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Small Loop Efficiency, Q & Bandwidth

A comprehensive guide to understanding the performance characteristics of small transmitting loop antennas, including efficiency, Q-factor, and bandwidth relationships.

What is a Small Transmitting Loop?

A small transmitting loop (STL) is a loop antenna where the circumference is significantly smaller than one wavelength (typically less than 0.1λ). These compact antennas are popular among amateur radio operators for:

  • Portability - Easy to transport and deploy
  • Small footprint - Ideal for apartments, HOAs, and limited spaces
  • Low noise - Excellent rejection of local electrical noise
  • Directional pattern - Useful null for QRM rejection
Definition: Small Loop

A loop is considered "small" when its circumference is less than 0.1λ (lambda). For example, at 14 MHz (20m band), a loop with a 1-meter diameter has a circumference of about 3.14m, which is approximately 0.047λ (4.7% of a wavelength).

Understanding Efficiency

Antenna efficiency (η) is the ratio of radiated power to input power. In a small loop, efficiency is determined by the relationship between radiation resistance and loss resistance.

The Efficiency Formula

η = Rr / (Rr + Rloss)

Antenna Efficiency

Where:

  • η (eta) = Efficiency (as a decimal, 0 to 1)
  • Rr = Radiation resistance (ohms)
  • Rloss = Total loss resistance (ohms)

Radiation Resistance

For a small loop, radiation resistance is very small and depends on the loop area and frequency:

Rr ≈ 31,171 × (A/λ²)²

Small Loop Radiation Resistance

Where:

  • A = Loop area (m²)
  • λ = Wavelength (m)
  • Rr = Radiation resistance (Ω)
Key Insight

Radiation resistance increases with the fourth power of frequency for a fixed loop size. This is why small loops work better on higher frequencies - a 1m loop at 28 MHz has 4× the radiation resistance of the same loop at 14 MHz!

Loss Resistance

Loss resistance comes primarily from two sources:

  1. Conductor losses - RF resistance of the loop conductor (skin effect)
  2. Capacitor ESR - Equivalent series resistance of the tuning capacitor

Total Loss Resistance

Rloss = Rconductor + Rcapacitor

Where both components increase with frequency due to skin effect and capacitor Q degradation.

Worked Example: 20m Loop

Let's calculate the efficiency of a typical 20-meter magnetic loop:

Parameters:

  • Frequency: 14.2 MHz (20m band)
  • Loop shape: Circular
  • Loop diameter: 1.0 meter
  • Conductor: Copper tubing, 12mm diameter
  • Capacitor: Air variable, ESR = 0.05Ω

Step 1: Calculate wavelength

  • λ = c/f = 299,792,458 m/s / 14,200,000 Hz = 21.1 m

Step 2: Calculate loop area

  • A = π × (d/2)² = π × (0.5)² = 0.785 m²

Step 3: Calculate radiation resistance

  • Rr = 31,171 × (0.785 / 21.1²)² = 31,171 × (0.001765)² = 0.097Ω

Step 4: Estimate conductor loss

  • Skin depth at 14.2 MHz in copper ≈ 17.4 μm
  • RF resistance per meter ≈ 0.034 Ω/m
  • Loop circumference = πd = 3.14 m
  • Rconductor = 0.034 × 3.14 = 0.107Ω

Step 5: Total loss resistance

  • Rloss = 0.107 + 0.05 = 0.157Ω

Step 6: Calculate efficiency

  • η = 0.097 / (0.097 + 0.157) = 0.097 / 0.254 = 38.2%
Typical Efficiency Range

Small transmitting loops typically achieve 20-50% efficiency on HF bands. While this might seem low, the compact size and excellent noise rejection often make the trade-off worthwhile. A well-designed loop can still deliver excellent performance!

Understanding Q-Factor

The Q-factor (quality factor) of a small loop is extremely high, which has both advantages and disadvantages.

Q-Factor Formula

Q = XL / Rtotal = 2πfL / (Rr + R loss)

Q-Factor

Where:

  • XL = Inductive reactance of the loop (Ω)
  • Rtotal = Total resistance (radiation + loss)
  • f = Operating frequency (Hz)
  • L = Loop inductance (H)

What Q-Factor Means

  • High Q (200-400): Very narrow bandwidth, excellent selectivity, difficult to tune
  • Medium Q (100-200): Balanced bandwidth and tuning ease
  • Low Q (<100): Wider bandwidth, easier tuning, but typically larger loop

Our example loop at 20m typically has a Q around 300, which is considered very high.

Bandwidth Relationships

Bandwidth and Q-factor are inversely related:

BW = f0 / Q

Bandwidth Formula

For our 20m loop example:

  • Q = 300
  • f0 = 14.2 MHz
  • BW = 14,200,000 / 300 = 47.3 kHz (2:1 SWR bandwidth)
Practical Implications

With a 47 kHz bandwidth, this loop covers most of the 20m band (14.0-14.35 MHz = 350 kHz) but requires retuning when changing frequency by more than ~25 kHz. This is typical for small loops and why a quality variable capacitor is essential!

Trade-offs and Design Considerations

Size vs. Efficiency

| Loop Diameter | Efficiency @ 20m | Q-Factor | Bandwidth | | ------------- | ---------------- | -------- | --------- | | 0.5m | ~15% | ~400 | ~35 kHz | | 1.0m | ~38% | ~300 | ~47 kHz | | 1.5m | ~58% | ~220 | ~65 kHz |

Doubling the loop diameter approximately quadruples the radiation resistance and more than doubles the efficiency. However, portability decreases significantly.

Conductor Diameter vs. Loss

Larger conductor diameter reduces RF resistance:

| Conductor Size | Rconductor @ 20m | Effect on Efficiency | | -------------- | --------------------------- | -------------------- | | 6mm copper | 0.21 Ω/m | ~30% efficient | | 12mm copper | 0.11 Ω/m | ~38% efficient | | 25mm copper | 0.05 Ω/m | ~45% efficient |

Capacitor Quality

The capacitor ESR directly impacts efficiency:

| Capacitor Type | Typical ESR | Loss Impact | | --------------- | ----------- | ------------------- | | Air variable | 0.02-0.05Ω | Minimal (~2% loss) | | Vacuum variable | 0.01-0.02Ω | Negligible | | Butterfly | 0.05-0.10Ω | Moderate (~5% loss) |

High Voltage Warning

Small loops concentrate high RF voltage across the tuning capacitor. At 100W transmit power with Q=300, the capacitor can see 1,500V RMS or more! Always use a capacitor rated for at least 3-5kV for QRP operation, and 10kV+ for 100W+.

Practical Design Guidelines

For Maximum Efficiency

  1. Make the loop as large as practically possible
  2. Use the largest diameter conductor you can manage
  3. Choose a high-quality capacitor (vacuum or air variable)
  4. Minimize all connections and joints
  5. Consider higher frequencies (better efficiency at 20m than 40m)

For Maximum Bandwidth

  1. Reduce the Q by using thicker conductors
  2. Accept slightly lower efficiency
  3. Consider a larger loop diameter
  4. Use lossier (but voltage-safe) coupling methods

For Portability

  1. Accept lower efficiency (20-30% is still usable)
  2. Target QRP power levels (5-25W)
  3. Use collapsible or sectional construction
  4. Balance weight vs. conductor size

Verification and Testing

After building a loop, verify performance with:

  1. SWR meter - Check resonance and bandwidth
  2. Dummy load + multimeter - Measure capacitor voltage (check Q calculation)
  3. On-air testing - Compare signal reports to other antennas
  4. Professional antenna analyzer - Measure impedance, Q, and efficiency
Try the Calculator

Want to see how these formulas work in practice? Use our

Magnetic Loop Calculator

to experiment with different designs and see real-time efficiency, Q, and bandwidth calculations!

Further Reading

Academic References

  • AA5TB Magnetic Loop Calculator - Steve Yates, AA5TB (comprehensive formula derivations)
  • ARRL Antenna Book - Chapter on Small Transmitting Loops
  • "Small Transmitting Loop Antennas" by Dr. Brian Austin, VP9MA

Online Resources

Related HamCalc Articles

Summary

Small transmitting loops offer excellent performance in compact packages, with these key characteristics:

  • Efficiency: 20-50% typical on HF, determined by Rr vs. Rloss
  • Q-Factor: Very high (200-400), leading to excellent selectivity
  • Bandwidth: Narrow (30-70 kHz at 20m), requiring retuning across bands
  • Trade-offs: Size vs. efficiency vs. portability

Understanding these relationships allows you to design a loop that meets your specific needs, whether that's maximum efficiency, maximum portability, or a balanced compromise.

Ready to design your own loop? Try the Magnetic Loop Calculator to experiment with parameters and optimize your design!