Advanced: fields, flux, losses, and real inductors
An ideal inductor has only inductance. A real one also has winding resistance, core loss, parasitic capacitance, leakage flux, temperature dependence, nonlinear permeability, and finite insulation strength. Its behavior therefore depends on current, frequency, temperature, waveform, and physical layout.
Flux linkage and inductance
A useful definition of inductance is the ratio of flux linkage to current:
λ = N Φ
- L is inductance.
- λ is flux linkage.
- N is the number of turns.
- Φ is magnetic flux through one turn.
For a linear magnetic structure, doubling turns tends to increase flux linkage strongly; many simple magnetic-circuit approximations make inductance proportional to the square of the turn count.
Magnetic circuits
Magnetic circuits are often analyzed using an analogy to Ohm's law:
Here NI is magnetomotive force and ℜ is magnetic reluctance. For a uniform core:
This analogy is useful but imperfect because magnetic materials are nonlinear, have hysteresis, and may saturate.
Inductor voltage
For a nonlinear core, L itself changes with current. Designers may distinguish between apparent, secant, incremental, and differential inductance depending on what the circuit calculation requires.
Stored magnetic energy
In gapped magnetic structures, a large portion of the usable energy is stored in the magnetic field of the air gap rather than in the high-permeability core. The gap also reduces effective permeability and allows greater DC bias before saturation.
Faraday's law
The minus sign expresses Lenz's law: the induced voltage has a polarity that opposes the change responsible for it.
Core saturation
Magnetic flux density B does not increase indefinitely in proportion to field strength H. As the material approaches saturation, incremental permeability falls and the coil's inductance decreases. In a switching converter this can cause current slope to rise rapidly:
If L collapses, di/dt increases. A control circuit that was stable at normal inductance may then encounter excessive peak current.
Hysteresis and core loss
Magnetic materials do not follow exactly the same B-H path while magnetizing and demagnetizing. The resulting hysteresis loop represents energy lost each cycle. Eddy currents and other material processes add frequency-dependent loss. Manufacturers therefore characterize core loss as a function of frequency, flux swing, waveform, and temperature.
Copper loss: DC and AC
At higher frequency, current crowds toward the surface of a conductor (skin effect) and is redistributed by magnetic fields from nearby conductors (proximity effect). Effective AC resistance can therefore be much larger than DCR.
Litz wire divides a conductor into many individually insulated strands arranged to share current more evenly. It is useful in selected high-frequency, high-current magnetics, although strand size and construction must be matched to frequency.
Parasitic capacitance and self-resonance
Adjacent turns, winding layers, terminals, and the circuit board form unwanted capacitance. A first approximation treats the coil as inductance in parallel with a small capacitance:
Below self-resonance, the part is normally inductive. At resonance its impedance can peak. Above resonance, capacitive behavior may dominate.
Quality factor
For a simplified series-loss inductor this becomes approximately Q = ωL/R. Q is frequency dependent because both reactive and loss terms change with frequency.
Mutual inductance
Two magnetic circuits can exchange flux:
The coupling coefficient k ranges from nearly zero for weakly coupled coils toward one for tightly coupled windings. Mutual inductance is central to transformers and coupled inductors, and is also an important source of unwanted crosstalk.
Inductor impedance model
A simple broadband model may include series DCR and inductance, with a parallel capacitance and frequency-dependent loss. At still higher frequencies, package, layout, and distributed effects make lumped-element models less accurate, so measured impedance or S-parameter data may be preferable.
Power-inductor selection at operating conditions
Inductance should be checked at the actual DC bias current, temperature, and test frequency. Two inductors with the same nominal L can behave quite differently because one saturates sharply, another soft-saturates, one has lower DCR, or one has much greater AC core loss.
| Parameter | Why it matters |
|---|---|
| L versus current | Shows saturation and usable energy-storage behavior. |
| DCR | Sets DC copper loss and voltage drop. |
| Core loss | Can dominate heating at high frequency and ripple. |
| IRMS | Relates current to thermal rise under stated conditions. |
| Isat | Indicates loss of inductance with peak current. |
| Q | Indicates loss relative to reactance in RF and resonant circuits. |
| SRF | Defines the region where parasitic capacitance changes behavior. |
| Temperature | Affects resistance, permeability, loss, and reliability. |
EMI and field control
A switching inductor can radiate a time-varying magnetic field into nearby circuitry. Shielded packages, orientation, board spacing, smaller current-loop area, and careful placement reduce coupling. Sensitive magnetic sensors, high-gain analog circuits, and other inductors deserve extra clearance.
Sources and further reading
- Coilcraft — Introduction to inductor specifications
- Coilcraft — Basics of inductor selection
- Coilcraft — Measuring self-resonant frequency
- For actual component limits, use the exact manufacturer's L-versus-current, impedance, loss, thermal, and reliability data.