Governing Equations for Utility Transformer
Governing equations for Utility Transformer:
- For sinusoidal flux of peak magnitude ‘Φm’ and frequency ‘f’ linking the transformer windings, the emf generated per turn of the winding having a rms magnitude ‘Et’ given by:
- The peak flux through the core is the product of peak flux density (Bm) and the core area (Ac), i.e.
- The windings are placed around the core and are accommodated in the window of the transformer. The transformer window area (Aw) is related with the winding’s current rating and the number of turns. For a single-phase transformer the relation between them is given by:
where kw is the window utilization factor and δ is the current density through the cross-sectional area of the transformer windings.
- Window utilization factor, WUF roughly varies between 0.35 to 0.6 and is dependent on the insulation requirements of the windings.
- A typical figure for the current density through copper conductors of naturally cooled transformers is 3X106 amps per square meter.
- If the current density through primary and secondary windings is taken identical, they occupy equal window-space of the transformer. Sometimes the current densities through the two windings may differ depending on their physical ability to dissipate heat.
- The VA rating of a single phase transformer (= N Et I) can now be found from the above equations as:
- For the given operating frequency (f) the product ‘Ac Aw’, known as area product is roughly proportional to the VA rating of the transformer as other parameters have nearly fixed magnitudes.
Derivation of Design Equations for SMPS Transformer:
- The nature of voltage and flux waveforms in SMPS transformers is different from that of utility transformer. Moreover SMPS circuits of different topologies generate different kinds of winding voltages (and hence the flux-linked waveforms) and need to be considered separately.