4.4.4. Notion of the ideal and the real transformers

A transformer is a device for the transfer of energy from one part of an electrical circuit to another, based on the use of the phenomenon of mutual induction. The transformer consists of a primary coil (winding), terminals of which are connected to the source of energy, and one or more secondary inductors coils (windings), terminals of which are connected to the loads. Network of double-wound transformer is shown in fig. 4.26. For the loops I and II we can write down in the complex form.

(4.188)

Here voltages of mutual induction j M I , j M I of the first and second windings are taken with a minus sign, because the winding are opposite connected ( currents i and i differently oriented relative to the same name terminals marked with dot) .

Fig. 4.26

Considering the transformer without core, that is, as the line element, we consider, as noted above , M = M = M. Then

(4.189)

From (4.189) it is obvious the current I of secondary winding is directly connected with current I of the primary winding. Thus if the current I of the secondary winding increase at a constant voltage U of primary winding, from the first equation in (4.189), current I of the primary winding is also increase. Physically it can be explained by the fact that the magnetic flux of mutual induction , created by the current I of the secondary winding, directed opposed to the magnetic flux of self-induction , created by the current I of the primary winding and, is deducted from , reduces the total magnetic flux of the primary winding. The result is a reduced full flux linkage (4.151) and equivalent inductance on the part of the primary winding. Therefore, its inductive reactance decrease and at constant voltage U current I is increase.

Using (4.189) you can build the equivalent circuit of a transformer without magnetic coupling of coils (Fig.4.27). This network is applied to the analysis and calculation of transformers.

Fig. 4.27

Let us consider the mode of no-load of transformer (I = 0, Z , fig.4.26, 4.27). We obtain from (4.189, a)

(4.189.a)

Current I is called no-load current or current of magnetization (Fig. 4.27). Under load this current decreases, and with an increase of the primary coil inductance L (L ) tends to zero. From (4.189) in no-load regime we get

(4.189.b)

For the analysis it is convenient to introduce the concept of the ideal transformer, for which there are no active losses (r = r = 0) and the coefficient of coupling (4.159) is equal to unit

(4.190)

Then, from (4.189.b)

(4.191)

The value n is called the coefficient of transformation. As of (4.154), (4.155)

(4.192)

and = F = 0, which follows from (4.190), then from (4.191)

(4.193)

Let the terminals of the secondary winding of the ideal transformer load impedance Z is connected (Fig. 4.26 and Fig. 4.27), that is

(4.193,a)

Then, from (4.189), (4.193,a) we get

(4.193,b)

Hence

(4.193,c)

(4.194)

Let us consider the relation

(4.195)

or with the account of (4.190)

(4.196)

Usually Z << j L . Then, with the account of (4.190), we obtain

(4.197)

Using (4.189), (4.193.a) we can build a vector diagram of the transformer under load (Fig. 4.28). Here with inductive load Z ( > 0) vector of the load voltage U at first put, then the vectors of voltage drop ocross the resistance R (in-phase with the current I ) and across the inductance L (at an angle to the current I ). Since the sum of the voltages in the loop of the secondary winding of the (4.189) equals to zero, then, connecting the origin of coordinates with the end of the vector j L I , we obtain the vector of mutual induction voltage j M I , which lags behind the current I to the angle . Hence we construct the vector of primary current I , the voltage drop across the resistance R and inductance L . The voltage vector mutual induction voltage j M I is built at an angle of " " to the vector of secondary current I . Combining the beginning of the origin coordinate with the end of the vector j M I , we obtain the vector of the primary voltage U .

Fig. 4.28

Let us consider the input resistance of the primary winding of transformer

(4.198)

That is the ideal transformer changes the impedance of the load in n times without changes of the argument of impedance. This property is used for the load match with the internal resistance of the power supply on the primary side of the transformer.