How Much Do Magnet Temperature Coefficients Actually Affect Motor Efficiency and Torque Ripple?

A motor's torque and efficiency at 120 °C rarely match the numbers measured on the bench at 25 °C. Winding resistance, the drive, and the lamination stack all move with temperature as well. A large and often under-accounted part of the torque-constant change is the permanent magnet: remanence and intrinsic coercivity shift with temperature, and they do not shift at the same rate. When a design team has to choose between an NdFeB grade that looks strong at room temperature and a heavy-rare-earth NdFeB or SmCo grade that costs more, the decision hinges on two coefficients — ${\alpha }_{Br}$, the reversible temperature coefficient of remanence, and ${\beta }_{Hcj}$, the reversible temperature coefficient of intrinsic coercivity — and on whether the hot operating point is still above the knee. This article is a selection comparison for that decision, not a material catalog and not a report of a particular tested machine.

The decision in front of you is which magnet grade holds your torque band across the operating temperature swing

Every permanent-magnet motor is designed around a flux level. That flux sets the torque constant $K_t$ and the back-EMF constant $K_e$. In SI units the two are numerically equal when the same peak or RMS convention is used; in other unit systems they differ by a constant. When the magnet heats and stays on a reversible recoil line, remanence falls, air-gap flux falls, and $K_t$ falls with it. For the same shaft torque the drive must then supply more current. Two customer-visible effects follow, and they should not be collapsed into one:

• Efficiency falls because copper loss scales with current squared, and because copper resistivity itself rises with winding temperature. The magnet coefficient controls only the first of those terms.
• Torque ripple does not have a universal hot-condition increase. In the linear regime, magnet flux harmonics scale with remanence, and cogging torque typically scales closer to the square of air-gap flux, so cogging amplitude usually falls as the magnet warms. Ripple becomes a temperature problem when the hot point nears the knee, when the extra current changes saturation, or when a compensation table tuned on a cold bench no longer matches the hot spectrum.

An illustrative servo axis makes the decision concrete. It is not a reported field failure. The axis runs a long duty and the magnet soaks from ambient toward roughly 100–120 °C. Cold-bench ripple meets the spec. Hot, $K_t$ has fallen, current for the same torque has risen, efficiency is worse, and a ripple compensation tuned at 20 °C is mistuned. Blaming the controller or the encoder first is a common misdiagnosis. The magnet's reversible loss, and a load line that was never checked at the hot current, are the items to close before the control loop is rewritten.

The decision the rest of this article addresses is: **given a defined magnet hot-spot temperature and a defined hot torque and ripple specification, which magnet family and which operating point keep both $K_t$ and the ripple spectrum inside tolerance — and what does that choice cost in material, mass, and thermal margin?**

The right comparison axis is coefficient-and-condition, not just grade name

A table of $B_r$, $H_{cj}$, $B{H}_{max}$, and a catalog maximum temperature says little about duty-cycle behavior. It omits the coefficients, the knee relative to the load line, and the thermal path. Catalog letter codes (N, SH, UH, EH, and the SmCo temperature class) are manufacturer operating-temperature classes under that manufacturer's stated permeance. They are not the knee temperature on your load line. Both numbers have to be checked.

The bands below are typical commercial catalog ranges for these material families. They are not measurements on a motor, and they are not specifications of any AIC Engineering product. Use the vendor curve for the actual grade and for the temperature interval you will run. Coefficients are average slopes over a stated interval; they are not constants to be extrapolated far outside that interval.

Read the table as a decision map, not a spec sheet. The useful split is this: **families with similar ${\alpha }_{Br}$ and different hot $H_{cj}$ fail differently.** A standard NdFeB grade that only loses flux reversibly recovers when it cools. A grade that crosses the knee loses flux permanently, and the ripple complaint becomes a warranty return. Heavy-rare-earth NdFeB is bought for that knee margin. It is not bought for a large reduction in reversible $K_t$ drift. SmCo is the option when the reversible drift itself, over a wide $\Delta T$, is the constraint.

A one-percent-per-hundred-degrees check stops a common extrapolation. In the linear model, a 0.01 %/°C difference in ${\alpha }_{Br}$ changes flux by about 1% over a 100 °C rise. That cannot overturn an 8% cold-$B_r$ advantage inside the NdFeB family. A ranking reversal on hot flux is plausible only against a much lower-$|{\alpha }_{Br}|$ family such as SmCo, and even then it is only a few percent unless the cold gap was already small. Evaluate the magnet at the hot corner, including the hot current, and do not treat a room-temperature $B_r$ sort as the answer.

How a coefficient error becomes a torque error and a copper-loss error

Two relations size the reversible part of the problem. They are not a motor loss model. First, the linear reversible change of remanence:

Here ${\alpha }_{Br}$ is in per °C, not in %/°C. A catalog value of −0.11 %/°C is ${\alpha }_{Br}=-0.0011{/}^{\circ }C$. Inserting −0.11 into the formula is a unit error, not a conservative estimate. The coefficient is an average over the vendor's stated interval. Do not reuse a 20–100 °C slope, unchanged, at 200 °C.

Second, for a magnet-torque-dominated machine still above the knee, with fixed geometry and armature reaction neglected, torque constant follows flux. At fixed winding resistance, copper loss for a fixed shaft torque then follows the current:

The second expression holds resistance constant. It also assumes torque is essentially magnet torque (surface-magnet or $I_d=0$ control). In an interior-magnet machine the reluctance term does not scale with $B_r$, so $K_t$ does not fall in the same proportion.

What the arithmetic says, and what it does not. Take ${\alpha }_{Br}=-0.11\%{/}^{\circ }C$ from 20 °C to 120 °C. Remanence falls by 11%: $1+(-0.0011)(100)=0.89$. If $K_t$ falls by 11%, current for the same shaft torque rises by $1/0.89-1\approx 12\%$. At unchanged resistance, $I^{2}R$ rises by about 26%. That 26% is the magnet-driven current term only. It is not a measured efficiency, and it is not an efficiency-class result. Copper resistivity rises by roughly 0.4 %/°C of winding temperature, which is a separate and often larger term, and iron loss usually falls as flux falls. Those terms can move total loss either way. IEC 60034-30-1 efficiency classes apply to line-operated AC motors under that standard's test method; they are not a servo ripple limit and they are not determined by this two-line estimate.

Two further limits belong in the same pass. The extra current increases armature reaction, so the hot load line is not the cold load line. A knee check at cold current is non-conservative. And ${\beta }_{Hcj}$ does not appear in the flux equation at all. Together with absolute $H_{cj}$, it decides whether the linear equation is still valid at the hot corner. Using ${\alpha }_{Br}$ alone, near the knee, is the error that turns a reversible estimate into a field demagnetization.

For ripple, the coefficient enters only as a scale factor while the magnet stays linear: every magnet-flux harmonic drops with $B_r$, and geometric phase is set by pole arc, skew, and slotting, not by ${\alpha }_{Br}$. Relative harmonic content and phase shift when saturation changes, when local demagnetization distorts the magnet, or when a cold current-harmonic injection is left in place. Specify and measure residual ripple at the hot corner. The cold-bench figure is the wrong acceptance number if the customer limit applies at temperature.

A recommended design window for temperature-driven torque stability

The following is a starting procedure for a motor with a defined hot soak. It is not a product specification, not a code limit, and not a universal flux-loss budget.

• Define magnet hot-spot temperature from the thermal model, not from ambient or housing temperature. In an enclosed machine the magnet can run well above the housing or coolant temperature. The offset is design-specific. Do not assume a fixed delta.
• Set the reversible-loss budget from the hot specification, not from a fixed percentage of cold flux. Compute hot $B_r$ from the vendor ${\alpha }_{Br}$ over the actual $\Delta T$, then accept the grade only if hot $K_t$ and hot ripple still meet the spec at the current required. Positioning duty is usually tighter than fan or pump duty, because ripple closes the control loop; the allowable drop is still the customer's number, not a default band.
• Keep the load line above the knee at the hot corner, at the hot current, with a margin you can state as a number. The margin is the distance between the worst-case permeance coefficient and the knee on the hot demagnetization curve, using minimum $H_{cj}$ from the production tolerance, not the nominal catalog point. The factor itself is application-dependent. "Comfortable" is not a margin.
• Write the ripple specification at the hot corner. If the limit applies at temperature, the cold-bench target has to be set with the hot spectrum in mind, including any compensation that will actually ship.
• **If $\Delta T$ is large — on the order of 100 °C — and cooling is limited, put SmCo or a high-$H_{cj}$ NdFeB grade on the shortlist before tooling. Which of the two belongs there depends on whether the constraint is reversible drift or knee margin. Learning that after the tool is cut is the expensive version of this calculation.
• Treat magnet temperature as a design output.** A rotor with poor conduction to the shaft or housing will run a hotter magnet than the average rotor temperature used in a lumped model. The coefficient correction cannot repair a low magnet temperature.

Where this approach fails, and where it should not be used at all

Temperature coefficients describe reversible change of remanence and coercivity while the operating point remains on a recoil line above the knee. They are the wrong tool, or an incomplete one, in the following cases.

• Irreversible loss past the knee. Once the hot operating point crosses the knee, flux is lost permanently. No ${\alpha }_{Br}$ maps that loss back when the machine cools, because the coefficient describes the original remanence, not a demagnetized magnet. A decision made on coefficient values alone, near the knee, is unsafe.
• Fault demagnetization. A short circuit or a drive fault at high temperature can drive a marginal magnet past the knee during the fault. The design variable is hot coercivity margin under the fault current, not remanence drift. The load line must be the fault load line.
• Extra armature reaction from the torque-recovery current. Holding torque as $K_t$ falls moves the operating point toward the knee. A cold-current knee check understates that risk.
• Eddy-current heating inside the magnet. In high electrical frequency or high slot-count machines, magnet eddy loss can raise local magnet temperature above the bulk rotor temperature. The coefficient is then being applied at the wrong temperature, and the loss is understated.
• Ripple that is not magnet ripple. Inverter dead time, current-loop bandwidth, and mechanical resonance are not cured by a magnet-family change. Identify the ripple source before spending on material.
• Little or no thermal swing. A well-cooled, lightly loaded machine that moves only about 20 °C has no meaningful coefficient problem. Specifying SmCo there buys stability the duty cycle will not use.
• Binder-limited magnets. For bonded NdFeB, the polymer system has its own temperature rating and its own mechanical drift. A magnetic coefficient table does not clear the part for continuous hot duty.
• Retention, coating, and adhesive limits. A grade can be magnetically acceptable while the coating, bond line, sleeve, or containment was only qualified cold. Hot shear strength, interference fit, and overspeed retention are separate checks. They are not given by ${\alpha }_{Br}$.
• Production spread. Nominal $B_r$ and $H_{cj}$ are not the parts you will receive. Knee margin and hot $K_t$ should be stated at the vendor's minimum, not at the catalog typical.

How AIC Engineering supports this design window in practice

AIC Engineering works this class of problem from the magnetic circuit outward, so the hot-corner operating point is on the table before tooling is committed. The capabilities that bear on temperature-driven torque and efficiency are:

• Magnetic-circuit and magnetic-application structural design — load line, knee margin, and hot-corner flux are set with the thermal path, not from a room-temperature catalog sort.
• Permanent-magnet components for special motors — multi-pole rings, radially oriented rings, and Halbach arrays, where the rotor needs a defined pole pattern or flux distribution to meet the flux and ripple targets. The array name does not cancel ${\alpha }_{Br}$. Temperature stability still comes from grade, load line, and thermal path.
• Permanent-magnet product quality inspection — magnetic and dimensional inspection so the as-built magnet can be compared with the design intent. The hot-corner check itself has to be in the design-verification plan; inspection does not replace the hot load-line calculation.
• Rapid prototyping in 3 to 7 days — a short prototype loop so magnet temperature, torque, and ripple can be measured on the duty cycle before the tool is cut.

For an engineer auditing a design, the review list is:

1. Magnet hot-spot temperature taken from the thermal model, not assumed from ambient or housing temperature.
2. ${\alpha }_{Br}$ and ${\beta }_{Hcj}$ taken from the vendor's data for the actual temperature interval, not from a room-temperature rule of thumb, and not extrapolated past that interval.
3. Load line plotted on the demagnetization curve at the hot corner and at the hot current, including the fault current, with the margin stated as a number at minimum $H_{cj}$.
4. Irreversible-loss check done for the worst-case field event at temperature.
5. Magnet eddy-current loss estimated separately from bulk rotor temperature.
6. Torque ripple specified and measured at the hot corner, not only on a cold bench. Cold compensation, if used, rechecked hot.
7. Thermal path from magnet to housing assessed, and coating, adhesive, and retention checked at the same hot-spot temperature.
8. Hot $K_t$ and knee margin stated with production tolerance, not at the nominal catalog point.

Next steps

1. Measure before you redesign. Instrument a representative build and record magnet temperature, torque, and ripple across a full duty cycle. One clean data set settles most coefficient arguments.
2. Re-plot the load line at the hot corner and at the hot current for the grade you are using, and state the knee margin as a number at minimum $H_{cj}$. If you cannot state it, you do not have a margin.
3. Compare alternatives on total cost, on the axis that is actually binding. A high-$H_{cj}$ NdFeB grade is the comparison when the knee is the risk. A lower-$|{\alpha }_{Br}|$ family is the comparison when reversible drift over a wide $\Delta T$ is the risk. Include magnet material, rotor mass, and the efficiency and ripple you must deliver.
4. Bring the thermal and magnetic models into one conversation. Visit https://www.aicmagnetics.com to request a free consultation and discuss a custom magnetic circuit design and rapid prototyping program built around your actual duty cycle and temperature window.

References

1. IEC 60034-1, Rotating electrical machines — Part 1: Rating and performance. Relevant as the rating and temperature-rise framework for machine performance. It does not define magnet temperature coefficients.
2. Furlani, E. P., Permanent Magnet and Electromechanical Devices: Materials, Analysis, and Applications, Academic Press. Background for operating points, load lines, and demagnetization curves.
3. Gieras, J. F., Permanent Magnet Motor Technology: Design and Applications, CRC Press. Background for cogging, ripple sources, and temperature effects in PM machines.
4. Krishnan, R., Permanent Magnet Synchronous and Brushless DC Motor Drives, CRC Press. Background for the torque-constant, back-EMF, and $I^{2}R$ relations used, under the assumptions stated above.
5. Coey, J. M. D., Magnetism and Magnetic Materials, Cambridge University Press. Background on the temperature dependence of remanence and coercivity in rare-earth permanent magnets. The table bands are typical commercial catalog ranges, not values taken from this book as a datasheet.