6. Microstrip and Planar Transmission Media

6.1 The microstrip line

Waveguides are heavy, expensive, and rigid. For PCB-level integration we need something flat and printable. Enter microstrip: a flat conductor on top of a dielectric substrate, with a continuous ground plane on the back.

          ─────────────  W
        ┌───────────────┐    ← top conductor (signal)
        │     copper    │
        ├───────────────┤
        │               │
        │  dielectric   │    h    ← substrate, ε_r
        │               │
        ├───────────────┤
        │     copper    │    ← bottom ground plane
        └───────────────┘

This structure is the foundation of every modern RF PCB. Wi-Fi routers, cell phones, GPS receivers, radar boards, satellite ground equipment, lab signal generators all use microstrip to route GHz signals between chips.

6.2 Quasi-TEM mode

Unlike a hollow waveguide, microstrip does have two conductors (the trace and the ground), so a TEM-like mode exists. But because the field exists partly in the dielectric (below the trace) and partly in the air (above the trace), the boundary at the air-dielectric interface bends the field lines, and a small longitudinal field component appears. The result is quasi-TEM: mostly transverse, with a small longitudinal admixture that grows with frequency.

The quasi-TEM mode is the workhorse mode you analyze with transmission-line theory (Chapter 9) plus a few corrections.

6.3 Effective dielectric constant

The wave does not see "the dielectric" or "air"; it sees a weighted average. Define

$\varepsilon_{eff} = \frac{\varepsilon_r + 1}{2} + \frac{\varepsilon_r - 1}{2}\cdot\frac{1}{\sqrt{1 + 12 h/W}}$

a Hammerstad-Jensen formula widely used in PCB layout tools. For very narrow traces ($W \ll h$), $\varepsilon_{eff} \to (\varepsilon_r+1)/2$, the average. For wide traces ($W \gg h$), more of the field is in the dielectric, and $\varepsilon_{eff} \to \varepsilon_r$. The wave's phase velocity inside the microstrip is $c/\sqrt{\varepsilon_{eff}}$.

For FR-4 ($\varepsilon_r \approx 4.4$) with $h = 1.6$ mm and $W = 3$ mm: $\varepsilon_{eff} \approx 3.32$. Phase velocity $\approx c/1.82 \approx 1.65 \times 10^8$ m/s. A 10 cm trace at 1 GHz is $0.6\lambda_g$, well into transmission-line territory.

6.4 Characteristic impedance

The Hammerstad-Jensen approximations also give $Z_0$. For wide strips ($W/h \geq 1$),

$Z_0 = \frac{120\pi}{\sqrt{\varepsilon_{eff}}\left[W/h + 1.393 + 0.667\ln(W/h + 1.444)\right]}$

For narrow strips ($W/h < 1$),

$Z_0 = \frac{60}{\sqrt{\varepsilon_{eff}}}\ln\left(\frac{8h}{W} + \frac{W}{4h}\right)$

Designers solve these inversely: given $Z_0 = 50$ Ω target, $\varepsilon_r$ and $h$ from the substrate, what $W$? The answer for FR-4 with 1.6 mm thickness is $W \approx 3$ mm. For 0.8 mm thickness it is $W \approx 1.5$ mm. Modern PCB tools (Altium, KiCad, Cadence) build this into the trace-width calculator and use it automatically.

6.5 Substrates: FR-4 versus Rogers

The substrate material matters enormously at microwave.

• FR-4 ($\varepsilon_r \approx 4.3$ at low freq, drifts to 4.0 at 10 GHz; $\tan\delta \approx 0.02$): the cheap general-purpose PCB substrate. Fine for digital up to a few GHz, marginal for serious RF above 5 GHz. Loss is dominated by dielectric loss at GHz, and $\varepsilon_r$ varies enough across a board that controlled-impedance tolerances are loose. Used in everything affordable: Wi-Fi routers, motherboards, IoT modules.
• Rogers RO4350B ($\varepsilon_r = 3.48$, $\tan\delta = 0.0037$): low-loss FR-4-compatible RF substrate. Compatible with standard PCB processing but with much better high-frequency behavior. Used in mid-range RF: cellular base stations, automotive radar.
• Rogers RT/duroid 5880 ($\varepsilon_r = 2.20$, $\tan\delta = 0.0009$): PTFE-glass laminate. Extremely low loss, very stable $\varepsilon_r$, but expensive and a different processing flow. The reference substrate for high-end RF: deep-space transponders, mm-wave radar, scientific instruments.

The choice between them is a cost-versus-loss tradeoff. For a 10 cm microstrip at 10 GHz, FR-4 might dissipate 2 dB; Rogers 4350 about 0.5 dB; RT/duroid 5880 about 0.15 dB. At 30 GHz the gaps widen further.

6.6 Microstrip losses

Three loss mechanisms compete:

• Conductor loss $\alpha_c$: copper resistance, scales as $\sqrt{f}$ via skin effect.
• Dielectric loss $\alpha_d$: substrate $\tan\delta$, scales linearly with $f$.
• Radiation loss $\alpha_r$: discontinuities and bends radiate, scales rapidly with $f$.

At low GHz, conductor loss dominates; at high GHz, dielectric loss takes over. Above 30 GHz on FR-4, dielectric loss alone can be a dB per inch and the substrate becomes useless.

Radiation loss is what makes microstrip useful for patch antennas (Chapter 13) and bad for through-traces. The two purposes cannot coexist on the same trace; you either deliberately radiate or deliberately confine. Stripline (next subsection) eliminates radiation loss entirely by burying the trace.

6.7 Stripline and other planar media

A few planar variants worth knowing:

• Stripline: signal trace sandwiched between two ground planes, surrounded by dielectric. No air interface, so the mode is pure TEM, no radiation, more isolated. Used in high-density RF backplanes and some military radar interconnects. Tradeoff: harder to mount components on (the trace is buried), so via fences are needed to connect to surface mounts.
• Coplanar waveguide (CPW): signal trace flanked by ground planes on the same surface. Field is mostly between the trace and adjacent grounds. Useful for chip-level interconnect because no via to a back ground is needed. Used heavily in MMICs (monolithic microwave ICs).
• Slotline: a slot etched in a single ground plane on a substrate. Carries quasi-TE mode. Used in some specialty circuits (balun structures, magnetic-coupled mixers).
• Substrate-integrated waveguide (SIW): rows of vias forming "walls" in a PCB substrate, mimicking a rectangular waveguide on a flat board. Behaves like waveguide above its cutoff but is fabricated with normal PCB processes. Used in mm-wave automotive radar to replace bulky waveguide with PCB-integrated equivalents.

Each variant trades off field confinement, radiation, ease of fabrication, and density.