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Design of Power Magnetic Devices: A Multi-Objective Design Approach
Chapter 11: AC Conductor Losses S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.1 Skin Effect in Strip Conductors
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.1 Skin Effect in Strip Conductors
We can show Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.1 Skin Effect in Strip Conductors
Derivation (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.1 Skin Effect in Strip Conductors
We can also show Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.1 Skin Effect in Strip Conductors
Derivation (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.1 Skin Effect in Strip Conductors
Material relationships We can now show S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.1 Skin Effect in Strip Conductors
Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.1 Skin Effect in Strip Conductors
The current density is given by S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.1 Skin Effect in Strip Conductors
Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.1 Skin Effect in Strip Conductors
Derivation (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.1 Skin Effect in Strip Conductors
We can also show that So that S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.1 Skin Effect in Strip Conductors
Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.1 Skin Effect in Strip Conductors
Derivation (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.1 Skin Effect in Strip Conductors
Derivation (continued again) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.1 Skin Effect in Strip Conductors
Example 11.1A: Consider a strip conductor w = 51 mm 2R = 3.21 mm Plot current density as a distance from center of the strip at a frequency of 5 kHz S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.1 Skin Effect in Strip Conductors
Result S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.1 Skin Effect in Strip Conductors
Impedance of strip conductor. It can be shown that Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.1 Skin Effect in Strip Conductors
Derivation (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.1 Skin Effect in Strip Conductors
Example 11.1B. Consider the impedance of the strip conductor of Example 11.1A as the frequency is varied from 10 Hz to 10 kHz S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.1 Skin Effect in Strip Conductors
Result S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.2 Skin Effect in Cylindrical Conductors
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.2 Skin Effect in Cylindrical Conductors
We still have (derivation same) We can show S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.2 Skin Effect in Cylindrical Conductors
Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.2 Skin Effect in Cylindrical Conductors
Next from We obtain S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.2 Skin Effect in Cylindrical Conductors
Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.2 Skin Effect in Cylindrical Conductors
Bessel equation Solution S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.2 Skin Effect in Cylindrical Conductors
Zero-order Bessel functions S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.2 Skin Effect in Cylindrical Conductors
We can show that by defining We obtain S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.2 Skin Effect in Cylindrical Conductors
Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.2 Skin Effect in Cylindrical Conductors
Thus, our current density may be expressed This reduces to We can also show S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.2 Skin Effect in Cylindrical Conductors
Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.2 Skin Effect in Cylindrical Conductors
Derivation (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.2 Skin Effect in Cylindrical Conductors
Next, the impedance may be expressed Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.2 Skin Effect in Cylindrical Conductors
Derivation (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.2 Skin Effect in Cylindrical Conductors
Example 11.2A. Let us find the ac resistance of a segment of #8 AWG wire. Nominal diameter is 3.23 mm S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.2 Skin Effect in Cylindrical Conductors
Predicted and measured resistance S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.3 Proximity Effect in a Single Conductor
Consider a rectangular conductor We can show S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.3 Proximity Effect in a Single Conductor
Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.3 Proximity Effect in a Single Conductor
Derivation (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.3 Proximity Effect in a Single Conductor
For a round conductor We can show S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.3 Proximity Effect in a Single Conductor
Suppose Then S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.3 Proximity Effect in a Single Conductor
Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.4 Independence of Skin and Prox. Effects
Consider a symmetric conductor S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.4 Independence of Skin and Prox. Effects
First we can show Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.4 Independence of Skin and Prox. Effects
Derivation (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.4 Independence of Skin and Prox. Effects
Next we define Thus S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.4 Independence of Skin and Prox. Effects
Let us represent current density as We can show S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.4 Independence of Skin and Prox. Effects
Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.4 Independence of Skin and Prox. Effects
Derivation continued S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.4 Independence of Skin and Prox. Effects
Symmetry conditions S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.4 Independence of Skin and Prox. Effects
It follows that S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.4 Independence of Skin and Prox. Effects
Thus S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.5 Proximity Effect in a Group of Conductors
We can show For rectangular conductors For round conductors S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.5 Proximity Effect in a Group of Conductors
Derivation – recall for rectangular conductors Derivation – recall for round conductors S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.5 Proximity Effect in a Group of Conductors
Since We conclude S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.5 Proximity Effect in a Group of Conductors
Defining Our expression becomes S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.5 Proximity Effect in a Group of Conductors
For round conductors, we may write Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.5 Proximity Effect in a Group of Conductors
Derivation (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.5 Proximity Effect in a Group of Conductors
Dynamic resistance in multi-winding systems Consider a two winding system Define S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.5 Proximity Effect in a Group of Conductors
We can show Where S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.5 Proximity Effect in a Group of Conductors
Derivation (1/3) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.5 Proximity Effect in a Group of Conductors
Derivation (2/3) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.5 Proximity Effect in a Group of Conductors
Derivation (3/3) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.5 Proximity Effect in a Group of Conductors
Finally, clearly for a single-winding system S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.6 Relating M.S.F. and Leakage Permeance
If region of proximity effect matches a leakage inductance, we can show S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.6 Relating M.S.F. and Leakage Permeance
Derivation (1/2) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.6 Relating M.S.F. and Leakage Permeance
Derivation (2/2) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.7 M.S.F. for Select Geometries
Exterior Adjacent Conductors S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.7 M.S.F. for Select Geometries
Exterior Isolated and Non-Gapped Closed Slot Conductors S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.7 M.S.F. for Select Geometries
Open Slot Conductors S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.7 M.S.F. for Select Geometries
Gapped Closed Slot Conductors S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.7 M.S.F. for Select Geometries
Boundary radius based on horizontal field S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.7 M.S.F. for Select Geometries
Boundary radius based on vertical field S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.7 M.S.F. for Select Geometries
Calculation of boundary point S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.7 M.S.F. for Select Geometries
Next we can show S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.7 M.S.F. for Select Geometries
Loss constants are given by S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.7 M.S.F. for Select Geometries
Finally S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.8 Conductor Losses in Rotating Machinery
Consider the geometry with two phases in a slot We can show S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.8 Conductor Losses in Rotating Machinery
First we find From which we obtain S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.8 Conductor Losses in Rotating Machinery
Consider S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.8 Conductor Losses in Rotating Machinery
Next Manipulating S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.8 Conductor Losses in Rotating Machinery
Manipulating (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.8 Conductor Losses in Rotating Machinery
Thus we obtain Now suppose S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.8 Conductor Losses in Rotating Machinery
Calculating the proximity effect loss …. S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.8 Conductor Losses in Rotating Machinery
Continuing … S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.8 Conductor Losses in Rotating Machinery
Continuing … S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.8 Conductor Losses in Rotating Machinery
Example 11.8A. Consider a slot Slot length = 78 cm Slot width = 6.93 mm N=24 Conductor radius = 0.51 mm Conductivity is 59.6 MS Let us compute the apparent resistance S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.8 Conductor Losses in Rotating Machinery
Computation of resistance S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.8 Conductor Losses in Rotating Machinery
Computation of resistance (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.8 Conductor Losses in Rotating Machinery
Result S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.8 Conductor Losses in Rotating Machinery
Example 11.8B Consider Design 38 from Chapter 9 P=12, Ss=72 l=3.34 cm, wsi=5.00 mm, dw=1.80 cm rc = 1.15 mm, s = 59.6 MS Slot 3: Na = 9, Nb = -2 Total a-phase conductors: 264 Phase current: 23 A rms at 500 Hz S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.8 Conductor Losses in Rotating Machinery
Example 11.8B (loss based on dc resistance) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.8 Conductor Losses in Rotating Machinery
Example 11.8B (loss based on ac resistance) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.8 Conductor Losses in Rotating Machinery
Example 11.8B (proximity effect loss) From proximity effect, total loss goes from 566 W to 590 W, an increase of 4% S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.9 Conductor losses in a UI core inductor
Consider a UI core inductor S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.9 Conductor losses in a UI core inductor
Assumed current waveform S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.9 Conductor losses in a UI core inductor
Assumed current waveform S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.9 Conductor losses in a UI core inductor
Example system: a buck converter S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.9 Conductor losses in a UI core inductor
DC loss S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.9 Conductor losses in a UI core inductor
Skin effect loss S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.9 Conductor losses in a UI core inductor
Proximity effect We can show S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.9 Conductor losses in a UI core inductor
Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.9 Conductor losses in a UI core inductor
Thus we have Finally S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.9 Conductor losses in a UI core inductor
Example 11.9A Consider Design 50 of UI core inductor example of Section 3.4 Current waveform imn = 9.5 A imx = 10.5 A Rise time: 90% of period Fall time: 10% of period Implications as we vary frequency S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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11.9 Conductor losses in a UI core inductor
Example 11.9A S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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