Sam Ben-Yaakov in this video explains definitions of ‘Inductance’, ‘AC inductor’, ‘DC inductor’ and needs for an air gap. He is also correcting some common misconceptions.
Understanding Inductance, AC Inductors and DC Inductors in Practical Power Electronics
Inductors are fundamental energy‑storage elements in power electronics, but their definitions and classifications are often presented in a way that confuses practicing engineers. This article clarifies what inductance actually is, how different definitions arise in nonlinear magnetic cores, and why the distinction between “AC inductors” and “DC inductors” as commonly presented is misleading and can result in oversized or incorrectly designed magnetics.
The discussion follows and expands on a short lecture by Prof. Sam Ben‑Yaakov, reframing it for design engineers and component purchasers working with real inductor parts in switched‑mode power supplies and related applications.
Inductance: From Faraday’s Law to Practical L
The starting point is Faraday’s law, which relates the induced voltage in a winding to the time variation of magnetic flux.
Formally, inductance can be defined in two ways that are often taught as separate concepts but actually stem from the same physical law:
- Flux‑based definition:
- L=N⋅ΦIL = \dfrac{N \cdot \Phi}{I}
- NN is the number of turns, Φ\Phi is the magnetic flux in the core, and IIIis the current through the winding.
- Terminal‑based (state‑equation) definition:
- v=Ldidtv = L \dfrac{\mathrm{d}i}{\mathrm{d}t}
- Here, LL is treated as the proportionality constant between voltage and the current derivative.
In a simple single‑winding inductor with a ferromagnetic core of permeability μ\mu, cross‑section area AeA_{\mathrm{e}} and magnetic path length lel_{\mathrm{e}}, the flux can be written as:
- Φ=μ⋅Ae⋅N⋅Ile\Phi = \mu \cdot A_{\mathrm{e}} \cdot \dfrac{N \cdot I}{l_{\mathrm{e}}}
Substituting this into Faraday’s law and rearranging shows that the inductance can be expressed as:
- L=N2⋅Ae⋅μleL = \dfrac{N^{2} \cdot A_{\mathrm{e}} \cdot \mu}{l_{\mathrm{e}}}
This expression is key for design engineers: it shows that inductance is determined by the core geometry (Ae,leA_{\mathrm{e}}, l_{\mathrm{e}}), the number of turns NN, and the effective permeability μ\mu of the magnetic path.
Nonlinear Cores and Multiple “Inductances”
Real ferromagnetic cores (ferrites, iron powder, etc.) do not have a constant permeability. Their BB–HH curve is nonlinear and saturates at higher flux densities. This nonlinearity implies that μ\mu depends on excitation level, so the apparent inductance depends on current.
From this, at least two useful inductance concepts arise:
- Total (or large‑signal) inductance:
- Based on total permeability and the ratio N⋅Φ/IN \cdot \Phi / I at a specific operating point.
- Corresponds closely to the textbook definition L=NΦ/IL = N \Phi / I when evaluated at a given current.
- Differential (local, small‑signal) inductance:
- Defined from the slope of the II–Φ\Phi (or BB–HH) curve at a given operating point.
- Can be written as:
- Ldiff=dΦdI⋅NL_{\mathrm{diff}} = \dfrac{\mathrm{d} \Phi}{\mathrm{d} I} \cdot N
- This is the inductance that relates a small AC perturbation to the resulting voltage at a given DC bias.
In practice, manufacturers typically provide “inductance versus DC current” curves that implicitly represent the differential (small‑signal) inductance, even if this is not explicitly stated. The usual measurement method is:
- Inject a DC bias current through the inductor.
- Superimpose a small AC excitation.
- Measure AC voltage and current and compute LL from the small‑signal response.
For design, especially in switch‑mode power supplies, the most relevant quantity is usually this differential inductance at the intended operating current rather than a single “nominal” inductance at zero current.
Summary of inductance notions
| Inductance type | Definition perspective | Typical use in design |
|---|---|---|
| Total inductance | L=N⋅ΦIL = \dfrac{N \cdot \Phi}{I} | Conceptual analysis, low‑current or linearized models |
| Differential L | L=dΦdI⋅NL = \dfrac{\mathrm{d}\Phi}{\mathrm{d}I} \cdot N | Small‑signal AC behavior around a DC operating point |
There is therefore no single unique inductance value for a nonlinear magnetic component. What matters is clearly specifying which definition is used and at which operating point, especially when comparing parts or interpreting datasheets.
Why “AC inductor” vs “DC inductor” Is Misleading
In engineering discussions (including social media, videos, and even some papers) one often encounters the idea that:
- “DC inductors” are components that handle a DC current with superimposed AC ripple and therefore need an air gap.
- “AC inductors” carry only AC current with zero DC component and supposedly do not require an air gap and follow a different design approach.
This distinction is fundamentally flawed.
The energy stored in an inductor is given by:
- W=12LI2W = \dfrac{1}{2} L I^{2}
For a given inductance and current, the stored energy scales with the square of the current. The physical size of an inductor is therefore closely related to the maximum energy it must store, not to whether the current is purely AC or has a DC component.
If a core is used without an air gap, its high initial permeability means that the magnetic flux density will rise quickly with current, and the core will reach its saturation limit at relatively low current. Because ferromagnetic materials are nonlinear, the usable BBB range is limited, which restricts how much current (and thus energy) can be handled before saturation.
Introducing an air gap reduces the effective permeability of the magnetic path. This has several consequences:
- For the same flux density limit, the allowable magnetizing field HH (and thus current) is much higher.
- The core can store more energy for a given size before reaching saturation.
- The inductance decreases for a given number of turns, but the usable current and energy increase dramatically.
For a given required energy storage, an inductor with a gapped core will therefore be smaller than a core without a gap. The presence or absence of a DC component in the current does not change the fact that energy storage is proportional to I2I^{2}.
The conclusion for design engineers is that any inductor required to store significant magnetic energy—whether carrying DC plus ripple or a large AC current with zero average—will generally benefit from a gapped core design. Treating so‑called “AC inductors” as if they can be built without an air gap simply because their average current is zero is a serious design mistake that tends to produce bulky components or lead to unintended saturation.
DC‑biased inductors (often labeled “DC inductors”)
These are inductors where a DC current with superimposed ripple flows through the component:
- Output chokes in buck, boost and buck‑boost converters.
- PFC boost inductors in power factor correction stages.
- Input inductors in many DC/DC topologies with continuous conduction.
For these components:
- A gapped core is almost always used to avoid saturation at the peak DC current.
- The relevant inductance is the differential inductance at the maximum operating current.
- Inductance‑versus‑current curves are essential to verify that the inductance does not collapse excessively at the design current.
Inductors with predominantly AC current (often labeled “AC inductors”)
These include:
- Inductors in resonant tanks, where the average current may be close to zero but the peak AC current is high.
- Filter inductors in AC filters where the waveform is symmetrical and has negligible DC bias.
Even though the average current is zero, the peak current (and hence stored energy) can be large:
- A gapped core is usually still beneficial or necessary if space is limited and the inductor must store energy.
- Designing these parts as if the absence of DC current made an air gap unnecessary will typically result in saturation or require an unnecessarily large core.
In both categories, the energy storage requirement and peak current drive the decision about core gapping and size. The DC versus AC label is much less important than the real operating current waveform and the permissible flux density.
Technical Highlights for Design and Purchasing
While the original lecture does not present a specific catalog part or detailed electrical ratings, it conveys several important technical points that are directly relevant when specifying and evaluating inductors.
Structural parameters and inductance
The approximate inductance of a simple core and winding configuration can be expressed as:
- L=N2⋅Ae⋅μleL = \dfrac{N^{2} \cdot A_{\mathrm{e}} \cdot \mu}{l_{\mathrm{e}}}
This highlights the main design knobs:
- Increasing NN (number of turns) increases LL but also copper losses and winding resistance.
- Increasing AeA_{\mathrm{e}} (core cross‑section) increases LL and allows higher flux without saturation, but results in a larger and heavier component.
- Reducing lel_{\mathrm{e}} (magnetic path length) increases LL, but is constrained by available core geometries.
- Adjusting effective permeability μ\muμ through the use of air gaps allows control of inductance and energy storage capability.
Core nonlinearity and inductance curves
Because permeability is not constant, the effective inductance depends on current:
- At low flux, the core exhibits high permeability, giving high inductance.
- As flux approaches saturation, permeability drops and inductance decreases.
- The small‑signal inductance at a given DC current is what manufacturers typically provide as “L versus I” plots.
When selecting parts:
- Always relate the specified inductance to the operating current listed in the datasheet.
- Use the inductance value at the actual operating current, not the nominal small‑signal value at near zero current.
Energy storage and size
The stored energy is:
- W=12LI2W = \dfrac{1}{2} L I^{2}
For a required energy per switching cycle, there is a trade‑off between:
- Higher inductance with lower current.
- Lower inductance with higher current.
However, the magnetic component’s physical size is largely dictated by the maximum stored energy and allowable flux density. Gapped cores enable more energy storage in a smaller size by allowing higher magnetizing field before reaching the flux limit.
Design‑In Notes for Engineers
Based on the clarified definitions, the following practical guidelines can help avoid common pitfalls in inductor selection and custom design.
1. Always specify the inductance type and operating point
- When specifying or interpreting an inductance value, state whether it refers to:
- Differential (small‑signal) inductance at a particular DC current.
- A low‑current, near‑linear inductance measurement.
- Ensure that the inductance value used in calculations matches the actual operating region of the device.
2. Think in terms of energy storage, not “AC vs DC”
- For any inductor, determine the maximum current and required energy storage per cycle.
- Use W=12LI2W = \dfrac{1}{2} L I^{2} to understand the magnetic energy requirement.
- Select a core and air gap combination that can store this energy without exceeding the allowed flux density.
3. Treat “AC inductors” with high AC current as energy‑storage components
- If the inductor must sustain large AC current (even with zero DC), treat it similarly to a DC‑biased choke from the standpoint of core selection and gapping.
- Avoid assuming that “no DC means no gap”. Instead, check peak current, flux swing and the resulting energy.
4. Use manufacturer curves correctly
- Pay attention to inductance versus current curves in datasheets.
- For a given design point, read off the inductance at that current rather than assuming the nominal low‑current value.
- If such curves are missing or incomplete, treat detailed inductance behavior as “according to manufacturer datasheet” and engage the supplier for clarification where necessary.
5. Size and gap the core for realistic flux density limits
- Do not design up to the formal saturation flux density. Use a more conservative limit to maintain acceptable permeability and avoid sharp drops in inductance.
- Adjust the air gap to keep flux within this limit at peak current while delivering the required inductance.
These rules apply across many common topologies—buck/boost converters, PFC stages, resonant tanks, filters and more—and help ensure that inductors behave as expected in real circuits rather than only in idealized calculations.
6. Purchasing and cross‑selection considerations
For purchasing teams and engineers working together:
- When comparing inductors from different suppliers, ensure that inductance and current ratings are based on comparable definitions and test conditions.
- If vendor A quotes inductance at zero current and vendor B at a given DC bias, normalize these values to the actual operating point in the application.
- Treat phrases like “AC inductor” or “DC choke” as high‑level labels and verify the underlying magnetic design (gapped versus ungapped, core material, energy rating).
Conclusion
Inductance is not a single fixed property but a parameter that depends on geometry, material and operating conditions. The traditional definitions L=NΦ/IL = N \Phi / I and v=L di/dtv = L \, \mathrm{d}i/\mathrm{d}t both originate from Faraday’s law, but in nonlinear cores they lead to different practical notions of total and differential inductance.
For energy‑storage inductors, the key engineering quantity is the differential inductance at the actual operating current, not just a nominal low‑current value. The common separation between “AC inductors” that supposedly do not need a gap and “DC inductors” that do is misleading; what really matters is the required energy storage and peak current, regardless of whether the current has a DC component.
By adopting an energy‑centric view, using inductance‑versus‑current curves correctly and specifying the inductance definition and operating point, design engineers and purchasers can select inductors that perform reliably, avoid saturation and achieve optimal size and cost in modern power electronics designs.
Source
This article is based on a lecture by Prof. Sam Ben‑Yaakov discussing the definitions of inductance, AC inductors and DC inductors, and clarifying common misconceptions from a practical power‑electronics perspective.
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