Approximate Equivalent Circuit From Open- and Short-Circuit Tests: 16-Mechatronics-B10, December 2019, Q3

Question 3: Approximate Equivalent Circuit From Open- and Short-Circuit Tests (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Rating 75 kVA, 2200/220 V, 60 Hz; OC test run on the 220 V (LV) side: V_OC=220 V, I_OC=9.6 A, P_OC=710 W; SC test run on the 2200 V (HV) side: V_SC=42 V, I_SC=57 A, P_SC=1030 W.

Find. The approximate (cantilever) equivalent-circuit elements R_eq, X_eq, R_c, X_m referred to the 2200 V side, and a sketch of the circuit.

Approach. The OC test (rated LV voltage, secondary open) isolates the shunt exciting branch; the SC test (reduced HV voltage, secondary shorted) isolates the series branch. Compute each on the side it was tested and refer the OC-side result to the HV side by the turns ratio squared.

OC test — exciting admittance on the test (LV) side. $$\begin{aligned} Y_{OC}&=\frac{I_{OC}}{V_{OC}}=\frac{9.6}{220}=0.04364\ \text{S}\\ \theta_{OC}&=\cos^{-1}\!\left(\frac{P_{OC}}{V_{OC}I_{OC}}\right)=\cos^{-1}\!\left(\frac{710}{2112}\right)=70.36^{\circ} \end{aligned}$$ $$G_{OC}=Y_{OC}\cos\theta_{OC}=0.01467\ \text{S}\ \Rightarrow\ R_{c,LV}=\frac{1}{G_{OC}}=68.17\ \Omega$$ $$B_{OC}=Y_{OC}\sin\theta_{OC}=0.04110\ \text{S}\ \Rightarrow\ X_{m,LV}=\frac{1}{B_{OC}}=24.33\ \Omega$$
Refer the shunt branch to the HV side. With a = 2200/220 = 10, impedances scale by a² = 100: $$R_c=68.17(100)=\boxed{6817\ \Omega},\qquad X_m=24.33(100)=\boxed{2433\ \Omega}$$
SC test — series impedance, already on the HV side. $$\begin{aligned} Z_{SC}&=\frac{V_{SC}}{I_{SC}}=\frac{42}{57}=0.7368\ \Omega\\ \theta_{SC}&=\cos^{-1}\!\left(\frac{P_{SC}}{V_{SC}I_{SC}}\right)=\cos^{-1}\!\left(\frac{1030}{2394}\right)=64.52^{\circ} \end{aligned}$$
(a) Series R and X. $$R_{eq}=Z_{SC}\cos\theta_{SC}=\boxed{0.317\ \Omega},\qquad X_{eq}=Z_{SC}\sin\theta_{SC}=\boxed{0.665\ \Omega}$$

(b) Approximate (cantilever) equivalent circuit referred to the 2200 V (HV) side: series R_eq+jX_eq from the SC test, shunt R_c ∥ jX_m (drawn at the input terminals, the standard "approximate" simplification) from the OC test referred by a².

Quantity	Value (referred to 2200 V / HV side)
R_eq	0.317 Ω
X_eq	0.665 Ω
R_c	6817 Ω
X_m	2433 Ω

Concept & Theory

Topic: Building the approximate (cantilever) equivalent circuit from open- and short-circuit test data.

Key relations: Y_OC=I_OC/V_OC, θ_OC=cos⁻¹(P_OC/V_OCI_OC) → G_OC/B_OC → R_c/X_m; Z_SC=V_SC/I_SC, θ_SC=cos⁻¹(P_SC/V_SCI_SC) → R_eq/X_eq; refer any test-side quantity to the desired side by ×a² (impedance) with a the ratio FROM the test side TO the target side.

Why this works: the OC test is run at rated voltage with the far winding open, so the small measured current is almost pure exciting current — negligible drop across the (comparatively tiny) series branch lets V_OC, I_OC, P_OC characterize the shunt branch in isolation. The SC test is the mirror image: the low applied voltage keeps the exciting current negligible, so V_SC, I_SC, P_SC characterize the series branch alone.

Common pitfall: referring the OC-side quantities the wrong way (dividing by a² instead of multiplying) — always refer FROM the side the test was run TOWARD the side the answer is wanted; also drawing the shunt branch on the LV terminals after already referring its value to the HV side.

Source: Chapman, Electric Machinery Fundamentals, Ch. 2 §2.7 (transformer testing, approximate equivalent circuit).

Question 3 of 5: Approximate Equivalent Circuit From Open- and Short-Circuit Tests

Question 3: Approximate Equivalent Circuit From Open- and Short-Circuit Tests (20 marks)