9.4 Mutually Coupled Coils

Consider the inductor (coil) shown in Figure 9.4.

Figure 9.4: Magnetic lines linking a coil

There are many magnetic lines of flux linking the coil L ₁ with N ₁ turns but for simplicity, only two are shown in Figure 9.4. The current i ₁ produces a magnetic flux ? _{1 1}. Then by (9.4) and (9.5), we obtain

and by Faraday s law of (9.6), in terms of the self-inductance L ₁,

Next, suppose another coil L ₂ with N ₂ turns is brought near the vicinity of coil L ₁ and some lines of flux are also linking coil L ₂ as shown in Figure 9.5.

Figure 9.5: Lines of flux linking two coils

It is convenient to express the flux ? ₁₁ as the sum of two fluxes ? _L1 and ? ₂₁, that is,

where the linkage flux ? _L1 is the flux which links coil L ₁ only and not coil L ₂, and the mutual flux ? ₂₁ is the flux which links both coils L ₁ and L ₂. We have assumed that the linkage and mutual fluxes ? _L1 and ? ₂₁ link all turns of coil L ₁ and the mutual flux ? ₂₁ links all turns of coil L ₂.

The arrangement above forms an elementary transformer where coil L ₁ is...