8.1 Transmission Line Characteristics, Impedance, and Velocity Factor

8.1 Transmission Line Characteristics, Impedance, and Velocity Factor

In direct current (DC) and low-frequency alternating current (AC) circuits, interconnecting conductors are routinely treated as ideal equipotential nodes. According to Kirchhoff's current and voltage laws, the current entering one end of a wire instantaneously equals the current exiting the other end, and the potential difference along the wire is assumed to be negligible. This lumped-element approximation holds true only as long as the physical length of the circuit remains negligibly small compared to the wavelength ($\lambda$) of the electromagnetic energy being transported—typically when the physical length is less than $\lambda / 16$ or $0.1\lambda$.

In radio frequency (RF) engineering, where frequencies span from high frequency (HF, $3\text{ to }30\text{ MHz}$) through very high frequency (VHF, $30\text{ to }300\text{ MHz}$) and ultra-high frequency (UHF, $300\text{ to }3000\text{ MHz}$), operational wavelengths shrink from $100\text{ meters}$ down to mere centimeters. Under these conditions, the physical length of the interconnecting cable between a transceiver and an antenna is often several wavelengths long. The transit time of the electromagnetic wave across the line becomes a substantial fraction of the RF wave's period, causing voltage and current to vary continuously in magnitude and phase along the conductor. Such an interconnection can no longer be modeled as a simple lumped wire; it must be analyzed as a distributed parameter transmission line.

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The Distributed Parameter Model (Primary Line Constants)

A transmission line consists of two or more parallel conductors guiding electromagnetic energy from a source (such as a transmitter) to a load (such as an antenna). Because the inductance, capacitance, resistance, and dielectric leakage are distributed continuously along the entire length of the cable rather than concentrated in discrete components, engineers model the line as an infinite cascade of infinitesimally short segments of length $\Delta x$.

            R' dx         L' dx
upper o----[ R ]---------[ L ]----+------------+----o upper
                                  |            |
                              [ C' dx ]    [ G' dx ]
                                  |            |
return o--------------------------+------------+----o return

V(x) is measured upper-to-return at the left.
V(x+dx) is measured upper-to-return at the right.

Each infinitesimal segment $\Delta x$ is governed by four primary line constants:

1. Distributed Series Resistance ($R'$): Measured in ohms per meter ($\Omega/\text{m}$), representing the combined ohmic resistance of both conductors per unit length. At radio frequencies, $R'$ is not constant; it increases with the square root of frequency ($\sqrt{f}$) due to the skin effect, where alternating current crowds into a shallow depth along the conductor surface.
2. Distributed Series Inductance ($L'$): Measured in henrys per meter ($\text{H/m}$), representing the total self-inductance of the conductors and the magnetic flux linkage established in the space between and around them per unit length.
3. Distributed Shunt Conductance ($G'$): Measured in siemens per meter ($\text{S/m}$), representing the leakage current flowing through the imperfect insulating dielectric separating the two conductors per unit length. In high-grade RF lines, $G'$ is extremely small but increases with frequency due to dielectric hysteresis losses.
4. Distributed Shunt Capacitance ($C'$): Measured in farads per meter ($\text{F/m}$), representing the electrostatic capacitance established between the conductors across the intervening dielectric per unit length.

The Telegrapher's Equations and Wave Propagation

Applying Kirchhoff's voltage and current laws to the infinitesimal segment of length $\Delta x$ as $\Delta x \to 0$ yields the foundational Telegrapher's Equations for sinusoidal steady-state RF signals ($e^{j\omega t}$):

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

where $\omega = 2\pi f$ is the angular frequency in radians per second. Differentiating these first-order coupled equations and substituting one into the other yields the second-order uncoupled wave equations:

$$\frac{d^2 V(x)}{dx^2} = \gamma^2 V(x)$$ $$\frac{d^2 I(x)}{dx^2} = \gamma^2 I(x)$$

where $\gamma$ is the complex propagation constant:

$$\gamma = \alpha + j\beta = \sqrt{(R' + j\omega L')(G' + j\omega C')}$$

• The real part, $\alpha$, is the attenuation constant (measured in nepers per meter, convertible to $\text{dB/m}$ via $1\text{ Np} \approx 8.686\text{ dB}$), representing signal amplitude decay per unit distance due to heat dissipation.
• The imaginary part, $\beta$, is the phase constant (measured in radians per meter, $\beta = 2\pi / \lambda$), representing the spatial phase rotation of the wave as it travels along the line.

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Characteristic Impedance ($Z_0$)

The characteristic impedance ($Z_0$), also termed the surge impedance, is defined as the ratio of the forward-travelling voltage wave to the forward-travelling current wave at any point on a uniform line of infinite length, or on a line of finite length terminated in an impedance equal to $Z_0$:

$$Z_0 = \frac{V^+(x)}{I^+(x)} = \sqrt{\frac{R' + j\omega L'}{G' + j\omega C'}}$$

The RF Low-Loss Approximation

In practical amateur radio installations operating at HF, VHF, and UHF, high-quality feedlines use conductors with high electrical conductivity (copper or silver-plated copper) and low-loss dielectrics (polyethylene, foam, or air). At these operating frequencies, the reactive terms far exceed the loss terms:

$$\omega L' \gg R' \quad \text{and} \quad \omega C' \gg G'$$

Under this low-loss condition, $R'$ and $G'$ can be neglected in the impedance formula:

$$Z_0 \approx \sqrt{\frac{j\omega L'}{j\omega C'}} = \sqrt{\frac{L'}{C'}}$$

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Physical Geometries and Geometric Impedance Formulas

Transmission lines in amateur radio fall into two primary physical categories: coaxial cables (unbalanced lines) and parallel open-wire / ladder lines (balanced lines).

1. Coaxial Cable Geometry

A coaxial cable consists of a central inner conductor of diameter $d$ concentric with a cylindrical outer shielding conductor of inner diameter $D$, separated by a dielectric medium having relative permittivity $\varepsilon_r$ (dielectric constant).

                            Coaxial Cable Cross-Section

                                  /-------------\
                                /   Dielectric    \
                               |     (ε_r)         |
                               |     /-------\     |
                               |    | Center  |    |
                               |    | Conductor    |
                               |    |  (d)    |    |
                               |     \-------/     |
                               |                   |
                                \   Outer Shield  /
                                  \-- (D) -------/

The distributed inductance and capacitance per unit length are:

$$L' = \frac{\mu_0 \mu_r}{2\pi} \ln\left(\frac{D}{d}\right) \quad (\text{H/m})$$ $$C' = \frac{2\pi \varepsilon_0 \varepsilon_r}{\ln(D/d)} \quad (\text{F/m})$$

Substituting $L'$ and $C'$ into $Z_0 = \sqrt{L'/C'}$ (assuming non-magnetic dielectric $\mu_r = 1$):

$$Z_0 = \frac{1}{2\pi} \sqrt{\frac{\mu_0}{\varepsilon_0 \varepsilon_r}} \ln\left(\frac{D}{d}\right) = \frac{60}{\sqrt{\varepsilon_r}} \ln\left(\frac{D}{d}\right) = \frac{138}{\sqrt{\varepsilon_r}} \log_{10}\left(\frac{D}{d}\right)$$

where $\sqrt{\mu_0 / \varepsilon_0} \approx 376.73\ \Omega$ is the intrinsic impedance of free space.

The Historical Origin of 50-Ohm and 75-Ohm Standards

Why did $50\ \Omega$ and $75\ \Omega$ become ubiquitous standards across RF systems?

• Minimum Attenuation ($77\ \Omega$): For an air-dielectric coaxial line, mathematical minimization of conductor skin loss yields an optimal diameter ratio of $D/d \approx 3.591$. Substituting this ratio into the impedance formula gives $Z_0 = 60 \ln(3.591) \approx 76.7\ \Omega$. This was rounded to $75\ \Omega$, which became the worldwide standard for low-loss receiving systems and telecommunications distribution (e.g., cable TV, RG-6, RG-11).
• Maximum RF Power Handling ($30\ \Omega$): In high-power transmitters, dielectric breakdown caused by peak electric field gradient ($E_{\text{max}}$) limits voltage capacity. Peak power handling is maximized when $D/d = \sqrt{e} \approx 1.649$, yielding $Z_0 = 60 \ln(1.649) \approx 29.9\ \Omega$.
• The 50-Ohm Compromise: In 1929, Bell Laboratories demonstrated that $50\ \Omega$ represents the ideal engineering compromise between the power-handling capability of $30\ \Omega$ and the low attenuation of $77\ \Omega$. Furthermore, when solid polyethylene ($\varepsilon_r \approx 2.26$) is used, the minimum loss ratio shifts downwards toward $52\ \Omega$, firmly establishing $50\ \Omega$ as the universal standard for amateur transceivers, commercial two-way radios, and RF test equipment.

2. Parallel Two-Wire Line Geometry (Open-Wire and Twin-Lead)

A parallel two-wire line consists of two identical conductors of diameter $d$, separated center-to-center by spacing $S$, immersed in a dielectric medium of relative permittivity $\varepsilon_r$.

                        Parallel Two-Wire Line Geometry

                             <------- S ------->
                             (•)               (•)
                             <-d->             <-d->

When the spacing is large compared to wire diameter ($S \gg d$):

$$Z_0 = \frac{120}{\sqrt{\varepsilon_r}} \cosh^{-1}\left(\frac{S}{d}\right) \approx \frac{120}{\sqrt{\varepsilon_r}} \ln\left(\frac{2S}{d}\right) = \frac{276}{\sqrt{\varepsilon_r}} \log_{10}\left(\frac{2S}{d}\right)$$

For an open-wire line suspended in air ($\varepsilon_r \approx 1.0$), with wire diameter $d = 2.0\text{ mm}$ and spacing $S = 150\text{ mm}$:

$$Z_0 = 276 \log_{10}\left(\frac{2 \times 150}{2.0}\right) = 276 \log_{10}(150) = 276 \times 2.176 \approx 600.6\ \Omega$$

This explains why classic open-wire ladder lines built by radio amateurs naturally possess characteristic impedances of $450\text{ to }600\ \Omega$.

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Standard Amateur Radio Feedlines

Radio amateurs select transmission lines based on power rating, physical environment, mechanical flexibility, and acceptable attenuation across specific frequency bands.

Use the selected manufacturer's data sheet for velocity factor, attenuation at the operating frequency and allowable RF power. Average thermal power and peak voltage limits are different ratings. Observe the stated temperature, VSWR and installation conditions; connectors can impose a lower limit than the cable.

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Velocity Factor ($VF$): Electrical vs. Physical Length

In a vacuum, electromagnetic waves propagate at the speed of light, $c \approx 2.9979 \times 10^8\text{ m/s}$ (routinely approximated as $3.00 \times 10^8\text{ m/s}$). When an electromagnetic wave travels through a dielectric medium (insulator), its propagation velocity ($v$) slows down according to the relative permittivity ($\varepsilon_r$) of the material:

$$v = \frac{c}{\sqrt{\varepsilon_r}}$$

The Velocity Factor ($VF$), also called the velocity of propagation, is the ratio of the phase velocity in the transmission line to the speed of light in vacuum:

$$VF = \frac{v}{c} = \frac{1}{\sqrt{\varepsilon_r}}$$

Because all practical dielectrics have a relative permittivity greater than unity ($\varepsilon_r > 1.0$), the velocity factor is always less than $1.0$:

• Air: $\varepsilon_r\approx1.0006\implies VF\approx0.9997$, effectively $1$ for ordinary calculations. Spacers and other dielectric supports can reduce a practical open-wire line's velocity factor.
• Cellular (Foam) Polyethylene: $\varepsilon_r \approx 1.38 - 1.64 \implies VF \approx 0.78 - 0.85$
• Polytetrafluoroethylene (PTFE / Teflon): $\varepsilon_r \approx 2.05 \implies VF \approx 0.70$
• Solid Polyethylene (PE): $\varepsilon_r \approx 2.26 \implies VF = \frac{1}{\sqrt{2.26}} \approx 0.66$

Calculating Physical Length from Electrical Length

Because the wave travels slower in the cable, the wavelength within the transmission line ($\lambda_{\text{line}}$) is compressed compared to the free-space wavelength ($\lambda_0$):

$$\lambda_{\text{line}} = \frac{v}{f} = \frac{VF \times c}{f} = VF \times \lambda_0$$

Consequently, when cutting transmission lines for resonant stubs, phasing sections, or quarter-wave transformers, the physical length ($l_{\text{phys}}$) required to achieve a specific electrical fraction ($n$) of a wavelength is:

$$l_{\text{phys}} = n \times \lambda_{\text{line}} = n \times VF \times \lambda_0 = n \times VF \times \left(\frac{c}{f}\right)$$

Step-by-Step Worked Example: Quarter-Wave Matching Section

An amateur radio operator needs to construct a quarter-wave ($\lambda/4$) matching transformer for an antenna array operating at $f = 144.20\text{ MHz}$. Calculate the physical length required if using:

1. Solid polyethylene coaxial cable ($VF = 0.66$)
2. Gas-injected foam polyethylene cable ($VF = 0.85$)
3. Open-wire line with a measured velocity factor of $0.98$

Step 1: Compute the free-space wavelength ($\lambda_0$):

$$\lambda_0 = \frac{c}{f} = \frac{2.9979 \times 10^8\text{ m/s}}{144.20 \times 10^6\text{ Hz}} \approx 2.0790\text{ m}$$

The free-space quarter-wavelength is:

$$\frac{\lambda_0}{4} = \frac{2.0790\text{ m}}{4} \approx 0.5197\text{ m} = 51.97\text{ cm}$$

Step 2: Apply the Velocity Factor for each cable type:

• Solid PE ($VF = 0.66$): $l_{\text{phys}} = 0.5197\text{ m} \times 0.66 = 0.3430\text{ m} = 34.30\text{ cm}$
• Foam PE ($VF = 0.85$): $l_{\text{phys}} = 0.5197\text{ m} \times 0.85 = 0.4417\text{ m} = 44.17\text{ cm}$
• Open-wire line (specified $VF=0.98$): $l_{\text{phys}} = 0.5197\text{ m} \times 0.98 = 0.5093\text{ m} = 50.93\text{ cm}$

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Transmission Line Loss Mechanisms

Power entering a transmission line is not transferred with $100\%$ efficiency. Power dissipation along the line occurs primarily through three distinct physical mechanisms:

1. Conductor Ohmic Losses ($I^2 R'$)

Conductor loss results from the finite electrical conductivity ($\sigma$) of the metal conductors. As alternating current flows, electrical energy is converted into heat according to $P_{\text{loss}} = I^2 R'$. At radio frequencies, this loss is severely exacerbated by the skin effect.

Alternating current flowing through a conductor induces internal magnetic flux that opposes current flow in the center of the conductor. Consequently, the current density ($J$) concentrates exponentially in a thin annular layer at the outer periphery of the conductor. The skin depth ($\delta$), defined as the depth at which current density drops to $1/e$ ($\approx 36.8\%$) of its surface value, is:

$$\delta = \sqrt{\frac{\rho}{\pi f \mu}} = \sqrt{\frac{1}{\pi f \mu \sigma}}$$

where $\rho$ is electrical resistivity ($\Omega\cdot\text{m}$), $\mu$ is magnetic permeability, and $\sigma$ is electrical conductivity ($\text{S/m}$). For pure copper at $20^\circ\text{C}$:

• At $f = 3.5\text{ MHz}$ (80m band): $\delta \approx 35.1\ \mu\text{m}$
• At $f = 14.0\text{ MHz}$ (20m band): $\delta \approx 17.5\ \mu\text{m}$
• At $f = 144\text{ MHz}$ (2m band): $\delta \approx 5.48\ \mu\text{m}$
• At $f = 432\text{ MHz}$ (70cm band): $\delta \approx 3.16\ \mu\text{m}$

Because the effective cross-sectional area through which RF current flows is restricted to this tiny skin layer, the effective RF resistance ($R_{\text{RF}}$) increases proportionally to the square root of frequency:

$$R_{\text{RF}} \propto \sqrt{f}$$

Conductor attenuation ($\alpha_c$) in decibels per unit length therefore scales as:

$$\alpha_c \propto \frac{\sqrt{f}}{d}$$

This explains why thin coaxial cables (such as RG-58, with center conductor diameter $d \approx 0.9\text{ mm}$) exhibit dramatically higher attenuation than heavy cables (such as RG-213 with $d \approx 2.25\text{ mm}$ or LMR-400 with $d \approx 2.74\text{ mm}$).

2. Dielectric Losses ($V^2 G'$)

Dielectric loss occurs when the alternating electric field between the conductors repeatedly realigns the molecular dipoles of the insulating material. The microscopic molecular friction converts electrical energy directly into thermal heat. Dielectric loss is governed by the material's dissipation factor or loss tangent ($\tan \delta$):

$$\alpha_d\approx27.3\,\frac{\sqrt{\varepsilon_r}\,f\tan\delta}{c}\quad(\text{dB/m})$$

3. Radiation and Induction Losses

In balanced open-wire lines, radiation loss occurs if the conductor spacing ($S$) approaches an appreciable fraction of a wavelength ($S > 0.05\lambda$) or if current unbalance exists between the two wires. Coaxial shielding confines normal differential fields mostly inside the cable. Real shields and connectors have finite leakage, and common-mode current on the outside can also radiate.

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Waveguides: Microwave Transmission Media

At frequencies above $1\text{ to }2\text{ GHz}$, conventional coaxial cables face severe operational bottlenecks: skin-effect resistance on the tiny center conductor becomes enormous, dielectric dissipation causes heavy signal loss, and high RF voltages can cause dielectric breakdown. To overcome these limitations, microwave systems utilize waveguides.

                     Rectangular Waveguide Geometry (TE_10 Mode)

                              +--------------------------+
                             /|                         /|
                            / |                        / |
                           +--------------------------+  |
                           |  |                       |  |
                         b |  |                       |  | b (Narrow Dimension)
                           |  |                       |  |
                           |  +-----------------------|--+
                           | /                        | /
                           |/                         |/
                           +--------------------------+
                                        a (Broad Dimension)

A waveguide is a hollow, highly conductive metallic pipe—typically rectangular or circular in cross-section—that confines and directs electromagnetic waves through its enclosed dielectric (usually dry air or pressurized inert gas). Because waveguides possess no center conductor and use air dielectric, they exhibit neither center-conductor skin loss nor solid dielectric dissipation, achieving exceptionally low attenuation and multi-kilowatt power handling at microwave frequencies.

Modes of Propagation and Cutoff Frequency

Electromagnetic waves cannot propagate through a single hollow pipe in the Transverse Electro-Magnetic (TEM) mode that characterizes coaxial and two-wire lines. TEM propagation requires two electrically isolated conductors. Instead, waveguides support two families of propagation modes:

1. Transverse Electric ($TE_{m,n}$ or $H_{m,n}$ Modes): The electric field ($E$) is everywhere purely transverse (perpendicular) to the direction of wave propagation along the guide ($E_z = 0$), while a longitudinal magnetic field ($H_z$) exists.
2. Transverse Magnetic ($TM_{m,n}$ or $E_{m,n}$ Modes): The magnetic field ($H$) is everywhere purely transverse ($H_z = 0$), while a longitudinal electric field ($E_z$) exists.

The integers $m$ and $n$ represent the number of half-wave field variations across the broad dimension ($a$) and narrow dimension ($b$) respectively.

The Dominant $TE_{10}$ Mode

In rectangular waveguides where $a > b$ (typically $a = 2b$), the mode possessing the lowest cutoff frequency is the $TE_{10}$ mode, known as the dominant mode. For the $TE_{10}$ mode, the cutoff wavelength ($\lambda_c$) and cutoff frequency ($f_c$) are:

$$\lambda_c = 2a$$ $$f_c = \frac{c}{\lambda_c} = \frac{c}{2a}$$

• High-Pass Filter Behavior: The waveguide functions as an intrinsic high-pass filter. Signals with frequencies below the cutoff frequency ($f < f_c$) cannot propagate down the guide; they are attenuated exponentially as evanescent waves.
• Operating Frequency Band: To ensure single-mode propagation and prevent multimode dispersion (where multiple modes propagate at different velocities, distorting signals), waveguides are operated within a recommended passband between $1.25 f_c$ and $1.90 f_c$.

In amateur radio, standard rectangular waveguide designations include:

• WR-90 (WG-16): Broad dimension $a = 0.900\text{ in} = 22.86\text{ mm}$. Cutoff frequency $f_c = 3.00 \times 10^8 / (2 \times 0.02286) \approx 6.56\text{ GHz}$. Designed for the $10.0\text{ to }10.5\text{ GHz}$ amateur microwave band.
• WR-42 (WG-20): Broad dimension $a = 0.420\text{ in} = 10.67\text{ mm}$. Cutoff frequency $f_c \approx 14.05\text{ GHz}$. Designed for the $24\text{ GHz}$ amateur band.