ln(b/a),

where a is inner-conductor radius and b is inner radius of the outer conductor. This expression assumes a uniform, lossless, quasi-TEM structure. At sufficiently high frequencies, higher-order modes and connector discontinuities limit performance.

Microstrip

Microstrip consists of a conducting strip above a dielectric substrate and a ground plane. It is widely used in printed microwave circuits because it supports low-cost integration. Its mode is quasi-TEM because fields occupy both dielectric and air. Effective permittivity, dispersion, conductor roughness, substrate tolerances, and radiation must be considered in precision designs.

Rectangular Waveguide

Rectangular waveguide offers high power handling and low conductor loss relative to many planar structures, particularly at centimetre and millimetre wavelengths. It does not support a TEM mode with a single hollow conductor system; its modes are TE or TM. Waveguide transitions to coaxial or microstrip lines are common sources of mismatch and radiation.

11. Applied Case: A 10 GHz Radar Interconnect

Background

Consider an illustrative design case involving a 10 GHz radar front end. A signal travels from a transceiver module to an antenna through a 50 Ω microstrip route on a printed-circuit board. The route length is 30 mm, and the board has an effective relative permittivity of approximately 3.0.

Problem

The design team initially treats the route as an ideal wire and observes unexpected amplitude reduction, phase error, and variation between prototypes. The objective is to determine whether a lumped model is adequate and identify the main physical causes.

Analysis

The approximate wavelength in the guided structure is

λg ≈ λ0/√εeff

= (3.00 × 10^8 / 10^10)/√3

≈ 17.3 mm.

The 30 mm route is approximately 1.73 guided wavelengths long, so it is clearly not electrically short. Even if the effective permittivity estimate is approximate, the route must be treated as a distributed transmission line.

The propagation delay is approximately

td ≈ l√εeff/c

= 0.03√3/(3.00 × 10^8)

≈ 173 ps.

A small dimensional change can create a significant phase change. The phase shift is approximately 360° l/λg, or about 625° for the nominal route, equivalent to 265° modulo 360°.

The team also identifies copper surface roughness, via transitions, a connector launch, and a sharp bend. At 10 GHz, copper skin depth is well below a micrometre, so current is confined near surfaces. Roughness comparable with relevant electromagnetic penetration dimensions increases effective loss. The connector launch introduces a discontinuity, while the via changes the current-return path and adds inductive and capacitive parasitics.

Approach and Solution

The route is redesigned as a controlled-impedance microstrip with a continuous ground-return path. The bend is replaced by a mitered or curved geometry, and the connector launch is optimized using a three-dimensional electromagnetic simulation. S-parameters are measured with a calibrated vector network analyzer using appropriate calibration planes and fixtures.

The design process includes:

• extracting or measuring substrate permittivity and loss tangent;
• specifying trace width, dielectric thickness, copper thickness, and roughness;
• maintaining a reference-plane continuity;
• using electromagnetic models for the launch and via transition;
• checking insertion loss, return loss, phase, and group delay across the required bandwidth;
• validating simulation with fabricated test coupons.

Outcome

The revised interconnect exhibits lower reflection and more repeatable phase. The system-level radar calibration becomes stable because the interconnect is no longer represented by an unrealistic zero-length wire. The case also demonstrates that a nominally small physical distance can be electrically long when wavelength is only a few millimetres.

Learning Points

The appropriate model is determined by electrical length, not physical length alone. At microwave frequencies, packaging, reference planes, current-return paths, surface finish, and transitions are electromagnetic components. Measurement calibration and model validation are as important as circuit design.

12. Design Practice, Limitations, and Contemporary Directions

Reliable microwave design begins with a clearly defined frequency range, power level, bandwidth, environment, and interface standard. Material data should be obtained at the actual frequency and temperature of operation rather than copied from low-frequency datasheets. Simulations should be checked against analytical limits and measured structures. Mesh convergence, port definitions, boundary conditions, and calibration uncertainty require explicit documentation.

Important limitations include uncertainty in dielectric properties, manufacturing tolerances, conductor roughness models, thermal expansion, connector repeatability, and the validity of ideal boundary assumptions. At millimetre-wave and sub-terahertz frequencies, atmospheric attenuation, surface roughness, packaging, and measurement access become increasingly important.

Modern electromagnetic design uses full-wave numerical methods such as finite-element, finite-difference time-domain, method-of-moments, and spectral-domain techniques. These methods can be integrated with optimization and surrogate modeling. Machine learning and AI can assist in parameter exploration, inverse design, defect detection, and calibration modeling, but they do not replace Maxwell-based validation or physical constraints. Training data may be narrow, and an apparently accurate surrogate can fail outside its design space.

Professional practice also involves regulatory and ethical responsibilities. Spectrum allocation, radiated-power limits, exposure requirements, electromagnetic compatibility, interference control, and measurement standards must be respected. Radar and sensing systems raise privacy and safety considerations, while high-power microwave systems require controls against exposure, arcing, and unintended interference. Engineers should document assumptions, uncertainty, compliance evidence, and foreseeable misuse.

Chapter Summary

Microwave engineering operates where wavelength is comparable with the dimensions of practical circuits and structures. Maxwell’s equations provide the fundamental description of electric and magnetic fields, while constitutive relations connect fields to material behavior. Boundary conditions determine reflection, transmission, and field behavior at interfaces.

Uniform plane waves exhibit orthogonal electric and magnetic fields, with propagation governed by phase velocity and intrinsic impedance. Polarization determines how antennas couple energy. The Poynting vector describes electromagnetic power flow, and Poynting’s theorem accounts for storage, transport, and dissipation.

Finite conductivity produces skin effect, reducing the effective current-carrying depth and increasing conductor loss with frequency. Dielectric loss, radiation, mismatch, and material dispersion further influence system performance. Waveguides are dispersive structures with cutoff frequencies, distinct phase and group velocities, and mode-dependent behavior.

Lumped-circuit models become inadequate when structures are not electrically small or when parasitic coupling, distributed phase, resonance, and loss become significant. Transmission lines, waveguides, microstrip, coaxial lines, S-parameters, and full-wave simulation provide appropriate high-frequency tools. Effective microwave engineering combines analytical field theory, careful materials and geometry selection, calibrated measurement, regulatory compliance, and validated computational models.

Key Terms

Microwave: Electromagnetic-frequency region conventionally extending from about 300 MHz to 300 GHz.

Wavelength: Spatial period of a wave, related to frequency by λ = v/f.

Permittivity: Material property relating electric flux density to electric field.

Permeability: Material property relating magnetic flux density to magnetic field.

Conductivity: Material property describing conduction-current response to electric field.

Maxwell’s equations: Four equations describing electromagnetic field generation and coupling.

Boundary condition: Mathematical relation governing fields at an interface.

Intrinsic impedance: Ratio of electric- to magnetic-field amplitude in a plane wave.

Phase velocity: Velocity of a constant phase point, vp = ω/β.

Group velocity: Derivative dω/dβ describing wave-packet propagation under suitable conditions.

Polarization: Time-dependent orientation of the electric-field vector.

Poynting vector: Electromagnetic power-flow density S = E × H.

Skin depth: Depth over which field amplitude in a good conductor decreases to 1/e.

Surface resistance: Effective high-frequency resistance associated with conductor surfaces.

Dispersion: Frequency dependence of propagation characteristics such as β or phase velocity.

Cutoff frequency: Lowest frequency at which a specified waveguide mode propagates.

Lumped model: Circuit representation assuming negligible spatial variation within components.

Distributed model: Representation accounting for voltage, current, and fields varying with position.

Scattering parameters: Traveling-wave ratios used to characterize high-frequency networks.

Loss tangent: Ratio representing dielectric dissipative behavior, commonly tan δ = ε''/ε'.

Review Questions

Short Answer Questions

1. Define microwave engineering and explain why the microwave range has no universally fixed boundary.

2. State Maxwell’s four equations in differential form.

3. What is meant by polarization, and under what conditions is a wave circularly polarized?

4. Define skin depth and state how it varies with frequency for a good conductor.

5. Why is a transmission-line model often required at microwave frequencies?

Long Answer Questions

1. Derive the source-free electromagnetic wave equation from Maxwell’s equations and state the assumptions used.

2. Explain the electromagnetic boundary conditions at a dielectric interface and discuss their significance for reflection and transmission.

3. Develop the relationship among phase velocity, group velocity, and cutoff frequency for the dominant mode of a rectangular waveguide.

4. Explain the Poynting vector and Poynting’s theorem, including their application to microwave loss analysis.

5. Compare coaxial line, microstrip, and rectangular waveguide in terms of modes, integration, loss, power handling, and applications.

Analytical / Application Questions

1. A 5 GHz plane wave in free space has a peak electric-field amplitude of 6 V/m. Calculate its average power density and magnetic-field amplitude.

2. A 20 mm interconnect operates at 12 GHz on a substrate with effective relative permittivity 2.25. Estimate its guided wavelength and determine whether the route satisfies the conservative l < λ/10 lumped-model guideline.

3. A waveguide has a dominant-mode cutoff frequency of 8 GHz and operates at 10 GHz in air. Calculate β relative to the free-space wavenumber and explain qualitatively why dispersion is important near cutoff.

Multiple Choice Questions with Answers

Question 1

Which Maxwell equation expresses the absence of magnetic monopoles?

A. Faraday’s law

B. Ampère–Maxwell law

C. Gauss’s electric law

D. Gauss’s magnetic law

Correct Answer: D. Gauss’s magnetic law

Question 2

The intrinsic impedance of free space is approximately:

A. 50 Ω

B. 120π Ω

C. 1 Ω

D. 377 kΩ

Correct Answer: B. 120π Ω

Question 3

For a uniform plane wave in a lossless medium, the electric field, magnetic field, and propagation direction are:

A. Parallel to one another

B. Mutually perpendicular

C. Always antiparallel

D. Independent of one another

Correct Answer: B. Mutually perpendicular

Question 4

The average power density of a free-space plane wave with peak electric-field amplitude E0 is:

A. E0/η0

B. E0²/η0

C. E0²/(2η0)

D. 2η0E0²

Correct Answer: C. E0²/(2η0)

Question 5

For a good conductor, skin depth varies with frequency approximately as:

A. f

B. 1/f

C. √f

D. 1/√f

Correct Answer: D. 1/√f

Question 6

A waveguide operating below the cutoff frequency of a particular mode will have that mode:

A. Propagate without attenuation

B. Become TEM automatically

C. Be evanescent rather than propagating

D. Travel faster with no phase change

Correct Answer: C. Be evanescent rather than propagating

Question 7

Which condition is required for ideal linear polarization in two orthogonal field components?

A. Equal amplitudes and 90° phase difference

B. Relative phase of 0 or 180°

C. Random relative phase

D. One component must be magnetic

Correct Answer: B. Relative phase of 0 or 180°

Question 8

A 30 mm interconnect at 10 GHz with guided wavelength 17.3 mm is best modeled as:

A. An ideal zero-length wire

B. A lumped resistor only

C. A distributed transmission line

D. A DC short circuit

Correct Answer: C. A distributed transmission line

Question 9

The electric-field reflection coefficient at normal incidence from medium 1 to medium 2 is:

A. (η1 + η2)/(η1 − η2)

B. (η2 − η1)/(η2 + η1)

C. η1η2/(η1 + η2)

D. η2/η1

Correct Answer: B. (η2 − η1)/(η2 + η1)

Question 10

Which quantity directly represents electromagnetic power-flow density?

A. E + H

B. E · H

C. E × H

D. ∇ · E

Correct Answer: C. E × H