Problems solved about Real Transformer

Problema 92-2 Source: Problem developed by the site author.

Consider a single-phase 50 kVA transformer with a rated voltage of 13,200 / 220 volts and that underwent a test where the measuring instruments were located in the primary and were obtained the following readings:

Test Type	Voltage (V)	Current (A)	Real Power (W)
Open Circuit	13,200	0.32	900
Short Circuit	850	3.79	280

a) Determine the transformer resistances and reactances referred to the primary.

b) Draw a diagram with the values obtained.

Solution of the Problem 92-2

Item a

The transformation ratio of the transformer is:

a = V₁ / V₂ = 13,200 / 220 = 60

Having the values of V_cc and I_cc, then by eq. 92-09:

Z'_eq = V_cc / I_cc = 850 / 3.79 = 224.27 ohms

Note that we are using values referred to the primary and we make this explicit using Z'_eq. On the other hand, using eq. 92-10 let's calculate the value of R'_eq using the other given data, that is, the power W_cc consumed by the transformer when in short circuit. Soon:

R'_eq = W_cc / I_cc² = 280 / (3.79)² = 19.50 ohms

We also use R'_eq to indicate that the calculated value is referred to the primary. So, with the Z'_eq and R'_eq values calculated, we can calculate the value of X'_eq using eq. 92-11.

X'_eq² = Z'_eq² - R'_eq² = 224.27² - 19.50² = 49,916.8

Extracting the square root we get

X'_eq = 223.42 ohms

To find the values of R₁ and X₁ let's use the equations eq. 92-14 and eq. 92-15. In this way we have:

R₁ = R'_eq/ 2 = 9.75 ohms

X₁ = X'_eq/ 2 = 111.71 ohms

With these calculated values we can calculate the values of R₂ and X₂, these values are already referenced to the secondary, using the relation below.

R₂ = R₁/ a² = 9.75 / 3600 = 0.00271 ohms

X₂ = X₁/ a² = 111.71 / 3600 = 0.031 ohms

Note that with the short circuit test we can determine the main parameters of the transformer. Now, the open circuit test will allow us to find the values of the parameters that make up the transformer magnetization circuit. Then, using the data provided along with the equation eq. 92-04 we can calculate the value of R_h, or

R_h = V_nom²/ W₀ = 13,200² / 900 = 193,600 Ω

And using eq. 92-05 we can calculate the value of Z_ex, or

Z_ex = V_nom/ I_p = 13,200 / 0.32 = 41,250 ohms

With the values of R_h and Z_ex, using eq. 92-06, repeated below, we can calculate the value of X_m, or

eq. 92-06

Substituting the numerical values and performing the calculation, we get

X_m = 42,220 ohms

Item b

In Figure 92-02.1 we show the schematic of the transformer seen from the high voltage side.

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