9. Things to Try

A curated set of exercises, simulations, and mental experiments to lock the chapter in.

1. Compute the speed of light in FR-4 ($\varepsilon_r = 4.5$). Should land near $1.4\times 10^8$ m/s, about 14 cm/ns. Implication: a 14 cm trace on a 1 GHz signal is one full wavelength; you cannot ignore distributed effects.

2. Skin depth at your frequency. For copper, $\delta = 66\,\mu\text{m}/\sqrt{f_{\text{MHz}}}$. At 100 MHz, 6.6 µm. At 5 GHz, 0.93 µm. Now contemplate why you should not use thin nickel plating on RF connectors.

3. VSWR of 50 to 100 Ω. $|\Gamma| = (100-50)/(100+50) = 1/3$. VSWR = $(1 + 1/3)/(1 - 1/3) = 2$. About 11% of incident power reflects.

4. Quarter-wave transformer design. Match 50 Ω to 75 Ω. The transformer impedance is $\sqrt{50\times 75}\approx 61.2$ Ω, the closest standard line. At 1 GHz on FR-4, the quarter-wavelength is 3.5 cm. Lay it out and simulate in your favorite EM solver.

5. Smith chart by hand. Draw a normalized impedance $z = 0.5 + j0.5$ on a Smith chart. Read off $\Gamma$. Move toward generator by $\lambda/8$ and see where you land. Compare with the algebraic answer using the input-impedance formula.

6. Simulate a transmission line in SPICE. Drive a 50 Ω lossless line with a 1 ns step into 100 Ω termination. Watch reflections bouncing back and forth. The SPICE T-element does this analytically. Compare to the same simulation with a matched 50 Ω terminator. The difference is everything PCB signal-integrity engineers worry about.

7. TDR thought experiment. You measure a coax cable with a TDR. The reflection arrives at $t = 100$ ns. The dielectric is polyethylene ($\varepsilon_r = 2.3$). How long is the cable? Velocity is $c/\sqrt{\varepsilon_r}\approx 1.97\times 10^8$ m/s. One-way time is 50 ns, so length is about 9.85 m.

8. Faraday cage at home. Place a phone in a metal cookie tin (lid closed) and try to call it. Service drops to nothing. Now wrap a piece of kitchen aluminum foil around the phone, leaving one corner open. Service usually returns through the gap. Demonstrates how field leakage scales with aperture.

9. EM probe a microcontroller. Wind a small coil (5-10 turns of magnet wire on a 2 mm form), connect to an oscilloscope's high-impedance input via a coax. Hover over a running Arduino. You should see clean spikes at the SPI clock rate. With a faster scope, the data is recoverable. Now you understand the simplest near-field side-channel setup.

10. Estimate the wavelength inside your microwave oven. Frequency 2.45 GHz, $\varepsilon_r$ of air essentially 1, so $\lambda = c/f = 12.2$ cm. The hot spots in a microwave (where it cooks a marshmallow row deeply) are spaced at half this wavelength, around 6 cm. Look for them next time you reheat something.