Electromagnetic fields for links: Maxwell in practice
At 1 GHz the current in a copper wire flows in the first 2 microns of its surface, and the energy does not travel through the copper but through the insulator around it.
01From physics to the link
Maxwell's equations already appeared in Physics II, on the page on Maxwell's equations, as the synthesis of all electricity and magnetism. In Transmission Media they are used for a concrete purpose: carrying a signal from one point to another, through the air, along a cable, through a waveguide or along a fiber. Every medium in this course is a solution of the same four equations with different boundary conditions.
Since communication signals are almost always sinusoidal, or sums of sinusoids, it is convenient to write the equations in phasor form, where the time derivative is replaced by \( j\omega \):
02Conductor or dielectric: it depends on the frequency
The ratio between the conduction current and the displacement current, \( \sigma/\omega\varepsilon \), decides how a material behaves. If it is much greater than 1, the material is a good conductor; if it is much less, a good dielectric. The same material can be both, depending on the frequency:
| Material | \( \sigma \) (S/m) | \( \varepsilon_r \) | Behaves as a conductor up to |
|---|---|---|---|
| Copper | \( 5.8 \times 10^7 \) | 1 | Optical frequencies |
| Seawater | 4 | 81 | About 900 MHz |
| Moist soil | \( 10^{-2} \) | 15 | About 12 MHz |
| Dry soil | \( 10^{-3} \) | 4 | About 4.5 MHz |
| FR-4 (circuit board) | Negligible | 4.4 | It is a dielectric; it loses through \( \tan\delta \approx 0.02 \) |
Seawater is the classic example: at low frequencies it is a conductor that does not let waves through —which is why submerged submarines communicate on very low frequencies and with enormous antennas—, and above a gigahertz it behaves as a lossy dielectric.
03Boundary conditions
What distinguishes a waveguide from a line or an antenna is the surfaces that bound it. At the boundary between two media, Maxwell's equations impose:
| Component | Between two dielectrics | On a perfect conductor |
|---|---|---|
| Tangential \( \mathbf{E} \) | Continuous | Zero |
| Tangential \( \mathbf{H} \) | Continuous (no surface current) | Equal to the surface current |
| Normal \( \mathbf{D} \) | Continuous (no surface charge) | Equal to the surface charge |
| Normal \( \mathbf{B} \) | Continuous | Zero |
The condition \( E_{\text{tan}} = 0 \) on the metal is the one that fixes the modes of a waveguide: only the fields that vanish at the walls survive.
04The skin effect
Inside a good conductor, an alternating field decays exponentially from the surface. The current concentrates in a layer of characteristic thickness \( \delta \), the penetration depth (skin depth):
The consequences are everywhere: the resistance of a wire grows with frequency, Litz wire —many thin insulated strands twisted together— is used in radio-frequency coils and in switching power supplies, high-frequency conductors are silver-plated because only the surface matters, and a shield a few \( \delta \) thick attenuates almost the entire field.
Penetration depth as a function of frequency for several materials, and the increase in the resistance of a round wire calculated with the exact solution (Kelvin functions). The horizontal line is the radius of the wire: when δ drops below it, the current stops using the center of the conductor.
05Where the energy travels
The power density carried by an electromagnetic field is the Poynting vector:
Applied to a coaxial cable, it gives a surprising result: the electric field runs from the center conductor to the shield and the magnetic field encircles the center conductor, so the power travels through the dielectric between the two, not through the copper. The conductors guide the energy and take the losses, but the energy is carried by the field. This is the idea that connects transmission lines with waveguides and antennas.
06Where it appears in electronics
| Case | The phenomenon |
|---|---|
| Inductor of a switching power supply at 200 kHz | Skin and proximity effects: Litz wire is used |
| Traces of a high-speed board | Conductor losses through \( R_s \) and dielectric losses through \( \tan\delta \) |
| Shielding of an instrument | Thickness of several \( \delta \) at the frequency of the interference |
| Communication with submarines | Very low frequencies because of the conductivity of seawater |
| Microwave oven | The mesh in the door reflects 2.45 GHz because its holes are much smaller than the wavelength |
07In the lab
With an LCR meter, measure the series resistance of a piece of 1 mm copper wire at 1 kHz, 10 kHz and 100 kHz, and compare it with the lab value. Repeat with Litz wire of the same cross-section.
With a coil driven by a generator and another one as a pickup, measure the signal with a 0.1 mm aluminum foil between the two at 1 kHz, 100 kHz and 1 MHz. Explain the results with the penetration depth.
Measure the capacitance of a parallel-plate capacitor formed by the two copper faces of an FR-4 board and deduce its relative permittivity.
08Common mistakes
- Using the DC resistance at high frequency. The skin effect multiplies it.
- Classifying a material as a conductor or a dielectric without looking at the frequency. It is decided by \( \sigma/\omega\varepsilon \).
- Thinking that the power travels through the copper. It travels through the field, between the conductors.
- Shielding against low-frequency magnetic fields with a thin foil. A thickness of several \( \delta \) is needed, or a magnetic material.
09Self-assessment
What is the penetration depth of copper at 100 kHz?
\( 66 \text{ µm} \times \sqrt{10} \approx 209 \) µm.
At what frequency does seawater stop behaving as a conductor?
When \( \sigma = \omega\varepsilon \): \( f = 4/(2\pi \times 81 \varepsilon_0) \approx 890 \) MHz.
What condition does a perfect conductor impose on the electric field?
That its tangential component be zero at the surface.
Why are radio-frequency conductors silver-plated?
Because the current flows in a surface layer of thickness δ, and it is enough for that layer to be a good conductor.
10Further reading
- David K. Cheng. Field and Wave Electromagnetics. 2nd ed., Addison-Wesley, 1989. The classic text on fields and waves for engineering, with the skin effect and boundary conditions.
- Simon Ramo, John R. Whinnery and Theodore Van Duzer. Fields and Waves in Communication Electronics. 3rd ed., Wiley, 1994. Maxwell applied to the problems of communications electronics.
- Howard Johnson and Martin Graham. High-Speed Signal Propagation: Advanced Black Magic. Prentice Hall, 2003. The skin effect and dielectric losses in real boards and cables, with measurements.