Assignment 4

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Michigan State University *

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320

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Electrical Engineering

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Dec 6, 2023

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ECE 320 sec.001 Spring 2023 Assignment #4 DUE: 3 Mar Directions Only submit the report for the Flux2D exercise. 1. An amplifier is to be connected to a loudspeaker by means of a two-winding transformer. The loud- speaker can be considered as a resistive load of 8Ω. The amplifier can be modeled as a source of audio frequency voltage in series with a resistance of 1000Ω. Assuming an ideal transformer, determine the turns ratio that will provide maximum power into the loudspeaker. (Answer: N 1 N 2 = 11 . 2 ) 2. A small toroidal ferrite core has a circular cross section with a radius of 7 . 5 mm and an inner diameter of 30 mm . The primary winding has 1000 turns and the secondary has 400 turns. The total thickness of these two windings is 5 mm . Approximately 40% of the total volume occupied by the winding is copper of resistivity 1 . 72 × 10 - 8Ω - m. (a) Estimate the resistances of the two windings. (Answer: R 1 = 3 . 72Ω , R 2 = 2 . 35Ω ) (b) Develop an equivalent circuit of the transformer with all the winding resistances referred to the primary side. (Answer: R e, 1 = 18 . 4Ω ) (c) If the core flux density is not to exceed 0 . 35 T , determine the maximum value of sinusoidal rms voltage that can be induced in the primary winding at a frequency of 400 Hz . (Answer: V 1 = 110 . 1 V ) (d) If the rms current density in the winding conductors is to be 5 A/mm 2 , estimate the ratio of the voltage drop across the effective resistance to the induced voltage of part 2c . (Answer: V w V q = 0 . 21 ) (e) Assuming the core in the transformer has a relative permeability of 300, estimate the value of the magnetizing inductance of the transformer as seen from the secondary. (Answer: L m, 2 = 76 mH ) 3. A transformer has the following parameters: R 1 = 0 . 5Ω, R 2 = 30Ω, L 1 L = 7 . 1 mH , L 2 m = 54 mH , N 1 N 2 = 0 . 12. Draw an equivalent circuit with all parameters referred to: (a) the high-potential side (Answer: R e 2 = 64 . 7Ω , L l 2 = 493 mH , L m 2 = 54 mH ) (b) the low-potential side. (Answer: R e 1 = 932 m Ω , L l 1 = 7 . 1 mH , L m 1 = 780 μH ) 4. A 200 kV A , 2300 : 230 V , 60 Hz single-phase transformer has a primary magnetizing current of 1 . 8 A . The winding resistances are 0 . 16Ω for the primary and 1 . 5 m Ω for the secondary. The leakage reactance at rated frequency as seen from the primary side is 1 . 8Ω. (a) Using rated nameplate quantities as base values, determine the base voltage, current, impedance, inductance, and complex power for both high- and low-voltage sides of the transformer. (Answer: S 1 b = 200 kV A , S 2 b = 200 kV A , V 1 b = 2 . 3 kV , V 2 b = 230 V , I 1 b = 87 A , I 2 b = 870 A , Z 1 b = 26 . 5Ω , Z 2 b = 260 m Ω , L 1 b = 70 . 2 mH , L 2 b = 690 μH ) (b) Determine the per unit values of the equivalent winding resistance, the leakage inductance, the magnetizing impedance, and the magnetizing inductance. (Answer: R e 1 ,pu = 0 . 012 , L l 1 ,pu = 0 . 068 , X m,pu = 4 . 83 , L m 1 ,pu = 4 . 83 ) (c) Determine the per unit value of the magnetizing current. (Answer: I m,pu = 0 . 021 ) 5. A 1 . 5 kV A , 220 : 110 V , 60 Hz single-phase transformer gave the following test results: (a) Open-circuit test, low-voltage winding excited: V 2 = 110 V, I 2 = 0 . 4 A, P 2 = 25 W, V 1 = 220 V (b) Short-circuit test, high-voltage winding excited: V 1 = 16 . 5 V, I 1 = 6 . 8 A, P 1 = 40 W 1

(c) Direct-current measurement of winding resistances: R 1 = 0 . 412Ω , R 2 = 0 . 112Ω Draw an equivalent circuit for this transformer referred to the high-voltage side and calculate the values of its parameters. (Answer: R e 1 = 865 m Ω , R 0 c 2 = 1 . 94 k Ω , X l 1 = 2 . 27Ω , X 0 m 2 = 1 . 33 k Ω ) 6. Complete the following Flux2D exercise. Write a short report ( 1 page) explaining how the frequency and current impacts hysteresis and eddy current loss. Be sure to include figures and data from Flux2D (screenshots are fine) to support your discussion. Submit to D2L by due date. 2

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© Altair Engineering, Inc. Proprietary and Confidential. All rights reserved. • This simulation presents the Characterization of magnetic material • This document explains about • Running the python script to obtain geometry, mesh, and physics setup • Plot the B-H characteristics of the core material and compare it with data sheet • Determine the hysteresis and Eddy current losses of the core (add procedure how to iron losses) • F1 Hz, F2 Hz, F3 Hz • I1 A, I2 A, I3 A at F1 Hz OVERALL VIEW ON THE EXERCISE 2

© Altair Engineering, Inc. Proprietary and Confidential. All rights reserved. • Run the python script “Torroidal_core_2D.py” CREATE GEOMETRY, MESH, AND PHYSICS 4 Step Action 1 Open Flux supervisor 2 Select the [2D] simulation context 3 Click on [Python scripts] 4 Select the [script] 5 Click on [Run the selected script] 2 3 5 4

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© Altair Engineering, Inc. Proprietary and Confidential. All rights reserved. • Go to solving < Solving scenario < Edit • Scenario_1 will open • Frequency of current input is varied from 50 Hz to 1000 Hz • You can change the number of variations of Frequency (3 steps in the example) CHECK THE SCENARIO CREATED 6

© Altair Engineering, Inc. Proprietary and Confidential. All rights reserved. • Go to Solving < Select Solve • Select the scenario to be solved, then Click Ok • Window with solving progress appears • After solving, information about the scenario appears at the bottom right of the window SOLVE THE MODEL 7

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© Altair Engineering, Inc. Proprietary and Confidential. All rights reserved. • To plot the BH curve, • Goto Curve< 2D curve (Material curve)< New B(H) 2D curve (Material) • Give the Minimum and Maximum bound for H (A/m) • Click OK PLOT BH CURVE OF CORE MATERIAL 9

© Altair Engineering, Inc. Proprietary and Confidential. All rights reserved. • To compute iron losses, Go to Computation< Computation of iron losses< Deprecated versions< Bertotti losses • Select the frequency in Scenario (on the left bottom), the compute Iron losses COMPUTATION OF IRON LOSSES 10 Quantity Value Hysteresis loss coeff. 130.246 Ws/T 2 /m 3 Classical loss coeff. 1923077 S/m Loss in excess coeff. 0.357 W/(T/s) 1.5 /m 3 Thickness of lamination 0.35e-3 m

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© Altair Engineering, Inc. Proprietary and Confidential. All rights reserved. • Characterization of magnetic material • Goals are : • Plot the B-H characteristics of the core material • Determine the hysteresis and Eddy current losses of the core • 50 Hz, 475 Hz, 1 kHz • 50 A, 100 A, 200 A at 50 Hz TORROIDAL CORE 11 Plot of Flux density on a point Torroidal core Problem 2

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