ENERGY CONVERSION ONE (Course 25741)

1
ENERGY CONVERSION ONE (Course 25741)

• Chapter one
• Electromagnetic Circuits
• continued

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Hysteresis Losses

• As I of coil slowly varying in a coil energy
  flows to coil-core from source
• However, Energy flowing in gt Energy returns
• The net energy flow from source to coil is the
  heat in core (assuming coil resistance
  negligible)
• The loss due to hysteresis called
• Hysteresis Loss
• hysteresis loss Size of hysteresis loop
• Voltage e across the coil eN df/dt

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Hysteresis Losses

• Energy transfer during t1 to t2 is
• VcoreA l, volume of core
• Power loss due to hysteresis in core PhVcore Wh
  f
• f freq. of variation of i
• Steinmetz of G.E. through large no. of experiment
  for machine magnetic materials proposed a
  relation
• Area of B-H loop
• Bmax is the max flux density

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Hysteresis Losses

• n varies from 1.5 to 2.5,
• K is a constant
• Therefore the hysteresis power loss
• PhKh (Bmax)n f
• Kh a constant depends on
• - ferromagnetic material and
• - core volume

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EDDY CURRENT LOSS

• Another power loss of mag. Core is due to rapid
  variation of B (using ac source)
• In core cross section, voltage induced
• and ie passes, resistance of core cause
• Pe ie2 R (Eddy Current loss)
• this loss can be reduced as follows when
• a- using high resistive core material, few
  Si
• b- using a laminated core

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EDDY CURRENT LOSS

• Application of Laminated Core
• Eddy current loss PeKeBmax2 f2
• Ke constant depends on material lamination
• thickness which varies from 0.01 to 0.5 mm

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CORE LOSS

• PcPhPe
• If current I varies slowly eddy loss negligible
• Total core loss determined from dynamic B-H loop
• Using a wattmeter core loss
• can be measured
• However It is not easy to know
• what portion is eddy hysteresis

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Eddy Current Core Loss Sl St

• Effect of lamination thickness (at 60 Hz)

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Eddy Current Core LossSl St

• Effect of Source Frequency

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Sinusoidal Excitation

• Example
• A square wave voltage E100 V f60 Hz applied
  coil on a closed iron core, N500
• Cross section area 0.001 mm2, assume coil has no
  resistance
• a- max value of flux sketch V f vs time
• b- max value of E if Blt1.2 Tesla

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Sinusoidal Excitation

• a - e N df/dt gt N.?fE.?t
• E constant gt 500(2fmax)Ex1/120
• Fmax100/(1000x120)Wb0.833x10-3 Wb
• b - Bmax1.2 T (to find maximum value of E)
• FmaxBmax x A1.2 x 0.0011.2 x10-3 Wb
• N(2fmax)E x 1/120
• Emax 120x500x2x1.2x10-3144 V

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Exciting CurrentUsing ac Excitation

• Current which establish the flux in the core
• The term If if B-H nonlinear, non-sinusoid
• a - ignoring Hysteresis
• B-H curve ? f-i curve (or the rescaled one)
• Knowing sine shape flux, exciting current
  waveform by help of f-i curve obtained
• The current non-sinusoidal, if1 lags V 90
• no loss (since Hysteresis neglected)

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Exciting Current

• Realizing Hysteresis, Exciting Current
  recalculated
• Now if determined from multi-valued f-I curve
• Exciting current nonsinusoid nonsymmetric
• It can split to 2 components ic in phase with e
  (represents loss), im in ph. With f symmetric

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Simulation of an RL Cct with Constant Parameters

• Source sinusoidal ? iIm . sin ?t
• V L di/dt R i
• ? v dt ?L.di ? Ri.dt
• ?L ?di R ?i . dt
• L Im sin?t R/? Im cos?t
• Now drawing ? versus i
• However with magnetic core
• L is nonlinear and saturate
• Note Current sinusoidal

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Wave Shape of Exciting Currenta- ignoring
hysteresis

• From sinusoidal flux wave f-i curve for mag.
  System with ferromagnetic core, if determined
• if as expected nonsinusoidal in phase with f
• and symmetric w.r.t. to e
• Fundamental component if1 of exciting current
  lags voltage e by 90? (no loss)
• F-i saturation characteristic exciting current

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Wave Shape of Exciting Currentb- Realizing
hysteresis

• Hysteresis loop of magnetic system with
  ferromagnetic core considered
• Waveform of exciting current obtained from
  sinusoidal flux waveform multivalued f-i curve
• Exciting current nonsinusoidal nonsymmetric

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Wave Shape of Exciting Current

• It can be presented by summation of a series of
  harmonics of fundamental power frequency
• ie ie1 ie3 ie5 A
• It can be shown that main components are the
  fundamental the third harmonic

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Equivalent Circuit of an Inductor

• Inductor is a winding around a closed magnetic
  core of any shape without air gap or with air gap
  
• To build a mathematical model we need realistic
  assumptions to simplify the model as required,
  and follow the next steps
• Build a System Physical Image
• Writing Mathematical Equations

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Equivalent Circuit of an Inductor

• Assumptions for modeling an Ideal Inductor
• 1- Electrical Fields produced by winding can
  be
• ignored
• 2- Winding resistance can be ignored
• 3- Magnetic Flux confined to magnetic core
• 4- Relative magnetic permeability of core
• material is constant
• 5- Core losses are negligible

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Equivalent Circuit of an InductorIdeal Inductor

• v e d? / dt Volts
• ? L ie Wb
• v L d ie /dt Volts
• realizing winding resistance in practice
• v L d ie /dt Rw ie Volts

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Equivalent Circuit of an Inductor

• Realizing the core losses and simulating it by a
  constant parallel resistance Rc with L

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Equivalent Circuit of an Inductor

• In practice Inductors employ magnetic cores with
  air gap to linearize the characteristic
• f fm fl
• N fm Lm ie Wb
• N fl Ll ie Wb
• ? N f Lm ie Ll ie
• e d?/dt
• Lm die/dt Ll die/dt

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Equivalent Circuit of an Inductor

• Example A inductor with air gap in its magnetic
  core has N2000, and resistance of Rw17.5 O.
  When ie passes the inductor a measurement search
  coil in air gap measures a flux of 4.8 mWb, while
  a search coil close to inductors winding
  measures a flux of 5.4 mWb
• Ignoring the core losses determine the equivalent
  circuit parameters

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Equivalent Circuit of an Inductor

• f 5.4 mWb, fm 4.8 mWb
• fl f fm 0.6 mWb
• LmN fm/ ie 2000x4.8/0.713.7 H
• LlN fl/ ie 2000 x 0.6 / 0.71.71 H