Calculating and Compensating Process Variations in PCBs with Embedded Resistors

1. Resistance Calculation

• Core Formula:
  $R=\rho \cdot \frac{L}{W\cdot T}(\text{Unit: }\mathrm{\Omega })$
  

  • $\rho$: Sheet resistance (Ω/□)

  • $L$: Length (mm)

  • $W$: Width (mm)

  • $T$: ThICkness (μm)

• Process Compensation:
  $R_{actual}=R_{design}\cdot (1+\alpha \cdot \mathrm{\Delta }W+\beta \cdot \mathrm{\Delta }T)$
  

  • $\alpha$: Width error coefficient (0.8-1.2)

  • $\beta$: Thickness error coefficient (1.5-2.0)

2. Material Properties

3. Error Sources & Compensation

3.1 Key Variations

3.2 Design Compensation

• Pre-scaling Method:

• Redundant Structure:
  Main Resistor + laser-trimmable micro-resistor (±25% range)

• Thermal Layout:
  +20% width in high-power areas

4. Process Control Parameters

5. Trimming & Validation Flow

5.1 Laser Trimming Specs

6. Process Variation Model

• Multi-factor Equation:
  $\frac{\mathrm{\Delta }R}{R}=a\cdot \frac{\mathrm{\Delta }\rho }{\rho }+b\cdot \frac{\mathrm{\Delta }L}{L}-c\cdot \frac{\mathrm{\Delta }W}{W}-d\cdot \frac{\mathrm{\Delta }T}{T}$

  • Coefficients: $a=1.0,b=0.1,c=1.2,d=1.8$

• SPC Control Limits:

  Control Line	3σ Range	Resistance Deviation
  UCL	+4.5%	Upper alarm
  LCL	-4.5%	Lower alarm

7. Application Cases

• High-Speed Backplane:

  • Requirement: 100Ω diff. impedance (±2%)

  • Solution:

    • RuO₂ material (1kΩ/□)

    • Dimensions: L=1.2mm, W=0.15mm

    • Pre-compensation: +8μm width

  • Result: CPK=1.65 (99.5±1.2Ω)

• Automotive ECU:

  • Issue: 12% drift (-40℃~125℃)

  • Compensation:

    • Parallel PTC carbon resistor (+400ppm/°C)

    • Drift reduced to ±2%