CableDataSheet, Cable and Wire Technical Consulting Service

1. DC Resistance
- Definition: DC resistance refers to the opposition a conductor offers to the flow of direct current (steady current).
- Formula: For a conductor of length L, cross-sectional area A, and resistivity ρ:
    R_{DC} = \frac{\rho L}{A}
  - Characteristics:
  - Only depends on material properties (ρ), length (L), and cross-sectional area (A).
  - Current distribution is uniform across the cross-section of the conductor since there is no variation in current over time.

2. AC Resistance
- Definition: AC resistance refers to the effective resistance of a conductor when alternating current flows through it.
- Formula: AC resistance is often expressed as:
    R_{AC} = R_{DC} \times (1 + \text{skin effect factor} + \text{proximity effect factor})
  - Characteristics:
  - Higher than DC resistance because of the skin effect and proximity effect.
  - Depends on the frequency (f) of the AC signal, as these effects increase with frequency.

3. Key Factors Influencing AC Resistance

a. Skin Effect
- What is it?
  - At higher AC frequencies, current tends to flow near the surface of the conductor rather than uniformly throughout the cross-section. This reduces the effective area available for current flow, increasing resistance.
- Why does it happen?
  - Alternating current generates a changing magnetic field, which induces eddy currents inside the conductor. These eddy currents oppose the flow of the primary current, forcing it towards the surface.
- Effect on Resistance:
  - As frequency increases, the depth of current penetration (skin depth δ) decreases:
        \delta = \sqrt{\frac{2\rho}{\mu \omega}}
        Where:
    - ρ: resistivity of the material
    - μ: permeability of the material
    - ω = 2πf: angular frequency
  - Smaller δ leads to higher AC resistance.

b. Proximity Effect
- What is it?
  - When multiple conductors are close to each other (e.g., in cables or transformers), the magnetic fields produced by the currents interact. This causes the current distribution in each conductor to become uneven, concentrating it in certain regions.
- Why does it happen?
  - The magnetic fields from nearby conductors induce eddy currents within the conductor, distorting the current flow.
- Effect on Resistance:
  - Further increases AC resistance due to uneven current distribution.

4. Comparison of AC and DC Resistance

[th]Parameter[/th][th]DC Resistance[/th][th]AC Resistance[/th]
Current Type	Steady current	Alternating current
Current Distribution	Uniform	Non-uniform (skin effect and proximity effect)
Frequency Dependency	Independent	Increases with frequency
Formula	Code Select Expand R_{DC} = \frac{\rho L}{A}	Code Select Expand R_{AC} = R_{DC} \times (1 + \text{effects})
Value	Lower	Higher

5. Mitigating High AC Resistance
To reduce the impact of skin and proximity effects:
1. Litz Wire:
   - Made of many thin insulated strands woven together.
   - Ensures current is distributed more evenly across the conductor.
2. Hollow Conductors:
   - Used in high-frequency or high-power applications (e.g., antennas, busbars).
   - Eliminates unused central material since current flows on the surface.
3. Material Selection:
   - Use conductors with low resistivity (e.g., copper or silver) to minimize resistance.
4. Frequency Management:
   - Design circuits to operate at lower frequencies when possible.

6. Applications and Importance
Understanding the relationship between AC and DC resistance is crucial for:
- Power Transmission:
  - AC systems require conductors that account for higher resistance at high frequencies.
- Transformer Design:
  - The proximity effect is significant in windings.
- RF (Radio Frequency) Applications:
  - Skin effect dominates at high frequencies, requiring specialized conductors.