Series and Parallel Inductors

An inductor is passive circuit element. Let us find out the equivalent inductance of series connected and parallel connected inductors.

Series Connected Inductors

Let us consider n number of inductors connected in series as shown below.Let us also consider that,
the inductance of inductor 1 and voltage drop across it are L₁ and v₁ rspectively,
the inductance of inductor 1 and voltage drop across it are L₂ and v₂ rspectively,
the inductance of inductor 1 and voltage drop across it are L₃ and v₃ rspectively,
the inductance of inductor 1 and voltage drop across it are L₄ and v₄ rspectively,
the inductance of inductor 1 and voltage drop across it are L_n and v_n rspectively.

Now, applying, Kirchhoff's Voltage Law, we get, total voltage drop (v) across the series combination of the inductors,The votage drop across an inductor of inductance L can be expressed as,Where, i is the instanteous current through the inductor. As all inductors of the combinations are connected in series, here, the current through each of the inductors is same, and say also it is i. So, from above KVL equation, we get,This equation can be rewritten as,Where, L_eq is equivalent inductance of the series combined inductors. Hence,

Equivalent inductance of series connecetd inductors is simply arithmetic sum of the inductance of individual inductors.

Parallel Connected Inductors

Let us consider n number of inductors connected in parallel as shown below.Let us also consider that,
the inductance of inductor 1 and current through it are L₁ and i₁ rspectively,
the inductance of inductor 1 and current through it are L₂ and i₂ rspectively,
the inductance of inductor 1 and current through it are L₃ and i₃ rspectively,
the inductance of inductor 1 and current through it are L₄ and i₄ rspectively,
the inductance of inductor 1 and current through it are L_n and i_n rspectively.

Now, applying, Kirchhoff's Current Law, we get, total current (i) entering in the parallel combination of the inductors,The current throgh an inductor of inductance L can be expressed as,Where, v is the instanteous voltage across the inductor. As all inductors of the combinations are connected in parallel, here, the voltage drop across each of the inductors is same, and say also it is v. So, from above KCL equation, we get,This equation can be rewritten as,Where, L_eq is equivalent inductance of the parallel combined inductors. Hence,

Reciprocal of equivalent inductance of parallel connecetd inductors is simply arithmetic sum of the reciprocal of inductance of individual inductors.