Chapter 2: transformers  (Page 4/6)

$Z_{\text{sc}}=R_1+{\text{jX}}_{1_1}+\frac{Z_{\varphi }(R_2+{\text{jX}}_{1_2})}{Z_{\varphi }+R_2+{\text{jX}}_{1_2}}$ (2.27)

$Z_{\text{sc}}\approx R_1+{\text{jX}}_{1_1}+R_2+{\text{jX}}_{1_2}=R_{\text{eq}}+{\text{jX}}_{\text{eq}}$ (2.28)

Typically the instrumentation will measure the rms magnitude of the applied voltage $V_{\text{sc}}$ , the short-circuit current $I_{\text{sc}}$ , and the power $P_{\text{sc}}$ . The circuit parameters (referred to the primary) can be found as (2.29)-(2.31).

$\mid Z_{\text{eq}}\mid =\mid Z_{\text{sc}}\mid =\frac{V_{\text{sc}}}{I_{\text{sc}}}$ (2.29)

$R_{\text{eq}}=R_{\text{sc}}=\frac{P_{\text{sc}}}{I_{\text{sc}}^2}$ (2.30)

$X_{\text{eq}}=X_{\text{sc}}=\sqrt{\mid Z_{\text{sc}}{\mid }^2-R_{\text{sc}}^2}$ (2.31)

• The equivalent impedance can be referred from one side to the other.
• Approximate values of the individual primary and secondary resistances and leakage reactances can be obtained by assuming that $R_1=R_2=0\text{.}{\mathrm{5R}}_{\text{eq}}$ and $X_{l_1}=X_{l_2}=0\text{.}{\mathrm{5X}}_{\text{eq}}$ when all impedances are referred to the same side.
• Note that it is possible to measure $R_1$ and $R_2$ directly by a dc resistance measurement on each winding. However, no such simple test exists for $X_{l_1}$ and $X_{l_2}$ .

• Open-Circuit Test

• The test is used to find the equivalent shunt impedance $R_c\text{//}{\text{jX}}_m$ .
• The test is performed with the secondary open-circuited and rated voltage impressed on the primary. If the transformer is to be used at other than its rated voltage, the test should be done at that voltage.
• An exciting current of a few percent of full-load current is obtained.
• See Fig. 2.16. Note that $Z_{\varphi }=R_c\text{//}{\text{jX}}_m$ .

Figure 2.13 Equivalent circuit with open-circuited secondary. (a) Complete equivalent circuit.(b) Cantilever equivalent circuit with the exciting branch at the transformer primary.

$Z_{\text{oc}}=R_1+{\text{jX}}_{1_1}+Z_{\varphi }=R_1+{\text{jX}}_{1_1}+\frac{R_c({\text{jX}}_m)}{R_c+{\text{jX}}_m}$ (2.32)

$Z_{\text{oc}}\approx Z_{\varphi }=\frac{R_c({\text{jX}}_m)}{R_c+{\text{jX}}_m}$ (2.33)

• Typically the instrumentation will measure the rms magnitude of the applied voltage $V_{\text{oc}}$ , the open-circuit current $I_{\text{oc}}$ , and the power $P_{\text{oc}}$ . The circuit parameters (referred to the primary) can be found as (2.34)-(2.36).

$R_c=\frac{V_{\text{oc}}^2}{P_{\text{oc}}}$ (2.34)

$\mid Z_{\varphi }\mid =\frac{V_{\text{oc}}}{P_{\text{oc}}}$ (2.35)

$X_m=\frac{1}{\sqrt{(1/\mid Z_{\varphi }\mid {)}^2-(1/R_c{)}^2}}$ (2.36)

• The open-circuit test can be used to obtain the core loss for efficiency computations and to check the magnitude of the exciting current.

• Note the term “Voltage Regulation” which is to be discussed in Example 2.6.

§2.6 Autotransformers; Multiwinding Transformers

• Two-winding  Other winding configurations.

§2.6.1 Autotransformers

• Autotransformer connection: Fig. 2.14.

Figure 2.14 (a) Two-winding transformer. (b) Connection as an autotransformer.

• The windings of the two-winding transformer are electrically isolated whereas those of the autotransformer are connected directly together.
• In the transformer connection, winding ab must be provided with extra insulation.
• Autotransformer have lower leakage reactances, lower losses, and smaller exciting current and cost less than two-winding transformers when the voltage ration does not differ too greatly from 1:1.
• The rated voltages of the transformer can be expressed in terms of those of the two-winding transformer as

$V_{L_{\text{rated}}}=V_{1_{\text{rated}}}$ (2.37)

$V_{H_{\text{rated}}}=V_{1_{\text{rated}}}+V_{2_{\text{rated}}}=\frac{N_1+N_2}{N_1}V_{L_{\text{rated}}}$ (2.38)

• The effective turns ratio of the autotransformer is thus $(N_1+N_2)/N_1$ .
• The power rating of the autotransformer is equal to $(N_1+N_2)/N_2$ times that of the two winding transformer.

§2.6.2 Multiwinding Transformers

• Transformers having three or more windings, known as multiwinding or multicircuit transformers, are often used to interconnect three or more circuits which may have different voltages.