Transformer Design Calculator

The electromagnetic sizing equations are derived here from Faraday's law and a bare-copper window budget. The textbook framework is McLyman, Transformer and Inductor Design Handbook, 4th edition (2011), particularly the chapters on window utilization, transformer trade-offs, efficiency and power-transformer design. No proprietary tables or empirical sizing constants are copied.

For a sinusoid, $V_{p,\mathrm{rms}}=\sqrt{2}\pi f N_p A_e B_{pk}$. For balanced bridge pulses, $\Delta B=V_p q/(2fN_pA_e)$ and $B_{pk}=\Delta B/2$. Here $q$ includes both active pulses. The primary is sized at the maximum allowed active fraction; secondary turns are rounded up, then operating duty is set to $q_{op}=hV_sN_p/(V_pN_s)$. This preserves the chosen no-load voltage allowance $h$.

At unity power factor $I_o=P_o/V_s$. Bridge load-component currents are $I_s=I_o\sqrt{q_{op}}$ and $I_{p,L}=I_sN_s/N_p$, assuming secondary rectifier freewheeling during zero-voltage intervals. Mains uses $q=1$. Enter the measured or conservatively estimated open-secondary primary RMS current $I_0$, including core-loss and magnetizing current, at the operating voltage, frequency and duty. The RMS triangle inequality gives the conservative sizing bound $I_{p,size}=I_{p,L}+I_0$. This bound sizes the primary conductor and its Joule loss without assuming a phase relationship or sinusoidal excitation current.

Neglecting excitation, rounding, voltage allowance and losses, the two winding copper areas give

where $k_w=\sqrt2\pi$ for sine and $4/q$ for the bridge. The actual fill is $(N_pA_{cu,p}+N_sA_{cu,s})/A_w$, so it includes the selected wire and integer turns.

The preset sine-loss laws use $p_v=p_{v0}(f/f_0)^\alpha(B_{pk}/B_0)^\beta$. Steel uses SIJ Acroni SIWATT M400-50A, 1/1-2023-EN (attached s6): typical mass-loss points on p.1 and density on p.2; its frequency exponent 1.5 is an assumption. N87 uses a local fit to the first three loss points on attached TDK datasheet s4, p.2, at 100 °C. Its fourth point is outside this fit.

For bridge excitation, Venkatachalam et al. (2002), equations 9, 10 and 15, give the no-minor-loop iGSE. Analytic integration for this waveform multiplies sine loss by $K_{tri}q_{op}^{1-\alpha}$, with $K_{tri}=2^{2\alpha}/[(2\pi)^{\alpha-1}C_\alpha]$ and $C_\alpha=2\sqrt\pi\,\Gamma((\alpha+1)/2)/\Gamma((\alpha+2)/2)$. Peak and peak-to-peak flux are reported separately.

Copper uses the IACS resistivity $1/58$ ohm·mm²/m and resistance coefficient 0.00393/K at 20 °C (NBS Circular 31, 4th edition). Multiply DC resistance by the entered AC/DC factor. With fixed core loss and one hot-spot temperature, the thermal equation is solved algebraically:

Nonpositive feedback denominator is rejected. With $I_{p,size}$, the conservative winding-drop estimate is $\Delta V=\sqrt q(I_sR_{s,hot}+nI_{p,size}R_{p,hot})$, where $n=N_s/N_p$. Primary drop is also flagged when it exceeds 10% of the primary RMS voltage, because flux sizing neglects that drop. Efficiency is a nominal-power loss-budget estimate using conservative winding loss.

Two necessary energy checks reject an inconsistent excitation estimate: $V_p\sqrt q I_0\ge P_{core}$ and $V_p\sqrt q I_{p,size}\ge P_o+P_{loss}$. These are apparent-power lower bounds, not a prediction of no-load power factor. Low-power properties include the 1 W case.

• Two windings, uniform effective flux, no DC bias and balanced volt-seconds. The steel geometry defaults are a custom sizing example, not a named core.
• Mains load current is sinusoidal and in phase with voltage. The ideal load component is reported separately. Primary conductor and loss calculations add the entered no-load RMS current as a conservative bound; they do not predict its waveform or power factor. Inrush remains excluded.
• Bridge mode supports balanced full or half bridges using the actual winding pulse voltage, a bridge secondary rectifier, and continuous output-inductor current. Each winding carries no load current during freewheel intervals. Phase-shift bridges with circulating current require different RMS-current estimates.
• Automatic wire sizing is continuous, not an AWG selection. Bare strand diameters exclude enamel; a fill allowance reserves the remaining window. Equal-length strands share current equally. Parallel strands only act as litz when suitably transposed.
• The skin-depth sizing rule, 10% copper-area allowance, current densities, flux ceilings, AC resistance factors and thermal resistances are design choices. AC factors are not derived by a Dowell or field model.
• Core loss is held at its calibration temperature. Ferrite N87 data refer to 100 °C; this is flagged when the predicted hot spot differs substantially. The steel datasheet does not specify a loss-test temperature.
• Both windings use one mean turn length and one estimated temperature. Thermal resistance is an input, not a guaranteed property of the example geometry.
• No-load RMS defaults of 0.05 A for mains and 0.10 A for the bridge are illustrative design assumptions, not core predictions. Replace them with measured or conservatively estimated values at the operating excitation; an estimate below the core-loss energy minimum is rejected.
• The fixed-winding load sweep changes the load component only; excitation current and its winding loss remain at zero output. Automatic conductor sizing is not repeated along that chart.

Use this as a transparent sizing and winding check, followed by prototype measurements. The mains range is 45–65 Hz; the steel flux fit is most useful around 0.8–1.5 T. The local N87 fit is intended for 25–300 kHz and approximately 0.05–0.2 T; errors can be appreciable away from its calibration points. Custom Steinmetz data must cover the actual frequency, flux and temperature. A local power-law fit is not a material guarantee.

iGSE here covers only the stated balanced major-loop waveform. It omits minor loops, DC bias, flux walking, relaxation loss and temperature dependence. Do not apply this mode to flyback, forward, push-pull centre-tapped or resonant waveforms without deriving their winding currents and volt-seconds separately.

Window fill does not verify creepage, clearance, winding layout, dielectric strength or insulation class. The copper model is limited to 200 °C; a much lower practical limit usually applies. A positive thermal-feedback denominator alone does not establish an acceptable temperature. AC resistance, thermal resistance and core loss need confirmation for the construction and cooling arrangement.

Loaded voltage uses a conservative resistance-only drop. Magnetizing and core-loss currents are represented by the entered no-load RMS bound, not by an equivalent-circuit solution. Leakage reactance, semiconductor drops, control dynamics, fringing and manufacturing tolerances are excluded. Calculate worst-case voltage and frequency corners separately. A chosen flux ceiling is an engineering limit, not an automatic saturation prediction. Mains wiring and installation must be carried out by a qualified person; check against the applicable standard in force.