“Ideal Transformer”

E’P , E’S = Voltages induced in the primary, secondary

E’HS , E’LS = Voltages induced at High-side, Low-Side

VHS , VLS = Nameplate voltages of High-side, Low-Side ECE 441 1

“Ideal Transformer” (continued)

• Define the turns ratio as “a”

– If the primary is the High-Side,

' '

' '

P HS HS HS

S LS LS LS

E E N VaE E N V

'

'

P P

S S

E NaE N

ECE 441 2

Input Impedance, Z’in

Z’in = Impedance looking into the primary = E’P / IP

Turns ratio a = E’P / E’S >>>>> E’P = aE’SECE 441 3

Input Impedance, Z’in (continued)

• For an IDEAL transformer, VAin =VAout

• E’PI*P = E’SI*

S

• I*P = E’SI*

S / E’P = I*S / a

• IP = IS / a

' ' '' 2 2

1P S S

in LOAD

P SS

E aE EZ a aZI II

a

ECE 441 4

“Real” Transformer

EP = 4.44NPfΦP ES = 4.44NSfΦS

ΦP = ΦM + Φlp ΦS = ΦM – Φls

EP = E’P + Elp ES = E’S - Els ECE 441 5

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

IP = Ife + IM + IP LOAD

“Real” Transformer (continued)

ECE 441 6

Equivalent Circuit of a Real Transformer

VT = EP + IPRP = E’P+ Elp +IPRPECE 441 7

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

E’S = Els + ISRS + VLOAD

“Real” Transformer (continued)

ECE 441 8

Equivalent Circuit of a Real Transformer

VT = EP + IPRP = E’P+ Elp +IPRP

E’S = Els + ISRS + VLOADECE 441 9

Equivalent Circuit using Leakage Reactance

ECE 441 10

Looking into the primary of the Ideal Transformer

Z’in = a2ZLOAD = a2(E’S / IS)

IS = E’S /(RS +jXls + ZLOAD) >>> E’S / IS = RS + jXls + ZLOAD

Z’in = a2RS + ja2Xls + a2ZLOAD ECE 441 11

Equivalent CircuitParameters Referred to Primary

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

ZeqP = RP + a2RS +jXlp +ja2XlsECE 441 12

Equivalent CircuitParameters Referred to Secondary

ZeqS = RS + RP/a2 +j(Xls + Xlp/a2)

ECE 441 13