Transformer Sizing & Specification

\[ S_{\text{req}} = S_{\text{load}} \times k_{\text{growth}} \]

Where \(S_{\text{load}}\) is the maximum demand (kVA) and \(k_{\text{growth}}\) is the future growth factor (default 1.2). The next standard rating \(S_{\text{rated}} \geq S_{\text{req}}\) is selected from the IEC preferred series.

With overload capability:

\[ S_{\text{req}} = \max\!\left( S_{\text{load}} \times k_{\text{growth}},\; \frac{S_{\text{load}}}{0.8} \right) \]

Keeps the present load at no more than 80% of the nameplate rating, i.e. the transformer can carry 125% of the base load before reaching its rating. The capacity check still compares \(S_{\text{load}} \times k_{\text{growth}}\) with the (derated) rating.

\[ \text{Utilization} = \frac{S_{\text{load}}}{S_{\text{rated}}} \times 100 \;\; (\%) \]

\[ Z_{\%} = Z_{\text{base}} \times k_{\text{voltage}} \]

Where \(Z_{\text{base}}\) depends on transformer rating (4.5% for <500 kVA, 6.0% for 500–2,500 kVA, 7.5% for 2,500–10,000 kVA, 8.5% for ≥10,000 kVA) and \(k_{\text{voltage}}\) adjusts for maximum winding voltage (1.0 for <11 kV, 1.1 for 11–33 kV, 1.2 for 33–66 kV, 1.3 for ≥66 kV). The result is rounded to the nearest 0.5%, halves rounding up (e.g. 7.5 × 1.1 = 8.25% → 8.5%).

No-load losses (core losses):

\[ P_0 = S_{\text{rated}} \times 1000 \times k_{P_0} \quad (\text{W}) \]

Where \(k_{P_0}\) is a percentage factor dependent on rating (1.8% for <100 kVA, 1.2% for 100–500 kVA, 0.8% for 500–2,500 kVA, 0.5% for 2,500–10,000 kVA, 0.4% for ≥10,000 kVA). Dry-type and cast resin transformers apply a 0.85× multiplier.

Load losses (winding losses):

\[ P_k = S_{\text{rated}} \times 1000 \times k_{P_k} \quad (\text{W}) \]

Where \(k_{P_k}\) is a percentage factor dependent on rating (2.5% for <100 kVA, 1.8% for 100–500 kVA, 1.3% for 500–2,500 kVA, 1.0% for 2,500–10,000 kVA, 0.8% for ≥10,000 kVA). Dry-type and cast resin transformers apply a 1.1× multiplier.

\[ \eta = \frac{S_{\text{rated}} \times n \times \cos\phi \times 1000} {S_{\text{rated}} \times n \times \cos\phi \times 1000 \;+\; P_0 + P_k \times n^2} \times 100 \;\; (\%) \]

Where \(n\) is the per-unit loading (e.g., 0.75 for 75% load), \(\cos\phi\) is the load power factor, \(P_0\) is no-load loss (W), and \(P_k\) is load loss at rated current (W). Core losses are constant; winding losses scale with the square of load.

\[ \text{Reg}_{\%} \approx R_{\%} \cos\phi + X_{\%} \sin\phi \]

Where the impedance is split into resistive and reactive components. When the nameplate load loss \(P_k\) is given, the resistive part follows from it (the load loss is the winding I²R loss at rated current):

\[ R_{\%} = \frac{P_k}{S_{\text{rated}} \times 1000} \times 100, \qquad X_{\%} = \sqrt{Z_{\%}^2 - R_{\%}^2} \]

Otherwise a typical X/R ratio of 5 is used:

\[ R_{\%} = \frac{Z_{\%}}{\sqrt{1 + (X/R)^2}}, \qquad X_{\%} = R_{\%} \times (X/R) \]

A nameplate load loss that gives \(R_{\%} \geq Z_{\%}\) is rejected as inconsistent.

Three-phase symmetrical fault current:

\[ I_{sc} = \frac{S_{\text{rated}} \times 1000}{\sqrt{3} \times V_2 \times 1000} \times \frac{100}{Z_{\%}} \quad (\text{A}) \]

Where \(V_2\) is the secondary voltage (kV) and \(Z_{\%}\) is the transformer impedance. For single-phase, the \(\sqrt{3}\) factor is omitted.

Peak short-circuit current (IEC 60909-0):

\[ \kappa = 1.02 + 0.98\, e^{-3R/X}, \qquad i_p = \kappa \sqrt{2}\, I_{sc} \]

\(R/X\) is the transformer's own ratio from the impedance split above (nameplate \(P_k\), or the typical X/R of 5, giving \(\kappa = 1.558\) and \(i_p = 2.20 \times I_{sc}\)). With an infinite source the transformer is the whole fault path, so its R/X sets the DC offset. Use \(i_p\) for switchgear making (peak withstand) ratings and \(I_{sc}\) for breaking ratings.

\[ V_{\text{tap}} = V_{\text{nom}} \times \left(1 + \frac{\Delta_{\%}}{100}\right) \]

Where the percentage offset from nominal for each tap position is:

\[ \Delta_{\%} = (t - t_{\text{nom}}) \times \frac{2 \times \text{Range}_{\%}}{N - 1} \]

\(t\) is the tap number (1 to \(N\)), \(t_{\text{nom}}\) is the centre tap, and \(\text{Range}_{\%}\) is the ±% range.

\[ k_{\text{alt}} = 1 - \frac{(h - 1000)}{100} \times 0.005 \]

Applied when altitude \(h > 1000\) m. Reduces the effective rating by 0.5% for every 100 m above 1,000 m.

\[ S_{\text{derated}} = S_{\text{rated}} \times k_{\text{alt}} \]

The standard rating is selected before derating. If \(S_{\text{load}} \times k_{\text{growth}} > S_{\text{derated}}\) the capacity check fails and the next larger rating is needed.

\[ T_{\text{max}} = T_{\text{ambient}} + \Delta T_{\text{rise}} \] \[ \text{Margin} = T_{\text{insulation\;limit}} - T_{\text{max}} \]

A warning is issued if the margin falls below 10 K.

pytest calcs/components/c0024_transformer_sizing/ -v
pytest calcs/test_c0024* -v