Antenna Factor Calculation for Probe Calibration¶

⚠️ IMPORTANT USAGE NOTE:
When changing the S21 measurement value:

  1. Update S21_dB in the 'Setup Parameters' section below
  2. Use Cell → Run All to recalculate everything correctly
  3. Do NOT run cells individually - this will cause incorrect results

This notebook calculates the antenna factor (AF) of a probe using the standard site method.

Measurement Setup¶

  • Frequency: 150 MHz
  • Transmit Antenna: Half-wave dipole at 2m height above ground
  • Receive Probe: Half-wave dipole at 150 MHz
  • Horizontal Separation: 2m
  • Measurement: S21 from calibrated VNA (50Ω)
  • Assumption: Negligible mismatch error

Theory¶

The antenna factor relates electric field strength to terminal voltage:

$$AF = \frac{E}{V}$$

where:

  • AF is in units of m⁻¹ (or dB/m when expressed in dB)
  • E is the electric field strength (V/m)
  • V is the voltage at the antenna terminals (V)

In decibels: $$AF_{dB} = E_{dB\mu V/m} - V_{dB\mu V}$$

Speed of light: 2.998e+08 m/s
Free space impedance: 376.73 Ω

Setup Parameters¶

Frequency: 150.0 MHz
Wavelength: 1.999 m
Half-wavelength: 0.999 m

Geometry:
  Tx height: 2.0 m
  Rx height: 2.0 m
  Separation: 2.0 m

Measured S21: -16.30 dB

Half-Wave Dipole Characteristics¶

For a half-wave dipole:

  • Gain: ~2.15 dBi
  • Effective length: λ/π
  • Radiation resistance: ~73Ω
Dipole gain: 2.15 dBi (1.641 linear)
Radiation resistance: 73 Ω
Effective length: 0.636 m

Ground Reflection Model¶

With both antennas above a ground plane, we need to account for:

  1. Direct path: Straight line from transmitter to receiver
  2. Reflected path: From transmitter to reflection point on ground, then to receiver

For specular reflection, the reflection point is at the midpoint on the ground between the two antennas (by Snell's law, angle of incidence = angle of reflection).

The reflected path length is calculated as:

  • Distance from Tx to reflection point: $\sqrt{(d/2)^2 + h_{tx}^2}$
  • Distance from reflection point to Rx: $\sqrt{(d/2)^2 + h_{rx}^2}$
  • Total reflected path: Sum of these two segments

For 'good earth' at VHF frequencies:

  • Reflection coefficient $\Gamma \approx -1$ (phase inversion)
  • Ground acts as nearly perfect conductor

Note: The "image method" gives the same result - imagine a mirror image antenna at height $-h_{tx}$ below ground. The distance from this image to the receiver equals the reflected path length.

Direct path length: 2.000 m

Reflected path:
  Reflection point at: 1.0 m horizontal
  Tx to reflection point: 2.236 m
  Reflection point to Rx: 2.236 m
  Total reflected path: 4.472 m
  (Image method check: 4.472 m)

Path difference: 2.472 m
Path difference in wavelengths: 1.237 λ

Calculate Electric Field at Receive Location¶

The electric field from a dipole with power P_t and gain G_t at distance r is:

$$E = \sqrt{\frac{30 \cdot P_t \cdot G_t}{r^2}}$$

For the two-ray model with ground reflection: $$E_{total} = E_{direct} + E_{reflected}$$

We'll work with available power at the VNA port (P_avail = 0 dBm reference).

Reference power: 0 dBm = 1.000 mW

Electric field components:
  Direct path amplitude: 0.110925 V/m (100.90 dBμV/m)
  Reflected path amplitude: 0.049607 V/m (93.91 dBμV/m)
  Amplitude ratio (direct/reflected): 2.24 (7.0 dB)

Phase analysis:
  Path difference: 2.472 m = 1.237 λ
  Phase from path difference: 445.3°
  Phase from reflection (Γ=-1): 180.0°
  Total phase difference: 625.3°

Resultant field:
  Total field (phasor sum): 0.117737 V/m (101.42 dBμV/m)
  Change from direct path only: 0.52 dB

Calculate Received Voltage from S21¶

The S21 parameter relates the incident wave at port 1 to the transmitted wave at port 2:

$$S_{21} = \frac{b_2}{a_1}$$

For matched conditions (50Ω system):

  • $a_1 = \sqrt{P_{avail}/Z_0}$ (incident wave at port 1)
  • $b_2 = \sqrt{P_{delivered}/Z_0}$ (transmitted wave at port 2)

The voltage at the receive antenna terminals is: $$V_{rx} = \sqrt{2 \cdot P_{rx} \cdot Z_{sys}}$$

S21 linear: 1.531087e-01
Received power: 2.344229e-05 W = -16.30 dBm
Received voltage (RMS): 4.841724e-02 V = 93.70 dBμV

Calculate Antenna Factor¶

The antenna factor is: $$AF = \frac{E}{V}$$

In dB: $$AF_{dB} = E_{dB\mu V/m} - V_{dB\mu V}$$

============================================================
ANTENNA FACTOR RESULTS
============================================================

Electric field at receive location: 101.42 dBμV/m
Voltage at receive antenna: 93.70 dBμV

Antenna Factor: 2.43 m⁻¹
Antenna Factor: 7.72 dB/m
============================================================

Theoretical Antenna Factor for Half-Wave Dipole¶

For comparison, the theoretical antenna factor of a half-wave dipole can be calculated from:

$$AF = \frac{1}{L_{eff}} \cdot \sqrt{\frac{Z_{sys}}{R_{rad}}}$$

In dB: $$AF_{dB} = 20\log_{10}\left(\frac{f_{MHz}}{9.73}\right)$$

This is for free-space conditions without ground effects.

Theoretical AF (free space): 1.30 m⁻¹
Theoretical AF (free space): 23.76 dB/m

Difference from measured: -16.04 dB
(Difference due to ground reflection effects)

Visualization: Ground Reflection Effects¶

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Figure saved as 'antenna_factor_measurement.png'

Summary Table¶

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MEASUREMENT SUMMARY
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                   Parameter         Value
                   Frequency     150.0 MHz
                  Wavelength       1.999 m
       Horizontal Separation         2.0 m
             Antenna Heights         2.0 m
          Direct Path Length       2.000 m
       Reflected Path Length       4.472 m
                Measured S21     -16.30 dB
              Electric Field 101.42 dBμV/m
            Received Voltage    93.70 dBμV
              Antenna Factor     7.72 dB/m
AF (theoretical, free space)    23.76 dB/m
============================================================

Notes¶

  1. Ground Reflection: The measurement includes ground reflection effects which modify the field strength at the receive location compared to free-space conditions.

  2. Mismatch: As specified, mismatch errors are assumed negligible. In practice, you should verify that both antennas are well-matched at the measurement frequency.

  3. Calibration Validation: The antenna factor should be within a reasonable range for a half-wave dipole at this frequency. Typical values are 20-30 dB/m at VHF frequencies.

  4. Height Effects: The specific heights and separation distance create a particular interference pattern due to ground reflections. Different geometries will yield different results.

  5. To use with your data: Simply replace the S21_dB value in the setup parameters section with your measured value from the VNA.