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Antenna

Loop Antenna Calculator

Calculate loop antenna radiation resistance, Q factor, bandwidth, and gain. Design magnetic loop antennas for HF and VHF bands. Free, instant results.

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Formula

Rrad=320π4⋅(A/λ2)2R_rad = 320π⁴·(A/λ²)²
R_rad— Radiation resistance (Ω)
A— Loop area (π·(D/2)²) (m²)
λ— Wavelength (c/f, c = 299 792 458 m/s) (m)
Q— Quality factor
BW— Bandwidth (f/Q) (Hz)

How It Works

Loop antenna calculator computes radiation resistance, efficiency, and directivity for small and large loops — amateur radio operators, direction-finding engineers, and IoT designers use loops for compact installations and low-noise reception — amateur radio operators, broadcast engineers, and EMC test technicians rely on loops for their predictable patterns and noise-rejection properties. Small loops (circumference < 0.1*lambda) behave as magnetic dipoles with figure-eight pattern and very low radiation resistance R_rad = 320*pi^4*(A/lambda^2)^2 ohms, per Balanis's 'Antenna Theory' (4th ed.) and Kraus's 'Antennas'.

For a 1-meter diameter circular loop at 7 MHz (lambda = 42.9 m), A = 0.785 m^2 yields R_rad = 320*pi^4*(0.785/1841)^2 = 0.0018 ohms — extremely low compared to conductor loss, limiting efficiency to < 1% without high-Q tuning. Small transmitting loops (STL or magnetic loops) use tuning capacitors to create high-Q resonance (Q = 200-500), achieving 10-50% efficiency in a compact package. Receiving loops need not be resonant — they capture magnetic field component, rejecting local electric-field noise from appliances.

Full-wave loops (circumference = lambda) achieve approximately 1 dBd gain with approximately 100-ohm feed impedance. The delta loop (triangular) and quad loop (square) are popular HF antennas providing 1-2 dB advantage over dipoles with lower-angle radiation. Loop gain increases with size: 2-lambda circumference provides approximately 3 dBd, making loops attractive for limited-space installations where vertical space is available but horizontal span is restricted.

Worked Example

Problem: Design a small transmitting magnetic loop for 40 meters (7 MHz) fitting within a 3-meter span.

Design per STL methodology:

  1. Loop circumference: C = pi D = pi 1.0 m = 3.14 m (fits 3 m constraint as octagonal)
  2. Wavelength: lambda = 300/7 = 42.86 m
  3. Electrical size: C/lambda = 3.14/42.86 = 0.073 (small loop, << 0.1*lambda)
Radiation resistance calculation:
  1. Loop area: A = pi r^2 = pi 0.5^2 = 0.785 m^2
  2. R_rad = 320 pi^4 (A/lambda^2)^2
R_rad = 320 97.4 (0.785/1837)^2 = 31170 * (4.27e-4)^2 = 0.0057 ohms

Conductor loss (22 mm diameter copper tubing):

  1. Skin depth at 7 MHz: delta = 25 um (copper)
  2. Conductor resistance: R_loss = rho C / (pi d * delta)
R_loss = 1.7e-8 3.14 / (pi 0.022 * 25e-6) = 0.031 ohms

Efficiency and Q:

  1. Radiation efficiency: eta = R_rad / (R_rad + R_loss) = 0.0057 / 0.0367 = 15.5%
  2. Total loop inductance: L = mu_0 D (ln(8*D/d) - 2) = 4.1 uH
  3. Required tuning capacitance: C = 1/(4*pi^2*f^2*L) = 126 pF (use 15-150 pF variable)
  4. Operating Q: Q = omega*L / R_total = 2*pi*7e6*4.1e-6 / 0.0367 = 4900
  5. Bandwidth: BW = f/Q = 7e6/4900 = 1.4 kHz (very narrow, requires retuning for frequency changes)
Capacitor voltage rating:
  1. At 100 W input, loop current I = sqrt(P/(R_rad+R_loss)) = sqrt(100/0.0367) = 52 A
  2. Capacitor voltage: V_cap = I / (2*pi*f*C) = 52 / (2*pi*7e6*126e-12) = 9.4 kV!
  3. Use vacuum variable capacitor rated for 10+ kV, or split capacitor configuration
Performance summary: 15% efficiency (-8 dB), 1.4 kHz bandwidth, 9.4 kV capacitor voltage at 100 W.

Practical Tips

  • ✓For receiving, untuned loops are preferred — they provide consistent figure-eight pattern for direction finding without retuning; efficiency is irrelevant since the receiver has plenty of gain
  • ✓For transmitting small loops, use vacuum variable capacitors or wide-gap air variables — voltage ratings of 5-15 kV are required at 100 W power levels; butterfly capacitors double the voltage handling
  • ✓Consider ferrite-loaded loops for VLF/LF applications — ferrite increases effective area by mu_rod factor (10-100x), dramatically improving efficiency and reducing physical size

Common Mistakes

  • ✗Expecting high efficiency from small loops without understanding R_rad physics — a 1 m loop at 7 MHz has R_rad = 0.006 ohms; 50% efficiency requires R_loss < 0.006 ohms, achievable only with heavy copper tubing (25+ mm diameter) or superconductors
  • ✗Using inadequate capacitor voltage rating — loop current at resonance is I = sqrt(P/R_total); with R_total = 0.05 ohms and 100 W, I = 45 A; capacitor sees V = I/(omega*C) which can exceed 10 kV at HF frequencies
  • ✗Ignoring conductor loss in efficiency calculations — at HF, skin effect concentrates current in outer 20-30 um; use thick-wall tubing (> 10 mm diameter) and minimize joints to reduce R_loss
  • ✗Assuming small loops reject all noise — small loops reject electric-field noise (from sparking contacts, appliances) but remain sensitive to magnetic-field noise (power lines, motors); proper location away from noise sources is still essential

Frequently Asked Questions

Three main advantages per Kraus: (1) Noise rejection — small loops respond to magnetic field component, rejecting electric-field noise from nearby sources (motors, power lines, electronics). SNR improvement of 10-20 dB versus vertical whips in noisy urban environments. (2) Compact size — small transmitting loops fit in apartments/patios where dipoles are impractical; a 1 m loop works on 40 m (42 m wavelength) with appropriate tuning. (3) Predictable pattern — figure-eight pattern with sharp nulls enables direction finding; rotating loop locates transmitter bearing within 2-5 degrees accuracy.
Size relative to wavelength determines behavior: Small loop (C < 0.1*lambda): Magnetic dipole, R_rad extremely low (milliohms), efficiency < 50% even with high-Q tuning, narrow bandwidth. Pattern is figure-eight perpendicular to loop plane. Resonant loop (C = lambda): Full-wave loop, R_rad approximately 100 ohms, efficiency > 90%, gain approximately 1 dBd. Pattern broadside to loop plane with some directivity. Large loop (C > lambda): Multi-wavelength loops have complex multi-lobe patterns, higher gain (3+ dBd), useful for limited-space directional arrays. Practical tradeoff: small loops sacrifice efficiency for size; full-wave loops match dipole performance in different form factor.
Yes, with caveats: Full-wave loops transmit efficiently (> 90%) like dipoles. Small transmitting loops (STL) achieve 10-50% efficiency with high-Q resonant tuning — bandwidth is very narrow (1-10 kHz at HF), requiring retuning when changing frequency more than a few kHz. Power limits depend on capacitor voltage rating and conductor heating: 100 W is practical with vacuum variable capacitor and heavy copper tubing; 1 kW requires extreme care due to 30+ kV capacitor voltages. STLs are popular for apartment/patio operation where full-size antennas are prohibited.
Radiation resistance R_rad represents power radiated as electromagnetic waves: P_rad = I^2 * R_rad. For small loops: R_rad = 320*pi^4*(A/lambda^2)^2 — fourth power dependence on (A/lambda) means R_rad drops extremely fast as loop shrinks. A 1 m loop at 7 MHz has R_rad = 0.006 ohms; at 3.5 MHz (lambda doubled), R_rad = 0.0004 ohms. Efficiency eta = R_rad/(R_rad + R_loss) — when R_rad << R_loss, most power dissipates as heat in conductor resistance. This is why small loops require: thick conductors (minimize R_loss), high-Q tuning (concentrates current at resonance), and modest power expectations.

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