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N 1 i 1 N 2 i 2 = 0 size 12{N rSub { size 8{1} } i rSub { size 8{1} } - N rSub { size 8{2} } i rSub { size 8{2} } =0} {} (2.11)

N 1 i 1 = N 2 i 2 size 12{N rSub { size 8{1} } i rSub { size 8{1} } =N rSub { size 8{2} } i rSub { size 8{2} } } {} (2.12)

i 1 i 2 = N 2 N 1 size 12{ { {i rSub { size 8{1} } } over {i rSub { size 8{2} } } } = { {N rSub { size 8{2} } } over {N rSub { size 8{1} } } } } {} (2.13)

  • An ideal transformer transforms currents in the inverse ratio of the turns in its windings.
  • From (2.10) and (2.13),

v 1 i 1 = v 2 i 2 size 12{v rSub { size 8{1} } i rSub { size 8{1} } =v rSub { size 8{2} } i rSub { size 8{2} } } {} (2.14)

    • Instantaneous power input to the primary equals the instantaneous power output from the secondary.
  • Impedance transformation properties: Fig. 2.8.

Figure 2.8 Three circuits which are identical at terminals ab when the transformer is ideal.

v ˆ 1 = N 1 N 2 v ˆ 2 and { v ˆ 2 = N 2 N 1 v ˆ 1 size 12{ { hat {v}} rSub { size 8{1} } = { {N rSub { size 8{1} } } over {N rSub { size 8{2} } } } { hat {v}} rSub { size 8{2} } " and {" hat ital {v}} rSub { size 8{2} } = { {N rSub { size 8{2} } } over {N rSub { size 8{1} } } } { hat {v}} rSub { size 8{1} } } {} (2.15)

I ˆ 1 = N 1 N 2 I ˆ 2 and { I ˆ 2 = N 2 N 1 I ˆ 1 size 12{ { hat {I}} rSub { size 8{1} } = { {N rSub { size 8{1} } } over {N rSub { size 8{2} } } } { hat {I}} rSub { size 8{2} } " and {" hat ital {I}} rSub { size 8{2} } = { {N rSub { size 8{2} } } over {N rSub { size 8{1} } } } { hat {I}} rSub { size 8{1} } } {} (2.16)

V ˆ 1 I ˆ 1 = N 1 N 2 2 V ˆ 2 I ˆ 2 size 12{ { { { hat {V}} rSub { size 8{1} } } over { { hat {I}} rSub { size 8{1} } } } = left ( { {N rSub { size 8{1} } } over {N rSub { size 8{2} } } } right ) rSup { size 8{2} } { { { hat {V}} rSub { size 8{2} } } over { { hat {I}} rSub { size 8{2} } } } } {} {} (2.17)

Z 2 = V ˆ 2 I ˆ 2 size 12{Z rSub { size 8{2} } = { { { hat {V}} rSub { size 8{2} } } over { { hat {I}} rSub { size 8{2} } } } } {} (2.18)

Z 1 = N 1 N 2 2 Z 2 size 12{Z rSub { size 8{1} } = left ( { {N rSub { size 8{1} } } over {N rSub { size 8{2} } } } right ) rSup { size 8{2} } Z rSub { size 8{2} } } {} (2.19)

  • Transferring an impedance from one side to the other is called “referring the impedance to the other side.” Impedances transform as the square of the turns ratio.
    • Summary for the ideal transformer:
  • Voltages are transformed in the direct ratio of turns.
  • Currents are transformed in the inverse ratio of turns.
  • Impedances are transformed in the direct ratio squared.
  • Power and voltamperes are unchanged.

§2.4 Transformer Reactances and Equivalent Circuits

  • A more complete model must take into account the effects of winding resistances, leakage fluxes, and finite exciting current due to the finite and nonlinear permeability of the core.
  • Note that the capacitances of the windings will be neglected.
  • Method of the equivalent circuit technique is adopted for analysis.
  • Development of the transformer equivalent circuit
  • Leakage flux: Fig. 2.9.

Figure 2.9 Schematic view of mutual and leakage fluxes in a transformer.

  • L 1 1 size 12{L rSub { size 8{1 rSub { size 6{1} } } } } {} = primary leakage inductance, X 1 1 size 12{X rSub { size 8{1 rSub { size 6{1} } } } } {} = primary leakage reactance

X 1 1 = fL 1 1 size 12{X rSub { size 8{1 rSub { size 6{1} } } } =2π ital "fL" rSub {1 rSub { size 6{1} } } } {} (2.20)

  • Effect of the primary winding resistance: R 1 size 12{R rSub { size 8{1} } } {}
  • Effect of the exciting current:

N 1 I ˆ ϕ = N 1 I ˆ 1 N 2 I ˆ 2 = N 1 ( I ˆ ϕ + I ˆ 2 ' ) N 2 I ˆ 2 alignl { stack { size 12{N rSub { size 8{1} } { hat {I}} rSub { size 8{ϕ} } =N rSub { size 8{1} } { hat {I}} rSub { size 8{1} } - N rSub { size 8{2} } { hat {I}} rSub { size 8{2} } } {} #" "=N rSub { size 8{1} } \( { hat {I}} rSub { size 8{ϕ} } + { hat {I}} sup { ' } rSub { size 8{2} } \) - N rSub { size 8{2} } { hat {I}} rSub { size 8{2} } {} } } {} (2.21)- (2.22)

I ˆ 2 ' = N 2 N 1 I ˆ 2 size 12{ { hat {I}} sup { ' } rSub { size 8{2} } = { {N rSub { size 8{2} } } over {N rSub { size 8{1} } } } { hat {I}} rSub { size 8{2} } } {} (2.23)

  • L m = size 12{L rSub { size 8{m} } ={}} {} magnetizing inductance, X m = size 12{X rSub { size 8{m} } ={}} {} magnetizing reactance

X m = fL m size 12{X rSub { size 8{m} } =2π ital "fL" rSub { size 8{m} } } {} (2.24)

  • Ideal transformer:

E ˆ 1 E ˆ 2 = N 1 N 2 size 12{ { { { hat {E}} rSub { size 8{1} } } over { { hat {E}} rSub { size 8{2} } } } = { {N rSub { size 8{1} } } over {N rSub { size 8{2} } } } } {} (2.25)

  • Secondary resistance, secondary leakage reactance
  • Equivalent-T circuit for a transformer:

X ˆ 1 2 = N 1 N 2 2 X 1 2 , { R 2 ' = N 1 N 2 2 R 2 , { V 2 ' = N 1 N 2 2 V 2 size 12{ { hat {X}} rSub { size 8{1 rSub { size 6{2} } } } = left ( { {N rSub {1} } over { size 12{N rSub {2} } } } right ) rSup {2} size 12{X rSub {1 rSub { size 6{2} } } } size 12{" , {" ital {R}} sup { ' } rSub {2} } size 12{ {}= left ( { {N rSub {1} } over { size 12{N rSub {2} } } } right ) rSup {2} } size 12{R rSub {2} } size 12{" , {" ital {V}} sup { ' } rSub {2} } size 12{ {}= left ( { {N rSub {1} } over { size 12{N rSub {2} } } } right ) rSup {2} } size 12{V rSub {2} }} {} (2.26)

  • Steps in the development of the transformer equivalent circuit: Fig.2.10
  • The actual transformer can be seen to be equivalent to an ideal transformer plus external impedances
  • Refer to the assumptions for an ideal transformer to understand the definitions and meanings of these resistances and reactances.

Figure 2.10 Steps in the development of the transformer equivalent circuit.

§2.5 Engineering Aspects of Transformer Analysis

  • Approximate forms of the equivalent circuit:

Figure 2.11 Approximate transformer equivalent circuits.

  • Two tests serve to determine the parameters of the equivalent circuits of Figs. 2.10 and 2.11.
  • Short-circuit test and open-circuit test
  • Short-Circuit Test
  • The test is used to find the equivalent series impedance R eq + jX eq size 12{R rSub { size 8{ ital "eq"} } + ital "jX" rSub { size 8{ ital "eq"} } } {} .
  • The high voltage side is usually taken as the primary to which voltage is applied.
  • The short circuit is applied to the secondary
  • Typically an applied voltage on the order of 10 to 15 % or less of the rated value will result in rated current.
  • See Fig. 2.12.Note that Z ϕ = R c // jX m size 12{Z rSub { size 8{ϕ} } =R rSub { size 8{c} } "//" ital "jX" rSub { size 8{m} } } {} .

Figure 2.12 Equivalent circuit with short-circuited secondary. (a) Complete equivalent circuit.(b) Cantilever equivalent circuit with the exciting branch at the transformer secondary.

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Source:  OpenStax, Electrical machines. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10767/1.1
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