“Real” Transformer

EP = 4.44NPfΦP ES = 4.44NSfΦS

ΦP = ΦM + Φlp ΦS = ΦM – Φls

EP = E’P + Elp ES = E’S - Els

KVL @ primary VT = EP + IPRP = E’P+ Elp +IPRP

IP = Ife + IM + IP LOAD

“Real” Transformer (continued)

Equivalent Circuit of a Real Transformer

VT = EP + IPRP = E’P+ Elp +IPRP

KVL @ secondary ES = ISRS + VLOAD = E’S – Els

E’S = Els + ISRS + VLOAD

“Real” Transformer (continued)

Equivalent Circuit of a Real Transformer

VT = EP + IPRP = E’P+ Elp +IPRP

E’S = Els + ISRS + VLOAD

s sls

s

N I

lR

s sls

s

N I

lR

Flux in the “Leakage coils”

Voltage generated by flux through the window of a coil

max,max

2

,max max

2

, ,

2

2

2

2

(2 )( )

p pp p

p

pp p

p

pp rms p rms

p

p p p

p p p

N IE fN

NE f I

NE f I

NL

E fL I

E I X

ll

ll

ll

l l

l l

R

R

R

R

Similar expression for the secondary

Equivalent Circuit using Leakage Reactance

Equivalent Impedance of a Transformer

'' 2 Sin

S

EZ a

I

'

'

' 2

' 2 2 2

( )

s

SS

S s load

SS s load

S

in S s load

in S load

EI

R jX Z

ER jX Z

I

Z a R jX Z

Z a R ja X a Z

l

l

l

l

Equivalent CircuitParameters Referred to the Primary

2 2,

, , ,

( )eq P P S p s

eq P eq P eq P

Z R a R j X a X

Z R jX

l l

Define ZeqP = The “equivalent impedance” with all parameters referred to the primary

2 2,

, , ,

( )eq P P S p s

eq P eq P eq P

Z R a R j X a X

Z R jX

l l

Equivalent CircuitParameters Referred to Secondary

, 2 2

, , ,

( )pPeq S S s

eq S eq S eq S

XRZ R j X

a aZ R jX

ll

Note that the excitation branch has been moved to the left! It’s OK as long as the transformer is operated near rated conditions!

Equivalent Circuit for Step-Down Operation Using High-Side Values

Use this equivalent circuit to determine the input impedance of the transformer and load.

Equivalent Circuit for Step-Down Operation Using Low-Side Values

Use this equivalent circuit to determine the no-load voltage and the voltage regulation.

Equivalent Circuit for Step-Up Operation Using Low-Side Values

Use this equivalent circuit to determine the input impedance of the transformer and load.

Equivalent Circuit for Step-UpHigh-Side Values

Use this equivalent circuit to determine the no-load voltage and the voltage regulation.