Technology / Technical guide
Magnetic Energy and Coenergy: Fixed Variables in Torque
Magnetic energy and coenergy are related descriptions of an electromagnetic state. They are not two separate stores of energy that can be added together. Their usefulness depends on which electrical variable is held fixed while a mechanical coordinate changes.
Choosing the fixed variable determines the torque derivative.
A reversible single-excitation example shows why a constant-current coenergy derivative and a constant-linkage energy derivative give the same mechanical result.
01 / Model boundary
Start with one electrical excitation and one angle.
Consider an ideal lossless magnetic system with one winding, no permanent-magnet bias and a mechanical angle θ.
Consider an ideal lossless magnetic system with one winding, no permanent-magnet bias and a mechanical angle θ. Assume a reversible, single-valued relationship between current i and flux linkage λ at each fixed angle. Hysteresis and additional excitation sources are outside this initial example.
Take positive electromagnetic torque Te to do positive mechanical work Te dθ. Electrical work entering the winding is i dλ. The balance is i dλ = dW + Te dθ, where W is stored magnetic energy. This declaration fixes the sign interpretation before derivatives are introduced.
02 / Two descriptions
Integrate along the same fixed-angle state relationship.
With a common zero reference at the unexcited state, magnetic energy is W(λ,θ) = ∫ i dλ from zero to the stated linkage at fixed θ.
With a common zero reference at the unexcited state, magnetic energy is W(λ,θ) = ∫ i dλ from zero to the stated linkage at fixed θ. Coenergy is W′(i,θ) = ∫ λ di from zero to the stated current at that same fixed angle.
For this reversible zero-biased model, W′ = iλ − W. Both quantities use joules. COMSOL explains the complementary energy descriptions and their virtual-work conditions. Coenergy is a useful mathematical representation; counting W and W′ as two independent physical stores would double-count the state.
03 / Torque derivatives
Hold linkage or current fixed, not both by assumption.
From the declared work balance, Te = −(∂W/∂θ)λ.
From the declared work balance, Te = −(∂W/∂θ)λ. The subscript means linkage is held fixed during the partial derivative. Using the complementary representation gives Te = +(∂W′/∂θ)i, with current held fixed. The signs accompany the selected positive mechanical work direction.
These expressions do not instruct a controller to maintain a variable during an actual transient. They compare neighbouring mathematical states for virtual work. Taking a total derivative along a changing operating trajectory adds electrical-state changes and is not the same operation.
04 / Linear example
Equivalent derivatives give one torque.
For a passive linear example λ = L(θ)i, W′ = ½L(θ)i² and W = λ²/[2L(θ)].
For a passive linear example λ = L(θ)i, W′ = ½L(θ)i² and W = λ²/[2L(θ)]. At fixed current the torque becomes Te = ½i² dL/dθ. At fixed linkage, differentiating the energy and applying the minus sign gives the identical result after λ = Li is substituted.
Hypothetically let i = 2 A and the local derivative dL/dθ = 0.001 H per radian. The model gives 0.002 N m in the positive angle direction. These invented quantities illustrate the derivative; they identify no generator inductance, current or torque capability.
05 / Beyond the example
Nonlinearity and permanent magnets change the state description.
In a nonlinear but reversible zero-biased system, the integrals remain distinct areas under the state relationship and are generally unequal.
In a nonlinear but reversible zero-biased system, the integrals remain distinct areas under the state relationship and are generally unequal. Hysteresis instead introduces history and dissipation, so a simple single-valued stored-energy function cannot represent every effect without additional treatment.
A permanent-magnet machine also requires its bias field and other energy sources to be included correctly. Zero winding current does not establish zero magnetic energy or zero cogging torque. The saliency guide describes one model contribution, rather than reducing the whole machine to this coil example.
06 / Interpretation scope
Torque differentiation and power accounting answer different questions.
The instantaneous-energy guide accounts for power entering, leaving and changing storage within an electrical boundary.
The instantaneous-energy guide accounts for power entering, leaving and changing storage within an electrical boundary. This page derives a mechanical interaction from changes in the magnetic state. Te times mechanical angular speed relates torque to mechanical power, but does not replace the energy derivative.
FD14-50KW’s confirmed PMSG architecture supplies no actual magnetic-energy function, saliency map or cogging result. The example explains fixed variables and signs. It supplies no torque-control procedure, magnetic design values or performance claim for a MARSTON machine.
Related technical guides: PMSG Saliency: Directional Inductance and Reluctance Torque · Generator Armature Reaction: How Load Current Changes the Field · Two-Inertia Torsional Vibration in Wind Generators.
General background: COMSOL: Energy, Coenergy and Virtual Work · MIT OpenCourseWare: Electromagnetic Forces. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| Energy W | ∫ i dλ at fixed θ | Reversible single excitation and declared zero reference. |
| Coenergy W′ | ∫ λ di at fixed θ | Complementary description, not another independent store. |
| Fixed linkage | Te = −(∂W/∂θ)λ | Positive mechanical work convention stated. |
| Fixed current | Te = +(∂W′/∂θ)i | Partial derivative differs from operating-trajectory derivative. |
| Linear limit | W = W′ = ½Li² | Zero bias and a linear relation through the origin. |
Engineering questions.
Can energy and coenergy be added as separate stores?
No. They describe the same magnetic state under complementary variables.
Why do the torque formulas have different signs?
The fixed variable and complementary state description differ under the declared work convention.
Does fixed current mean a real machine must use current control?
No. It is a condition of the virtual-work derivative.
Are energy and coenergy always equal?
They are equal in the stated zero-biased linear model, but generally differ for nonlinear response.
Does zero current imply no permanent-magnet field energy?
No. The simple coil reference does not represent permanent-magnet bias.
Is the torque example a MARSTON measurement?
No. Its current and inductance derivative are explicitly hypothetical.
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Magnetic energy and coenergy are related descriptions of an electromagnetic state. Share the model and applicable operating or design information for an engineering discussion.