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Beginner

**Copper Fill Factor**

Inductor winding is made of a conductor material which may be a single round wire or a unique multi-stranded conductor known as Litz wire. Litz wire has the advantage of reduced skin effect. Let the cross-sectional area for the conductor be A_{C} and say it is making N number of turns. Then, the total cross-section covered by the conductor is N A_{C}.

Let the window size for the winding be A_{W} which includes the area of the space between the conductors and the area covered by the insulation.

Conductor fill factor,

$$F_{C}=\frac{V_{C}}{V_{W}}=\frac{NA_{C}}{A_{w}}$$

where V_{C }is the volume occupied by the conductor and V_{W} is the volume for the window.

Practically, F_{C} is 0.3 for Litz wire and 0.5 to 0.6 for round conductors.

Let d be the diameter of the round conductor, then $$δ ≥ \frac{d}{2};$$ while the skin effect is negligible.

**Winding Loss and Current Density**

The winding loss due to the DC resistance of the conductor winding per unit conductor volume is given by the following expression:

$$P_{C}=ρ_{C}(\frac{I_{RMS}}{A_{C}})^{2}$$

Here, $$ρ_{C}$$ is the specific resistivity of the conductor. It can also be written as

$$P_{C}=ρ_{C}(J_{C})^{2}$$

where J_{C} is the current density in the conductor.

If we want to express the power loss per unit window volume, then it can be written as,

$$P_{W}=F_{C}ρ_{C}(J_{C})^{2}$$

If the skin effect is considered, the formula will be,

$$P_{W}=\frac{R_{AC}}{R_{DC}}F_{C}ρ_{C}(J_{C})^{2}$$

Now the following approximations are considered to make the design simpler and to derive the constraints’ equations for the design of the inductor.

Let the inductor L be carrying the worst-case current I_{m} without saturating the core. This causes an operating core flux density at maximum value of B_{m}. It must be less than the flux density at saturation i.e. B_{m }< B_{SAT}.

Copper Loss $$=P_{C}=(I_{RMS})^{2}R$$

We also know that resistivity of the conductor’s core is much smaller that of the resistivity of the air gap, so let the resistivity of the core be neglected:

$$NI≈φR_{g}$$

$$\Rightarrow NI≈BA_{C'}R_{g}$$

$$\Rightarrow NI_{m}≈B_{m}A_{C'}R_{g}=\frac{B_{m}A_{C'}l_{g}}{μ_{0}}$$ [Equation 1]

It shows the relationship between the turns ratio and the air gap length which put a constraint on these parameters.

Resistance of Winding, $$R_{W}=\frac{ρl_{WR}}{A_{C}}$$

where ρ = resistivity of material for conductor; A_{C }= conductor’s cross-sectional area; l_{WR}=N l_{meanT} is the length of the wire where l_{meanT} is the mean length of the wire per turn.

Thus, $$R_{W}=\frac{ρl_{WR}}{A_{C}}=\frac{ρ N l_{meanT}}{A_{C}}$$ [Equation 2]

This is also a constraint equation.

Another constraint equation according to the basic inductance equation is given below.

$$L=\frac{N^{2}}{R_{g}}=\frac{µ_{0}A_{C'}N^{2}}{l_{g}}$$ [Equation 3]

This relationship put the constraint on the design by proposing the relation between the turns ratio, N; area of the core, A_{C'}; and the length of air gap, I_{g}.

The last constraint is according to the window size. Area of the window must be larger than the cumulative area occupied by the conductors, thus,

$$F_{C}A_{W}≥NA_{c}$$ [Equation 4]

The overall summary of the constraints for the design of the filter inductor are,

$$NI_{m}≈B_{m}A_{c'}R_{g}=\frac{B_{m}A_{c'}l_{g}}{μ_{0}}$$

$$R_{W}=\frac{ρl_{WR}}{A_{C}}=\frac{ρ N l_{meanT}}{A_{C}}$$

$$L=\frac{N^2}{R_{g}}=\frac{µ_{0}A_{C'}N^2}{l_{g}}$$

$$F_{C}A_{W}≥NA_{C}$$

In these constraints equations,

A_{C'}, A_{W}, and l_{mean} depend on the geometry of the core.

I_{m}, B_{m}, μ_{0}, L, F_{C}, ρ and R are known parameters which are given according to the specification of the particular inductor.

The value of the number of turns N, length of air gap I_{g}, and the cross-sectional area of the conductor, A_{C} of the inductor are all unknowns here.

One of the most obvious things about the design of the inductor is the choice of material. Inductor core is made up of various materials which shows different B-H curve. The temperature of the core rises due to eddy current losses especially when high frequencies are used in the converter. Thus, we generally use the laminated sheets or granulated iron in cases where iron must be used as the core material. The ferrites which have large thermal conductivity are also commonly used as core materials to limit eddy current losses. The inductor temperature is dependent on the convective and radiative transfer of heat. A prototype of the desired inductor should be operated at the rated values and the temperatures of it should be measured. If the temperature is excessively high or low, the design should be changed consequently.

One of the most fundamentals in the design of inductor is to know the stored energy in the inductor. We know that the total flux linkage for the inductor is given by

$$Nφ_{core}=LI$$

$$F_{C}A_{W}≈NA_{C}$$

$$φ_{core}=A_{C'}B$$

$$J_{RMS}=\frac{I_{RMS}}{A_{C}} $$

$$\Rightarrow \frac{F_{C}A_{W}}{A_{C}}A_{C'}BI_{RMS}=L I I_{RMS}$$

$$\Rightarrow F_{C}A_{W}A_{C'}BJ_{RMS}=L I I_{RMS}$$ [Equation 5]

B and J_{RMS} depend on the selection of material while parameters F_{C}, A_{C} and A_{C'} depend on the geometry of the winding and core. These given equations above relates all these mentioned parameters to the input parameters of the inductor: L,I and I_{RMS}.

Now the step-by-step procedure is given for the design of inductor below. It is designed for the allowable temperature and power density of the inductor.

If the complete parameters for the core characteristics are not available, then we can find the proper values with the help of iteration. Initially, an estimated value of the current density J_{RMS} ≈ 2 to 4 $$\frac{A}{mm^{2}}$$ is considered. Also, the magnetic flux density can be found using the expected value of the core loss and finding the corresponding value of the operating flux density using the graph between core loss and B. Then core size can be decided based on Equation 5. The corresponding iterative process is shown below.

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