5. Transmission Lines: When Wires Become Waveguides

Now we descend from free space to the cables, traces, and connectors of real hardware.

5.1 When is a wire a transmission line?

A wire becomes a transmission line when its length is comparable to a wavelength of the signal it is carrying. The rule of thumb: if the line length is more than $\lambda/10$, treat it as a transmission line. Below that, lumped circuit analysis (Chapter 2) is fine.

Some examples in FR-4 ($v \approx 14$ cm/ns, so $\lambda = v/f$):

A 2 cm trace at 100 MHz is fine to treat as a wire. The same trace at 5 GHz is unambiguously a transmission line, and ignoring that fact will produce reflections, ringing, and signal-integrity disasters.

5.2 Types of transmission lines

   ┌────────────────────────┐
   │  Coax (shielded)       │
   │   ●←inner               │
   │  ───────────            │
   │   ◯ outer (shield)      │
   ├────────────────────────┤
   │  Microstrip             │
   │  ━━━━ trace             │
   │                          │
   │  ════════ ground plane   │
   ├────────────────────────┤
   │  Stripline              │
   │  ════════ ground         │
   │  ━━━━ trace (centered)   │
   │  ════════ ground         │
   ├────────────────────────┤
   │  CPW (coplanar)         │
   │  ═══━━━━═══              │
   │  ─────────────           │
   │  ════════ optional gnd   │
   ├────────────────────────┤
   │  Twisted pair           │
   │  ⊗⊙⊗⊙⊗⊙⊗⊙ two wires      │
   └────────────────────────┘

• Two-wire (open-wire): old phone lines, ladder-line antenna feeders. High impedance (300-600 Ω), moderate loss, no shielding.
• Coaxial cable (coax): inner conductor inside an outer shield. Standard impedances 50 Ω (RF, test) and 75 Ω (TV, video). The shield contains the field, providing excellent immunity. In every cable TV install, every oscilloscope probe, every base-station antenna.
• Microstrip: a signal trace on top of a PCB above a ground plane, with the dielectric between. On every Wi-Fi router PCB, every smartphone radio frontend. Easy to fabricate, mostly TEM, but radiates a bit because half its field is in air.
• Stripline: trace sandwiched between two ground planes inside the PCB stackup. Better shielding than microstrip, slightly slower (all field is in dielectric). Used for sensitive analog or where coupling to top components is a concern.
• CPW (coplanar waveguide): signal and ground all on the same top layer. Easy probe access, uniform geometry, popular in millimeter-wave chips and test fixtures.
• Twisted pair: two wires twisted around each other. Differential signaling rejects common-mode noise and limits radiation. Ethernet, USB, telephone, HDMI internal pairs.
• Waveguide: hollow metal tube, no inner conductor. Field propagates inside without a center wire. Cannot support TEM mode; carries only TE and TM modes above a cutoff frequency. We meet these in Chapter 18 (microwave engineering).

5.3 The distributed equivalent circuit

A real transmission line, no matter what type, can be modeled as a chain of infinitesimal sections. Each tiny length $\Delta z$ has:

   ─┬─[R Δz]──[L Δz]─┬──
    │                 │
   [G Δz]           [C Δz]
    │                 │
   ─┴─────────────────┴──

• $R$ (Ω/m): series resistance per unit length, from conductor losses.
• $L$ (H/m): series inductance per unit length, from magnetic energy stored around the conductors.
• $G$ (S/m): shunt conductance per unit length, from dielectric losses.
• $C$ (F/m): shunt capacitance per unit length, from electric energy stored between conductors.

These four are the primary line parameters. They are derived from the line's geometry and material properties, by exactly the kind of Gauss's-law and Ampere's-law calculations we did in Sections 1 and 2. For coax, for instance:

$L = \frac{\mu_0}{2\pi}\ln(b/a), \quad C = \frac{2\pi\varepsilon}{\ln(b/a)}$

Plug in $a = 0.45$ mm and $b = 1.5$ mm for RG-58, with $\varepsilon_r = 2.3$ for polyethylene, and you get about $L \approx 240$ nH/m and $C \approx 100$ pF/m.

5.4 The telegraph equations

Apply KVL around the loop of one $\Delta z$ section:

$V(z) - V(z + \Delta z) = (R\,\Delta z) I + (L\,\Delta z)\frac{\partial I}{\partial t}$

Apply KCL at the node:

$I(z) - I(z + \Delta z) = (G\,\Delta z) V + (C\,\Delta z)\frac{\partial V}{\partial t}$

Divide by $\Delta z$, take the limit, switch to phasors at frequency $\omega$:

$\frac{dV}{dz} = -(R + j\omega L)I$ $\frac{dI}{dz} = -(G + j\omega C)V$

These are the telegraph equations (so-named because they were derived to model long telegraph lines in the 19th century). Differentiate the first and substitute the second:

$\frac{d^2 V}{dz^2} = (R + j\omega L)(G + j\omega C)\,V = \gamma^2 V$

A 1-D wave equation again, this time for voltage on the line. The general solution is

$V(z) = V_+ e^{-\gamma z} + V_- e^{+\gamma z}$

A forward-traveling wave plus a backward-traveling wave. Each travels at the line's wave velocity, with attenuation given by $\alpha = \text{Re}(\gamma)$.

5.5 Characteristic impedance derivation

Substitute the voltage solution into the first telegraph equation and solve for $I(z)$:

$I(z) = \frac{1}{Z_0}\left(V_+ e^{-\gamma z} - V_- e^{+\gamma z}\right)$

with characteristic impedance

$Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}}$

For a lossless line ($R = 0, G = 0$):

$Z_0 = \sqrt{L/C}$

This is one of the most important formulas in all of high-speed digital and RF design. It tells you that the impedance of a transmission line is set by geometry and dielectric, and not by length. A 1-meter and a 100-meter coax of the same construction have the same characteristic impedance.

For our RG-58 numbers:

$Z_0 = \sqrt{240\text{ nH/m}\,/\,100\text{ pF/m}} = \sqrt{2400}\approx 49\text{ Ω}$

There is your 50 Ω.

Standard impedances in real systems:

Why 50 Ω specifically? It is a compromise: minimum loss in air-dielectric coax happens around 77 Ω, maximum power-handling at 30 Ω, the geometric mean is around 50 Ω, and Bell Labs settled on it in the 1930s. Inertia and standardization did the rest.

5.6 Lossless and distortionless lines

A lossless line has $R = 0$ and $G = 0$. Then $\gamma = j\omega\sqrt{LC}$ is purely imaginary; $\alpha = 0$ and $\beta = \omega\sqrt{LC}$. Phase velocity $v_p = 1/\sqrt{LC}$, independent of frequency: every component of a signal travels at the same speed, so the signal shape is preserved.

A distortionless line is a real line with losses, but with $R/L = G/C$. Plugging in:

$\gamma = \sqrt{(R + j\omega L)(G + j\omega C)} = \sqrt{LC}(R/L + j\omega)$

so $\alpha = R\sqrt{C/L}$ (real, frequency-independent) and $\beta = \omega\sqrt{LC}$ (also independent of frequency). The signal attenuates uniformly across all frequencies but does not disperse: pulses come out smaller but with the same shape. Telegraph engineers used to add loading coils along long lines to satisfy $R/L = G/C$ artificially. Submarine telegraph cables in the late 1800s were loaded this way until cables shifted to repeated amplifiers and eventually fiber.

5.7 Reflection coefficient on a terminated line

Now suppose the line of impedance $Z_0$ is terminated at the far end ($z = 0$, say) by a load impedance $Z_L$. The boundary condition there is $V/I = Z_L$ at $z = 0$. Plug in the wave solution:

$Z_L = Z_0\frac{V_+ + V_-}{V_+ - V_-}$

Solve for the reflection coefficient at the load:

$\Gamma_L = \frac{V_-}{V_+} = \frac{Z_L - Z_0}{Z_L + Z_0}$

Identical in form to the Fresnel reflection coefficient at a dielectric boundary. Same physics. Special cases:

• $Z_L = Z_0$: $\Gamma = 0$, perfectly absorbed, no reflection.
• $Z_L = 0$ (short circuit): $\Gamma = -1$, full reflection with sign inversion.
• $Z_L = \infty$ (open circuit): $\Gamma = +1$, full reflection same sign.
• $Z_L = jX$ (pure reactance): $|\Gamma| = 1$, full reflection at some phase.

5.8 Standing waves and VSWR

When $\Gamma \neq 0$, the forward and reflected waves interfere along the line. At points where they add in phase, voltage maximum. Where they oppose, voltage minimum. The pattern is stationary in space (it does not propagate, hence "standing wave") but oscillates at $\omega$ in time.

   |V(z)|
        ╱╲      ╱╲      ╱╲      ╱╲
       ╱  ╲    ╱  ╲    ╱  ╲    ╱  ╲
   ───╱────╲──╱────╲──╱────╲──╱────╲─── z
            ╲╱      ╲╱      ╲╱      ╲╱
   ↑                                    ↑
   load                              source

The ratio of maximum to minimum voltage is the Voltage Standing Wave Ratio (VSWR):

$\text{VSWR} = \frac{V_{max}}{V_{min}} = \frac{1 + |\Gamma|}{1 - |\Gamma|}$

Hams and base-station engineers think in VSWR. PCB and signal-integrity engineers think in return loss $RL = -20\log_{10}|\Gamma|$ in dB. They are equivalent, just different units. A return loss of 20 dB corresponds to $|\Gamma| = 0.1$, VSWR 1.22, 1% of incident power reflected.

Why VSWR matters in real life. A high VSWR on a transmitter's antenna feedline means power is bouncing back into the power amplifier, sometimes catastrophically. Modern transmitters fold back power or shut down if VSWR exceeds ~3:1. On a digital trace, high VSWR means reflections that ring at every transition; eyes close, bits get mangled, your DDR4 fails timing margin.

5.9 Input impedance of a terminated line

What does the source see when it looks into a transmission line of length $\ell$, terminated in $Z_L$? Compute the ratio of $V$ to $I$ at the input:

$Z_{in}(\ell) = Z_0\frac{Z_L + jZ_0\tan(\beta\ell)}{Z_0 + jZ_L\tan(\beta\ell)}$

This is the input impedance. The line transforms impedance, in a way that depends on length, frequency, and load. Some particularly useful special cases:

• Half-wavelength line ($\ell = \lambda/2$): $\tan(\beta\ell) = 0$, so $Z_{in} = Z_L$. The half-wave line repeats the load impedance unchanged. Useful for moving an antenna's impedance across a wall or rooftop without changing it.
• Quarter-wavelength line ($\ell = \lambda/4$): $\tan(\beta\ell) = \infty$, and the ratio collapses to:

$Z_{in} = \frac{Z_0^2}{Z_L}$

The quarter-wave line is an impedance transformer.

• Short-circuited stub ($Z_L = 0$): $Z_{in} = jZ_0\tan(\beta\ell)$. Pure reactance, varying from inductive ($\ell < \lambda/4$) through infinite ($\ell = \lambda/4$) to capacitive ($\lambda/4 < \ell < \lambda/2$) and back. A short-circuited stub is a tunable reactance you can build out of nothing but copper trace.
• Open-circuited stub ($Z_L = \infty$): $Z_{in} = -jZ_0\cot(\beta\ell)$. Same flexibility, dual behavior.

5.10 Quarter-wave transformer

To match a 50 Ω source to a 200 Ω load, insert a quarter-wavelength of line whose characteristic impedance is the geometric mean:

$Z_T = \sqrt{Z_0 Z_L} = \sqrt{50\times 200} = 100\text{ Ω}$

Then $Z_{in} = Z_T^2/Z_L = 10000/200 = 50$ Ω. Perfect match. At one frequency.

The quarter-wave transformer is narrowband. Above and below the design frequency, the match degrades because the line is no longer exactly a quarter-wavelength. Multi-section transformers (two or more cascaded quarter-wave sections of carefully chosen impedances) trade complexity for bandwidth. The stripline impedance transitions in your microwave radio's RF front-end are typically a chain of multi-section quarter-wave transformers.

   Source        λ/4 of Z_T          Load
   50 Ω ───[ ━━━━━━━━━━━━━ ]─── 200 Ω
              Z_T = 100 Ω

5.11 Stub matching: single and double

What if you cannot break the line to insert a quarter-wave transformer? Place a stub in parallel with the main line at a chosen position, with a chosen length, and let it cancel out the imaginary part of the load admittance.

Single-stub match procedure:

1. Move along the line from the load until the admittance $Y_d$ has the right real part, $Y_d = Y_0 + jB$.
2. Place a parallel stub at that point whose susceptance is $-B$. The combination has admittance $Y_0$, matched.

The Smith chart (Section 5.13) makes this two-step procedure visual and three minutes of work.

Double-stub match: two stubs at fixed positions (often $\lambda/8$ apart) with adjustable lengths. Avoids the need to slide the stub position along the line, at the cost of being unable to match every load.

These techniques are the bread and butter of every RF engineer matching antennas, RF amplifiers, mixers, and filters. On a real Wi-Fi board, you can sometimes spot the matching stubs as little stubby pads or short trace fingers near the front-end IC.

5.12 Time-domain reflectometry

Send a step pulse down a line and watch the reflected waveform as a function of time. Each impedance discontinuity along the line produces a partial reflection arriving at a delay $2t$, where $t$ is the one-way travel time to the discontinuity. The amplitude of each reflection tells you the magnitude (and sign) of the impedance jump.

This is time-domain reflectometry (TDR), and it is hugely useful:

• Find a fault in a buried cable from one end. The reflection time tells you exactly how far down the fault is.
• Detect a tap on a network cable. A tap is a small impedance discontinuity, and TDR sees it as a small reflection at the tap location. This is one of the standard hardware-tampering detection methods, used everywhere from data-center cable monitoring to secure networking.
• Validate PCB stackups. Run TDR on a high-speed trace to confirm the impedance matches the design across the trace.
• Evaluate connectors and vias. Each is a small discontinuity, and TDR turns each into a measurable reflection.

The companion technique, time-domain transmission (TDT), watches what comes out the far end of the line for clues about loss and dispersion.

5.13 The Smith chart

The Smith chart is a graphical device, invented by Phillip H. Smith in 1939, that maps the complex reflection coefficient $\Gamma$ onto a chart with circles of constant resistance and constant reactance. It collapses the messy algebra of impedance transformation onto a single picture you can manipulate by inspection.

The mathematical idea: define normalized impedance $z = Z/Z_0 = r + jx$. The reflection coefficient is

$\Gamma = \frac{z - 1}{z + 1}$

This bilinear transformation maps the right half of the complex $z$-plane (passive impedances, $r \geq 0$) onto the unit disk in the $\Gamma$-plane. Constant-$r$ vertical lines in the $z$-plane become circles in the $\Gamma$-plane. Constant-$x$ horizontal lines become arcs.

   Smith chart sketch (axes labeled)
 
                  +jx
                   ↑
              ╲   │   ╱
        x=1 →╲ ╱╲│╱╲ ←x=1
         x=0.5  │  x=0.5
            ────┼────  r=∞ (right edge)
        x=0.5↗│↘x=0.5
        x=1 ↗ ╲│╱ ↘x=1
              ╱╱│╲╲
                ↓ -jx
 
   Center:   Γ = 0,  perfect match
   Right:    Γ = +1, open
   Left:     Γ = -1, short
   Outer rim: |Γ| = 1, total reflection

Things you can do on the Smith chart by inspection:

• Read off VSWR. Distance from center is $|\Gamma|$. The outermost circle has VSWR $\infty$; concentric circles inward correspond to lower VSWR.
• Move along a transmission line. Translation along a lossless line is a rotation about the center of the Smith chart, by $2\beta\ell$ radians (a full $\lambda/2$ goes around once). Clockwise toward the source, counterclockwise toward the load.
• Convert between $z$ and admittance $y$. Reflect through the center.
• Design a single-stub match. Rotate the load until $g = 1$ on the admittance chart, then cancel the residual susceptance with a parallel stub.

Modern engineers use software like ADS, Microwave Office, or Python (scikit-rf) for all of this. Yet every RF engineer still draws Smith charts on a whiteboard when designing matching networks. The Smith chart is the compass-and-straightedge of RF engineering, and an active tool in industry, not a museum piece.

5.14 Why coax does not radiate

By Gauss's law, the field inside a coax cable is entirely between inner conductor and shield. Outside the shield, the enclosed charge is zero (Section 1.4). By Ampere's law, the magnetic field around the cable is zero too, because the inner conductor's current is exactly canceled by the equal-and-opposite return current on the inside of the shield.

So coax is, by construction, a perfectly contained electromagnetic structure. It does not leak outward, and it cannot pick up external interference (which would have to set up a current on the outside of the shield, decoupled from the signal current on the inside). This is why coax has been the workhorse of broadcast TV, professional audio, RF test equipment, base-station antenna feeds, and any application that demands signal integrity across long distances.

Practical caveat: coax is only this good if the shield is solid and well-grounded. Cheap cables with foil-only shields (no braid) leak at high frequencies. Connectors with bad ground contacts leak at the connection. Cable installers will tell you a coax is only as shielded as its worst connector.