__Operation of a Single-Phase Transformer __

__Operation of a Single-Phase Transformer__

In sketch form, the arrangement of a single-phase transformer is shown in Fig. 5 a. On the secondary side, where the winding has *W _{2}* turns, the transformer acts as a supply source for a load

whose resistance is *r*_{2,load} . On the primary side, where the winding has *W _{1}* turns, it acts as a load with respect to its own power supply.

To begin with, suppose that the secondary circuit of the transformer is open and that the supply voltage *V _{1}* =

*e*causes a current

*i*to flow in its primary winding. The mmf

_{1}*i*induces in the core a magnetic flux whose positive direction is determined by the corkscrew rule (see Fig. 1 a). This flux induces an emf of self-induction, e

_{1}w_{1}*, in the primary and an emf of mutual induction,*

_{L1}*e*in the secondary. (Neither of the emfs is shown in the diagram.) When the secondary circuit is completed. the emf of mutual induction

_{M2}*e*gives rise to a current,

_{M2}*i*, in the load resistance

_{2 }*r*

_{2,load}·

For the polarity of the primary and secondary windings indicated in Fig. 5 and the assumed positive directions for the currents it and *i _{2 }*the magnetic flux induced in the core by the mmf

*i*opposes the magnetic flux due to the mmf

_{2}W_{2}*i*In consequence, the primary and secondary windings are connected' in opposition. which fact is stated by assigning the same lapels to the winding terminals as were used in this Figure .

_{1}W_{1}Therefore the total mmf of the primary and secondary windings is equal to *i _{1}W_{1}* –

*i*. This mmf induces a common magnetic flux Φ in the core. Also, in analysing the operation of the transformer, we should include the leakage flux linkage of the primary, ψ

_{2}W_{2 }*, and of the secondary, ψ*

_{leak,l}*, each proportional to the respective current,*

_{leak,2}*i*or

_{1}*i*.

_{2}Figure 5 b shows an equivalent circuit of the transformer; it is made up of the resistances *r _{1}* and

*r*of the primary and secondary windings, and of the respective leakage inductances,

_{2}*L*= = ψ

_{leak,l}*/*

_{leak,l}*i*and

_{1}*L*= ψ

_{leak,2}*(similarly to Fig. 4).*

_{leak,2}It is useful to introduce the concept of an ideal transformer, that is, one in which the primary and secondary windings have neither resistance nor leakage flux linkage. In Fig. 5 b it is enclosed in the dashed box. The positive directions for *e*_{1} and it in its primary are the same as they are in an iron-cored coil (see Fig. 4) to which the transformer reduces when its secondary circuit is opened. Since the primary emf

*e _{1} = -W_{1} *dΦ

*/*d

*t*

and the secondary emf

*e*_{2} = -*W*_{2} dΦ/d*t*

are induced by the same magnetic flux Φ in the core, their positive directions relative to the like terminals of the two windings are the same. While in the primary winding e_{1} and i_{l} are in the same direction, in the secondary e_{2} and i2 are assigned opposite directions. This corresponds to the physical interpretation of the role played by the emf: in the former case it opposes any change in the current; in the latter case, it gives rise to a current.

__The Equation of an Ideal Single-Phase Transformer __

__The Equation of an Ideal Single-Phase Transformer__

To begin with, we will consider an ideal single-phase transformer in which the core is fabricated from a ferromagnetic material having a linear *B/H* relation, *B* = *μ _{r}μ_{0}*H (see Fig. 6 c).

As has been noted in Sec. 6.3, the magnetic field in a core is not the same over its cross-sectional area S. For simplicity, we will ignore this fact and assume that both the magnetic induction and the magnetic intensity in the transformer core are everywhere the same as they are at the central magnetic line. If so, the electric circuit of the transformer is linear, and it can be analysed using the complex-number method.

in Fig. 6 an ideal single-phase transformer is shown connected between a source of emf *Ė* and a load of complex impedance __z___{2} ∠ φ_{2} ·

Let us write the values of *Ė _{ l}* and

*Ė*induced in the primary and secondary windings of the ideal transformer by the magnetic flux Φ̇ .

_{ 2}in the core. By the law of electromagnetic induction in complex form, Eq. (2.33)

*Ė _{ l}* = -jɷ

*w*

_{1}Φ̇ = - jɷ

*w*

_{1}

*Ḃ*S = -jɷ

*w*

_{l}μ

_{r}μ

_{0}

*ḢS*(8.1a)

*Ė _{ 2}* = -jɷ

*w*

_{2}Φ̇ = - jɷ

*w*

_{2}

*Ḃ*S = -jɷ

*w*

_{2}μ

_{r}μ

_{0}

*ḢS*(8.1b)

where *Ḃ* is the complex magnetic induction, and *Ḣ* is the complex magnetic intensity.

If the complex currents in the primary and secondary windings of an ideal single-phase transformer are *İ*_{1} and *İ*_{2 }, then, by virtue of Eq. (6.2), the average magnetic intensity, that is, one at the centre line of the core, is

*Ḣ* = *İ*_{1}*w*_{1}/*Ɩ*_{av} - *İ*_{2}*w*_{2}/*Ɩ*_{av} (8.2)

By definition, the source emf is *Ė* = *V̇*_{1} the primary emf is

*Ė*_{1} = - *V̇*_{1} and the secondary emf is *Ė*_{2} = - *V̇*_{2} (see Fig. 6). Therefore, subject to Eqs. (8.1) and (8.2),

*V̇*_{1} = -jɷ*w*_{1}Φ̇ = jɷ*w*_{l}μ_{r}μ_{0}*S* (*İ*_{1}*w*_{1}/*Ɩ*_{av} -* İ*_{2}*w*_{2}/*Ɩ*_{av}) (8.3a)

*V̇*_{2} = Z_{2}*İ*_{2}= jɷ*w*_{2}Φ̇ (8.3b)

Notably, at no-load, or in the open-circuit condition, (which implies that the secondary circuit is open and *İ _{2}* = 0)

*V̇*_{1} = jɷ*w*_{l}μ_{r}μ_{0}*S* (*İ*_{1,oc}*w*_{l}/* Ɩ*_{av}) (8.3c)

where *İ*_{1,oc} is the *open-circuit* (or *no-load*) current also called the *magnetizing current*.

Since the source emf , *Ė*_{1} = *V̇*_{1} is a constant quantity, then, by virtue of Eqs. (8.3a) and (8.3c)

*İ*_{1}*w*_{1}/*Ɩ*_{av} - *İ*_{2}*w*_{2}/*Ɩ*_{av} = *İ*_{1,oc}*w*_{1}/*Ɩ*_{av} = const (8.4)

On dividing Eq. (8.3b) by (8.3a) termwise, we get

*V _{2}/V_{1}* =

*w*

_{2}/

*w*

_{1}=

*n*

_{21}(8.5)

which is the turns (or *transformation*) ratio of an ideal single-phase transformer. on substituting for Φ̇ from (8.3b) into (8.3a), we get

*V̇*_{1} = * Z_{2}* (

*w*

_{1}/

*w*

_{2})

*İ*

_{2}(8.6)

Now we multiply and divide the right-hand side of Eq. (8.6)

by *w*_{1}/*w*_{2 }

*V̇*_{1 }=* Z_{2 }*(

*w*

_{1}/

*w*

_{2})

^{2}(

*w*

_{2}/

*w*

_{1})

*İ*

_{2}=

__Z'___{2}İ'_{2 }(8.7)

where

__Z'__* _{2}* =

*(*

__Z___{2}*w*

_{1}/

*w*

_{2})

^{2}=

__Z___{2}/

*n*(8.8)

^{2}_{21 }is the complex impedance of the secondary circuit referred (or transferred) to the primary side or, simply, the referred (transferred) impedance, and

*İ'*_{2} = (*w*_{2}/*w*_{1}) *İ*_{2} = *n _{21}İ*

_{2 }(8.9)

is the complex current in the secondary circuit, referred (or transferred) to the primary side or, simply, the referred (transfer ret!) current.

By invoking the concepts of transferred current and impedance we may re-cast Eqs. (8.4) and (8.3) as

*I _{1 }*-

*İ'*

_{2 }=

*İ*

_{1,oc }(8.10a)

*V̇'*_{2 }= (*w*_{1}/*w*_{2}) *V̇*_{2 }= _{}*V̇*_{2}/*n _{21}*

_{ }=

__Z'___{2}İ'_{2 }(8.10c)

where *w ^{2}*

_{2}μ

_{r}μ

_{0}

*S*/

*Ɩ*

_{av}=

*L*is the inductance of the primary winding in an ideal single-phase transformer, and

_{1}*V̇'*

_{2}is the complex secondary voltage of an ideal single-phase transformer, referred (or transferred) to the primary side or, Simply, the

*referred (transferred) voltage.*

Equations (8.10) correspond to the equivalent circuit shown in Fig. 7 in which the ideal transformer model is enclosed in a dashed box.

A further assumption made for an ideal transformer is that at μ_{r} = ∞ , the inductive reactance

* _{L1 }*is infinitely large and the magnetizing current is

*İ*

_{1,oc}= 0 .

With the secondary circuit opened, an ideal single-phase transformer reduces to an ideal iron-cored coil. As a consequence, an ideal single-phase transformer at no-load has the same equivalent circuit as an ideal iron-cored coil (see Fig. 4) on the condition that the coil and the primary winding of the transformer have the same number of turns and the same cores.