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✅ When copper-wire properties start to matter.

1. Inductance becomes noticeable around ~10 kHz and up

A typical straight copper wire has about:

L ≈ 0.6 to 1 uH per meter

Inductive reactance:

X_L = 2 * pi * f * L

Example for L = 1 uH:

• At 1 kHz → X_L = 0.006 ohms
• At 10 kHz → X_L = 0.06 ohms
• At 1 MHz → X_L = 6.28 ohms (significant)

So:

Inductance matters from ~10 kHz

Dominates above ~1 MHz

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2. Parasitic capacitance matters from ~100 kHz+

Typical wire-to-environment capacitance:

C ≈ 10 pF to 100 pF

Capacitive reactance:

X_C = 1 / ( 2 * pi * f * C )

Example for C = 10 pF:

• At 100 kHz → X_C = 160 kohms
• At 10 MHz → X_C = 1.6 kohms
• At 100 MHz → X_C = 160 ohms

So:

Capacitance becomes important above ~1 to 10 MHz

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3. Transmission line effects start around ~30 MHz

A wire behaves like a transmission line when:

length >= ( wavelength / 10 )

Wavelength:

lambda = c / f

For typical insulated wire or PCB trace, signal speed:

v ≈ 0.7 * c

This gives:

• At 30 MHz → lambda ≈ 7 m → 1 m wire ≈ lambda/7 (borderline)
• At 100 MHz → lambda ≈ 2.1 m → 10–20 cm wires behave as transmission lines
• At 1 GHz → even a few millimeters matter

So:

Transmission line behavior matters above ~30 MHz

Critical above ~100 MHz

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4. Skin effect becomes important above ~100 kHz

Skin depth:

delta = sqrt( 2 / ( omega * mu * sigma ) )

where:

• omega = 2 * pi * f
• mu = magnetic permeability
• sigma = conductivity of copper

Typical values:

• 60 Hz → delta ≈ 8 mm
• 100 kHz → delta ≈ 0.2 mm
• 10 MHz → delta ≈ 20 micrometers

So:

Skin effect matters above ~100 kHz

Dominates in MHz and GHz ranges

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5. Wire self-resonant frequency (SRF)

A wire’s inductance (L) and capacitance (C) form a natural LC resonant circuit:

f_resonance = 1 / ( 2 * pi * sqrt(L * C) )

Long wires (meters) → SRF in tens of MHz
Short wires (cm) → SRF hundreds of MHz
PCB traces → often in GHz

So:

Self-resonance usually between ~30 MHz and several GHz

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📌 Summary Table (ASCII)

Effect                     | Frequency where it matters

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Inductance                 | ~10 kHz+

Skin effect                | ~100 kHz+

Parasitic capacitance      | ~1–10 MHz+

Transmission line effects  | ~30–100 MHz+

Self resonance             | ~30 MHz – GHz

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