Chapter 12. Basic Magnetics Theory

Home , Copper loss, Eddy current, Leakage inductance, Magnetic circuit, Magnetic core, Magnetic flux

12.1. Review of basic magnetics 12.1.1. Basic relations 12.1.2. Magnetic circuits 12.2. Transformer modeling 12.2.1. The ideal transformer 12.2.3. Leakage inductances 12.2.2. The magnetizing inductance 12.3. Loss mechanisms in magnetic devices 12.3.1. Core loss 12.3.2. Low-frequency copper loss 12.4. Eddy currents in winding conductors 12.4.1. The skin effect 12.4.4. Power loss in a layer 12.4.2. The proximity effect 12.4.5. Example: power loss in a transformer winding 12.4.3. MMF diagrams 12.4.6. PWM waveform harmonics

Fundamentals of Power Electronics2 Chapter 12: Basic Magnetics Theory 12.1. Review of basic magnetics 12.1.1. Basic relations

Faraday's law v(t) B(t), Φ(t)

terminal core characteristics characteristics

i(t) H(t), F(t) Ampere's law

Fundamentals of Power Electronics3 Chapter 12: Basic Magnetics Theory Basic quantities

Magnetic quantities Electrical quantities

length l length l magnetic field H electric field E

x1 x2 x1 x2 +–+– MMF voltage F = Hl V = El

surface S surface S with area Ac with area Ac total flux Φ total current I flux density B { current density J {

Fundamentals of Power Electronics4 Chapter 12: Basic Magnetics Theory Magnetic field H and magnetomotive force F

Magnetomotive force (MMF) F between points x1 and x2 is related to the magnetic field H according to

x2 F = H ⋅ dl x1

Example: uniform magnetic Analogous to electric field of field of magnitude H voltagestrength (EMF) E, which induces V: length l length l magnetic field H electric field E x1 x2 x1 x2 +– +– MMF voltage F = Hl V = El

Fundamentals of Power Electronics5 Chapter 12: Basic Magnetics Theory Flux density B and total flux Φ

Φ The total magnetic flux passing through a surface of area Ac is related to the flux density B according to

Φ = B ⋅ dA

surface S

Example: uniform flux density of Analogous to electrical conductor magnitude B current density of magnitude J, Φ = BA which leads to total conductor c current I: surface S surface S with area A with area Ac c total flux Φ total current I { flux density B { current density J

Fundamentals of Power Electronics6 Chapter 12: Basic Magnetics Theory Faraday’s law

Voltage v(t) is induced in a loop of wire by change in area Ac the total flux Φ(t) passing through the interior of the loop, according to dΦ(t) v(t)= flux Φ(t) dt { For uniform flux distribution, – Φ (t) = B(t)Ac and hence v(t) + dB(t) v(t)=A c dt

Fundamentals of Power Electronics7 Chapter 12: Basic Magnetics Theory Lenz’s law

The voltage v(t) induced by the changing flux Φ(t) is of the polarity that tends to drive a current through the loop to counteract the flux change.

induced current Example: a shorted loop of wire i(t) ¥Changing flux Φ(t) induces a voltage v(t) around the loop ¥This voltage, divided by the impedance of the loop flux Φ(t) shorted conductor, leads to current i(t) loop ¥This current induces a flux Φ’(t), which tends to oppose changes in Φ(t) induced flux Φ'(t)

Fundamentals of Power Electronics 8 Chapter 12: Basic Magnetics Theory Ampere’s law

The net MMF around a closed path is equal to the total current passing through the interior of the path:

H ⋅ dl = total current passing through interior of path

closed path

Example: magnetic core. Wire through core window. carrying current i(t) passes H i(t) ¥ Illustrated path follows magnetic path magnetic flux lines length lm around interior of core ¥ For uniform magnetic field strength H(t), the integral (MMF)

is H(t)lm. So

F (t)=H(t)lm =i(t)

Fundamentals of Power Electronics9 Chapter 12: Basic Magnetics Theory Ampere’s law: discussion

¥ Relates magnetic field strength H(t) to winding current i(t) ¥ We can view winding currents as sources of MMF

¥ Previous example: total MMF around core, F(t) = H(t)lm, is equal to the winding current MMF i(t) ¥ The total MMF around a closed loop, accounting for winding current MMF’s, is zero

Fundamentals of Power Electronics10 Chapter 12: Basic Magnetics Theory Core material characteristics: the relation between B and H

Free space A magnetic core material B B µ B = 0 H

µ H H 0

µ 0 = permeability of free space Highly nonlinear, with hysteresis = 4π á 10-7 Henries per meter and saturation

Fundamentals of Power Electronics11 Chapter 12: Basic Magnetics Theory Piecewise-linear modeling of core material characteristics

No hysteresis or saturation Saturation, no hysteresis B B

Bsat B = µ H µ µ µ = r 0 µ µ µ µ = r 0

H H

– Bsat

µ 3 5 Typical Bsat = 0.3-0.5T, ferrite Typical r = 10 - 10 0.5-1T, powdered iron 1-2T, iron laminations

Fundamentals of Power Electronics12 Chapter 12: Basic Magnetics Theory Units

Table 12.1. Units for magnetic quantities quantity MKS unrationalized cgs conversions µ µ µ core material equation B = 0 r HB = r H B Tesla Gauss 1T = 104G H Ampere / meter Oersted 1A/m = 4π⋅10-3 Oe Φ Weber Maxwell 1Wb = 108 Mx 1T = 1Wb / m2

Fundamentals of Power Electronics13 Chapter 12: Basic Magnetics Theory Example: a simple inductor

Faraday’s law: Φ For each turn of core area wire, we can write i(t) Ac + Φ n d (t) v(t) vturn(t)= turns dt – core permeability µ Total winding voltage is core dΦ(t) v(t)=nv (t)=n turn dt

Φ Express in terms of the average flux density B(t) = (t)/Ac dB(t) v(t)=nA c dt

Fundamentals of Power Electronics 14 Chapter 12: Basic Magnetics Theory Inductor example: Ampere’s law

Choose a closed path H which follows the average magnetic field line around i(t) the interior of the core. magnetic path n length lm Length of this path is turns calledpath length the mean magnetic lm. For uniform field strength H(t), the core MMF

around the path is H lm. Winding contains n turns of wire, each carrying current i(t). The net current passing through the path interior (i.e., through the core window) is ni(t). From Ampere’s law, we have

H(t) lm = n i(t)

Fundamentals of Power Electronics15 Chapter 12: Basic Magnetics Theory Inductor example: core material model

Bsat ≥ µ Bsat for H Bsat / µ µ µ B = H for H < Bsat / ≤ µ ± Bsat for H Bsat / H

– B Find winding current at onset of saturation: sat µ substitute i = Isat and H = Bsat/ into equation previously derived via Ampere’s law. Result is

Bsat lm Isat = µ n

Fundamentals of Power Electronics16 Chapter 12: Basic Magnetics Theory Electrical terminal characteristics

We have: B for H ≥ B / µ dB(t) sat sat v(t)=nA µ µ c dt H(t) lm = n i(t) B = H for H < Bsat / ≤ µ ± Bsat for H Bsat /

Eliminate B and H, and solve for relation between v and i. For | i | < Isat, dH(t) µ 2 v(t)=µnA n Ac di(t) c dt v(t)= lm dt which is of the form di(t) µ 2 v(t)=L n A c dt with L = lm —an inductor

For | i | > Isat the flux density is constant and equal to Bsat. Faraday’s law then predicts dB v(t)=nA sat =0 —saturation leads to short circuit c dt

Fundamentals of Power Electronics17 Chapter 12: Basic Magnetics Theory 12.1.2. Magnetic circuits

length l Uniform flux and +–MMF F area magnetic field inside Ac a rectangular element: flux Φ { MMF between ends of core permeability µ element is F=Hl l H R=µ Ac Since H = B / µ and Β = Φ / Ac, we can express F as l Φ F= µ with R= l A c µ A c A corresponding model: +–F R = reluctance of element Φ R

Fundamentals of Power Electronics 18 Chapter 12: Basic Magnetics Theory Magnetic circuits: magnetic structures composed of multiple windings and heterogeneous elements

¥ Represent each element with reluctance ¥ Windings are sources of MMF ¥ MMF → voltage, flux → current ¥ Solve magnetic circuit using Kirchoff’s laws, etc.

Fundamentals of Power Electronics19 Chapter 12: Basic Magnetics Theory Magnetic analog of Kirchoff’s current law

Physical structure node Divergence of B = 0 Φ Φ 1 3 Flux lines are continuous and cannot end Φ Total flux entering a node 2 must be zero

Magnetic circuit Φ Φ Φ node 1 = 2 + 3

Φ Φ 1 3

Φ 2

Fundamentals of Power Electronics20 Chapter 12: Basic Magnetics Theory Magnetic analog of Kirchoff’s voltage law

Follows from Ampere’s law:

H ⋅ dl = total current passing through interior of path

closed path

Left-hand side: sum of MMF’s across the reluctances around the closed path Right-hand side: currents in windings are sources of MMF’s. An n-turn winding carrying current i(t) is modeled as an MMF (voltage) source, of value ni(t). Total MMF’s around the closed path add up to zero.

Fundamentals of Power Electronics21 Chapter 12: Basic Magnetics Theory Example: inductor with air gap

core µ permeability c Φ cross-sectional i(t) area Ac + n v(t) turns air gap – lg

magnetic path length lm Ampere’s law:

F c + F g = n i

Fundamentals of Power Electronics22 Chapter 12: Basic Magnetics Theory Magnetic circuit model

F core + c – µ permeability c Φ R c cross-sectional i(t) area Ac + + n v(t) turns air gap n i(t) + Φ(t) R F – lg – g g

magnetic path – length lm

l F + F = n i c c g R c = µ A c l Φ R = g ni= Rc +Rg g µ 0 A c

Fundamentals of Power Electronics23 Chapter 12: Basic Magnetics Theory Solution of model

core µ permeability c Fc Φ + – cross-sectional R i(t) c area Ac + n + v(t) turns air gap – lg n i(t) + Φ(t) – R g Fg magnetic path length lm –

dΦ(t) Φ R R Faraday’s law: v(t)=n ni= c + g dt l 2 di(t) c n R c = µ Substitute for Φ: v(t)= A c Rc +Rg dt lg R g = µ Hence inductance is 0 A c 2 L = n R c + R g

Fundamentals of Power Electronics24 Chapter 12: Basic Magnetics Theory Effect of air gap

Φ ni= Rc +Rg Φ = BA 2 c L = n R + R B A c g sat c 1 R Φ c sat = BsatAc 1 R + R BsatA c c g Isat = n R c + R g

Effect of air gap: nI nI ∝ sat1 sat2 ni Hc ¥ decrease inductance ¥ increase saturation current ¥ inductance is less dependent on core – BsatAc permeability

Fundamentals of Power Electronics25 Chapter 12: Basic Magnetics Theory 12.2. Transformer modeling

Two windings, no air gap: Φ i (t) i (t) lm 1 2 R = µ + A c n + v (t) 1 n2 1 turns turns v2(t) F c = n1 i1 + n2 i2 – –

Φ core R = n1 i1 + n2 i2

Φ Rc Magnetic circuit model: + – Fc n i + + n i 1 1 – – 2 2

Fundamentals of Power Electronics26 Chapter 12: Basic Magnetics Theory 12.2.1. The ideal transformer

In the ideal transformer, the core Φ Rc reluctance R approaches zero. + – Fc Φ MMF Fc = R also approaches + n1 i1 + n2 i2 zero. We then obtain – –

0=n1 i1 +n2 i2 Also, by Faraday’s law, dΦ v1 = n1 n : n i dt i1 1 2 2 Φ v = n d + + 2 2 dt Eliminate Φ : Φ v v v v d = 1 = 2 1 2 dt n1 n2 Ideal transformer equations: – – v1 v2 Ideal = and n1 i1 + n2 i2 =0 n1 n2

Fundamentals of Power Electronics27 Chapter 12: Basic Magnetics Theory 12.2.2. The magnetizing inductance

R For nonzero core reluctance, we obtain Φ c dΦ + – Φ R = n i + n i with v = n Fc 1 1 2 2 1 1 dt Eliminate Φ: n i + + n i 1 1 – – 2 2 n2 n v = 1 d i + 2 i 1 R 1 2 dt n1

n2 This equation is of the form i n i2 1 1 n1 : n2 i2 + + dimp v = L n2 1 mp i1 + i2 dt n1 2 2 n v n 1 v with L = 1 1 L mp = 2 mp R R n2 imp = i1 + i2 – – n1 Ideal

Fundamentals of Power Electronics28 Chapter 12: Basic Magnetics Theory Magnetizing inductance: discussion

¥ Models magnetization of core material ¥ A real, physical inductor, that exhibits saturation and hysteresis ¥ If the secondary winding is disconnected: we are left with the primary winding on the core primary winding then behaves as an inductor the resulting inductor is the magnetizing inductance, referred to the primary winding ¥ Magnetizing current causes the ratio of winding currents to differ from the turns ratio

Fundamentals of Power Electronics29 Chapter 12: Basic Magnetics Theory Transformer saturation

¥ Saturation occurs when core flux density B(t) exceeds saturation

flux density Bsat. ¥ When core saturates, the magnetizing current becomes large, the impedance of the magnetizing inductance becomes small, and the windings are effectively shorted out.

¥ Large winding currents i1(t) and i2(t) do not necessarily lead to saturation. If

0=n1 i1 +n2 i2

then the magnetizing current is zero, and there is no net magnetization of the core. ¥ Saturation is caused by excessive applied volt-seconds

Fundamentals of Power Electronics30 Chapter 12: Basic Magnetics Theory Saturation vs. applied volt-seconds

n Magnetizing current 2 i i1 n 2 n : n i depends on the integral 1 1 2 2 + + of the applied winding n2 i1 + i2 voltage: n1 n 2 v1 L = 1 v2 1 mp R imp(t)= v1(t) dt Lmp – – Flux density is Ideal proportional: Flux density befcomes large, and core B(t)= 1 v(t)dt 1 λ n1 Ac saturates, when the applied volt-seconds 1 are too large, where t λ 2 1 = v1(t) dt t1 limits of integration chosen to coincide with positive portion of applied voltage waveform

Fundamentals of Power Electronics31 Chapter 12: Basic Magnetics Theory 12.2.3. Leakage inductances

Φ M i1(t) i2(t) + + v (t) Φ Φ 1 l1 l2 v2(t) – –

Φ Φ l1 l2 Φ M i1(t) i2(t) + + v (t) 1 v2(t) – –

Fundamentals of Power Electronics32 Chapter 12: Basic Magnetics Theory Transformer model, including leakage inductance

i Ll1 Ll2 i 1 n1 : n2 2 + +

v1(t) L 11 L 12 d i1(t) n1 = v1 L mp = n L 12 v2 v2(t) L 12 L 22 dt i2(t) 2

– – mutual inductance Ideal

n1 n2 n2 L12 = = L mp R n1

primary and secondary L 22 effective turns ratio ne = self-inductances L 11 n1 L 11 = L l1 + L 12 L n2 coupling coefficient 12 n k = 2 L 11 L 22 L 22 = L l2 + L 12 n1

Fundamentals of Power Electronics33 Chapter 12: Basic Magnetics Theory 12.3. Loss mechanisms in magnetic devices

Low-frequency losses: Dc copper loss Core loss: hysteresis loss High-frequency losses: the skin effect Core loss: classical eddy current losses Eddy current losses in ferrite cores High frequency copper loss: the proximity effect Proximity effect: high frequency limit MMF diagrams, losses in a layer, and losses in basic multilayer windings Effect of PWM waveform harmonics

Fundamentals of Power Electronics34 Chapter 12: Basic Magnetics Theory 12.3.1. Core loss

Energy per cycle W flowing into n- Φ core area turn winding of an inductor, i(t) Ac + excited by periodic waveforms of n frequency f: v(t) turns – core permeability µ W = v(t)i(t)dt core one cycle Relate winding voltage and current to core B and H via Faraday’s law and Ampere’s law: dB(t) v(t)=nA c dt H(t) lm = n i(t) Substitute into integral: dB(t) H(t)l W = nA m dt c dt n one cycle

= A clm HdB one cycle

Fundamentals of Power Electronics35 Chapter 12: Basic Magnetics Theory Core loss: Hysteresis loss

Area W = Aclm HdB one cycle HdB one cycle

The term Aclm is the volume of H the core, while the integral is the area of the B-H loop.

(energy lost per cycle) = (core volume) (area of B-H loop)

PH = f A clm HdB Hysteresis loss is directly proportional one cycle to applied frequency

Fundamentals of Power Electronics36 Chapter 12: Basic Magnetics Theory Modeling hysteresis loss

¥ Hysteresis loss varies directly with applied frequency ¥ Dependence on maximum flux density: how does area of B-H loop depend on maximum flux density (and on applied waveforms)? Emperical equation (Steinmetz equation): α PH = K H fBmax(core volume)

α The parameters KH and are determined experimentally.

Dependence of PH on Bmax is predicted by the theory of magnetic domains.

Fundamentals of Power Electronics37 Chapter 12: Basic Magnetics Theory Core loss: eddy current loss

Magnetic core materials are reasonably good conductors of electric current. Hence, according to Lenz’s law, magnetic fields within the core induce currents (“eddy currents”) to flow within the core. The eddy currents flow such that they tend to generate a flux which opposes changes in the core flux Φ(t). The eddy currents tend to prevent flux from penetrating the core.

flux eddy Φ(t) Eddy current current loss i2(t)R i(t) core

Fundamentals of Power Electronics38 Chapter 12: Basic Magnetics Theory Modeling eddy current loss

¥ Ac flux Φ(t) induces voltage v(t) in core, according to Faraday’s law. Induced voltage is proportional to derivative of Φ(t). In consequence, magnitude of induced voltage is directly proportional to excitation frequency f. ¥ If core material impedance Z is purely resistive and independent of frequency, Z = R, then eddy current magnitude is proportional to voltage: i(t) = v(t)/R. Hence magnitude of i(t) is directly proportional to excitation frequency f. ¥ Eddy current power loss i2(t)R then varies with square of excitation frequency f. ¥ Classical Steinmetz equation for eddy current loss: 2 2 PE = K E f Bmax(core volume) ¥ Ferrite core material impedance is capacitive. This causes eddy current power loss to increase as f4.

Fundamentals of Power Electronics39 Chapter 12: Basic Magnetics Theory Total core loss: manufacturer’s data

Ferrite core 1 Empirical equation, at a material fixed frequency: 1MHz 500kHz 200kHz β 3 Pfe = K fe Bmax A c lm

100kHz Watts / cm Watts

0.1

50kHz Power loss density, Power loss density,

20kHz

0.01 0.01 0.1 0.3

Bmax, Tesla

Fundamentals of Power Electronics40 Chapter 12: Basic Magnetics Theory Core materials

Core type Bsat Relative core loss Applications

Laminations 1.5 - 2.0 T high 50-60 Hz transformers, iron, silicon steel inductors

Powdered cores 0.6 - 0.8 T medium 1 kHz transformers, powdered iron, 100 kHz filter inductors molypermalloy Ferrite 0.25 - 0.5 T low 20 kHz - 1 MHz Manganese-zinc, transformers, Nickel-zinc ac inductors

Fundamentals of Power Electronics41 Chapter 12: Basic Magnetics Theory 12.3.2. Low-frequency copper loss

DC resistance of wire l R = ρ b A w

where Aw is the wire bare cross-sectional area, and ρ lb is the length of the wire. The resistivity is equal to 1.724⋅10-6 Ω cm for soft-annealed copper at room temperature. This resistivity increases to i(t) 2.3⋅10-6ÊΩÊcm at 100ûC.

The wire resistance leads to a power loss of R

2 Pcu = I rms R

Fundamentals of Power Electronics42 Chapter 12: Basic Magnetics Theory 12.4. Eddy currents in winding conductors 12.4.1. The skin effect

wire

current Φ(t) density

eddy wire currents

i(t)

eddy currents i(t)

Fundamentals of Power Electronics43 Chapter 12: Basic Magnetics Theory Penetration depth δ

For sinusoidal currents: current density is an exponentially decaying function of distance into the conductor, with characteristic length δ known as the penetration depth or skin depth. ρ δ = π µ f Wire diameter 0.1 #20AWG penetration depth δ, cm

100˚C #30AWG

25˚C 0.01 #40AWG For copper at room temperature:

δ = 7.5 cm f 0.001 10kHz 100kHz 1MHz frequency

Fundamentals of Power Electronics44 Chapter 12: Basic Magnetics Theory 12.4.2. The proximity effect

Ac current in a conductor induces eddy currents in adjacent conductors by a effect layer 3i process called the proximity 3 . This causes significant –2i power loss in the windings of 2Φ high-frequency transformers 2i and ac inductors. layer 2 –i A multi-layer foil winding, with Φ d >> δ. Each layer carries net current i(t). i layer d 1 J density current

Fundamentals of Power Electronics45 Chapter 12: Basic Magnetics Theory Estimating proximity loss: high-frequency limit

Let P1 be power loss in layer 1: P = I 2 R d 1 rms dc δ layer 3i 3 –2i Power loss P2 in layer 2 is: 2Φ P = I 2 R d +2I 2 R d 2 rms dc δ rms dc δ 2i layer =5P 1 2 –i

Power loss P3 in layer 3 is: Φ 2 d 2 d P3 =2Irms Rdc δ +3Irms Rdc δ i layer d 1 =13P1

Power loss Pm in layer m is: J

2 2 density Pm = ((m Ð 1) + m ) P1 current

Fundamentals of Power Electronics46 Chapter 12: Basic Magnetics Theory Total loss in M-layer winding: high-frequency limit

Add up losses in each layer:

M M 2 Pw δ = Σ Pj = 2M +1 P1 d >> j =1 3 Compare with dc copper loss: If foil thickness were d = δ, then at dc each layer would produce

copper loss P1. The copper loss of the M-layer winding would be P = MP w,dc d = δ 1 For foil thicknesses other than d = δ, the dc resistance and power loss are changed by a factor of d/δ. The total winding dc copper loss is δ P = MP w,dc 1 d So the proximity effect increases the copper loss by a factor of P F = w d >> δ = 1 d 2M 2 +1 R d >> δ δ Pw,dc 3

Fundamentals of Power Electronics47 Chapter 12: Basic Magnetics Theory Approximating a layer of round conductors as an effective foil conductor

(a) (b) (c) (d) Conductor spacing factor: π n d η = d l 4 lw

Effective ratio of conductor thickness lw to skin depth: ϕ η d = δ

Fundamentals of Power Electronics48 Chapter 12: Basic Magnetics Theory 12.4.3. Magnetic fields in the vicinity of winding conductors: MMF diagrams

Two-winding transformer example

core primary layers secondary layers { {

i –i i –i 2i 2i –2i –2i 3i –3i

Fundamentals of Power Electronics49 Chapter 12: Basic Magnetics Theory Transformer example: magnetic field lines

F(x) m p ± ms i = F (x) H(x)= lw

–

F(x)

lw i i i –i –i –i

Fundamentals of Power Electronics50 Chapter 12: Basic Magnetics Theory Ampere’s law and MMF diagram

i i i –i –i –i MMF F(x)

3i 2i i 0 x

F(x) m p ± ms i = F (x) H(x)= lw

Fundamentals of Power Electronics51 Chapter 12: Basic Magnetics Theory MMF diagram for d >> δ

MMF i –i 2i –2i 3i –3i 2i –2i i –i F(x)

3i 2i i 0 x

Fundamentals of Power Electronics52 Chapter 12: Basic Magnetics Theory Interleaved windings

pri sec pri sec pri sec

ii–i i–i –i MMF F(x)

3i 2i i 0 x

Fundamentals of Power Electronics53 Chapter 12: Basic Magnetics Theory Partially-interleaved windings: fractional layers

secondaryprimary secondary

±3i ±3i i i i ±3i ±3i 4 4 4 4 MMF F(x)

1.5 i i 0.5 i 0 –0.5 i x –i –1.5 i

Fundamentals of Power Electronics54 Chapter 12: Basic Magnetics Theory 12.4.4. Power loss in a layer

Approximate computation of copper loss in one layer Assume uniform magnetic fields at surfaces of layer, of strengths H(0) and H(d). Assume that these fields are H(0) H(d)

parallel to layer surface (i.e., neglect layer fringing and assume field normal component is zero). The magnetic fields H(0) and H(d) are driven by the MMFs F(0) and F(d). Sinusoidal waveforms are assumed, and F(d) F(x) rms values are used. It is assumed that F(0) H(0) and H(d) are in phase.

0 d

Fundamentals of Power Electronics55 Chapter 12: Basic Magnetics Theory Solution for layer copper loss P

Solve Maxwell’s equations to find current density distribution within layer. Then integrate to find total copper loss P in layer. Result is ϕ 2 2 ϕ ϕ P = Rdc 2 F (d)+F (0) G1( )±4F(d)F(0) G2( ) nl where (MLT) n2 nl = number of turns in layer, R = ρ l dc η Rdc = dc resistance of layer, lw d (MLT) = mean-length-per-turn, sinh (2ϕ) + sin (2ϕ) G (ϕ)= or circumference, of layer. 1 cosh (2ϕ) ± cos (2ϕ)

sinh (ϕ) cos (ϕ) + cosh(ϕ) sin (ϕ) G (ϕ)= 2 cosh (2ϕ) ± cos (2ϕ) π n ϕ = η d η = d l δ 4 lw

Fundamentals of Power Electronics56 Chapter 12: Basic Magnetics Theory Winding carrying current I, with nl turns per layer

If winding carries current of rms magnitude I, then

F (d)±F(0) = nl I

Express F(d) in terms of the winding current I, as

F(d)=mnl I H(0) H(d) layer The quantity m is the ratio of the MMF F(d) to

the layer ampere-turns nlI. Then, F(0) = m ±1 F(d) m F(d) Power dissipated in the layer can now be written F(x) 2 ϕ ϕ F(0) P = I Rdc Q'( , m)

ϕ 2 ϕ ϕ Q'( , m)= 2m ±2m+1 G1( )±4mm±1 G2( ) 0 d

Fundamentals of Power Electronics57 Chapter 12: Basic Magnetics Theory Increased copper loss in layer

m = 15 1210 8 6 5 4 3 P ϕ ϕ 100 2 = Q'( , m) I Rdc

P 2 2 I Rdc 1.5

10 1

m = 0.5

1 0.1 1 10 ϕ

Fundamentals of Power Electronics58 Chapter 12: Basic Magnetics Theory Layer copper loss vs. layer thickness

m = 15 12 10 8 P = Q'(ϕ, m) 100 Pdc d = δ 6 P 5 P dc ϕ =1 4

3 10

2 Relative to copper 1.5 loss when d = δ 1 1

m = 0.5

0.1 0.1 1 10 ϕ

Fundamentals of Power Electronics59 Chapter 12: Basic Magnetics Theory 12.4.5. Example: Power loss in a transformer winding

Two winding primary layers secondary layers transformer { { Each winding consists of M layers n i n i n i n i n i n i Proximity effect p p p p p p increases copper m = F loss in layer m by n pi M the factor P ϕ ϕ 2 = Q'( , m) 2 I Rdc 1 Sum losses over all 0 primary layers: x P M pri 1 ϕ ϕ FR = = Σ Q'( , m) Ppri,dc M m =1

Fundamentals of Power Electronics60 Chapter 12: Basic Magnetics Theory Increased total winding loss Express summation in F = ϕ G (ϕ)+2 M2 ±1 G(ϕ)±2G(ϕ) closed form: R 1 3 1 2

number of layers M = 15 12 10 8 7 6 5 4 100 3

Ppri FR = 2 Ppri,dc 1.5

10 1

0.5

1 0.1 1 10 ϕ Fundamentals of Power Electronics61 Chapter 12: Basic Magnetics Theory Total winding loss

number of layers M = 15 12 100 10 P 8 pri 7 P 6 pri,dc ϕ =1 5 10 4 3

2 1.5 1 1

0.5

0.1 0.1 1 10 ϕ

Fundamentals of Power Electronics62 Chapter 12: Basic Magnetics Theory 12.4.6. PWM waveform harmonics

Fourier series: i(t) ∞ I ω pk i(t)=I0 +2Σ Ijcos (j t) j =1 with

2 I pk I = sin (jπD) I0 = DI pk j j π Copper loss: 0 DT T t 2 s s Dc Pdc = I 0 Rdc

P = I 2 R j ϕ G ( j ϕ )+2 M2 ±1 G( jϕ)±2G( jϕ) Ac j j dc 1 1 1 3 1 1 2 1

Total, relative to value predicted by low-frequency analysis:

ϕ ∞ 2 π Pcu 2 1 sin (j D) ϕ 2 2 ϕ ϕ = D + Σ G1(j1)+ M ±1 G1( j 1)±2G2( j 1) 2 π2 j=1 DIpk Rdc D jj 3

Fundamentals of Power Electronics63 Chapter 12: Basic Magnetics Theory Harmonic loss factor FH

Effect of harmonics: FH = ratio of total ac copper loss to fundamental copper loss

∞

Σ Pj j =1 FH = P1

The total winding copper loss can then be written

2 2 Pcu = I 0 Rdc + FH FR I 1 Rdc

Fundamentals of Power Electronics64 Chapter 12: Basic Magnetics Theory Increased proximity losses induced by PWM waveform harmonics: D = 0.5

10 D = 0.5 M = 10

FH 8

6 5 4

2 1.5 1

M = 0.5

1 0.1 1 10 ϕ 1

Fundamentals of Power Electronics65 Chapter 12: Basic Magnetics Theory Increased proximity losses induced by PWM waveform harmonics: D = 0.3

100

D = 0.3

M = 10 10 8 6 5 4 3 2 1.5 1 M = 0.5

1 0.1 1 10 ϕ 1 Fundamentals of Power Electronics66 Chapter 12: Basic Magnetics Theory Increased proximity losses induced by PWM waveform harmonics: D = 0.1

100

M = 10 D = 0.1 8 6 FH 5 4 3 2

1.5 10 1

M = 0.5

1 0.1 1 10 ϕ 1 Fundamentals of Power Electronics67 Chapter 12: Basic Magnetics Theory Summary of Key Points

1. Magnetic devices can be modeled using lumped-element magnetic circuits, in a manner similar to that commonly used to model electrical circuits. The magnetic analogs of electrical voltage V, current I, and resistance R, are magnetomotive force (MMF) F, flux Φ, and reluctance R respectively. 2. Faraday’s law relates the voltage induced in a loop of wire to the derivative of flux passing through the interior of the loop. 3. Ampere’s law relates the total MMF around a loop to the total current passing through the center of the loop. Ampere’s law implies that winding currents are sources of MMF, and that when these sources are included, then the net MMF around a closed path is equal to zero. 4. Magnetic core materials exhibit hysteresis and saturation. A core material

saturates when the flux density B reaches the saturation flux density Bsat.

Fundamentals of Power Electronics68 Chapter 12: Basic Magnetics Theory Summary of key points

5. Air gaps are employed in inductors to prevent saturation when a given maximum current flows in the winding, and to stabilize the value of inductance. The inductor with air gap can be analyzed using a simple magnetic equivalent circuit, containing core and air gap reluctances and a source representing the winding MMF. 6. Conventional transformers can be modeled using sources representing the MMFs of each winding, and the core MMF. The core reluctance approaches zero in an ideal transformer. Nonzero core reluctance leads to an electrical transformer model containing a magnetizing inductance, effectively in parallel with the ideal transformer. Flux that does not link both windings, or “leakage flux,” can be modeled using series inductors. 7. The conventional transformer saturates when the applied winding volt- seconds are too large. Addition of an air gap has no effect on saturation. Saturation can be prevented by increasing the core cross-sectional area, or by increasing the number of primary turns.

Fundamentals of Power Electronics69 Chapter 12: Basic Magnetics Theory Summary of key points

8. Magnetic materials exhibit core loss, due to hysteresis of the B-H loop and to induced eddy currents flowing in the core material. In available core

materials, there is a tradeoff between high saturation flux density Bsat and high core loss Pfe. Laminated iron alloy cores exhibit the highest Bsat but also the highest Pfe, while ferrite cores exhibit the lowest Pfe but also the lowest Bsat. Between these two extremes are powdered iron alloy and amorphous alloy materials. 9. The skin and proximity effects lead to eddy currents in winding conductors,

which increase the copper loss Pcu in high-current high-frequency magnetic devices. When a conductor has thickness approaching or larger than the penetration depth δ, magnetic fields in the vicinity of the conductor induce eddy currents in the conductor. According to Lenz’s law, these eddy currents flow in paths that tend to oppose the applied magnetic fields.

Fundamentals of Power Electronics70 Chapter 12: Basic Magnetics Theory Summary of key points

10. The magnetic field strengths in the vicinity of the winding conductors can be determined by use of MMF diagrams. These diagrams are constructed by application of Ampere’s law, following the closed paths of the magnetic field lines which pass near the winding conductors. Multiple-layer noninterleaved windings can exhibit high maximum MMFs, with resulting high eddy currents and high copper loss. 11. An expression for the copper loss in a layer, as a function of the magnetic field strengths or MMFs surrounding the layer, is given in Section 12.4.4. This expression can be used in conjunction with the MMF diagram, to compute the copper loss in each layer of a winding. The results can then be summed, yielding the total winding copper loss. When the effective layer thickness is near to or greater than one skin depth, the copper losses of multiple-layer noninterleaved windings are greatly increased.

Fundamentals of Power Electronics71 Chapter 12: Basic Magnetics Theory Summary of key points

12. Pulse-width-modulated winding currents of contain significant total harmonic distortion, which can lead to a further increase of copper loss. The increase in proximity loss caused by current harmonics is most pronounced in multiple-layer non-interleaved windings, with an effective layer thickness near one skin depth.

Fundamentals of Power Electronics72 Chapter 12: Basic Magnetics Theory