Wizards of Electromagnetism

There are many practical difficulties that have to be solved in a signal integrity project, but two difficulties have their roots deep in the fundamentals of Maxwell’s equations.

One is obvious: each 0 or 1 pulse is actually an electromagnetic wave that travels at finite speed. The speed is named after as that famous electromagnetic wave: light. In vacuum, this universal speed limit is about one inch per 84.7 picoseconds, or one centimeter per 33.4 picoseconds. For striplines in FR4 board it’s about half that. Although the wave is guided by the copper trace, the wave travels in the insulating material surrounding the trace. The speed is set by the dielectric properties called permeability and permittivity:

c = (μ₀μ_rε₀ε_r)^−0.5

(The wave does penetrate the copper somewhat, but it dies away exponentially from the metal surface, with a decay constant equal to the skin depth.)

But another fundamental limitation is also related to these dielectric material properties. The relative magnitudes of the electric field E and the magnetic intensity H are called the impedance, defined by Z=E/H. The impedance for free propagation is:

Z = (μ₀μ_r/ε₀ε_r)^0.5

For vacuum, this is about 377 Ω and for FR4 it’s about half that. (Hence the retro graphic at the head of this post: the old color code for a five-band 377 Ω resistor was orange-purple-purple-black) For a guided wave, the impedance is generally less than that for free propagation. You can divide the free propagation impedance by a unitless geometrical form factor that is a function of various dimensional ratios of the cross-section of the wave guide. Often the function contains a logarithm of the geometric ratios and its value varies only slowly with aspect ratio. For practical geometries, the form factor is a smallish number say between 1 and 10. FR4 and a form factor of 4 would lead to an impedance of about 50 Ω.

All very well, but why should you care about impedance?

Just as light partly reflects and partly transmits at the impedance change between air and window glass, so your signal will partly reflect of off impedance changes along your trace. (Actually, optical folk usually talk about refractive index, n = Z₀/Z, rather than the impedance itself.) The reflected energy will bounce up and down the line and arrive at your receiver later than the main, direct pulse. Like a latecomer at the theater, this energy will corrupt the flow of information. Communication engineers call this multipath effect “inter symbol interference” or ISI. To minimize reflections and interference, you have to match the output impedance of the transmitter to the interconnect impedance, keep the impedance constant throughout the length of the interconnect, and match the interconnect impedance to the input impedance of the receiver.

Analogy of digital pulse to light

 More info on this topic at our January 15th, 2009 webcast “Back to Basics: Measurement-Based Channel Modeling for Signal Integrity using Agilent ADS”

Hope to see you there!

When Dorothy arrives in Oz, she says to her dog, “Toto, I’ve a feeling we’re not in Kansas any more.”
 
Likewise, digital electronic engineers like yourselves are being thrown from your world of ones and zeros into a microwave “Oz” – with its splendid but puzzling reflections, impedances, and electromagnetics — and you’re saying “Toto, I’ve a feeling we’re not in binary any more.”
 
You’re right.

Hopefully this blog will prove a more reliable guide than the Lion, the Tin Man and the Scarecrow. We’ll try to help you become the Wizard of Electromagnetism.

In this first post, I’ll reference the posting that inspired this blog. It honored James Clerk Maxwell, born June 13th, 1831 in Edinburgh, Scotland, UK (also the birthplace, by the way, of Alexander Graham Bell). Maxwell synthesized previously unrelated observations, experiments, and equations of electricity, magnetism, and optics into a consistent theory and set of equations—Maxwell’s equations—demonstrating that electricity, magnetism and even light are all manifestations of the electromagnetic field.

James Clerk Maxwell with one of his colour wheels

Maxwell’s work has stood the test of time. Maxwell’s equations are consistent with the Lorentz transformation, and inspired Einstein’s special relativity. In this view, magnetism is not a separate force, but the simply the dynamics of the electric force with space-time distortion from charged bodies in relative motion (electrodynamics). If the electric field is interpreted as the probability of observing a photon, the equations are consistent with quantum mechanics (quantum electrodynamics). QED has even been extended to cover the weak nuclear force (responsible for beta decay) in the electroweak theory. Thus extended, the theory encompasses in principle all physical, chemical, and biological phenomena except gravity and the strong nuclear force (quarks and gluons and such). The bottom line is that you can take the solutions to the bank: unlike SPICE, there are no approximations.

EM solvers are now a standard signal integrity analysis tool. Although the equations are computationally expensive to solve, the results accurately reflect the distributed nature of multigigabit per second serial links, where the wavelength of the highest frequency component is shorter than the physical size of the backplane. This is in contrast to nodal solvers like SPICE, which use a lumped element approximation, and which ignore crosstalk due to magnetic induction in conductor loops (curl(E) = -dB/dt).

So this blog will pick up the thread from the post on my signal integrity blog to drill down on practicle applications of electromagnetism at microwave frequencies to digital electronics.