================================================================================
SCHEMATIC: Ugly Balun (Coiled Coax) and Bead-on-Coax Wideband Chokes
TM-CMC-001  Rev A
Ugly Balun: Self-Resonant Coiled Coax for HF Common-Mode Suppression
Bead-on-Coax: Multiple Ferrite Beads Threaded on Coax for 1–300 MHz
================================================================================

================================================================================
SECTION A — UGLY BALUN (COILED COAX AIR-CORE CHOKE)
================================================================================

THEORY
──────
The "ugly balun" (attributed to W7EL, Roy Lewallen) is simply coaxial cable
wound into a coil.  The inductance of the coil presents common-mode impedance
without affecting differential mode.

The coil is a simple air-core inductor.  Common-mode impedance:
  Z_CM = 2π × f × L_coil   (for f < self-resonant frequency)
  Z_CM = very high          (near self-resonant frequency f_SR)
  Z_CM decreases above f_SR (capacitive behavior dominates)

Self-resonant frequency:
  f_SR = 1 / (2π × √(L_coil × C_coil))
  where C_coil = distributed capacitance of wound coil

To be useful, f_SR should be within or just above the operating frequency band.
Maximum common-mode impedance occurs AT f_SR.

Inductance of a solenoid coil:
  L = (µ₀ × N² × A) / (l × (1 + 0.9 × r/l))
  Simplified Wheeler formula: L (µH) = (N² × r²) / (9r + 10l)
  where r = coil radius (inches), l = coil length (inches)

Example: 9 turns RG-58, 4.5" diameter (r=2.25"), 1.5" long (l=1.5"):
  L = (81 × 5.0625) / (9 × 2.25 + 10 × 1.5)
  L = 410.6 / (20.25 + 15) = 410.6 / 35.25 = 11.6 µH

At 14 MHz (20M): |Z| = 2π × 14×10⁶ × 11.6×10⁻⁶ = 1021 Ω
At  7 MHz (40M): |Z| = 510 Ω (marginal)
At  3.5 MHz (80M): |Z| = 255 Ω (insufficient alone)

Conclusion: Ugly balun is most effective at and above the design frequency.
For 20M-10M operation, design f_SR near 14–21 MHz.

================================================================================
DESIGN 1: 20M–10M UGLY BALUN (3" DIAMETER FORM)
================================================================================

MATERIALS:
  Coax:  RG-58 (5mm OD; total length ~2.5m)
  Form:  PVC pipe 3" OD (75mm); length 100mm; 2× cable tie grooves
  Turns: 9 (for f_SR near 21 MHz)

WINDING:
  1. Cut 2.5m RG-58.
  2. Form PVC pipe coil former: drill 4mm hole at each end for tie-wrap retention.
  3. Start from one end of former; wrap 9 turns tightly butted (turns touching).
  4. Secure with 2× cable ties.
  5. Calculated dimensions: OD = 85mm, length = 9 × 5mm = 45mm.
  6. Install PL-259 connectors on each end.

CALCULATED PERFORMANCE:
  Coil: N=9, r=42.5mm=1.67", l=45mm=1.77"
  L = (81 × 2.79) / (9 × 1.67 + 10 × 1.77) = 226 / (15.0 + 17.7) = 6.9 µH

  C_distributed (estimate for tightly wound RG-58): ~30 pF
  f_SR = 1 / (2π × √(6.9e-6 × 30e-12)) = 11.1 MHz

  Common-mode impedance (measured / calculated):

  f (MHz) | |Z_CM| (Ω) | Note
  --------|-----------|----------------------------------
    1.8   |    78      | Very low — 160M useless
    3.5   |   152      | 80M inadequate
    7.0   |   303      | 40M marginal
   10.1   |   438      | 30M marginal
   11.1   |   ∞ (SR)   | Peak: very high impedance here
   14.0   |  1500 est  | 20M useful (past SR, complex Z)
   21.0   |   800 est  | 15M moderate
   28.0   |   400 est  | 10M diminishing

  Note: Impedance behavior past f_SR is complex; actual Z depends on coil Q
  and cable loss.  Measurements vary considerably by construction.

IMPROVEMENT — SPACE WINDING:
  Increase coil length l while maintaining same N and r.
  Reduces C_distributed → raises f_SR to desired frequency.
  For f_SR = 14 MHz target: space turns to l ≈ 70mm (8mm pitch).

================================================================================
DESIGN 2: 40M–15M UGLY BALUN (6" DIAMETER FORM)
================================================================================

MATERIALS:
  Coax:  RG-8X (8.4mm OD; total length ~3.5m)
  Form:  PVC pipe 6" OD (150mm); length 150mm
  Turns: 7

WINDING:
  Coil: N=7, r=75mm=2.95", l=70mm=2.76"
  L = (49 × 8.70) / (9 × 2.95 + 10 × 2.76) = 426 / (26.6 + 27.6) = 7.9 µH
  C_distributed (loose wind, RG-8X): ~15 pF
  f_SR = 1 / (2π × √(7.9e-6 × 15e-12)) = 14.6 MHz  ← excellent for 20M peak

PERFORMANCE:
  f (MHz) | |Z_CM| est (Ω) | Note
  --------|--------------|------------------------------
    3.5   |   174         | 80M weak
    7.0   |   348         | 40M useful (low end)
   10.1   |   502         | 30M moderate
   14.6   |  >3000        | f_SR peak — 20M excellent
   21.0   |  ~2000        | 15M good
   28.0   |  ~1000        | 10M adequate

POWER HANDLING:
  At 1500W into 50Ω resistive load, I_CM = 0 (ideal), no heating.
  Under common-mode current (RFI condition): heating is in coax shield loss.
  RG-8X handles 1500W continuous above 14 MHz.  No derating for ugly balun.

================================================================================
UGLY BALUN DIAGRAM
================================================================================

Top view (coax wound on PVC pipe section):

     ╔══════════════════════════════════════════╗
     ║   ┌──────────────────────────────────┐   ║
     ║   │→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→│   ║  each row = 1 turn coax
     ║   │→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→│   ║
     ║   │→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→│   ║
     ║   │→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→│   ║
     ║   │→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→│   ║
     ║   │→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→│   ║
     ║   │→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→│   ║
     ║   └──────────────────────────────────┘   ║
     ╚══════════════════════════════════════════╝
              PVC pipe form (inside)

Side view:

  PL-259        Coiled coax        PL-259
  ─┤├─────────[OOOOOOOOOOOOOO]─────────┤├─
    ↑          7–12 turns            ↑
  TX/RX end    on form         Antenna end

================================================================================
COIL FORM DIMENSIONS TABLE
================================================================================

  Form OD (inches) | PVC nominal | Form ID | Coax    | N  | f_SR target
  -----------------|-------------|---------|---------|----|--------------
    2"              | 2" Sch 40   |  51 mm  | RG-174  | 12 | 50 MHz (6M)
    3"              | 3" Sch 40   |  76 mm  | RG-58   |  9 | 21 MHz (15M)
    4"              | 4" Sch 40   | 102 mm  | RG-58   |  8 | 14 MHz (20M)
    6"              | 6" Sch 40   | 150 mm  | RG-8X   |  7 | 14 MHz (20M)
    8"              | 8" Sch 40   | 200 mm  | RG-8    |  7 | 7 MHz (40M)

================================================================================
SECTION B — BEAD-ON-COAX WIDEBAND CHOKE
================================================================================

THEORY
──────
The bead-on-coax choke (also called "W7FG" or "pigtail" choke) threads the
coaxial cable through multiple small ferrite toroid cores.  Unlike snap-on
split beads, solid toroids provide higher Q and better impedance at high
frequencies.  Unlike wound chokes, they require no winding — just thread through.

For a single pass through one #61 toroid (FT-50-61, A_L = 36 nH/N²):
  L = 1² × 36 nH = 36 nH
  At 144 MHz: |Z| = 2π × 144×10⁶ × 36×10⁻⁹ = 32.6 Ω per bead

For adequate VHF choking, need many beads OR multiple passes.
10 beads in series: |Z| = 10 × 32.6 = 326 Ω (marginal)
20 beads: |Z| = 652 Ω (good)

Better approach: use FT-82-43 or FT-82-61 (larger OD, still passable for RG-58):
  FT-82-43: A_L = 557 nH/N², at 144 MHz: |Z| = 2π × 144M × 557n = 504 Ω per bead
  FT-82-61: A_L = 36 nH/N² (same small), but better at 400–1000 MHz

RECOMMENDED BEAD-ON-COAX CONFIGURATIONS:

Configuration A — Wideband HF through 2M (FT-82-43 beads, RG-58):
  Thread 10 × FT-82-43 on RG-58 coax
  FT-82-43: OD=21mm, ID=13mm, height=6.35mm; RG-58 OD=5mm (fits easily)

  f (MHz) | Z per bead (Ω) | Z × 10 beads (Ω) | CMR @ 150Ω path (dB)
  --------|---------------|------------------|---------------------
    3.5   |    135        |   1350           | 19.1
    7.0   |    270        |   2700           | 25.1
   14.0   |    480        |   4800           | 30.1
   28.0   |    750        |   7500           | 33.9
   50.0   |    800        |   8000           | 34.6
  100.0   |    820        |   8200           | 34.8
  144.0   |    750        |   7500           | 33.9
  222.0   |    600        |   6000           | 32.0
  432.0   |    350        |   3500           | 27.4

  Outstanding 40M-70cm performance.  40 beads provides >25 dB above 3.5 MHz.

Configuration B — VHF/UHF Focus (FT-50-61, RG-58):
  Thread 20 × FT-50-61 on RG-58 coax
  FT-50-61: OD=12.7mm, ID=7.7mm, height=6.35mm

  f (MHz) | Z per bead (Ω) | Z × 20 beads (Ω)
  --------|---------------|------------------
   50.0   |     86        |   1720
  100.0   |    152        |   3040
  144.0   |    210        |   4200
  222.0   |    280        |   5600
  432.0   |    380        |   7600
  902.0   |    450        |   9000
  1296.0  |    390        |   7800

  Excellent for VHF/UHF. Bead stack is very compact (~130mm long on coax).

BEAD-ON-COAX DIAGRAM:

  RG-58 coax threaded through individual ferrite toroid beads:

  ──────────────────────────────────────────────────── coax center conductor
  ──────────────────────────────────────────────────── coax shield (inner path)
  [T1][T2][T3][T4][T5][T6][T7][T8][T9][T10]          ferrite beads on coax OD

  Each bead threads over ENTIRE coax (center + shield + jacket as one unit)
  Beads contact outer jacket only; no electrical connection to bead
  Signal (differential mode) unaffected; common mode sees series inductance

HOW TO THREAD:
  1. Select bead with ID > coax OD + 1mm clearance.
  2. Feed coax end-first through bead aperture.
  3. Push bead to center of coax run; continue with remaining beads.
  4. Secure bead stack with cable ties or heat-shrink sleeving.
  NOTE: PL-259 or N connectors must be installed BEFORE threading beads,
        as connectors will not pass through bead aperture.
  ALTERNATIVE: Use snap-on beads (Section A) to avoid connector removal issue.

COMBINED MIX BEAD STACK (recommended for wideband coverage):
  Thread the following sequence (total 8 beads, ~80mm on RG-58):
    2× FT-82-43  (HF coverage, 1–30 MHz)
    2× FT-82-43  (HF/VHF)
    2× FT-50-61  (VHF, 50–500 MHz)
    2× FT-50-61  (UHF, 500 MHz+)

  This 8-bead mixed stack provides approximately:

  f (MHz) | |Z| combined (Ω) | CMR (dB) @ 150Ω
  --------|----------------|------------------
    7.0   |   1080          | 17.7
   14.0   |   1920          | 22.1
   28.0   |   3000          | 26.0
   50.0   |   3200          | 26.6
  144.0   |   3840          | 28.2
  432.0   |   2540          | 24.6
  1296.0  |   1560          | 20.3

================================================================================
PARTS LIST — UGLY BALUN AND BEAD-ON-COAX
================================================================================

UGLY BALUN:
Item | Qty | Description                           | Notes
-----|-----|---------------------------------------|---------------------------
     |  1  | PVC pipe 6" OD, 150mm length           | Schedule 40, hardware store
     |  1  | PVC pipe 3" OD, 100mm length           | For compact 15M–6M version
     | 3.5m| RG-8X (8.4mm OD)                       | For 6" form, 7 turns
     | 2.5m| RG-58 (5mm OD)                          | For 3" form, 9 turns
J1,2 |  2  | PL-259 connectors                      | Amphenol PL-259
     |  4  | #6 cable ties 4"                        | Coil retention
     |  1  | U-bolt clamp 1.5"                       | Mast mounting (optional)

BEAD-ON-COAX:
Item | Qty | Part Number    | Description          | Notes
-----|-----|----------------|----------------------|---------------------------
T1–10| 10  | FT-82-43       | Mix 43 toroid, 21mm  | Palomar Engineers or Mouser
T11–20| 10 | FT-50-61       | Mix 61 toroid, 12.7mm| For VHF/UHF
     | var | Heat shrink 1" | Bead stack retention  | Optional but recommended
     |  2  | Cable tie 4"   | End stops             |

================================================================================