Have you ever wondered how much power you can get through a given core? Wonder no longer. As long as you can find manufacturer data on the part, you can determine a reasonable upper limit, with respect to permissible temperature rise and operating frequency, and even find the quality factor (Q).

Core manufacturers often provide tools and resources describing how to choose their products, based on inductance or transformer size, but thermal limitations are less often considered. With the relatively high losses of many materials, this is not a small problem. As far as choosing a particular core by thermal limit, no manufacturer provides such a resource. The intrepid engineer is left to design by trial and error, picking cores at random until one seems to fit the design equations. The resulting selection is sub-optimal, often either too small and it overheats, or too large and overly expensive. Although the complete design process still can't be encapsulated in a single formula, the formulas in this paper should come in handy.

Note: MMKS units are used in this document. This is SI standard MKS units, with meters changed to milimeters; in most places, this pushes a factor of a million somewhere else convenient, for example: frequency is in MHz, capacitance in *μ*F, inductance in *μ*H, flux in *μ*Wb, etc. Turns are also treated as a unit, as consistently as possible.

Magnetic flux density (tesla) has the somewhat unwieldy expansion *μ*Wb mm^{-2 }t^{-1}, but the units serve nicely in dimensional analysis to remember which variables are correct.

The governing equations in this analysis come from basic magnetic and electric theory. First, let's consider what we want to determine: we need the reactive power density of a core material: the power handling of a given core. Without a consistent formula for the core losses (few materials have ideal Steinmetz losses), we fall back to the characteristic curves given by the manufacturer. These curves are usually in terms of power loss density p_{c}, as a function of flux density B_{max} for a selection of frequencies. So we shall select points from the graph and work from there.

Symbol | Value | Unit |
---|---|---|

O.D. | 17.5 | mm |

I.D. | 9.4 | mm |

Ht. | 6.35 | mm |

μ_{r} | 75 | |

A_{L} | 0.058 | μH t^{-2} |

l_{e} | 42.3 | mm |

A_{e} | 24.2 | mm^{2} |

v_{e} | 1030 | mm^{3} |

The next thing we need to know is how to get reactive power out of that: the best way is to use the core properties and solve for the voltage and current we would apply to an arbitrary inductor wound on that core.

Suppose we selected the T68-26A core from Micrometals. Its properties are shown in Table 1. Given the core parameters, we can also calculate A_{L}:

(1)

Which, taking μ_{0} = 0.00126 *μ*H mm^{-1}, gives 0.054 *μ*H t^{-2}, close enough to the published value. Wound with wire, it looks like Figure 2, a commercial piece found in the lab (the core can be identified by its yellow/white color code, a Micrometals Mix #26). With 33 t, evidently this is a L = A_{L} N^{2} = 63 *μ*H coil.

If we applied a voltage to this coil, a certain amount of flux density would be achieved in the core:

(2)

(The factor √2 π is the familiar "4.44" seen in transformer design.) Which, knowing the frequency as well, we can now find how much core loss it would dissipate. To find the maximum voltage at a given frequency, for a given core, invert the equation:

(3)

Since this coil is 63 *μ*H, if we applied a 1 V_{rms} sine wave at 0.1 MHz, it would have a reactance of X_{L} = 2πFL = 39.6 Ω, carry a current of I = V / R = 0.025 A, and a reactive power of 0.025 VA. Hopefully, the real power dissipated is less than 25 mW (that'd be a resistor..), but this coordinate (2.8 mT at 100 kHz) doesn't appear on the graph, so we can't tell how much is dissipated. Instead of guessing how much voltage to apply, we can apply (3).

How much loss is too much? A good rule of thumb is 1000 mW mm^{-3}, but this depends on core size, air flow and so on. A small core might tolerate much higher loss density, or a medium core with air flow, but a very large core might overheat internally, even with cooling. The T68-26A might dissipate one or two watts comfortably, maybe just one when copper losses are included in the final analysis. With a volume of nearly 1000 mm^{3}, this is a good ballpark figure to go with.

At 100 kHz, we see 1000 mW mm^{-3} is reached at 46 mT (Figure 1). Using (3), we find a maximum voltage of 16.3 V, which would be acceptable for, say, a low voltage buck or boost converter, running in continuous conduction mode (CCM). The AC current draw is 0.41 A, so the reactive power handled is 6.7 VA. The power dissipation was 1 W, so the quality factor Q is a paltry 6.7. Judging by the wire size, this coil should easily handle 5 A or so, DC (the bias will reduce inductance a little, but not too much). This suggests not just a CCM type converter, but one with an overall power capacity of ballpark 16 V × 5 A = 80 W, so even though the Q is crappy, it's more than made up for with the large DC component.

This is all well and good, but we still don't have a solution for picking cores yet. If we bring together all the circuit parameters (volts, amps, inductance) with all the physics (power and flux densities), we can eliminate some variables and finish this up:

Putting them all together, V, I and N drop out, so it's no longer dependent on circuit-level quantities! A_{e} and l_{e} appear together, suggesting they should be paired as A_{e} l_{e} = v_{e}. These are not necessarily the same quantities; fortunately, the product is very close to the published v_{e} on most cores, so it seems good enough to use.

(4)

Important features of this equation:

- It depends on B
_{max}^{2}, which makes sense (power goes as V^{2}, and V and B are proportional). It's also proportional to volume, which is sensible. - Ideal Steinmetz core loss, when solved for constant power, has F ∝ B
^{-2}; since our formula for S is proportional to F and B^{2}, such a core works just as well at any frequency (it gives constant Q). Real cores have different exponents for F and B, but since Q can be calculated at a typical point, we can find which materials have enough Q for the required application. - S is inversely proportional to μ
_{r}, which makes sense: higher permeability means more core material and less air gap, and it's the air gap that stores energy. Air also doesn't heat or saturate (at least, at experimentally known field intensities!). If you want to store a lot of energy, look for a low permeability material (or a large air gap, to get a low equivalent permeability). Low permeability materials also tend to have lower core losses, leading to higher Q and efficiency.

Usage: First, select a material. (Unfortunately, this formula doesn't help select a material; it's still up to the engineer's instinct, and the manufacturer's suggestions, as to which material to choose.) Within that material, determine the maximum flux density B_{max} for the chosen frequency of operation, based on either saturation or permissible core loss density. Knowing your application's reactive power demand, calculate the required core volume from:

(5)

Find a nearby core in the manufacturer's catalog and evaluate it further. You'll probably end up needing a slightly larger core (maybe 50% more VA capacity) to account for waveform, copper losses (remember that lower permeability cores take more copper) and safety margin.

To calculate the Q, we know that Q = S / P_{c} and P_{c} = p_{c} v_{e}, thus:

(6)

Which is now independent even of the core—purely a property of the material, frequency and core loss density, all of which can be read off the loss curves directly.

Ferrite datasheets often give a (small-signal) loss factor, or a plot of complex permeability (real μ′ and imaginary μ′′) versus frequency. Below the cutoff frequency (when μ′ > μ′′), we can easily relate Q to these parameters. In terms of permeability, Q = *μ*′ / *μ*′′. In terms of loss factor, 1 / Q = *μ*_{i} × (tan δ / *μ*_{i}) (why they give this peculiar unit, and not simply tan δ, without the "per μ_{i}" hanging on, I have no idea).

Magnetic design remains a tedious process for the design engineer, but these formulas will at least come in handy. Determine the reactive power S your inductor has to handle, and if your application requires a certain amount of efficiency, setting a minimum Q as well. Plug these into Equations (5) and (6) and find how big a core you need. Browse the manufacturer's catalog, find a core a few sizes above the v_{e} you found, and see if it checks out. Further winding and thermal analysis is required for a complete design, which is unfortunately beyond the scope of this paper.

Although this design flow is intended specifically for powdered iron cores, the same data is available for many materials. Ferrites are somewhat different: since they're often used for pulse transformers and EMI filters, complete loss data isn't always provided. Still, these formulas can be of help in designing high ripple, gapped ferrite cored chokes.

Description | Symbol | Units | Formula (if applicable) |
---|---|---|---|

Core Loss Density | p_{c} | W mm^{-3} | |

Core Loss | p_{c} | W | P_{c} = p_{c} v_{e} |

Effective Length | l_{e} | mm | |

Effective Area | A_{e} | mm^{2} | |

Effective Volume | v_{e} | mm^{3} | v_{e} ≈ A_{e} l_{e} |

Relative Permeability | μ_{r} | - | |

Magnetic Constant | μ_{0} | μH mm^{-1} | μ_{0} ≈ 0.00126 μH mm^{-1} |

Flux Density | B_{max} | T | (T ≡ μWb mm^{-2} t^{-1}) |

Turns | N | t | |

Frequency | F | MHz | |

Core Inductivity | A_{L} | μH t^{-2} | A_{L} = μ_{r}μ_{0} A_{e} / l_{e} |

Inductance | L | μH | L = A_{L} N^{2} |

Inductive Reactance | X_{L} | Ω | X_{L} = 2πFL |

Apparent Power | S | VA | S = |V| |I| |

See Table 2 for a summary of quantities and some short formulas used above. Notes:

- For the most part,
*v*is true within a few percent._{e}≈ A_{e}l_{e} - Turns: Number of loops the winding makes around the magnetic flux path, or vice versa. For a toroid, this is the number of passes through the center. A toroid on a straight wire is one turn, because the path itself is the first loop.
- Reactance (
*X*) is the imaginary component of impedance. When Ohm's law is used here (_{L}*V = IX*), there is an unwritten phase shift between_{L}*V*and*I*—we're only concerned about the magnitudes here. To prevent confusion, this is specified when calculating apparent power: the absolute value of*V*and*I*are taken.