# DC Wire Size Calculator

Created by Luis Hoyos
Last updated: Feb 18, 2023

Welcome to the DC wire size calculator, a tool that will help you to calculate the cable size of batteries, solar power systems, or any device where direct current (DC) appears.

With this tool, you can calculate the cable size of DC and single-phase AC systems. For three-phase systems, click on the advanced mode or use our more complete wire size calculator. If you previously know or calculated your DC cable size, you can also use this tool the other way and know your DC wire ampacity (maximum allowable current).

The following sections present the formulas to calculate your DC wire's area, diameter, and ampacity. Please refer to our wire gauge calculator to know how to convert those wire sizes into AWG (American Wire Gauge) units.

## Deriving the formula to calculate a DC cable size

The formula to calculate the cable size of a battery or any other DC system relies on Ohm's law and Pouillet's law.

Ohm's law states that the current between two points is directly proportional to the voltage difference across those points, where the inverse of the conductor resistance is the constant of proportionality:

I = V/R

where:

• I — Electrical current;
• V — Voltage difference; and
• R — Conductor wire resistance.

On the other side, Pouillet's law states that the resistance of a conductor is proportional to its length and inversely proportional to its cross-sectional area, being resistivity the constant of proportionality:

R = ρ(L/A)

where:

• ρ — Resistivity of the conductor material;
• L — Length of the conductor;
• A — Cross-sectional area of the conductor.

If we input Pouillet's law into Ohm's law and solve for A, we obtain:

A = IρL/V

We almost have our formula, but there are two additional aspects to consider:

• In this case, the conductor is the wire. One-way distance is the distance from the source voltage to the load, but we need another wire of the same length to return to the source voltage. Therefore, the cable length (L) is two times the one-way distance (D). In math terms L = 2D.
• Depending on the type of system, the area obtained with the formula has to be multiplied by a coefficient called phase factor (ϕ).
• For DC and AC single-phase systems, ϕ = 1.
• For three-phase systems, ϕ = √3/2.

Now we're ready to state the final version of the formula to calculate the DC wire size.

## Formula to calculate the DC wire size

Considering the information from the previous section, we can finally obtain the formula to calculate the wire size for DC systems:

A = 2IρD/V

where:

• A — Cross-sectional area of the DC wire, in squared meters (m2);
• I — Current through the DC wire, in amperes (A);
• ρ — Resistivity of the wire material, in ohms meters (Ω m);
• D — One-way distance (how far is the cable run from its source to the farther point), in meters (m); and
• V — Voltage drop across the wire, in Volts (V).

Knowing the area, we can easily calculate the wire diameter (d) with the following formula:

d = √(4A/π)

We can also calculate the DC wire ampacity by solving for current I:

I = AV/(2ρD)

### Voltage drop

The voltage drop across the wire equals the source voltage times the allowable percentage drop. For example, for a 12 V source voltage and a 3% permissible drop:

V = 12 V × 3% = 12 V × 0.03 = 0.36 V

### Resistivity

As mentioned before, resistivity is directly related to the wire resistance. The greater the resistivity, the greater wire resistance and, consequently, the required wire size. Resistivity depends on the wire material and operating temperature. The following formula models the resistivity of any material as a function of its temperature:

ρ = ρ1​[1 + α(T ​− T1​)]

where:

• T1 — Reference temperature corresponding to the resistivity ρ1 at that temperature;
• T — Target temperature at which you want to find the resistivity ρ; and
• α — Temperature coefficient, specific for each material.

🙋 We can express both temperatures in °C or Kelvin, as long the unit is the same for both temperatures.

For example, for copper, we know that the temperature coefficient of copper is 0.00404, and its resistivity at 20°C is 1.68 × 10−8 Ω m. We can use that information to know the copper resistivity at 75°C:

ρ = ρ1​[1 + α(T ​− T1​)] = 1.68 × 10−8[1 + 0.00404(75°C - 20°C)] Ω m
= 2.05 × 10−8 Ω m

We can do the same for aluminum, considering its temperature coefficient is 0.00404, and its resistivity at 20°C is 2.65 × 10−8 Ω m. Even so, by only inputting the maximum operating temperature, the calculator will find the resistivity for you.

And that's it for now! 🎉Hopefully, now you're prepared to calculate the cable size of batteries or any other DC systems. Don't forget to refer to the other related calculator for more information.

Luis Hoyos
Source voltage
V
Allowable voltage drop (V)
%
Conductor material
Copper
Current (I)
A
One-way distance (D)
ft
Maximum wire temperature
°F
Recommended wire size
Wire gauge
AWG
Wire cross-sectional area (A)
kcmil
Wire diameter (d)
in
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