Lecture 21 Magnetic g Circuits,, Materials ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc Magnetic Circuits and Transformers Understand magnetic fields and their interactions with moving charges Use the right-hand right hand rule to determine the direction of the magnetic field around a current-carrying wire or coil coil ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc Calculate forces on moving charges and current carrying wires due to magnetic fields fields Calculate the voltage induced in a coil by a changing magnetic flux or in a conductor cutting through a magnetic field field Use Lenz Lenz’ss law to determine the polarities of induced voltages ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc Apply magnetic-circuit concepts to determine the magnetic fields in practical devices Determine the inductance and mutual inductance of coils given their physical parameters Understand hysteresis, saturation, core loss, and d eddy dd currents iin cores composedd of magnetic materials such as iron ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc Understand ideal transformers and solve circuits that include transformers 10 Use the equivalent circuits of real g transformers to determine their regulations and power efficiencies ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc Magnetic g Field Lines Magnetic fields can be visualized as lines of flux that h form f closed l d paths The flux density vector B is tangent to the h li lines off flux fl B = Magnetic flux density ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc Magnetic g Fields • Magnetic g flux lines form closed ppaths that are close together where the field is strong and p where the field is weak farther apart • Flux lines leave the north-seeking g end of a magnet and enter the south-seeking end • When placed in a magnetic field, a compass indicates north in the direction of the flux lines ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc Right-Hand g Rule ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc Forces on Charges g Movingg in Magnetic Fields f = qu qu × B f = quB sin (θ ) ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc Forces on Current-Carrying y g Wires dl df = dqq × B dt dq = dl × B dt = idl × B Force on straight wire of length l in a constant magnetic field f = ilB sin i (θ ) ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc Exercise 15.9 Agap = (2cm + 1cm) x(2cm + 1cm) = x10 − m μ gap ≈ μ = 4πx10 R gap = −7 −2 1x10 m −7 −4 4πx10 8.75 x10 m = 8.842 x10 ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc Exercise 15.9 Rcore l (2 x8 + x6 − 1)cm = = μ A μ r μ (2cm)(2cm) = −2 27 x10 m −7 −4 (5000)(4πx10 ) x10 m = 107.4 x10 ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc Exercise 15.9 Rtotal = R gap + Rcore = 8.842 x10 + 0.107 x10 ≈ R gap φ = B gap Agap = (0.5T )(9 x10 − m ) = 0.45mWb Rtotal φ i= N (8.842 x10 + 0.107 x10 )(0.45 x10 −3 ) = 1000 = 4.027 A ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc A Magnetic Circuit with Reluctances in Series and Parallel Find the flux density in each gap ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc A Magnetic Circuit with Reluctances in Series and Parallel Rtotal t t l = Rc + 1 + Ra Rb Ni φc = Rtotal φa = Rb φc Ra + Rb (current divider ) Ra φb = φc Ra + Rb Ba = Ba = φa Aa φa Aa ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc Magnetic g Materials ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc Magnetic Materials The relationship between B and H is not linear for the types yp of iron used in motors and transformers B-H curves exhibits “h t “hysterysis” i” ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc Magnetic g Materials Residual alignment at H=0 T t l alignment Total li t Alignment 2-3 Linear 1-2 • Magnetic field of atoms within small domains are aligned • Magnetic fields of the small domains are initially randomly oriented • As the magnetic field intensity increases, the domains tend to align, leaving a residual alignment even when the applied field is reduced to zero ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc Energy gy Considerations t t ∫ ∫ W = vi dt = N 0 φ dφ i dt = Ni dφ dt ∫ Ni = Hl and dφ = AdB B B ∫ ∫ W = AlH dB = Vcore H dB 0 B ∫ W = WV = H dB V ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc Energy gy Considerations B W Wv = = ∫ H dB Al The area between the B-H curve and the B axis represents p the volumetric energy supplied to the core ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc Core Loss Power loss due to hysteresis y is pproportional p to frequency, assuming constant peak flux ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc Eddy-Current y Loss As the magnetic g field changes g in a material, it causes “eddy currents” to flow Power loss due y currents is proportional p p to the square q of to eddy frequency, assuming constant peak flux v P= R ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc Energy Stored in the Magnetic g Field B Wv = ∫ B B dB = μ 2μ ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R Hambley, ©2008 Pearson Education, Inc