4.4 Critical Frequency, Maximum Usable Frequency (MUF), and Fading

4.4 Critical Frequency, Maximum Usable Frequency (MUF), and Fading

Predicting HF propagation conditions requires understanding how operating frequency, elevation angle, and ionospheric ionisation levels interact. Radio operators must select frequencies between defined upper and lower limits to achieve reliable skywave communication.

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Measuring the Ionosphere: Ionosondes

The ionosphere is monitored using specialised radar systems called vertical incidence ionosondes. An ionosonde transmits short pulses of RF energy straight up into the ionosphere ($90^\circ$ elevation angle, vertical incidence) while sweeping across frequencies from 1 MHz to 20+ MHz.

By measuring the time delay of returned echoes, the ionosonde determines the virtual height of the ionospheric layers and generates a graphical plot known as an ionogram.

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Critical Frequency ($f_c$)

The Critical Frequency ($f_c$ or $f_o$) is the highest frequency that will be refracted back to Earth when transmitted vertically ($90^\circ$ elevation angle).

                Critical Frequency Concept
                  f > fc (Escapes to space)
                     |    ^  f = fc (Critical Threshold)
                     |   / 
                     |  /   f < fc (Refracted back)
       +-------------|-/--------------------+ Ionospheric F Layer
       |             |/                     |
       |             /                      |
       |            /                       |
       |           /                        |
  =====|==========/=========================|=====
                  Transmitter

• If a vertical signal frequency is below $f_c$, it is refracted back to Earth.
• If a vertical signal frequency is above $f_c$, the ionospheric electron density is insufficient to bend the wave back down, and the signal passes straight through into space.

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Maximum Usable Frequency (MUF)

In practical amateur communications, signals are transmitted at oblique angles (elevation angles below $90^\circ$) rather than vertically. When a wave enters the ionosphere obliquely at an angle of incidence $\theta$ (measured relative to the normal of the ionospheric layer), the wave undergoes greater total refraction over a longer path within the layer.

Consequently, the Maximum Usable Frequency (MUF) for an oblique path is significantly higher than the vertical critical frequency ($f_c$).

The Secant Law / MUF Formula

The mathematical relationship between Critical Frequency ($f_c$) and MUF for an angle of incidence $\theta$ is expressed by the Secant Law:

$\text{MUF} = \frac{f_c}{\cos\theta} = f_c \times \sec\theta$

Where:

• $f_c$ is the Critical Frequency for vertical incidence.
• $\theta$ is the angle of incidence at the ionospheric layer (relative to the normal/perpendicular).
• $\psi$ is the takeoff angle (radiation elevation angle above the Earth's horizon).

                    Angle Geometry at Ionosphere
                               | Normal (Perpendicular)
                               | 
                               | / Angle of Incidence (theta)
                               |/
  -----------------------------+----------------------------- Ionospheric Layer
                              / 
                             / Radiation Path
                            /
  ====(Takeoff Angle psi)==/================================= Earth's Surface

Impact of Radiation Takeoff Angle ($\psi$)

• Low Radiation Takeoff Angle ($\psi \to 0^\circ$): The signal strikes the ionosphere at a shallow glancing angle (large incidence angle $\theta \to 90^\circ$). Because $\cos\theta$ becomes small, the $\text{MUF}$ becomes very high, enabling maximum single-hop distances.
• High Radiation Takeoff Angle ($\psi \to 90^\circ$): The signal strikes the ionosphere nearly vertically ($\theta \to 0^\circ, \cos\theta \to 1$). The MUF drops down toward $f_c$.

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Worked Calculation: MUF

Problem: A vertical ionosonde measures a critical frequency $f_c = 7.0 \text{ MHz}$. Calculate the Maximum Usable Frequency (MUF) for an oblique propagation path where the angle of incidence $\theta = 60^\circ$.

Step 1: Determine $\cos(60^\circ)$. $\cos(60^\circ) = 0.5$

Step 2: Apply the MUF formula. $\text{MUF} = \frac{f_c}{\cos\theta} = \frac{7.0 \text{ MHz}}{\cos(60^\circ)} = \frac{7.0 \text{ MHz}}{0.5} = 14.0 \text{ MHz}$

Answer: The MUF for this path is 14.0 MHz (the 20-metre amateur band).

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Lowest Usable Frequency (LUF)

While the MUF establishes the upper boundary for HF communication over a given path, the Lowest Usable Frequency (LUF) defines the lower boundary.

• D-Layer Absorption: At lower frequencies, D-layer absorption increases rapidly ($A \propto 1/f^2$).
• Background Atmospheric Noise: Natural noise levels rise at lower frequencies.

If the operating frequency drops below the LUF, signal absorption in the D layer becomes so severe that received signal strength falls below the noise floor, rendering communication impossible.

Unlike the MUF (which is determined purely by ionospheric electron density and geometric path angle), the LUF depends on transmitter power and antenna gain. Increasing transmitter power or antenna gain can lower the effective LUF by overcoming absorption losses.

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Optimum Working Frequency (OWF / FOT)

Operating exactly at the MUF is risky. The ionosphere is dynamic, and minor fluctuations in electron density can cause the MUF to drop suddenly, causing the operating frequency to exceed the MUF and pass through into space.

To ensure reliable, stable skywave communication, operators choose the Optimum Working Frequency (OWF)—also known in international notation as FOT (Fréquence Optimale de Travail):

$\text{OWF} = \text{FOT} = 0.85 \times \text{MUF}$

Operating at 85% of the MUF provides an optimal compromise:

1. It remains comfortably below the MUF to prevent unexpected fade-outs.
2. It remains far enough above the LUF to avoid heavy D-layer absorption and background noise.

                  HF Propagation Frequency Window
  Frequency (MHz)
        ^
        |   Unusable (Passes into Space)
  ======+=================================================== MUF
        |   Fluctuating Buffer
  ------|--------------------------------------------------- OWF / FOT (85% of MUF)
        |   OPTIMAL COMMUNICATION WINDOW
        |   (Low absorption, high stability)
  ======+=================================================== LUF
        |   Unusable (Absorbed by D Layer)
        v

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Skywave Fading Mechanisms

Fading is the continuous variation in received signal strength over time. Skywave signals experience three primary fading mechanisms:

1. Multipath Fading (Interference Fading)

Multipath fading occurs when a signal travels from transmitter to receiver via two or more distinct paths of different lengths (e.g., a 1-hop path vs. a 2-hop path, or a low ray vs. a high Pedersen ray).

Because the path lengths differ, the arriving waves have relative phase differences. As ionospheric height fluctuates slightly, the arriving waves alternate between:

• Constructive Interference: Waves arrive in phase ($0^\circ$ phase difference), reinforcing each other (signal peak).
• Destructive Interference: Waves arrive out of phase ($180^\circ$ phase difference), canceling each other out (deep signal null).

2. Polarisation Rotation (Faraday Rotation)

As an EM wave travels through the ionised ionosphere in the presence of the Earth's magnetic field, the wave is split into ordinary and extraordinary modes that travel at slightly different phase velocities. This causes the wave's plane of polarisation to rotate continuously.

When the rotating wave arrives at a fixed linearly polarised receiving antenna (e.g., horizontal dipole), the signal strength drops whenever the polarisation vector becomes perpendicular to the receiving antenna.

3. Selective Fading

Selective fading affects specific frequencies within a signal's modulation bandwidth differently. For example, in an SSB or AM signal, the upper sideband may experience constructive interference while the lower sideband or carrier experiences destructive cancellation, causing severe audio distortion.