Magnetic Design Considerations

• Transformers– Used to step-up or step-down voltages

• Inductors– Storage during energy transfer– Carries a DC current while supplying current– Need to avoid saturation of the core

Transformer Primary Voltage in Terms of Flux

mmm

rms

mm

m

mm

fNfNE

E

NE

tEe

tNtNdt

dNe

44.42

2

2

)90sin(

)90sin()cos(

Transformer Powers and Efficiency

• Pt = apparent transformer power

• Pi = transformer input power

• Po = transformer output power

• η = transformer efficiency

Bridge Rectifier Example

Analysis

• Primary voltage V1

– V1 = KtfN1Φm (Kt = 4.44 for sinusoids)

• Apparent Power of the transformer– Pt = V1I1 + V2I2

• For N1 = N2 = N and I1 = I2 = I– Pt = VI = KtfNΦmI– Pt = KtfBmAcNI

• Bm = peak flux density• Ac = cross-sectional area of the flux path

Analysis (continued)

• Number of Ampere-Turns– NI = KuWaJ

• Ku = fill factor (between 4 and 6)

• Wa = window area

• Area Product– Ap = WaAc = Pt/(KtfBcKuJ)

• Current Density– J = KjAp

x

Core Configuration Constants

Area Product Ap

41

1410

cmKKfBK

PA

x

fumt

tp

Core Area for Various Core Types

Wa

Design of a TransformerExample 14.6

f = 60Hz V1 = 120V 60-Hz square-wave Vo = 40V, Io = 6.5A

Assume η = 95% and window factor Ku = 0.4

Use E-core

Solution

• Kt = 4 for a square wave

• Output Power– Po = VoIo = (40V)(6.5A) = 260 W

• Total Power– Pt = Po(1/η + 1) = 260(1/0.95 + 1) = 533.7 W

• Check Table 14.1 for E-core– Kj = 366 x = -0.14

• Choose Bm = 1.4

41

1410

cmKKfBK

PA

x

fumt

tp

4

4892.0

412.01

14

1.206

)2.115(

)366)(4.0)(4.1)(60)(4(

10537

cmA

cmA

cmA

p

p

p

Choosing the E-core

• Choose a type core2-138EI– Ap = 223.39 cm4

– core weight, Wt = 3.901 kg

– core area, Ac = 24.4 cm2

– mean length of a turn, lmt = 27.7 cm

Calculate the number of primary turns

132

)4.24)(4.1)(60)(44.4(

10120

10

4

41

1

11

p

p

cmtp

p

cmm

mt

N

N

AfBK

VN

NN

AB

fNKV

Calculate the number of secondary turns

44120

40132

1

1

s

s

ops

o

p

s

N

N

V

VNN

V

V

N

N

Calculate the primary current

AI

I

V

PPI

cmAJ

J

AKJ

ot

xpj

28.2120

2607.533

/6.173

7.206366

1

1

11

2

14.0

Calculate the primary wire size

2

2

1

013.0

/3.176

28.0

cmA

cmA

AA

J

IA

wp

wp

wp

Check wire-size Table B.2 in Appendix B

AWG #16 has cross-sectional area of 13.07x10-3cm2

Calculate the primary copper loss

WP

P

RIP

R

R

cm

NlR

p

s

sss

s

s

s

ssmls

1.2

)05.0()5.6(

05.0

)1037.41)(44)(7.27(

/37.41

10

2

2

6

6

Calculate the secondary wire size

2

2

037.0

/6.173

5.6

cmA

cmA

AA

J

IA

ws

ws

ows

Check wire-size Table B.2 in Appendix B

AWG #11 has cross-sectional area of 41.68x10-3cm2

Calculate the secondary copper loss

WP

P

RIP

R

R

cm

NlR

s

s

sss

s

s

s

ssmls

1.2

)05.0()5.6(

05.0

)1037.41)(44)(7.27(

/37.41

10

2

2

6

6

Push-Pull

Half Bridge with Center-Tap Transformer

Cores With 2 Permeability Regions