Problems solved about Real Transformer

Problem 92-5 Source: Problem developed by the site author.

A single-phase transformer of 250 KVA, with a primary voltage of 13,200 V and a secondary voltage of 440 V is put to the test, and the results are shown in the table below.

Test Type	Voltage (V)	Current (A)	Real Power (W)
Open Circuit	13,200	1.20	2,000
Short Circuit	620	18.94	5,000

a) In this condition, find the efficiency of the transformer.

b) Determine the equivalent parameters of the transformer electrical circuit referred to the primary.

Solution of the Problem 92-5

Item a

To determine the efficiency of the transformer we will use the eq. 92-21, reproduced below.

eq. 92-21

Note that V₂ I₂ cos φ₂ represents the rated power of the transformer and how the power factor is unity, so this value is equal to 250,000 W. The losses in iron have been given and their value is P_fe = 2,000 W. And the portion R"_eq I₂² is the copper losses, this value given by the short circuit test and, in this case, it is worth 5,000 W. Thus, substituting these values in the eq. 92-21, we find the transformer efficiency, or:

η = 250,000 / ( 250,000 + 2,000 + 5,000 ) = 0.9728

The usual thing is to write the efficiency in percent, or

η = 97.28%

Item b

To determine the transformer parameters, let's start with the open circuit test and calculate the values of R_h and X_m, using eq. 92-04, reproduced below.

eq. 92-04

Substituting for the numerical values, we find

R_h = 13,200² / 2,000 = 87,120 Ω

With the given data we can calculate the equivalent circuit of the transformer excitation circuit. Therefore, let's use the eq. 92-05, shown below, where I_p represents the current absorbed by the transformer in open circuit.

eq. 92-05

Substituting for the numerical values, we find:

Z_ex = 13,200 / 1.2 = 11,000 Ω

With the values found we are able to calculate the value of X_m, using eq. 92-06 shown below.

eq. 92-06

Substituting for the numerical values, we find:

X_m = 11.089 Ω

These values found are referenced to the transformer primary. Now, using the short-circuit test, we can determine the equivalent circuit for the transformer winding. So, using eq. 92-09 let's calculate the value of Z'_eq which is the value referenced to the primary, according to the adopted convention.

eq. 92-09

Substituting for the numerical values, we find:

Z'_eq = 620 / 18.94 = 32,73 Ω

We can find the value of R'_eq using eq. 92-10, shown below.

eq. 92-10

Substituting for the numerical values, we find:

R'_eq = 5.000 / 18.94² = 13.94 Ω

And finally we can calculate the value of X'_eq using eq. 92-11 shown below.

eq. 92-11

Substituting for the numerical values, we will find the value of:

X'_eq = 29.61 Ω

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