Electromagnetic Wave Lecture Notes

Low and High Frequency Approximations

Table 7-1: Expressions for \,\beta, \etac, up, and \, for Various Types of Media

• Any Medium

  • \alpha = \omega \sqrt{{\mu \epsilon' \over 2} \left[ \sqrt{1 + (\frac{\epsilon''}{\epsilon'})^2} - 1 \right]} (Np/m)
  • \beta = \omega \sqrt{{\mu \epsilon' \over 2} \left[ \sqrt{1 + (\frac{\epsilon''}{\epsilon'})^2} + 1 \right]} (rad/m)
  • \eta_c = \sqrt{\frac{\mu}{\epsilon'}} \left( 1 + \frac{\epsilon''}{\epsilon'} \right)^{-1/2}
  • u_p = \frac{\omega}{\beta}
  • \lambda = \frac{2\pi}{\beta} = \frac{u_p}{f}

• Lossless Medium (\sigma = 0)

  • \alpha = 0
  • \beta = \omega \sqrt{\mu \epsilon}
  • \eta_c = \sqrt{\frac{\mu}{\epsilon}}
  • u_p = \frac{1}{\sqrt{\mu \epsilon}}
  • \lambda = u_p/f

• Low-loss Medium (\frac{\epsilon''}{\epsilon'} << 1)

  • \alpha \approx \frac{\omega \mu \epsilon''}{2 \sqrt{\mu \epsilon'}} = \frac{\sigma}{2} \sqrt{\frac{\mu}{\epsilon'}}
  • \beta \approx \omega \sqrt{\mu \epsilon'}
  • \eta_c \approx \sqrt{\frac{\mu}{\epsilon'}}
  • u_p \approx \frac{1}{\sqrt{\mu \epsilon'}}
  • \lambda \approx u_p/f

• Good Conductor (\frac{\epsilon''}{\epsilon'} > 1)

  • \alpha = \sqrt{\pi f \mu \sigma}
  • \beta = \sqrt{\pi f \mu \sigma}
  • \eta_c = (1 + j) \sqrt{\frac{\pi f \mu}{\sigma}}
  • u_p = \omega/\beta
  • \lambda = 4\pi f/\mu \sigma

• Notes:

  • \epsilon' = \epsilon; \epsilon'' = \sigma/\omega
  • In free space: \epsilon = \epsilon0, \mu = \mu0
  • Low-loss medium: \epsilon''/\epsilon' = \sigma/\omega \epsilon < 0.01
  • Good conducting medium: \epsilon''/\epsilon' > 100

Effect of Skin Depth

• Lossless Medium, Good Conductor
• DC vs AC: Skin depth \delta = \frac{1}{\sqrt{\pi f \mu \sigma}}
• Frequency (f) increases, \, decreases.
• DC: Current density (J) is uniform.
• AC: Current density (J) is concentrated near the surface.

Shielding and RFID

• Effect of Skin Depth
  • E = H = 0 inside a good conductor due to shielding.
• RFID Communication
  • Low frequencies (125 kHz, 13.56 MHz): Coil inductors as magnetic antennas.
  • High frequencies (900 MHz, 2.54 GHz): Dipole antennas.

DC vs AC Current Flow in Conductors

• \vec{J} = \sigma \vec{E}
• Current (I) is the integral of current density (J) over a volume.
• I = \int_V \vec{J} \cdot d\vec{A}
• In AC, current density is concentrated within a skin depth \, near the surface.

Linear Conductor: DC vs AC

• DC: f = 0, \, = \infty, Current density ($\J_c$) is constant
• AC: f >> 0, \, << \infty, Current density ($\J_s$)is not constant

Linear Conductor

• For a conductor with \, at the surface:
  • \tilde{E}(z) = \hat{x} E_0 e^{-\alpha z} e^{-j\beta z}
  • \tilde{H}(z) = \hat{y} \frac{E0}{\etac} e^{-\alpha z} e^{-j\beta z}
  • \vec{J}(z) = \hat{x} J_x(z)
  • Jx(z) = \sigma E0 e^{-\alpha z} e^{-j\beta z} = J_0 e^{-\alpha z} e^{-j\beta z}

Total Current Crossing the y-z Plane

• Total current ($\I_T$) crossing the y-z plane:
  • IT = w \int0^{\infty} J_x(z) dz
  • IT = w \int0^{\infty} J0 e^{-(1+j)z/\deltas} dz = \frac{J0 w \deltas}{1 + j}

Surface Impedance

• Voltage across a length l at the surface:
  • V = E0 l = \frac{J0}{\sigma} l
• Impedance of a slab of width w, length l, and depth d = \,:
  • Z = \frac{V}{I} = \frac{1 + j}{\sigma \delta_s} \frac{l}{w}
• Surface impedance defined:
  • Zs = \frac{1 + j}{\sigma \deltas}
• Conductor is equivalent to a resistor in series with an inductor.

Power Density

• Instantaneous power density: P(d, t) = v(d, t) i(d, t)
• Average power density:
  • P{av}(d) = \frac{1}{T} \int0^T p(d, t) dt = \frac{V^2}{2Z_0}

Wave Propagation

• Transverse Electromagnetic (TEM) Wave
  • \vec{k} \propto \vec{E} \times \vec{H}

Power Density

• Poynting Vector (instantaneous):
  • \vec{S} = \vec{E} \times \vec{H} (W/m²)
• Total Power Intercepted by Area A:
  • P = \oint_A \vec{S} \cdot \hat{n} dA (W)
• Time-Average Power Density:
  • \vec{S}_{av} = \frac{1}{2} Re[\vec{E} \times \vec{H}^*] (W/m²)

Power Density Carried by Plane Wave

• The average power density carried by the wave is given. (formula missing in provided text)

Plane Wave in Lossy Medium

• Electric field decays by e^{-1} for every 1 meter it travels. \alpha is the attenuation constant (Np/m).

Attenuation

• 20 \log_{10} e^1 = 8.69 \text{ dB/m}
• 10 \log_{10} e^2 = 8.69 \text{ dB/m}

Example 7-6: Power Received by a Submarine Antenna

• A submarine at a depth of 200 m receives signals at 1 kHz.
• Given: \, = 4.44 (mV/m), \, = 0.126 (Np/m), \, = 0.044∠45° ($\Omega$)
• Average Power Density:
  • \text{Sav}(z) = \frac{|E0|^2}{2 |\etac|} e^{-2\alpha z} \cos(\theta_\eta) = \frac{(4.44 \times 10^{-3})^2}{2 \times 0.044} e^{-0.252z} \cos(45^\circ) = 20.16 e^{-0.252z} (\text{mW/m}^2)
• At z = 200 m:
  • \text{Sav} = 20.16 \times 10^{-3} e^{-0.252 \times 200} \cos(45^\circ) = 2.1 \times 10^{-26} (\text{W/m}^2)

2012 Test 1 Q2

• A uniform plane, time-harmonic electromagnetic wave travels in seawater at f = 5 MHz.
• Seawater parameters: \, = 72, \, = 1, \, = 4 S/m
  • a) Ratio of Conduction Current Density ($\Jc$) to Displacement Current Density ($\Jd$)
    • \vec{J_c} = \sigma \vec{E}
    • \vec{J_d} = j\omega \epsilon \vec{E} = \frac{\partial}{\partial t}(\epsilon \vec{E})
    • \frac{Jc}{Jd} = \frac{\sigma}{\omega \epsilon} = \frac{4}{2 \pi \times 5 \times 10^6 \times 72 \times 8.854 \times 10^{-12}} = 200
  • b) Attenuation Constant ($\alpha$):
    • Good conductor: \frac{\sigma}{\omega \epsilon} = 200 >> 1
    • \alpha = \sqrt{\pi f \mu \sigma} = 8.89 Np/m
  • c) Phase Constant ($\beta$):
    • \beta = \sqrt{\pi f \mu \sigma} = 8.89 \text{ rad/m}
  • d) Intrinsic Impedance ($\eta_c$):
    • \eta_c = \sqrt{\frac{j \omega \mu}{\sigma}} = (1 + j) \sqrt{\frac{\pi f \mu}{\sigma}} = (1+j) \alpha = 3.53 \angle 45^\circ \Omega
  • e) Phase Velocity ($\u_p$):
    • u_p = \frac{\omega}{\beta} = \frac{2 \pi \times 5 \times 10^6}{8.89} = 3.53 \times 10^6 \text{ m/s}
  • f) Skin Depth ($\delta$):
    • \delta = \frac{1}{\sqrt{\pi f \mu \sigma}} = \frac{1}{\alpha}= 0.112 \text{ m}
  • g) Time-Averaged Power Density Vector ($\vec{S}$):
    • \vec{S} = \frac{1}{2}Re[\vec{E} \times \vec{H}^*] = \frac{1}{2} \frac{|E0|^2}{\eta} e^{-2\alpha z} \hat{z} = \hat{z}(0.1125 |E0|^2/2\eta) 0.001 e^{-2\alpha z} \cos(45^\circ) \text{ W/m}^2
  • h) Depth at Which Power Density is Reduced by 30 dB:
    • 10 \log(\frac{S(z)}{S_0}) = -30 \Rightarrow z = 0.388 \text{ m}

Expressions for E(z,t) and H(z,t) at z = 0.8 m

• Given: \vec{E}(z, t) = \hat{x} E_0 e^{-\alpha z} \cos(\omega t - \beta z)
• \vec{E}(z, t) = \hat{x} 0.00082 \cos(10^7 \pi t - 7.11) \text{ V/m}
• \vec{H}(z, t) = \hat{y} 0.00026 \cos(10^7 \pi t - 7.11 - 45^\circ) \text{ A/m}

2013 Test 1 Q3

• International safety standard: \, < 10 \text{ mW/cm}^2
• Radiated power density from a GSM base station antenna at R = 1 m: \, = 1600 \text{ W/m}^2
• Electric field strength decays proportional to \, (1/R).
  • a) Radius of the Unsafe Region:
    • S = \frac{E^2}{R} => Safe distance = 5m
    • $$R_{safe} = \
  • b) Distance from the antenna at which the amplitude of the electric field is 1% of the value at R = 1 m:
    • 101m