Theoretical Calculation of Transfer Impedance (TI) for Copper Braided Shields

Theoretical Calculation of Transfer Impedance (TI) for Copper Braided Shields

Transfer Impedance (Z_t) measures the effectiveness of a shield (e.g., a copper braid) in protecting against electromagnetic interference (EMI). It combines contributions from DC resistance, AC resistance, and leakage inductance. Below is the theoretical framework for calculating TI.

---

1. General Transfer Impedance Formula

QuoteThe general formula for Transfer Impedance is:
\[ Z_t(f) = \frac{V}{I} \, \text{(per unit length)} \]
Where:

• V: Voltage induced inside the shield (volts)
• I: Current applied to the shield (amperes)
• f: Frequency (Hz)

---

2. Components of Transfer Impedance
Transfer Impedance is made up of three main contributions:

• (a) DC Resistance (\( R_{\text{DC}} \))
  \[R_{\text{DC}} = \frac{\rho}{n \cdot A_{\text{wire}} \cdot p}\]
  Where:
  
  • \( \rho \): Resistivity of copper (\(1.68 \times 10^{-8} \, \Omega \cdot \text{m}\))
  • \( n \): Number of wires in the braid
  • \( A_{\text{wire}} \): Cross-sectional area of each wire (\( \pi \cdot (d_{\text{wire}}/2)^2 \))
  • \( p \): Shield coverage (percentage of shield coverage)
• (b) AC Resistance (\( R_{\text{AC}} \))
  At higher frequencies, resistance increases due to the skin effect:
  \[R_{\text{AC}} = R_{\text{DC}} \cdot \frac{\delta}{d_{\text{wire}}} \cdot \coth\left(\frac{d_{\text{wire}}}{\delta}\right)\]
  Where:
  
  • \( \delta \): Skin depth (\( \delta = \sqrt{\frac{2 \rho}{\mu \cdot \omega}} \))
  • \( \mu \): Magnetic permeability of copper (\(4\pi \times 10^{-7} \, \text{H/m}\))
  • \( \omega \): Angular frequency (\( \omega = 2\pi f \))
• (c) Leakage Inductance (\( L_{\text{leak}} \))
  Leakage inductance accounts for the magnetic field inefficiency between the braid and internal conductor:
  \[L_{\text{leak}} = \mu_0 \cdot \ln\left(\frac{D_{\text{outer}}}{D_{\text{inner}}}\right)\]
  Where:
  
  • \( D_{\text{outer}} \): Outer diameter of the braid
  • \( D_{\text{inner}} \): Inner diameter of the braid
  • \( \mu_0 \): Magnetic permeability of free space (\( 4\pi \times 10^{-7} \, \text{H/m} \))

---

3. Combined Formula for Transfer Impedance
The total Transfer Impedance is:
\[Z_t(f) = R_{\text{DC}} + R_{\text{AC}} + j\omega L_{\text{leak}}\]

---

4. Example Calculation
Parameters:

• Shield coverage: \( p = 0.95 \)
• Wire diameter: \( d_{\text{wire}} = 0.2 \, \text{mm} \)
• Number of wires: \( n = 96 \)
• Frequency: \( f = 10 \, \text{MHz} \)
• Outer diameter: \( D_{\text{outer}} = 10 \, \text{mm} \)
• Inner diameter: \( D_{\text{inner}} = 9 \, \text{mm} \)

Step 1: Calculate DC Resistance (\( R_{\text{DC}} \))
\[R_{\text{DC}} = \frac{1.68 \times 10^{-8}}{96 \cdot \pi \cdot (0.2 \times 10^{-3}/2)^2 \cdot 0.95} \approx 6.37 \, \text{m}\Omega/\text{m}\]

Step 2: Calculate Skin Depth (\( \delta \))
\[\delta = \sqrt{\frac{2 \cdot 1.68 \times 10^{-8}}{4\pi \times 10^{-7} \cdot 2\pi \cdot 10^7}} \approx 0.021 \, \text{mm}\]

Step 3: Calculate AC Resistance (\( R_{\text{AC}} \))
\[R_{\text{AC}} \approx R_{\text{DC}} \cdot \frac{\delta}{d_{\text{wire}}} = 6.37 \cdot \frac{0.021}{0.2} \approx 0.67 \, \text{m}\Omega/\text{m}\]

Step 4: Calculate Leakage Inductance (\( L_{\text{leak}} \))
\[L_{\text{leak}} = 4\pi \times 10^{-7} \cdot \ln\left(\frac{10}{9}\right) \approx 1.32 \times 10^{-7} \, \text{H/m}\]

Step 5: Calculate Inductive Contribution (\( Z_{t,\text{ind}} \))
\[Z_{t,\text{ind}} = j\omega L_{\text{leak}} = j \cdot 2\pi \cdot 10^7 \cdot 1.32 \times 10^{-7} \approx j 8.29 \, \text{m}\Omega/\text{m}\]

Step 6: Total Transfer Impedance
\[Z_t \approx 6.37 + 0.67 + j 8.29 = 7.04 + j 8.29 \, \text{m}\Omega/\text{m}\]

---

5. Summary

• At low frequencies: \( R_{\text{DC}} \) dominates the impedance.
• At high frequencies: Leakage inductance (\( L_{\text{leak}} \)) and skin effect (\( R_{\text{AC}} \)) dominate.
• Proper shield geometry minimizes \( Z_t \), improving EMI performance.