Beginner

#### Recommended Level

**Introduction to Skin and Proximity Effects**

*Skin Effect:* When a DC current flows through a conductor, current is uniformly distributed across the section of the conductor. On the other hand, when an AC current flows through a conductor, outer filament of that conductor carries more current as compared to the filament closer to its center. This results in higher resistance to AC than to DC and is known as **skin effect**. This is due to more flux linkage per ampere to inner filaments as compared to outer side of the conductor.

This effect is more significant for bigger size conductors and for higher frequencies. Current density will decrease exponentially with respect to the depth of conductor from outside.

Consider an AC current that flows in copper conductor coils connected in winding.

Then, the penetration depth is given by

$$δ_{cu}=\sqrt{\frac{2}{µ_{0}\;\omega\;σ_{cu}}}$$

Where,

$$\omega$$ = 2π*f ;* σ_{cu }= conductivity of copper.

If the diameter of conductor is less than the value of depth of penetration, then the skin effect will be low.

*Proximity Effect: *The alternating flux in a conductor is caused by the current of the other nearby conductor. This flux produces a circulating current or eddy current in the conductor which results an apparent increase in the resistance of the wire and; thus, more power losses in the windings. This phenomenon is **proximity effect**.

Proximity effect can be reduced by selecting the core and number of turns that optimizes the number of layers.. An increased number of layers decreases the losses after the first selection. Foil winding layers reduces the losses more effectively as compared to round wires on a single layer. Interleaving the winding also reduces the proximity effect. Interleaving decreases the effective number of layers in each section of winding and; thus, the resulting field build up more uniformly than rising gradually in between.

Current produced is basically due to the saturation of core that contains PWM current having significant harmonic distortions. This can also lead to the increase of copper losses basically in multiple layers and non-interleaved windings as against the impedance of single layers which is almost negligible for such cases.

**Leakage Flux in Windings**

We have discussed earlier that leakage flux is represented by the leakage inductance in series with the supply for a transformer model. The leakage flux lines follows the three dimensional path as shown below.

**Figure 1. Leakage Flux 3D for a Concentric Winding**

Depending on the configuration of its winding, we can determine the value of it based on certain assumptions.

Consider the magnetic path is linear. Then the leakage flux configuration for the concentric winding will be as shown below. Magnetic field intensity will vary with the distance from the limb of transformer.

**Figure 2. Concentric Winding of a Transformer**

It is assumed that flux of each winding is filling their own volume with half of the space in between these two windings.

From the figure and basic equations for H field,

$$H_{xm}L_{c} = N_{2}I_{2}\frac{x}{a_{2}}$$

Now, for the distance a_{2} < x < a_{2} + δ

$$H_{xm}=\frac{(N_{2}I_{2})}{L_{C}}\;\frac{x}{a_{2}}≈\frac{(N_{2}I_{2})}{L_{C}}≈-\frac{(N_{1}I_{1})}{L_{C}}$$ (Due to ampere-turns balance)

Magnetic energy stored by the secondary winding is given by

$$E_{2}=R_{cf}\,W_{2}=R_{cf}\,\frac{1}{2}L_{2l}\,{i_{2}}^{2}$$

R_{cf} is the Rogowski’s coefficient which is used to correct the magnetic field path effect which is assumed linear. Its value should be greater than 1.

$$\Rightarrow$$ Leakage inductance for secondary winding is,

$$L_{2l}=\frac{(R_{cf2}\;2W_{2})}{{i_{2}}^{2}}=2\frac{R_{cf}}{{i_{2}}^{2}}(\frac{1}{2})μ_{0}\int_{0}^{a_{2}+\frac{δ}{2}}{H_{x}}^{2}π(D+2x)L_{C}dx$$

$$\Rightarrow L_{2l}=\frac{(R_{cf2}\,μ_{0}\,{N_{2}}^{2}\,π\,D_{2avg}\,a_{r2})}{L_{C}}$$

Where,

$$D_{2avg}=D+\frac{3a_{2}}{2}$$ and $$a_{r2}=\frac{a_{2}}{3}+\frac{δ}{2}$$

Similarly,

$$L_{1l}=\frac{(R_{cf1}\,μ_{0}\,{N_{1}}^{2}\,π\,D_{1avg}\,a_{r1})}{L_{C}}$$

Where, $$D_{1avg}=D+(a_{2}+d)+3\frac{a_{1}}{2}$$ which is a high-voltage winding place away from the core as compared to a low-voltage winding number 2.

Similar analysis can be done to find out the leakage inductance for alternate winding in different chamber of transformer or bi-concentric winding. But, nowadays finite element 3D analysis is carried out for the calculation of more precise value of leakage inductance.

**Foil Windings and Layers**

Foil windings are used to reduce the effective copper loss. They are vertical section of conducting plates which are symmetrically placed on both sides of the transformer. These windings have low eddy current losses for the magnetic field parallel to the foil. These are insulated from the conductors with varnish or insulation sheets. Filling factor for foil winding is dependent on the thickness of insulation. The major difficulty in manufacturing of foil winding is the labor for placing the foil winding to the transformer coils.

Filling factor is the ratio of cross-sectional area of the conductor and the window area of the core. Cross-sectional area of the conductor is given by the product of the number of turns and the wire cross-section area of a conductor.

Foil winding is especially popular for small transformers due to its simplicity. Finite element analysis can be used to analyze the eddy current loss distribution within the foil winding.

**Power Loss in a Transformer Winding**

Number of windings on the primary and secondary sides of the transformer is decided based on the voltage per turn. Next, we have to choose the proper current density of the primary and secondary windings for finding out the area requirement for the windings. The current density is dependent on the local heating and efficiency. It is important to decide the proper value of the current density for the design of windings as the load for maximum efficiency and copper losses is dependent on this choice. It is different for small-sized and big-sized transformers.

Area of the conductor = Current in that winding / Current density for that winding.

$$a_{p}=\frac{I_{p}}{δ_{p}}$$

$$a_{s}=\frac{I_{s}}{δ_{s}}$$

Let the specific copper loss per kilogram for copper conductor be ρ_{c}.

Volume of the conductor in primary = V_{P}

Volume of the conductor in secondary = V_{S}

Total volume of the conductors ≈ V_{P }+ V_{S }= V_{t }≈ constant

The copper loss in primary = ρ_{c }δ_{p}^{2 }V_{P }and the copper loss in secondary = ρ_{c }δ_{s}^{2 }V_{S}

Hence, the total copper loss

P_{tc }= ρ_{c }δ_{P}^{2 }V_{P }+ ρ_{c }δ_{S}^{2 }(V_{t }- V_{P})

Differentiating it with V_{P} and equating it to zero to get the condition for the minimum loss. We can get the same value of current density for the both conductors. However, current density for outer winding is somewhat larger than its inner winding due to better cooling condition.

**Interleaving the Windings**

This interleaving scheme is applicable to transformer but not to inductors. If the design of transformer winding is altered in such a way that one winding layer lies within another layer using the same wire. For instance, a four layer winding with pattern (P1, P2, S1, S2) changed to (P1, S1, P2, S2) or (S1, P1, P2, S2) or (P1, S1, S2, P2).

Interleaving can reduce the ohmic and eddy current losses by a factor of two. Also, the current-handling capacity of winding can be increased by a factor of $$\sqrt{2}$$.

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