10.3 Transmission Lines, Characteristic Impedance, Attenuation & SWR

10.3 Transmission Lines, Characteristic Impedance, Attenuation & SWR

A transmission line (or feedline) is the electrical conduit that transports radio frequency energy between a transmitter/receiver and an antenna system. Rather than behaving as simple DC hookup wires, transmission lines operate as distributed-parameter waveguides governed by electromagnetic field equations.

Selecting the correct transmission line, calculating its physical and electrical length, understanding line losses across frequency, and mastering the mechanics of Standing Wave Ratio (SWR) and wave reflections are critical skills tested on the FCC General Class exam. This section explores the electrical physics of coaxial cables, open-wire ladder lines, velocity factors, and power reflection dynamics.

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1. Transmission Line Architecture & Characteristic Impedance ($Z_0$)

Every transmission line possesses distributed electrical properties distributed continuously along its entire length: inductance per unit length ($L$) in henrys and capacitance per unit length ($C$) in farads.

+-----------------------------------------------------------------------------------------+
|                   EQUIVALENT DISTRIBUTED CIRCUIT OF A TRANSMISSION LINE                 |
|                                                                                         |
|           L/2          L/2                 L/2          L/2                             |
|      +---CCCC---------CCCC---+        +---CCCC---------CCCC---+                         |
|      |                       |        |                       |                         |
|   ---+                       +--------+                       +--->                     |
|             |          |                     |          |                               |
|            ---        ---                   ---        ---                              |
|          C ---      G ---                 C ---      G ---                              |
|             |          |                     |          |                               |
|   ---------------------------------------------------------------->                     |
|                                                                                         |
|   Characteristic Impedance:  Z0 = √(L / C)                                              |
+-----------------------------------------------------------------------------------------+

Mathematical Definition of $Z_0$

For a lossless transmission line, the characteristic impedance ($Z_0$) is the ratio of RF voltage to RF current of a single wave traveling down the line without reflections:

$Z_0 = \sqrt{\frac{L}{C}}$

Physical Determinants of $Z_0$

Characteristic impedance is determined solely by the physical geometry of the conductors and the dielectric constant ($\epsilon_r$) of the insulating material:

1. Coaxial Cable: $Z_0 = \frac{138}{\sqrt{\epsilon_r}} \log_{10}\left(\frac{D}{d}\right)$ Where $D$ is the inside diameter of the outer shield braid, $d$ is the outside diameter of the center conductor, and $\epsilon_r$ is the dielectric constant of the insulator.
2. Parallel Open-Wire Line: $Z_0 = \frac{276}{\sqrt{\epsilon_r}} \log_{10}\left(\frac{2S}{d}\right)$ Where $S$ is the center-to-center conductor spacing and $d$ is the wire diameter.

[!IMPORTANT] Exam Rule: The characteristic impedance ($Z_0$) of a transmission line depends only on conductor diameter, spacing, and dielectric material. It is completely independent of the length of the line and independent of operating frequency.

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2. Velocity Factor ($VF$) & Electrical vs. Physical Length

Electromagnetic waves travel at the speed of light ($c = 300\times 10^6\text{ m/s}$) only in a vacuum or in air. When an electromagnetic wave travels through a solid or foam dielectric insulating material inside a cable, its propagation velocity slows down.

The Velocity Factor ($VF$) is the ratio of wave velocity along the transmission line ($v_{\text{line}}$) to the velocity of light in free space ($c$):

$VF = \frac{v_{\text{line}}}{c} = \frac{1}{\sqrt{\epsilon_r}}$

+-----------------------------------------------------------------------------------------+
|                       DIELECTRIC MATERIALS & VELOCITY FACTORS                           |
|                                                                                         |
|   DIELECTRIC MATERIAL           TYPICAL VF       COMMON CABLE TYPES                     |
|   ---------------------------   ----------       ------------------------------------   |
|   Air Dielectric (Open Wire)    0.95 - 0.98      450 Ω Window Line, 600 Ω True Ladder   |
|   Foam Polyethylene (FPE)       0.80 - 0.85      LMR-400, RG-8X, RG-213 Foam            |
|   Solid Polytetrafluoroethylene 0.69 - 0.71      RG-142, RG-400 (Teflon)                |
|   Solid Polyethylene (PE)       0.66             RG-58, RG-8U, RG-213 Solid             |
+-----------------------------------------------------------------------------------------+

Calculating Physical Length of Transmission Line Stubs

Because the wave travels slower inside the cable, a quarter-wavelength or half-wavelength section of transmission line is physically shorter than its free-space counterpart:

$L_{\text{physical (feet)}} = L_{\text{free space}} \times VF = \left(\frac{984}{f_{\text{MHz}}} \times \text{Electrical Fraction}\right) \times VF$

Practical Stub Calculation Example:

Calculate the physical length of an electrical quarter-wavelength ($\lambda / 4$) matching stub at $14.100\text{ MHz}$ using solid polyethylene coaxial cable ($VF = 0.66$):

$L_{\text{free space (}\lambda/4\text{)}} = \frac{246}{14.100} = 17.447\text{ feet}$

$L_{\text{physical}} = 17.447 \times 0.66 = 11.51\text{ feet}$

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3. Coaxial Cable Types vs. Open-Wire / Ladder Line

Amateur operators choose between shielded coaxial cables and balanced parallel-conductor lines based on power levels, frequency, physical routing, and anticipated SWR.

+-----------------------------------------------------------------------------------------+
|                        COAXIAL CABLE VS. OPEN-WIRE WINDOW LINE                          |
|                                                                                         |
|   COAXIAL CABLE (UNBALANCED 50 Ω / 75 Ω):                                               |
|   - Shielded construction prevents RF radiation and allows routing near metal/conduits   |
|   - Higher dielectric loss at high frequencies                                          |
|   - High SWR dramatically increases internal cable heat and signal attenuation          |
|                                                                                         |
|   OPEN-WIRE / 450 Ω WINDOW LADDER LINE (BALANCED):                                      |
|   - Air-insulated dielectric provides exceptionally low loss                            |
|   - Handles very high SWR (10:1 or higher) with virtually zero added loss               |
|   - Must be suspended away from metal objects, gutters, and earth                       |
+-----------------------------------------------------------------------------------------+

Common Transmission Line Specifications Matrix

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4. Standing Wave Ratio (SWR) & Reflection Mechanics

When a transmission line of characteristic impedance $Z_0$ is terminated in a load impedance ($Z_L$) that matches $Z_0$ perfectly ($Z_L = Z_0$), all energy arriving at the load is absorbed and converted into radiated RF or heat. The line is said to be flat, and the voltage reflection coefficient is zero.

If $Z_L \ne Z_0$, boundary conditions dictate that the load cannot absorb all incident power. A portion of the incoming forward voltage wave ($V_{\text{fwd}}$) is reflected back toward the generator as a reflected voltage wave ($V_{\text{ref}}$).

+-----------------------------------------------------------------------------------------+
|                        FORWARD & REFLECTED WAVE INTERFERENCE                            |
|                                                                                         |
|   Forward Wave  ===========================================> (Load ZL ≠ Z0)             |
|   Reflected Wave <==========================================                            |
|                                                                                         |
|   Composite Standing Wave:                                                              |
|   Vmax = |Vfwd| + |Vref|   (Constructive Interference Node)                             |
|   Vmin = |Vfwd| - |Vref|   (Destructive Interference Node)                              |
|                                                                                         |
|   SWR = Vmax / Vmin                                                                     |
+-----------------------------------------------------------------------------------------+

The Voltage Reflection Coefficient ($\Gamma$)

The ratio of the reflected voltage wave to the forward voltage wave is the complex voltage reflection coefficient ($\Gamma$):

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

Calculating Standing Wave Ratio (SWR)

The superposition of forward and reflected traveling waves creates a stationary interference pattern along the line called a standing wave. SWR is defined as the ratio of maximum RF voltage to minimum RF voltage along the line:

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

Simplified Resistive SWR Calculation

For purely resistive antenna loads ($R_L$ with zero reactance), SWR is easily calculated by dividing the larger resistance by the smaller resistance:

$\text{SWR} = \frac{R_L}{Z_0} \quad (\text{if } R_L > Z_0) \qquad \text{or} \qquad \text{SWR} = \frac{Z_0}{R_L} \quad (\text{if } R_L < Z_0)$

Examples:

• If a $100\ \Omega$ non-reactive load is connected to a $50\ \Omega$ line: $\text{SWR} = 100 / 50 = \mathbf{2.0:1}$.
• If a $25\ \Omega$ non-reactive load is connected to a $50\ \Omega$ line: $\text{SWR} = 50 / 25 = \mathbf{2.0:1}$.
• If a $150\ \Omega$ non-reactive load is connected to a $50\ \Omega$ line: $\text{SWR} = 150 / 50 = \mathbf{3.0:1}$.

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5. Forward Power, Reflected Power & Net Power Transfer

A common misconception is that reflected power travels back into the transmitter and instantly destroys the output transistors, or that reflected power represents permanently lost energy.

Power Relationships & Percentage Reflected

The fraction of power reflected from the load is equal to the square of the magnitude of the reflection coefficient ($|\Gamma|^2$):

$\frac{P_{\text{reflected}}}{P_{\text{forward}}} = |\Gamma|^2 = \left(\frac{\text{SWR} - 1}{\text{SWR} + 1}\right)^2$

+-----------------------------------------------------------------------------------------+
|                       SWR VS. REFLECTED POWER REFERENCE TABLE                           |
|                                                                                         |
|   SWR RATIO       REFLECTION COEFF (|Γ|)      % REFLECTED POWER      % DELIVERED POWER  |
|   ---------       ----------------------      -----------------      -----------------  |
|   1.0 : 1         0.000                       0.0%                   100.0%             |
|   1.5 : 1         0.200                       4.0%                    96.0%             |
|   2.0 : 1         0.333                      11.1%                    88.9%             |
|   3.0 : 1         0.500                      25.0%                    75.0%             |
|   4.0 : 1         0.600                      36.0%                    64.0%             |
|   5.0 : 1         0.667                      44.4%                    55.6%             |
|   10.0 : 1        0.818                      66.9%                    33.1%             |
|   ∞ : 1 (Open)    1.000                     100.0%                     0.0%             |
+-----------------------------------------------------------------------------------------+

What Really Happens to Reflected Power?

In a real transmission line system:

1. The transmitter launches forward power ($P_{\text{fwd}}$) toward the antenna.
2. At the mismatched antenna feedpoint, a portion ($P_{\text{ref}}$) reflects back down the feedline toward the transmitter.
3. When the reflected wave reaches the source (or antenna tuner matching network), the impedance discontinuity re-reflects the wave back toward the antenna in phase with subsequent transmitter cycles.
4. Net Power Transfer: In a low-loss line (such as $450\ \Omega$ ladder line), nearly 100% of the net transmitter power is eventually radiated by the antenna, minus minor ohmic line heating losses after multiple reflections.
5. Why High SWR Hurts Coaxial Lines: In lossy coaxial lines (like RG-58), standing waves create extreme voltage and current peaks. Because $I^2R$ ohmic heating and dielectric absorption increase exponentially with current and voltage, each pass of reflected power undergoes additional attenuation. On VHF/UHF or long coaxial runs, high SWR can convert most of your RF power into heat inside the cable.