ECOE 427: Microwave Circuits and Optical Fibers
‫ دوائر الموجات المتناهية القصر واأللياف الضوئية‬:427 ‫هكت‬
Electromagnetic Theory
(Text Book: Microwave Engineering by David M. Pozar, 4th ED, 2012)
Dr. Doaa Abdelhameed
Aswan University, Faculty of Engineering
E-mail: doaa_abdelhameed@aswu.edu.eg
2025/3/10
1
Course / lecture map
• 10~12 Weeks,
• Total course degree →150,
• Final exam → 100,
• Midterm → 30,
• Quizzes / assignments→ 10,
• Class Works →10,
• Office hours : Sunday 13:00-14:00 (Office #2311),
2025/3/10
2
Course Content
• Microwave Bands and Applications (Chapter 1).
• TRANSMISSION LINE THEORY (Chapter 2).
• TRANSMISSION LINES AND WAVEGUIDES (Chapter 3).
• MICROWAVE NETWORK ANALYSIS (Chapter 4).
• MICROWAVE RESONATORS (Chapter 6).
• POWER DIVIDERS AND DIRECTIONAL COUPLERS (Chapter 7).
• MICROWAVE FILTERS (Chapter 8).
• OPTEICAL FIBERS (Lecture notes).
Why Microwave Engineering?
Why Microwave Engineering?
• Radio Frequency (RF) range from very high frequency (VHF) (30–300 MHz) to
ultra high frequency (UHF) (300–3000 MHz).
• Microwave is typically used for frequencies between 3 and 300 GHz, with a
corresponding electrical wavelength between λ = c/ f = 10 cm and λ = 1 mm,
respectively.
• Signals with wavelengths on the order of millimeters are often referred to as
millimeter waves.
• Because of the high frequencies (and short wavelengths), standard circuit theory
often cannot be used directly to solve microwave network problems.
• Standard circuit theory is an approximation, or special case, of the broader theory
of electromagnetics as described by Maxwell’s equations.
Why Microwave Engineering?
• The lumped circuit approximations of circuit theory may not be valid at high RF and microwave
frequencies.
• Distributed elements, where the phase of the voltage or current changes significantly over the
physical extent of the device because the device dimensions are on the order of the electrical
wavelength.
• At much lower frequencies the wavelength is large enough that there is insignificant phase
variation across the dimensions of a component (lumped circuit element).
• Solving Maxwell’s equations is essential for microwave engineering
Applications of Microwave Engineering
• Antenna gain is proportional to the electrical size of the antenna.
• At higher frequencies, more antenna gain can be obtained for a given physical antenna size.
• More bandwidth (directly related to data rate) can be realized at higher frequencies. A 1% bandwidth at 600
MHz is 6 MHz, which (with binary phase shift keying modulation) can provide a data rate of about 6 Mbps
(megabits per second), while at 60 GHz a 1% bandwidth is 600 MHz, allowing a 600 Mbps data rate.
• Microwave signals travel by line of sight and are not bent by the ionosphere as lower frequency signals.
• The effective reflection area (radar cross section) of a radar target is usually proportional to the target’s
electrical size. This fact, coupled with the frequency characteristics of antenna gain, generally makes
microwave frequencies preferred for radar systems.
• Various molecular, atomic, and nuclear resonances occur at microwave frequencies, creating a variety of
unique applications in the areas of basic science, remote sensing, medical diagnostics and treatment, and
heating methods.
Applications of Microwave Engineering
2.5kbps
100kbps
10Mbps
100Mbps
10Gbps
Maxwell’s Equations
In the Free space, the relations between the electric and magnetic field intensities and flux densities are given by:
where 𝜇0 = 4𝜋 × 10−7 henry/m is the permeability of free-space, and 𝜖0 = 8.854 × 10−12 farad/m is the permittivity of
free-space.
Maxwell’s Equations
• The continuity equation:
Maxwell’s Equations in Integral Forms
• By applying the divergence theorem to:
• where 𝑄 represents the total charge contained in the closed volume V
• Applying Stokes’ theorem to:
Faraday’s law
Kirchhoff’s voltage law
Maxwell’s Equations in Integral Forms
Maxwell’s Equations in Integral Forms
• Applying Stokes’ theorem to:
Ampere’s
law
Total electric current flow through the surface S
Maxwell’s Equations in Phasor Form
• The sinusoidal electric field polarized in the 𝑥ො direction of the form
• The phasor electric field
• where,
Maxwell’s Equations in Phasor Form
Fields in Media and Boundary Conditions
ത
• For a dielectric material, an applied electric field 𝐸causes
the polarization of the atoms or molecules of the
ഥ . This additional
material to create electric dipole moments that augment the total displacement flux, 𝐷
polarization vector is called 𝑃ഥ𝑒 , the electric polarization,
• In a linear medium the electric polarization is linearly related to the applied electric field as
• where 𝜒𝑒 , which may be complex, is the electric susceptibility. Then,
• Where,
Fields in Media and Boundary Conditions
• For a dielectric material, Maxwell’s equations can be written in phasor form as
• The constitutive relations are
Fields at a General Material Interface
Fields at a General Material Interface
• In the limit as ℎ → 0, 𝐷𝑡𝑎𝑛 = 0,
• In vector form, we can write
• A similar argument for 𝐵ത leads to the result that
The wave Equation and Basic Plane Wave Solution
• The Helmholtz Equation (Wave Equation):
⁃ In a source-free, linear, isotropic, homogeneous region
ഥ = 0,
𝐽ҧ = 𝑀
𝜎 = 0,
Where, 𝜇𝑟 , 𝜖𝑟 are constant over the homogenous region.
*Note: vector identity: ∇ × ∇ × 𝐴ҧ = ∇(∇. 𝐴)ҧ − ∇2 𝐴,ҧ ∇. 𝐸ത = 0 in a source-free region.
The wave Equation and Basic Plane Wave Solution
• The Helmholtz Equation (Wave Equation):
⁃ In a source-free, linear, isotropic, homogeneous region
∇2 𝐸ത + 𝜔2 𝜇 ε𝐸ത = 0,
ഥ + 𝜔 2 𝜇 ε𝐻
ഥ = 0,
∇2 𝐻
⁃ A constant 𝑘 = 𝜔 𝜇ε is defined and called the propagation constant (also known as the phase constant, or wave
number), of the medium; its units are 1/𝑚.
➢ The solution of wave equation in a Lossless Medium
➢ The solution of wave equation in a General Lossy Medium
➢ The solution of wave equation in a Good Conductor
Plane Waves in a Lossless Medium
𝑦ො
• In a lossless medium, 𝜖 and 𝜇 are real numbers, and so 𝑘 is real.
• For simplicity, consider an electric field with only an 𝑥ො component and uniform (no
variation) in the 𝑥 and 𝑦 directions.
𝜕
𝜕
=
= 0,
𝜕𝑥 𝜕𝑦
𝜕 2 𝐸𝑥
+ 𝑘 2 𝐸𝑥 = 0
2
𝜕𝑧
The solution of the wave equation in the phasor form given as:
𝐸𝑥 𝑧 = 𝐸 + 𝑒 −𝑗𝑘𝑧 + 𝐸 − 𝑒 𝑗𝑘𝑧
𝑧Ƹ
• In the time domain, this result is written as
𝜀𝑥 𝑧, 𝑡 = 𝐸 + cos 𝜔𝑡 − 𝑘𝑧 + 𝐸 − cos(𝜔𝑡 + 𝑘𝑧)
• The first term represents a wave traveling in the +𝑧 direction, the second term represents a wave
traveling in the negative 𝑧 direction—hence the notation 𝐸 + and 𝐸 − for these wave amplitudes.
𝑥ො
Plane Waves in a Lossless Medium
• Phase velocity:
𝜔𝑡 − 𝑘𝑧 = 𝑐𝑜𝑛𝑠𝑡𝑛𝑡,
𝜔𝑡 − 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑧=
,
𝑘
𝑑𝑧
𝑑 𝜔𝑡 − 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝜔
1
𝑣𝑝 =
=
= =
.
𝑑𝑡 𝑑𝑡
𝑘
𝑘
𝜇𝜖
*Note: In the free-space, 𝑣𝑝 = 𝑐 which is the speed of light.
• Wavelength:
𝜔𝑡 − 𝑘𝑧 − 𝜔𝑡 − 𝑘 𝑧 + λ
2𝜋 2𝜋𝑣𝑝 𝑣𝑝
λ=
=
= .
𝑘
𝜔
𝑓
= 2𝜋,
Plane Waves in a Lossless Medium
• The magnetic field
By applying Maxwell’s equation
then 𝐻𝑥 = 𝐻𝑧 = 0, and
𝜔𝜇
By putting,  = 𝑘 =
𝑗 𝜕𝐸𝑥
𝑘
𝐻𝑦 =
=
𝐸 + 𝑒 −𝑗𝑘𝑧 − 𝐸 − 𝑒 𝑗𝑘𝑧
𝜔𝜇 𝜕𝑧
𝜔𝜇
𝜇
is known as the intrinsic impedance of the medium.
𝜖
ഥ field components is seen to have units of impedance, known as the wave impedance;
• The ratio of the 𝐸ത and 𝐻
• For planes waves the wave impedance is equal to the intrinsic impedance of the medium.
ഥ vectors are orthogonal to each other and orthogonal to the direction of propagation
• Note that the 𝐸ത and 𝐻
(±𝑧);
Ƹ this is a characteristic of transverse electromagnetic (TEM) waves.
Plane Waves in a Lossless Medium
• Example:
A plane wave propagating in a lossless dielectric medium has an electric field given as 𝐸𝑥 = 𝐸0 cos(𝜔𝑡 − 𝛽𝑧) with a
frequency of 5.0 GHz and a wavelength in the material of 3.0 cm. Determine the propagation constant, the phase velocity,
the relative permittivity of the medium, and the wave impedance.
Solution:
The propagation constant 𝑘
The phase velocity 𝑣𝑝
The relative permittivity of the medium 𝜖𝑟
The wave impedance 
Plane Waves in a General Lossy Medium
• If the medium is conductive, with a conductivity 𝜎, Maxwell’s curl equations can be written as
• The wave equation is given as
• The complex propagation constant
• Where, 𝛼 is the attenuation constant and 𝛽 is the phase constant.
Plane Waves in a General Lossy Medium
• The wave equation reduces to
• Has a solution
• The associated magnetic field can be calculated as
• The intrinsic impedance of the conducting medium is now complex and equal to the wave impedance
Plane Waves in a Good Conductor
• A good conductor is a special case of the preceding analysis, where the conductive current is much greater
than the displacement current
• The skin depth, or characteristic depth of penetration, is defined as
• The intrinsic impedance inside a good conductor
Plane Wave Reflection from a Media Interface
• General Medium
Plane Wave Reflection from a Media Interface
• General Medium
• The incident fields for z < 0 is given as
• The reflected wave in region z < 0 given as
• where  is the unknown reflection coefficient of the reflected electric field
Plane Wave Reflection from a Media Interface
• General Medium
• The transmitted fields for z > 0 in the lossy medium can be written as
• where 𝑇 is the transmission coefficient of the transmitted electric field and 𝜂 is the intrinsic impedance
(complex) of the lossy medium in the region z > 0.
• The propagation constant is
Plane Wave Reflection from a Media Interface
• General Medium
• The two unknown constants  and 𝑇 are found by applying boundary conditions for 𝐸𝑥 and 𝐻𝑦 at 𝑧 = 0.
1−
𝑇
= ,
0

1 + =T,
• By solving these equations:
 − 0
=
,
 + 0
2
𝑇 = 1 + =
,
 + 0
Plane Wave Reflection from a Media Interface
• Lossless Medium
• If the region for 𝑧 > 0 is a lossless dielectric, then 𝜎 = 0, and 𝜇 and 𝜖 are real quantities.
• The propagation constant in this case is purely imaginary:
𝛾 = 𝑗𝛽 = 𝑗𝜔 𝜇𝜖 = 𝑗𝑘0 𝜇𝑟 𝜖𝑟
• The wavelength in the dielectric is
λ=
• The phase velocity is
2𝜋
2𝜋
λ0
=
=
𝛽
𝜔 𝜇𝜖
𝜇𝑟 𝜖𝑟
𝜔
1
𝑐
𝑣𝑝 = =
=
𝛽
𝜇𝜖
𝜇𝑟 𝜖𝑟
• The intrinsic impedance of the dielectric is
𝑗𝜔𝜇
=
=
𝛾
𝜇
𝜇𝑟
= 0
𝜖
𝜖𝑟
ഥ are in phase with each other in both regions.
• For this lossless case, 𝜂 is real, so both  and T are real, and 𝐸ത and 𝐻
Plane Wave Reflection from a Media Interface
• Good Conductor
• If the region for 𝑧 > 0 is a good (but not perfect) conductor, the propagation constant can be written as
𝜔𝜇𝜎 1 + 𝑗
𝛾 = 𝛼 + 𝑗𝛽 = (1 + 𝑗)
=
2
𝛿𝑠
• The intrinsic impedance of the conductor simplifies to
 = (1 + 𝑗)
𝜔𝜇 1 + 𝑗
=
2𝜎
𝜎𝛿𝑠
ഥ will be 45◦ out of phase, and  and 𝑇
• Now the impedance is complex, with a phase angle of 45◦ , so 𝐸ത and 𝐻
will be complex.
Plane Wave Reflection from a Media Interface
• Perfect Conductor
• Assume that the region z > 0 contains a perfect conductor, which mean that 𝜎 → ∞. Then, 𝛼 → ∞; 𝜂 → 0;
𝛿𝑠 → 0; 𝑇 → 0 and  = −1.
• The fields for 𝑧 > 0 thus decay infinitely fast and are identically zero in the perfect conductor.
• The perfect conductor can be thought of as “shorting out” the incident electric field.
Example:
• Consider a plane wave normally incident on a half-space of copper. If 𝑓 = 1𝐺𝐻𝑧,
compute the propagation constant, intrinsic impedance, and skin depth for the
conductor. Also compute the reflection and transmission coefficients.
• Solution: For copper, σ = 5.813 × 107 S/m,