The resistance of copper traces on printed circuit boards impacts power distribution, signal integrity and overall circuit performance. Accurately calculating trace resistance is therefore an important skill for PCB designers. This article will provide an overview of the key parameters and formulas used to determine trace resistance.

## Factors Affecting Trace Resistance

The resistance of a PCB trace depends on several factors:

**Trace Geometry**– Dimensions like length, width and thickness. Longer, narrower and thinner traces have higher resistance.**Copper Properties**– Resistivity and temperature coefficient depend on copper purity and alloy percentages.**Trace Shape and Path**– Meandering traces have more resistance than straight lines.**Copper Surface**– Surface roughness from etching impacts resistance.**Temperature**– Copper resistance increases with temperature.**Operating Current**– At high currents, resistance increases due to self-heating.

Accurately accounting for all these parameters helps predict the trace resistance seen in the actual PCB circuit.

## Trace Resistance Calculation Fundamentals

The resistance of a PCB trace depends on its resistivity, length and cross-sectional area as described by:

$$R_{trace} = \rho \frac{L}{A}$$

Where:

- Rtrace = Trace resistance in ohms (Ω)
- ρ = Resistivity of copper (Ω.m)
- L = Length of the trace (m)
- A = Cross-sectional area of trace (m2)

### Resistivity (ρ)

Resistivity is a material property indicating how strongly it opposes electric current flow. For copper, it is:

- 1.72 x 10<sup>-8</sup> Ω.m at 20°C
- Increases by 0.00393 per °C above 20°C due to temperature coefficient of copper.

### Length (L)

Length is the end-to-end distance travelled by the trace in meters. This should account for any meandering of the trace.

### Cross-sectional Area (A)

For a standard rectangular trace, the cross-sectional area is:

$$A = T \times W$$

Where:

- T = Trace thickness in meters
- W = Trace width in meters

This simple model allows us to approximate the resistance of basic straight trace geometries. However, real-world traces often have more complex shapes which require further considerations.

## Advanced Modelling for Trace Resistance

To account for various complexities in PCB traces, advanced modelling techniques are required:

### Accounting for Non-Straight Traces

For meandering traces, break the path into straight rectangular segments and calculate resistance piece-wise before summing.

### Accounting for Varying Widths

For traces with varying widths, break into segments of constant width and sum the resistances.

### Accounting for Internal Layers

Traces in inner layers have reduced surface roughness. Use adjusted resistivity values.

### Accounting for Self-Heating

At high currents, use incremental resistance ratios to model self-heating.

### Accounting for High Frequencies

Consider skin and proximity effects that redistribute current at high frequencies.

### Accounting for Non-Uniform Thickness

Use minimum expected thickness in calculations to account for fabrication variations.

### Accounting for Surface Roughness

Model the uneven copper surface as a correction factor to the area.

### Using 3D EM Simulation

For precision modelling, use 3D electromagnetic simulation of the entire trace shape.

By utilizing these advanced modelling techniques, very accurate estimation of trace resistances can be obtained.

## Calculating Single-Layer Trace Resistance

For simple rectangular traces on a single layer, we can apply the fundamental resistance equation:

**Example**

Calculate resistance of a 200mm long, 0.5mm wide trace in 1oz (35μm) copper.

**Known:**

- Length (L) = 0.2 m
- Width (W) = 0.0005 m
- Thickness (T) = 35 x 10<sup>-6</sup> m (1oz)
- Resistivity of Copper (ρ) = 1.72 x 10<sup>-8</sup> Ω.m

**Cross-sectional Area (A)**

A = T x W = (35 x 10<sup>-6</sup>) x (0.0005) = 17.5 x 10<sup>-6</sup> m<sup>2</sup>

**Applying resistance equation:**

Rtrace = (ρ x L) / A = (1.72 x 10<sup>-8</sup> x 0.2) / 17.5 x 10<sup>-6</sup> = 0.0196 Ω

Therefore, resistance of the 200mm long, 0.5mm wide trace in 1oz copper is 0.0196 Ω.

This approach allows quickly estimating single-layer rectangular trace resistances.

## Multi-Layer Trace Resistance Calculation

For traces passing between layers in a multi-layer PCB, adjustments are needed in the resistance calculations:

**Accounting for Via Resistance**

Add the resistance of vias connecting adjacent trace sections.

**Accounting for Inter-Layer Dielectric**

Account for dielectric thickness between layers when summing segment lengths.

**Using Layer Adjusted Resistivity**

Inner layers have smoother copper so lower resistivity. Outer layers are rougher.

**Example Multi-Layer Calculation**

- 200mm long trace with 100mm on top layer and 100mm on inner layer
- 2 vias connecting layers, each 0.1mm diameter, 0.2mm length
- 0.5mm trace width
- 1oz (35μm) copper on outer layers
- 1oz (35μm) + 18μm copper on inner layer
- FR-4 dielectric between layers, 0.2mm thick

**Via Resistance**

Rvia = 2 x ρCu x Lvia / Avia

= 2 x 1.72×10<sup>-8</sup> x 0.2 / (π x (0.1/2)<sup>2</sup>) = 0.0079 Ω

**Outer Layer Trace**

Rtop= ρCu x Ltop / Atop

= 1.72×10<sup>-8</sup> x 0.1 / (35×10<sup>-6</sup> x 0.0005) = 0.0098 Ω

**Inner Layer Trace**

Rinner = ρCu x Linner / Ainner = 1.52×10<sup>-8</sup> x (0.1+0.2) / (35×10<sup>-6</sup>+18×10<sup>-6</sup>) x 0.0005

= 0.0067 Ω

**Total Resistance**

Rtotal = Rtop + Rvia + Rinner = 0.0098 + 0.0079 + 0.0067 = 0.0244 Ω

This demonstrates how to account for vias and different layers when calculating overall trace resistance for a multi-layer PCB.

## Trace Resistance Calculation Tool

Manually applying the equations can get tedious. For convenience, online trace resistance calculators allow specifying all the parameters needed and provide the computed resistance.

Here is an example screenshot of a browser-based calculator:

These tools provide a quick and easy way to estimate trace resistances for your PCB designs.

## Effects of Temperature on Trace Resistance

Due to the positive temperature coefficient of copper, resistance of traces increases with temperature:

**Temperature Coefficient of Copper**

Around 0.00393 Ω/Ω/°C

**Temperature Adjusted Resistivity**

ρT = ρ20°C [1 + α (T – 20°C)]

Where:

- ρT = Resistivity at temperature T
- ρ20°C = Resistivity at 20°C (1.72 x 10<sup>-8</sup> Ω.m)
- α = Temperature coefficient (0.00393 for copper)
- T = Actual operating temperature in °C

This adjusted resistivity is then used in resistance calculations to account for temperature.

**Example**

A 50mm long, 0.25mm wide trace experiences 50°C temperature rise during operation. Initial resistance at 20°C is:

R20°C = ρ20°C x L / A

= 1.72 x 10<sup>-8</sup> x 0.05 / (0.035 x 0.00025) = 0.0049 Ω

Resistance at 70°C is:

ρ70°C = 1.72 x 10<sup>-8</sup> [1 + 0.00393 x (70 – 20)] = 2.012 x 10<sup>-8</sup> Ω.m

R70°C = ρ70°C x L / A

= 2.012 x 10<sup>-8</sup> x 0.05 / (0.035 x 0.00025) = 0.0058 Ω

The temperature rise has increased the trace resistance by 18%.

## Impact of Trace Resistance

The resistance of PCB traces has several important effects on circuit performance:

- Voltage drops along traces carrying high currents – can affect device operation.
- Power loss and heating due to current flow – impacts thermal design.
- Signal degradation and delays – limits maximum trace lengths for signals.
- Impedance discontinuities – affects signal integrity especially for high-speed signals.
- Noise pickup – higher resistance traces are more susceptible.

Hence considering trace resistance and mitigating its impact is crucial during PCB design to ensure proper functioning of circuits.

## Techniques to Reduce Trace Resistance

Here are some methods to minimize trace resistance on PCBs:

- Use thicker copper – 2oz and 3oz copper significantly reduce resistance.
- Increase trace widths for power traces carrying higher currents.
- Use shorter and straighter trace routing. Avoid meandering paths.
- Use inner PCB layers which have smoother copper.
- Use wider power/ground planes to distribute current.
- Maintain lower ambient operating temperatures.
- Coat traces with low-resistance silver, gold or tin alloys.

With careful design, the impact of inherent copper trace resistance can be mitigated, enabling high-performance PCB implementation.

## Conclusion

- The resistance of printed circuit board traces depends on resistivity of copper, trace length and cross-sectional area.
- Advanced modelling techniques are needed to account for complex real-world trace geometries and multi-layer boards.
- Trace resistance impacts power distribution, thermal design and signal integrity.
- Careful calculations coupled with mitigation techniques help overcome limitations of inherent copper resistivity.

## Top 5 FAQs on Calculating Trace Resistance

### Q1: How accurate are simple trace resistance calculations?

For straight rectangular traces, the simple resistance equation provides a good estimate. Real-world complex traces require advanced modelling for accuracy.

### Q2: Do wider traces always have lower resistance?

Yes, for the same thickness, increasing trace width reduces resistance. But wider traces have higher capacitance impacting signals.

### Q3: Does trace length include bends and meanders?

Yes, total end-to-end length following the entire trace path must be used, not just linear distance between endpoints.

### Q4: Can increasing copper thickness eliminate resistance issues?

Thicker copper helps reduce resistance but is limited by manufacturability and costs. Wide traces still provide lower resistance for power distribution.

### Q5: What precision of trace resistance calculation is needed?

1-5% accuracy is sufficient for most needs. Precise modelling is required where resistance impacts impedance matching, voltage drops or thermal management.