AC Resistance vs DC: Why Frequency Changes Wire Sizing and Voltage Drop

The Core Definition: What AC Resistance Actually Is

AC resistance ($R_{ac}$) is the effective opposition a conductor or component presents to alternating current that results in real power dissipation as heat, which is always higher than its DC resistance due to frequency-dependent electromagnetic effects. While DC resistance ($R_{dc}$) relies purely on the material's resistivity, length, and cross-sectional area, AC resistance accounts for the fact that alternating magnetic fields force current to redistribute within the conductor.

What it changes in a real circuit: Ignoring $R_{ac}$ leads to underestimated voltage drops, unexpected thermal loading in conductors and magnetic cores, and premature insulation failure in high-frequency applications. If you size a long feeder based solely on DC resistance, your equipment may suffer from brownouts under load.

What people commonly confuse it with: Impedance ($Z$). Impedance is the vector sum of resistance ($R$) and reactance ($X$). $R_{ac}$ is strictly the real, heat-producing resistive component of that total impedance, not the inductive or capacitive opposition.

The Physics: Why Frequency Changes the Rules

To understand why $R_{ac}$ is higher than $R_{dc}$, you have to look at how alternating magnetic fields interact with the flow of electrons. There are three primary culprits:

• Skin Effect: As AC frequency increases, the changing magnetic field inside the conductor induces eddy currents that oppose the flow of electrons in the center of the wire. This forces the majority of the current to travel only along the outer "skin" of the conductor. Think of a multi-lane highway where a magnetic force field suddenly blocks the middle lanes, forcing all traffic onto the shoulder; the usable width shrinks, and the traffic jam (resistance) increases. The skin depth ($\delta$) in copper at 60Hz is about 8.5mm, but at 10kHz, it shrinks to just 0.66mm.
• Proximity Effect: When multiple current-carrying conductors are close together (like in a tightly bundled conduit or transformer winding), their alternating magnetic fields distort each other's current distribution, further crowding electrons into smaller cross-sectional areas and spiking resistance.
• Core and Dielectric Losses: In inductors and capacitors, the alternating magnetic and electric fields cause physical friction at the molecular level in the core materials and insulation, adding to the effective $R_{ac}$ of the component.

For a deeper mathematical breakdown of these electromagnetic behaviors, the All About Circuits AC textbook provides excellent foundational theory on how alternating fields dictate conductor behavior.

Worked Example: 500 kcmil Copper Feeders at 60Hz

Let’s look at a real-world scenario where $R_{ac}$ bites you if you aren't paying attention. Suppose you are running a 400A continuous service feeder, 300 feet one-way, using 500 kcmil uncoated copper THHN in a steel conduit.

If we look at standard DC resistance (which is what many basic online voltage drop calculators use), the NFPA National Electrical Code (NEC) Chapter 9, Table 8 lists the DC resistance of 500 kcmil copper at 75°C as 0.0258 $\Omega$/1000 ft.

However, NEC Chapter 9, Table 9 provides the actual AC resistance for uncoated copper in a magnetic (steel) conduit at 60Hz: 0.029 $\Omega$/1000 ft.

The Math:

• Total wire length (out and back) = 600 ft (0.6 kft).
• Total $R_{dc}$ = $0.0258 \times 0.6 = 0.01548 \Omega$.
• Total $R_{ac}$ = $0.029 \times 0.6 = 0.0174 \Omega$.

The AC resistance is 12.4% higher than the DC resistance. Now, let's look at the real power loss ($I^2R$) at a 400A load:

• DC Power Loss: $400^2 \times 0.01548 = 2,476$ Watts.
• AC Power Loss: $400^2 \times 0.0174 = 2,784$ Watts.

That is 308 Watts of extra heat dissipated inside your conduit purely because the current is alternating. Over a long summer afternoon, that extra heat is the difference between your terminations holding steady at 65°C and creeping past the 75°C rating of your breaker lugs, triggering nuisance trips or degrading the insulation.

Where You Meet AC Resistance in Practice

You might think this only matters to utility engineers, but $R_{ac}$ shows up on the workbench and the jobsite in several critical ways:

1. Variable Frequency Drive (VFD) Output Cables
VFDs don't output a clean 60Hz sine wave; they output a Pulse Width Modulated (PWM) waveform with carrier frequencies typically between 2 kHz and 16 kHz. At 10 kHz, the skin depth in copper is less than a millimeter. A standard 2 AWG THHN wire will effectively act like a much smaller wire, causing massive $R_{ac}$ heating. This is why VFD installations require specialized symmetrical, shielded cables with multiple smaller ground wires rather than one massive solid conductor.

2. Switch-Mode Power Supplies (SMPS) and Induction Heating
In high-frequency magnetics, the proximity effect is devastating. If you wind a high-frequency transformer with thick, solid magnet wire, the outer turns will induce eddy currents in the inner turns, causing the winding to overheat rapidly. This is why high-frequency inductors use Litz wire—a specialized cable made of hundreds of individually enameled, micro-thin strands that are continuously woven so that each strand spends equal time on the outside and inside of the bundle, effectively defeating the skin and proximity effects.

3. Long Underground Feeders
When pulling long runs of large-gauge aluminum or copper for subpanels or agricultural buildings, always use NEC Table 9 (or your local equivalent) for voltage drop calculations, not Table 8. The steel conduit or aluminum armor adds magnetic hysteresis losses that further inflate the $R_{ac}$ compared to wires sitting in free air or PVC.