Transformer core and copper losses
Losses in a transformer are mainly divided into fixed core losses —caused by hysteresis and eddy currents— and variable winding losses due to Joule effect, which depend on the square of the load current. In a typical 500 kVA distribution transformer, total losses represent around 1.5 % of rated power, approximately split into 0.5 % core losses and 1 % copper losses at full load.
Core Losses (Iron)
Section titled “Core Losses (Iron)”Core losses of a 60 Hz transformer made with M4 grain-oriented silicon steel typically amount to 0.9 W / lb (1.98 W / kg) measured at 1.5 T induction. These losses are constant over the entire load range and are divided into two components: hysteresis losses and eddy current losses.
Hysteresis Losses
Section titled “Hysteresis Losses”They are caused by the energy required to reorient the magnetic domains of the ferromagnetic material in each cycle of alternating magnetization. The dissipated power is expressed as:
Ph = Kh · f · Bmax^n · V
| Variable | Description | SI Unit | Imperial Unit |
|---|---|---|---|
| Ph | Hysteresis loss power | W | W |
| Kh | Material hysteresis constant | J / (m³·Tⁿ) | BTU / (ft³·Gⁿ) |
| f | Operating frequency | Hz | Hz |
| Bmax | Maximum flux density | T | G (1 T = 10 000 G) |
| n | Steinmetz exponent (1.6–2.0 for silicon steels) | dimensionless | dimensionless |
| V | Core volume | m³ | ft³ |
The exponent n is usually taken as 1.6 for grain-oriented silicon steels, and the constant Kh varies between 0.001 and 0.003 W·s / (T¹·⁶·m³) (≈ 0.5 × 10⁻⁶ – 1.5 × 10⁻⁶ BTU·s / (G¹·⁶·ft³)).
Eddy Current Losses
Section titled “Eddy Current Losses”They are generated by the induced currents that circulate in the conductive core material when traversed by the alternating flux. The classic formula for this loss is:
Pe = Ke · f² · Bmax² · t² · V
| Variable | Description | SI Unit | Imperial Unit |
|---|---|---|---|
| Pe | Eddy current loss power | W | W |
| Ke | Eddy current constant | Ω⁻¹·m³·s² | Ω⁻¹·ft³·s² |
| f | Frequency | Hz | Hz |
| Bmax | Maximum flux density | T | G |
| t | Lamination thickness | m | ft (or mils) |
| V | Core volume | m³ | ft³ |
For M4 silicon steel with 0.27 mm (0.0106 in) lamination thickness and operating at 60 Hz / 1.5 T, the value of Ke is on the order of 5 × 10⁻⁵ Ω⁻¹·m³·s².
Copper Losses (Windings)
Section titled “Copper Losses (Windings)”In a 1000 kVA transformer with copper windings, the copper losses at full load represent approximately 1.2 % of the rated power, i.e., about 12 kW. This loss is proportional to the square of the load current and increases rapidly under overloads.
The Joule power dissipated in each winding is calculated using:
Pcu = I² · R
Where I is the RMS current flowing through the winding and R is the ohmic resistance of the conductor at operating temperature.
To include the effect of temperature on the resistance of copper or aluminum, the following is used:
R(T) = R₀ · [1 + α · (T – T₀)]
| Variable | Description | Typical value (Cu) | Typical value (Al) |
|---|---|---|---|
| α | Temperature coefficient | 0.00393 °C⁻¹ | 0.00403 °C⁻¹ |
| T₀ | Reference temperature | 20 °C / 68 °F | 20 °C / 68 °F |
Thus, a copper winding measuring 0.10 Ω at 20 °C (68 °F) reaches 0.14 Ω at 120 °C (248 °F), increasing losses by 40 %.
| Material | Resistivity at 20 °C | Density | α (at 20 °C) |
|---|---|---|---|
| Annealed copper | 1.68 × 10⁻⁸ Ω·m / 10.37 Ω·cmil / ft | 8960 kg / m³ / 559 lb / ft³ | 0.00393 °C⁻¹ |
| Hard aluminum | 2.65 × 10⁻⁸ Ω·m / 16.06 Ω·cmil / ft | 2700 kg / m³ / 169 lb / ft³ | 0.00403 °C⁻¹ |
Calculation of Total Losses
Section titled “Calculation of Total Losses”Total losses in a 2500 kVA transformer with no-load losses of 3.5 kW and load losses of 18 kW amount to 21.5 kW at full load, which represents 0.86 % of rated power. The general expression is:
Ptotal = Pcore + Pcopper
| Component | Dependency | Example value (2500 kVA) |
|---|---|---|
| Pcore (iron) | Constant (does not depend on load) | 3.5 kW / 4.69 hp |
| Pcopper at full load | ∝ I² (square of load current) | 18 kW / 24.14 hp |
| Ptotal at full load | Direct sum | 21.5 kW / 28.83 hp |
If the transformer operates at 50 % of its rated load, the copper loss is reduced to 25 % of the full load value (4.5 kW), while the core loss remains at 3.5 kW, resulting in a total of 8.0 kW (0.64 % of delivered power).
Factors Affecting Losses
Section titled “Factors Affecting Losses”The resistivity of copper at 20 °C is 1.68 × 10⁻⁸ Ω·m (10.37 Ω·cmil / ft); any increase in temperature raises this value and increases Joule losses. The main factors are summarized below:
| Factor | Effect on Pcore | Effect on Pcopper |
|---|---|---|
| Frequency | Ph ∝ f ; Pe ∝ f² | Negligible up to ≈400 Hz; then increases due to skin effect |
| Maximum induction (Bmax) | Ph ∝ Bmax¹·⁶ ; Pe ∝ Bmax² | No direct influence |
| Lamination thickness | Only affects Pe (Pe ∝ t²) | Not applicable |
| Temperature | Very slight (decreases Ph, increases Pe due to resistivity variation) | Increases R → increases Pcu (+0.39 % / °C in copper) |
| Load | Does not vary | Pcu ∝ I² → proportional to square of load |
Reduction of Losses in Transformers
Section titled “Reduction of Losses in Transformers”Using grain-oriented silicon steel with 0.23 mm (0.009 in) lamination thickness allows reducing eddy current losses by up to 40 % compared to laminations of 0.35 mm (0.014 in). The reduction strategies are grouped according to the nature of the loss:
| Method | Affects | Typical reduction achieved |
|---|---|---|
| Use grain-oriented silicon steel | Pcore (hysteresis + eddy current) | 20 %–30 % compared to non-oriented steel |
| Reduce lamination thickness (0.23 mm / 0.009 in) | Pcore (eddy current) | Up to 40 % compared to 0.35 mm / 0.014 in |
| Increase conductor cross-section | Pcopper | Proportional to reduction in R (e.g., 25 % more cross-section → 20 % less Pcu) |
| Use copper instead of aluminum | Pcopper | ≈ 40 % less losses for same geometry |
| Improve cooling (lower operating temperature) | Pcopper | ~2 % less Pcu for each 5 °C reduction |
Comparison Between Core and Copper Losses
Section titled “Comparison Between Core and Copper Losses”| Characteristic | Core losses (iron) | Copper losses (windings) |
|---|---|---|
| Physical nature | Hysteresis and eddy currents | Joule effect (I²R) |
| Load dependency | Constant over entire load range | Proportional to square of current (I²) |
| Frequency | Ph ∝ f , Pe ∝ f² | Independent up to ≈400 Hz; at high frequency increases due to skin effect |
| Materials affected | Magnetic core (silicon steel, ferrite, amorphous) | Conductors (copper or aluminum) |
| Typical value at full load (60 Hz, 1.5 T) | 0.9 W / lb (1.98 W / kg) | Depends on design: ≈1 %–2 % of rated power |
| Standard measurement | No-load test (open circuit) | Short-circuit test |
| Evolution with operating time | Practically constant if voltage and frequency are stable | Increases with contact degradation and temperature rise |
Frequently Asked Questions (FAQ)
Section titled “Frequently Asked Questions (FAQ)”How are copper losses in a transformer calculated?
Section titled “How are copper losses in a transformer calculated?”Joule’s law Pcu = I²·R is used, where I is the RMS current flowing through each winding at a given load and R is the conductor resistance at operating temperature. If high accuracy is required, the resistance is corrected for temperature using the material’s temperature coefficient α.
Why are core losses considered constant?
Section titled “Why are core losses considered constant?”Because the applied voltage and line frequency usually remain constant; therefore, the maximum flux density and frequency do not vary, and hysteresis and eddy current losses depend only on those parameters, not on the load current.
Which material reduces core losses the most?
Section titled “Which material reduces core losses the most?”Low-thickness grain-oriented silicon steels (M3, M4) offer the best cost-performance trade-off. For high frequencies, ferrites are used, and in very high-efficiency applications, amorphous cores are used, reducing losses by up to 70 % compared to conventional silicon steel.
Does frequency affect copper losses?
Section titled “Does frequency affect copper losses?”In power transformers (50/60 Hz) the effect is negligible. When frequency increases above about 400 Hz, skin and proximity effects increase the effective conductor resistance, raising I²R losses.
How does temperature affect transformer losses?
Section titled “How does temperature affect transformer losses?”Temperature increases copper resistivity (≈0.39 % / °C) and, consequently, copper losses for the same current. In the core, the effect is contradictory: it slightly decreases hysteresis loss but may increase eddy currents by changing the steel resistivity.
Can losses in a transformer be completely eliminated?
Section titled “Can losses in a transformer be completely eliminated?”No. Core losses are inherent to the alternating magnetization of the ferromagnetic material, and copper losses are unavoidable as long as current flows through a conductor with non-zero resistance. They can be minimized with optimized designs and materials, but never eliminated.
Sources Consulted
Section titled “Sources Consulted”- engineeringtoolbox.com: https://www.engineeringtoolbox.com/voltage-drop-d_1550.html
- electrical4u.com: https://www.electrical4u.com/hysteresis-eddy-current-iron-or-core-losses-and-copper-loss-in-transformer/