Power Electronics

Copper Loss

Q: How does copper loss differ from core loss in a transformer?

Copper loss is the I-squared-R dissipation in the winding conductors and scales with the square of the load current, so it is sometimes called the load loss or variable loss. Core loss is the hysteresis and eddy-current dissipation in the magnetic core; it depends on the peak flux density and frequency but is essentially independent of load current, so it is called the no-load loss. At rated load the two are often comparable, and the maximum efficiency point of a transformer occurs where copper loss equals core loss. A key practical difference: you can measure core loss with an open-circuit test at rated voltage and zero load current, and copper loss with a short-circuit test at rated current and reduced voltage.

Q: Why does copper loss increase at RF even though the conductor is unchanged?

At DC, current fills the entire conductor cross-section, so the resistance is rho times length divided by full area. As frequency rises, the skin effect forces current into a surface layer roughly one skin depth thick. Copper skin depth is about 2.1 micrometres at 1 GHz and only 0.66 micrometres at 10 GHz, so a 1 mm diameter wire that carries current through its whole 0.79 square-millimetre area at DC effectively uses only a thin annular shell at microwave frequencies. The AC resistance therefore rises in proportion to the square root of frequency, and the proximity effect from adjacent turns can multiply it further. A wire with 0.02 ohms of DC resistance can present several ohms of AC resistance at 1 GHz.

Q: How do litz wire and surface plating reduce copper loss?

Litz wire divides a conductor into many fine, individually insulated and transposed strands so that each strand is thinner than a skin depth and every strand spends equal time near the surface, which suppresses both skin and proximity effects. It is effective up to a few megahertz; above roughly 2 to 3 MHz the strand insulation and bundle complexity make it impractical and solid or tubular conductors are preferred. Silver plating helps because silver has slightly higher conductivity than copper, about 6.3 versus 5.96 by ten to the seven siemens per metre, but because RF current flows only in the skin, even a few micrometres of plating thickness carries nearly all the current. Surface roughness matters too: a rough plated surface lengthens the current path and can add 10 to 50 percent to copper loss at millimetre-wave frequencies.

/KOP-er laws/

Heat dissipated in the copper conductors and windings of inductors, transformers, and lines whenever current flows, equal to I²R where R is the conductor resistance. Often called I²R loss or winding loss, it is distinct from core loss because it scales with the square of load current rather than flux density. At radio frequencies the relevant R is the AC resistance, which the skin effect drives far above the DC value as current crowds into a thin surface shell. Minimizing copper loss is central to high-efficiency power converters, RF chokes, and matching-network inductors where every milliohm of conductor loss degrades the achievable Q.

Category: Power Electronics

Scales as: I²R (load current squared)

AC/DC ratio at 1 GHz: 10× to 100×

Where the Heat Goes in a Winding

Every real conductor has a finite resistance, and any current passing through it dissipates power as heat following the I²R relationship. In a power transformer or a DC-DC converter inductor, this is the dominant loss mechanism that determines temperature rise, copper utilization, and ultimately how small the magnetic part can be made before it overheats. Because the dissipation grows with the square of the current, doubling the load current quadruples the copper loss, which is why copper loss is also known as the variable or load loss in transformer terminology.

At low frequencies the resistance used in I²R is simply the DC resistance, R_DC = ρ × ℓ / A, set by the resistivity of copper (about 1.68 × 10⁻⁸ Ω·m at 20 °C), the conductor length, and its cross-sectional area. Copper resistivity also climbs roughly 0.39 percent per degree Celsius, so a winding sitting 60 °C above the 20 °C reference (an 80 °C conductor) dissipates about 23 percent more copper loss than the same winding at room temperature. This positive temperature coefficient creates a feedback path: more loss raises temperature, which raises resistance, which raises loss again, and the thermal design must guarantee this loop converges.

As operating frequency increases, the simple DC picture breaks down. The skin effect confines current to a surface layer one skin depth deep, and the proximity effect from the magnetic field of neighboring turns redistributes current still further, so the effective AC resistance can be many times the DC resistance. For an RF choke or a resonant-converter inductor switching at hundreds of kilohertz to several megahertz, the AC copper loss often dwarfs the DC term, and the conductor geometry, not just its cross-sectional area, governs the design.

Governing Equations

DC copper loss:
P_cu,DC = I_rms² × R_DC, R_DC = ρ × ℓ / A

Skin depth (copper):
δ = √(ρ / (π × f × μ)) ≈ 66 / √f_(Hz) mm ≈ 2.1 μm at 1 GHz

AC resistance and total loss:
R_AC = F_R × R_DC, P_cu = I_rms² × R_AC

Temperature correction:
R(T) = R₂₀ × [1 + 0.00393 × (T − 20)]

Where ρ = resistivity (≈ 1.68 × 10⁻⁸ Ω·m), ℓ = length, A = cross-section, μ = permeability, f = frequency, F_R = AC/DC resistance ratio (Dowell factor). Example: a wire with R_DC = 20 mΩ and F_R = 50 at 1 GHz presents R_AC = 1 Ω.

Loss Trade-Offs by Conductor Type

Conductor	Useful frequency	AC/DC ratio (typ.)	Relative cost	Best application
Solid round wire	DC to ~50 kHz	1 to 1.5×	Lowest	Line-frequency magnetics, DC chokes
Litz wire	50 kHz to ~3 MHz	1.05 to 1.3×	High	Resonant converters, induction heating
Foil / strap	10 kHz to ~1 MHz	1.2 to 3×	Medium	High-current low-turn windings
Tubular / pipe	1 MHz to ~100 MHz	near 1× (thin wall)	Medium	RF tank coils, high-power matching
Silver-plated conductor	100 MHz to 110 GHz	Surface-dominated	High	Microwave cavities, waveguide, RF lines

Common Questions

Frequently Asked Questions

How does copper loss differ from core loss in a transformer?

Copper loss is the I²R dissipation in the winding conductors and scales with the square of load current, so it is the load or variable loss. Core loss is the hysteresis and eddy-current dissipation in the magnetic core; it depends on peak flux density and frequency but is nearly independent of load current, so it is the no-load loss. Maximum efficiency occurs where copper loss equals core loss. You measure core loss with an open-circuit test at rated voltage and copper loss with a short-circuit test at rated current.

Why does copper loss increase at RF even though the conductor is unchanged?

At DC, current fills the whole cross-section. As frequency rises, the skin effect forces current into a surface layer about one skin depth thick: roughly 2.1 μm at 1 GHz and 0.66 μm at 10 GHz. A 1 mm wire that uses its full 0.79 mm² at DC effectively uses only a thin shell at microwave frequencies, so R_AC rises with the square root of frequency. Proximity effect from adjacent turns multiplies it further, turning 0.02 Ω of DC resistance into several ohms at 1 GHz.

How do litz wire and surface plating reduce copper loss?

Litz wire splits a conductor into many fine, individually insulated and transposed strands, each thinner than a skin depth, suppressing skin and proximity effects up to a few megahertz; above 2 to 3 MHz it becomes impractical and tubular conductors win. Silver plating helps because silver conducts slightly better than copper (6.3 vs 5.96 × 10⁷ S/m), and since RF current flows only in the skin, a few micrometres of plating carries nearly all of it. Surface roughness can still add 10 to 50 percent loss at millimetre-wave frequencies.

Related Terms

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