I need to find out how the equations shown below for V1 and

I need to find out how the equations shown below for V_1 and V_2 V_1 = jwL_1 I_1 + jwMI_2 V_2 = jwL_2 I_2 + jwMI_1 Are re arranged to become I_1 = (V(L_2 - M))/(jw(L_1 times L_2 - M^2)) I_2 = (V(L_1 - M))/(jw(L_1 times L_2 - M^2))

Solution

We have V₁ = jwL₁I₁ + jw MI₂ …(1) and V₂ = jwL₂I₂ + jw MI₁ …(2).

On multiplying the 1^st equation by L₂ and the 2^nd equation by M, we get V₁ L₂ = jwL₁ L₂ I₁ + jw M L₂ I₂ …(3) and M V₂ = jw ML₂I₂ + jw M² I₁ …(4). Now, on subtraction the 4^th equation from the 3^rd equation, we get V₁ L₂- M V₂ = jwL₁ L₂ I₁ + jw M L₂ I₂ – (jw ML₂I₂ + jw M² I₁) = jw(L₁ L₂ –M²) I₁ or, I₁ = (V₁ L₂- M V₂)/[ jw(L₁ L₂ –M²)] . Further, if V₁ and V₂ are same, then only , we have I₁ = [ V(L₂- M]) / [ jw(L₁ L₂ –M²)].

Similarly, on multiplying the 1^st equation by M and the 2^nd equation by L₁, we get MV₁ = jwML₁I₁ + jw M² I₂ …(5) and L₁V₂ = jw L₁ L₂I₂ + jw M L₁ I₁ …(6). Now, on subtraction the 5^th equation from the 6^th equation, we get L₁V₂ - MV₁ = jw L₁ L₂I₂ + jw M L₁ I₁ -(jwML₁I₁+ jw M²I₂) = jw( L₁ L₂ – M²)I₂ or, I₂ =(L₁V₂ - MV₁)/[ jw( L₁ L₂ – M²)]. Further, if V₁ and V₂ are same, then only , we have I₂ =[ V (L₁ - M)/[ jw( L₁ L₂ – M²)].