Plane Spreading Resistance

Rspread = ρ / (π × r)

Quick answer

This calculator solves Rspread = ρ / (π × r) for Spreading Resistance. With Resistivity (Ω·m) 0.5 Ω·m, Via/Pad Radius 10 mm, the result is 15.92 Ω. It solves in 3 directions, so you can enter the values you have and get the one you need.

Calculator

Spreading Resistance (mΩ)
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Formula

Rspread = ρ / (π × r)

Description

When current enters a large ground or power plane through a via or pad of radius r, it spreads radially outward. The spreading resistance represents the concentrated resistance near the via where current density is highest. For a 1 oz copper plane (35 µm thick), the spreading resistance from a typical 0.2mm radius via pad is in the low milliohm range. This resistance is important in power distribution network analysis because it limits how effectively a plane can deliver transient current to a specific location. Multiple vias reduce the effective spreading resistance.

Variables

  • Rspread — Spreading resistance (Ω)
  • ρ — Sheet resistivity of the plane (Ω·m)
  • r — Radius of the current injection point (m)

Practical Notes

This formula assumes a semi-infinite plane. For finite planes or near edges, the resistance increases. Copper sheet resistance for 1 oz (35µm): 0.5 mΩ/square. For PDN analysis, spreading resistance from capacitor mounting pads to IC power pins is a key contributor to the total impedance. Wider vias and via arrays reduce spreading resistance effectively.

Symbols and units

Symbol Meaning Default unit Accepted units
Rspread Spreading Resistance mΩ mΩ, Ω, kΩ, MΩ
rho Resistivity (Ω·m) — Ω·m
r Via/Pad Radius mm mm

Step by step

  1. 1. What you know: Resistivity (Ω·m) 0.5 Ω·m, Via/Pad Radius 10 mm
  2. 2. The relation: Rspread = ρ / (π × r)
  3. 3. Substitute: rho = 0.5 Ω·m, r = 10 mm
  4. 4. Result: Spreading Resistance = 15.92 Ω

Nearest standard component values

Series Nearest value Error
E12 15 Ω 5.8%
E24 16 Ω 0.53%

Worked examples

Minimum range

Given: Resistivity (Ω·m) 0.025 Ω·m, Via/Pad Radius 0.1 mm

Result: Spreading Resistance = 79.58 Ω

Low range

Given: Resistivity (Ω·m) 0.25 Ω·m, Via/Pad Radius 1 mm

Result: Spreading Resistance = 79.58 Ω

Typical

Given: Resistivity (Ω·m) 0.5 Ω·m, Via/Pad Radius 10 mm

Result: Spreading Resistance = 15.92 Ω

High range

Given: Resistivity (Ω·m) 0.8 Ω·m, Via/Pad Radius 100 mm

Result: Spreading Resistance = 2.546 Ω

Maximum range

Given: Resistivity (Ω·m) 8 Ω·m, Via/Pad Radius 1000 mm

Result: Spreading Resistance = 2.546 Ω

Solving for each variable

Solve for Spreading Resistance

Resistivity (Ω·m) 0.025 Ω·m, Via/Pad Radius 0.1 mm → 79.58 Ω

Solve for Resistivity (Ω·m)

Spreading Resistance 0.1 mΩ, Via/Pad Radius 0.1 mm → 3.14 × 10⁻⁸ Ω·m

Solve for Via/Pad Radius

Spreading Resistance 0.1 mΩ, Resistivity (Ω·m) 0.025 Ω·m → 79580 mm

How Spreading Resistance changes with Resistivity (Ω·m)

Resistivity (Ω·m) (input) Spreading Resistance (result)
0.05 Ω·m 1.592 Ω
0.125 Ω·m 3.979 Ω
0.25 Ω·m 7.958 Ω
0.375 Ω·m 11.94 Ω
0.5 Ω·m 15.92 Ω
0.75 Ω·m 23.87 Ω
1 Ω·m 31.83 Ω
2 Ω·m 63.66 Ω
3.5 Ω·m 111.4 Ω
5 Ω·m 0.1592 kΩ

The same result in other units

Unit Value
mΩ 15920
Ω 15.92
kΩ 0.01592

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Frequently asked questions

How do you calculate Plane Spreading Resistance?

Use Rspread = ρ / (π × r). With Resistivity (Ω·m) 0.5 Ω·m, Via/Pad Radius 10 mm, Spreading Resistance works out to 15.92 Ω.

What is the formula for Plane Spreading Resistance?

The formula is Rspread = ρ / (π × r). When current enters a large ground or power plane through a via or pad of radius r, it spreads radially outward.

How do you find Resistivity (Ω·m) when you know the other values?

Rearrange for Resistivity (Ω·m): with Spreading Resistance 0.1 mΩ, Via/Pad Radius 0.1 mm, Resistivity (Ω·m) is 3.14 × 10⁻⁸ Ω·m.

How do you find Via/Pad Radius when you know the other values?

Rearrange for Via/Pad Radius: with Spreading Resistance 0.1 mΩ, Resistivity (Ω·m) 0.025 Ω·m, Via/Pad Radius is 79580 mm.

What units does Plane Spreading Resistance use?

Spreading Resistance is shown in mΩ, Ω, kΩ. Values are held in SI base units internally, so you can mix prefixes without converting by hand.

Related Concepts

All PCB & Traces formulas →
Trace Current Capacity (IPC-2221) Trace Resistance Trace Voltage Drop Via Current Capacity (IPC-2221) Microstrip Impedance Stripline Impedance Differential Pair Impedance Near-End Crosstalk

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