Electromagnetics

Electromagnetics is the study of electric and magnetic fields and their interactions. It forms the foundation for understanding phenomena ranging from static electric charge to electromagnetic wave propagation, including radio waves, light, and X-rays. Maxwell's equations provide the unified mathematical framework for electromagnetic phenomena.

Maxwell's Equations

Differential Forms

\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \quad \text{(Gauss's Law for Electric Fields)}

\nabla \cdot \mathbf{B} = 0 \quad \text{(Gauss's Law for Magnetic Fields)}

\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \quad \text{(Faraday's Law)}

\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t} \quad \text{(Ampere-Maxwell Law)}

Integral Forms

\oint_S \mathbf{E} \cdot d\mathbf{a} = \frac{Q_{enc}}{\epsilon_0}

\oint_S \mathbf{B} \cdot d\mathbf{a} = 0

\oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi_B}{dt}

\oint_C \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc} + \mu_0\epsilon_0 \frac{d\Phi_E}{dt}

Constitutive Relations

\mathbf{D} = \epsilon\mathbf{E} = \epsilon_0\epsilon_r\mathbf{E}

\mathbf{H} = \frac{1}{\mu}\mathbf{B} = \frac{1}{\mu_0\mu_r}\mathbf{B}

\mathbf{J} = \sigma\mathbf{E} \quad \text{(Ohm's Law in differential form)}

Where:

$\epsilon_0 = 8.854 \times 10^{-12} \text{ F/m}$ (permittivity of free space)
$\mu_0 = 4\pi \times 10^{-7} \text{ H/m}$ (permeability of free space)
$\epsilon_r, \mu_r$ are relative permittivity and permeability
$\sigma$ is conductivity

Electrostatics

Coulomb's Law

\mathbf{F} = \frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r^2}\hat{r}

Electric Field

\mathbf{E} = \frac{\mathbf{F}}{q_{test}} = \frac{1}{4\pi\epsilon_0}\sum_i \frac{q_i}{r_i^2}\hat{r}_i

For continuous charge distributions:

\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|^2}\hat{r}_{\mathbf{r}\mathbf{r'}} \, d\tau'

Electric Potential

V = -\int_{\infty}^{\mathbf{r}} \mathbf{E} \cdot d\mathbf{l} = \frac{1}{4\pi\epsilon_0}\sum_i \frac{q_i}{r_i}

\mathbf{E} = -\nabla V

Laplace and Poisson Equations

\nabla^2 V = -\frac{\rho}{\epsilon_0} \quad \text{(Poisson's equation)}

\nabla^2 V = 0 \quad \text{(Laplace's equation for charge-free regions)}

Capacitance

C = \frac{Q}{V} = \epsilon \frac{A}{d} \quad \text{(parallel plate capacitor)}

Magnetostatics

Biot-Savart Law

d\mathbf{B} = \frac{\mu_0}{4\pi}\frac{I d\mathbf{l} \times \hat{r}}{r^2}

\mathbf{B} = \frac{\mu_0}{4\pi}I \oint_C \frac{d\mathbf{l} \times \hat{r}}{r^2}

Ampère's Law

\oint_C \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc}

Magnetic Vector Potential

\mathbf{B} = \nabla \times \mathbf{A}

\mathbf{A} = \frac{\mu_0}{4\pi} \int \frac{\mathbf{J}(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} \, d\tau'

Magnetic Dipole Moment

\mathbf{m} = I\mathbf{a} \quad \text{(loop of current area vector)}

Electromagnetic Waves

Wave Equation

Taking the curl of Faraday's law and substituting Ampère-Maxwell:

\nabla^2\mathbf{E} - \mu\epsilon\frac{\partial^2\mathbf{E}}{\partial t^2} = 0

\nabla^2\mathbf{B} - \mu\epsilon\frac{\partial^2\mathbf{B}}{\partial t^2} = 0

Plane Wave Solutions

\mathbf{E} = \mathbf{E}_0 e^{j(\mathbf{k} \cdot \mathbf{r} - \omega t)}

\mathbf{B} = \mathbf{B}_0 e^{j(\mathbf{k} \cdot \mathbf{r} - \omega t)}

Where $\mathbf{k}$ is the wave vector, and $k = |\mathbf{k}| = \omega/c$ in free space.

Wave Properties

Phase Velocity

v_p = \frac{\omega}{k} = \frac{1}{\sqrt{\mu\epsilon}}

Group Velocity

v_g = \frac{\partial \omega}{\partial k}

Intrinsic Impedance

\eta = \sqrt{\frac{\mu}{\epsilon}} = \sqrt{\frac{\mu_0}{\epsilon_0}} = 377 \Omega \quad \text{(for free space)}

Poynting Vector (Power Flow)

\mathbf{S} = \mathbf{E} \times \mathbf{H}

\langle\mathbf{S}\rangle = \frac{1}{2}\text{Re}(\mathbf{E} \times \mathbf{H}^*)

Transmission Lines

Telegrapher's Equations

For a differential length $dz$ of transmission line:

\frac{\partial V(z,t)}{\partial z} = -L\frac{\partial I(z,t)}{\partial t} - RI(z,t)

\frac{\partial I(z,t)}{\partial z} = -C\frac{\partial V(z,t)}{\partial t} - GV(z,t)

Where $R$ , $L$ , $C$ , $G$ are resistance, inductance, capacitance, and conductance per unit length.

Characteristic Impedance

Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}}

For lossless line ( $R = G = 0$ ):

Z_0 = \sqrt{\frac{L}{C}}

Propagation Constant

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

Where $\alpha$ is the attenuation constant and $\beta$ is the phase constant.

For lossless line:

\gamma = j\omega\sqrt{LC} = j\beta \quad \text{where } \beta = \omega\sqrt{LC}

Wave Velocity on Transmission Line

v = \frac{\omega}{\beta} = \frac{1}{\sqrt{LC}}

Waveguides

Rectangular Waveguide

Cutoff Frequency

f_{c,mn} = \frac{c}{2}\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2}

Where $a, b$ are dimensions of the waveguide cross-section.

Phase and Group Velocities

v_p = \frac{c}{\sqrt{1 - \left(\frac{f_c}{f}\right)^2}}

v_g = c\sqrt{1 - \left(\frac{f_c}{f}\right)^2}

Note: $v_p \cdot v_g = c^2$

TE and TM Modes

TE Modes (Transverse Electric)

E_z = 0, \quad H_z \neq 0

TM Modes (Transverse Magnetic)

H_z = 0, \quad E_z \neq 0

Antennas and Radiation

Antenna Parameters

Gain and Directivity

D = \frac{4\pi U_{max}}{P_{rad}}

G = \eta_r D

Where $U_{max}$ is maximum radiation intensity and $P_{rad}$ is total radiated power.

Effective Aperture

A_e = \frac{\lambda^2}{4\pi}G

Radiation Resistance

R_{rad} = \frac{2P_{rad}}{I_{rms}^2}

Near and Far Field Regions

Near Field

r < \frac{2D^2}{\lambda}

Where $D$ is the largest dimension of the antenna.

Far Field (Fraunhofer)

r > \frac{2D^2}{\lambda} \quad \text{and} \quad r > \frac{\lambda}{2\pi}

Electromagnetic Boundary Conditions

At Interface Between Two Media

Normal Components

D_{1n} - D_{2n} = \sigma_s \quad \text{(Surface charge density)}

B_{1n} - B_{2n} = 0 \quad \text{(No magnetic monopoles)}

Tangential Components

E_{1t} - E_{2t} = 0 \quad \text{(Conservative field)}

H_{1t} - H_{2t} = \mathbf{K}_s \times \mathbf{\hat{n}} \quad \text{(Surface current density)}

Applications in Engineering

Radar Systems

\text{Radar equation}: P_r = \frac{P_t G^2 \lambda^2 \sigma}{(4\pi)^3 R^4 L}

Where $\sigma$ is radar cross-section, $R$ is range, and $L$ is system losses.

Wireless Communications

\text{Friis equation}: \frac{P_r}{P_t} = G_t G_r \left(\frac{\lambda}{4\pi R}\right)^2

Shielding Effectiveness

SE = A + R + B \quad \text{(Absorption + Reflection + Multiple Reflection)}

Real-World Application: Antenna Design for 5G Communications

5G communications require sophisticated antenna design for millimeter-wave frequencies.

5G Antenna Analysis

# 5G antenna design parameters at 28 GHz
freq_5g = 28e9  # Hz (28 GHz band)
wavelength_5g = 3e8 / freq_5g  # meters
wavelength_mm = wavelength_5g * 1000  # mm

# Calculate Fresnel zone for antenna separation
antenna_distance = 0.5  # meters
first_fresnel_radius = math.sqrt((wavelength_5g * antenna_distance) / 4)  # meters

# Path loss calculation (Free space)
path_loss_fs = 20 * math.log10(4 * math.pi * antenna_distance / wavelength_5g)  # dB

# Calculate antenna parameters
antenna_diameter = 2 * wavelength_5g  # 2λ diameter for good directivity
beamwidth = 70 * wavelength_5g / antenna_diameter  # degrees (approximation)

# Aperture efficiency considerations
aperture_efficiency = 0.7  # Typical for well-designed antennas
physical_aperture = (antenna_diameter / 2)**2 * math.pi  # m²
effective_aperture = aperture_efficiency * physical_aperture  # m²

# Calculate gain assuming aperture efficiency
gain_directivity = 4 * math.pi * effective_aperture / wavelength_5g**2
gain_db = 10 * math.log10(gain_directivity)

# Calculate beamwidth-radius relationship
beamwidth_exact = 2 * math.asin(1.22 * wavelength_5g / antenna_diameter) * 180 / math.pi  # degrees

print(f"5G antenna design at {freq_5g/1e9:.0f} GHz:")
print(f"  Wavelength: {wavelength_mm:.2f} mm")
print(f"  Antenna diameter: {antenna_diameter*1000:.1f} mm")
print(f"  First Fresnel zone radius: {first_fresnel_radius*1000:.1f} mm")
print(f"  Free space path loss: {path_loss_fs:.1f} dB")
print(f"  Estimated beamwidth: {beamwidth:.2f}°")
print(f"  Exact beamwidth: {beamwidth_exact:.2f}°")
print(f"  Antenna gain: {gain_db:.1f} dBi")
print(f"  Effective aperture: {effective_aperture*1e4:.2f} cm²")

# Calculate number of antenna elements for beamforming
# In 5G, multiple elements are used for MIMO
elements_1d = 8  # Number of elements in one dimension
total_elements = elements_1d**2  # For 2D array
array_side = elements_1d * wavelength_5g  # Physical size of array

print(f"  Total array elements: {total_elements}x")
print(f"  Array physical size: {array_side*100:.1f} cm x {array_side*100:.1f} cm")

# Calculate beamforming gain
array_gain = total_elements * gain_directivity  # If all elements phased correctly
array_gain_db = 10 * math.log10(array_gain)

print(f"  MIMO array gain: {array_gain_db:.1f} dBi")

# Link budget considerations
transmit_power = 0.1  # W (20 dBm)
received_power = transmit_power / (10**(path_loss_fs/10))  # W
received_power_dbm = 10 * math.log10(received_power) + 30  # dBm

print(f"  Received power: {received_power_dbm:.1f} dBm")
print(f"  For successful communication, Rx power should be > -90 dBm (typical sensitivity)")

# Feasibility assessment
if received_power_dbm > -90:
    link_feasibility = "Feasible - sufficient signal strength"
    margin = received_power_dbm - (-90)
    print(f"  Link margin: {margin:.1f} dB")
else:
    link_feasibility = "Not feasible - insufficient signal strength"
    required_improvement = -90 - received_power_dbm
    print(f"  Required improvement: {required_improvement:.1f} dB")
    
print(f"  Link feasibility assessment: {link_feasibility}")

Millimeter-Wave Challenges

Addressing the unique challenges of high-frequency communications.

Your Challenge: Transmission Line Analysis

Analyze the behavior of a transmission line with various load impedances and determine the resulting voltage standing wave patterns.

Goal: Calculate impedance transformations, standing wave ratios, and reflection coefficients for different loading conditions.

Transmission Line Parameters

import math

# Coaxial transmission line analysis
line_params = {
    'characteristic_impedance': 50,  # Ohms (RG-58/U)
    'frequency': 1e9,              # Hz (1 GHz)
    'line_length': 0.5,            # meters
    'relative_permittivity': 2.3,  # For PTFE dielectric
    'attenuation_constant': 0.02,   # Neper/meter
    'load_impedance': 75 + 50j,    # Ohms (complex load)
    'source_impedance': 50,        # Ohms (generator impedance)
    'source_voltage': 10.0         # Volts (generator voltage)
}

# Calculate wavelength in the line
c = 3e8  # m/s (speed of light)
velocity_factor = 1 / math.sqrt(line_params['relative_permittivity'])
wavelength = c / line_params['frequency'] * velocity_factor

# Calculate electrical length
beta = 2 * math.pi / wavelength  # Phase constant
electrical_length = beta * line_params['line_length']  # radians

# Calculate propagation constant
gamma = line_params['attenuation_constant'] + 1j * beta  # Complex propagation constant

# Calculate reflection coefficient at load
ZL = line_params['load_impedance']
Z0 = line_params['characteristic_impedance']
reflection_load = (ZL - Z0) / (ZL + Z0)  # Complex reflection coefficient

# Calculate VSWR (Voltage Standing Wave Ratio)
magnitude_refl = abs(reflection_load)
if magnitude_refl < 1:
    vswr = (1 + magnitude_refl) / (1 - magnitude_refl)
else:
    vswr = float('inf')  # Perfectly matched if magnitude_refl = 0

# Calculate input impedance looking into the line
# Z_in = Z0 * (ZL + Z0 * tanh(gamma * length)) / (Z0 + ZL * tanh(gamma * length))
# For lossless line: Z_in = Z0 * (ZL + j*Z0*tan(beta*l)) / (Z0 + j*ZL*tan(beta*l))

# Simplified lossless calculation first
if line_params['attenuation_constant'] == 0:
    ZL_real = ZL.real
    ZL_imag = ZL.imag
    tan_bl = math.tan(electrical_length)
    
    numerator = complex(ZL_real, ZL_imag + Z0 * tan_bl)
    denominator = complex(Z0, ZL_imag * tan_bl)
    Z_input = Z0 * numerator / denominator
else:
    # More complex calculation with attenuation
    Z_input = Z0 * (ZL + Z0 * math.tanh(gamma * line_params['line_length'])) / (Z0 + ZL * math.tanh(gamma * line_params['line_length']))

# Calculate reflection coefficient at input
reflection_input = (Z_input - line_params['source_impedance']) / (Z_input + line_params['source_impedance'])

# Calculate power delivered to load
# Need to calculate voltage and current at load
source_Z = line_params['source_impedance']
source_V = line_params['source_voltage']

# For simplification, calculate power transfer efficiency
Z_load_conjugate_match = Z_input.conjugate()
power_transfer_efficiency = 4 * Z_load_conjugate_match.real * Z0 / abs(Z_load_conjugate_match + Z0)**2

# Calculate maximum power transfer when Z_load = Z_source (matched)
if abs(Z0 - line_params['source_impedance']) < 0.1:  # Approximately matched
    max_power_condition = "Matched"
    reflection_at_source = 0
else:
    max_power_condition = "Mismatched"
    reflection_at_source = (line_params['source_impedance'] - Z0) / (line_params['source_impedance'] + Z0)

Analyze the transmission line behavior for various loading conditions.

Hint:

Use transmission line equations to calculate input impedance
Consider the effects of line length and load impedance
Calculate reflection coefficients and standing wave ratios
Analyze power transfer efficiency

# TODO: Calculate transmission line parameters
input_impedance = 0+0j  # Complex input impedance in ohms
vswr = 0  # Voltage Standing Wave Ratio
reflection_coefficient_load = 0+0j  # Complex reflection coefficient at load
power_transfer_efficiency = 0  # Fraction (0-1 scale)
attenuation_db = 0  # Total line attenuation in dB
wavelength_in_line = 0  # Meters (wavelength in transmission line)

# Calculate wavelength in line
velocity_factor = 1 / math.sqrt(line_params['relative_permittivity'])
wavelength_in_line = (c / line_params['frequency']) * velocity_factor

# Calculate electrical length
beta = 2 * math.pi / wavelength_in_line
electrical_length = beta * line_params['line_length']

# Calculate input impedance (lossless line approximation)
ZL = complex(line_params['load_impedance'])  # Convert to complex if not already
Z0 = line_params['characteristic_impedance']

numerator = ZL + 1j * Z0 * math.tan(electrical_length)
denominator = Z0 + 1j * ZL * math.tan(electrical_length)
input_impedance = Z0 * (numerator / denominator)

# Calculate reflection coefficient at load
reflection_coefficient_load = (ZL - Z0) / (ZL + Z0)

# Calculate VSWR
refl_mag = abs(reflection_coefficient_load)
vswr = (1 + refl_mag) / (1 - refl_mag) if refl_mag < 1 else float('inf')

# Calculate power transfer efficiency
# For power transfer, we want to consider the source match
Z_source = line_params['source_impedance']
Z_in = input_impedance

# Reflection coefficient looking into the line from source
reflection_input = (Z_in - Z_source) / (Z_in + Z_source)
power_transfer_efficiency = 1 - abs(reflection_input)**2

# Calculate total attenuation in dB
attenuation_db = line_params['attenuation_constant'] * line_params['line_length'] * 8.686  # Convert from nepers to dB

# Print results
print(f"Line parameters:")
print(f"  Characteristic impedance: {Z0} Ω")
print(f"  Operating frequency: {line_params['frequency']/1e9} GHz")
print(f"  Line length: {line_params['line_length']} m")
print(f"  Load impedance: {line_params['load_impedance']} Ω")
print(f"  Wavelength in line: {wavelength_in_line*100:.2f} cm")

print(f"\nResults:")
print(f"  Input impedance: {input_impedance:.2f} Ω")
print(f"  Reflection coefficient (load): {reflection_coefficient_load:.3f}")
print(f"  VSWR: {vswr:.2f}")
print(f"  Attenuation: {attenuation_db:.2f} dB")
print(f"  Power transfer efficiency: {power_transfer_efficiency:.3f}")

# Analyze matching conditions
if vswr < 1.5:
    match_quality = "Excellent match"
elif vswr < 2.0:
    match_quality = "Good match"
elif vswr < 3.0:
    match_quality = "Fair match"
else:
    match_quality = "Poor match - significant reflections"

print(f"\n  Match quality: {match_quality}")

# Efficiency assessment
if power_transfer_efficiency > 0.9:
    efficiency_label = "High efficiency - minimal reflected power"
elif power_transfer_efficiency > 0.7:
    efficiency_label = "Moderate efficiency - some power loss"
else:
    efficiency_label = "Low efficiency - significant reflected power"
    
print(f"  Power efficiency: {efficiency_label}")

# Applications recommendation
if vswr < 2.0 and power_transfer_efficiency > 0.8:
    application = "Suitable for high-power applications"
elif vswr < 3.0:
    application = "Suitable for low-power applications with some caution"
else:
    application = "Requires impedance matching network"

print(f"  Recommended application: {application}")

How would you design an impedance matching network to transform the load impedance to match the characteristic impedance of the transmission line?

Electromagnetics

Electromagnetics

Maxwell's Equations

Differential Forms

Integral Forms

Constitutive Relations

Electrostatics

Coulomb's Law

Electric Field

Electric Potential

Laplace and Poisson Equations

Capacitance

Magnetostatics

Biot-Savart Law

Ampère's Law

Magnetic Vector Potential

Magnetic Dipole Moment

Electromagnetic Waves

Wave Equation

Plane Wave Solutions

Wave Properties

Phase Velocity

Group Velocity

Intrinsic Impedance

Poynting Vector (Power Flow)

Transmission Lines

Telegrapher's Equations

Characteristic Impedance

Propagation Constant

Wave Velocity on Transmission Line

Waveguides

Rectangular Waveguide

Cutoff Frequency

Phase and Group Velocities

TE and TM Modes

TE Modes (Transverse Electric)

TM Modes (Transverse Magnetic)

Antennas and Radiation

Antenna Parameters

Gain and Directivity

Effective Aperture

Radiation Resistance

Near and Far Field Regions

Near Field

Far Field (Fraunhofer)

Electromagnetic Boundary Conditions

At Interface Between Two Media

Normal Components

Tangential Components

Applications in Engineering

Radar Systems

Wireless Communications

Shielding Effectiveness

Real-World Application: Antenna Design for 5G Communications

5G Antenna Analysis

Millimeter-Wave Challenges

Your Challenge: Transmission Line Analysis

Transmission Line Parameters

ELI10 Explanation

Self-Examination