Linear Circuit Analysis


Inductor Combinations

Series

If two or more inductors $L_1$, $L_2$, ... $L_n$ are connected in series they are equivalent with a inductor with $$L_{eff}=L_1+L_2+...+L_n$$ That means we can replace one of the $n$ inductors with an equivalent inductor with inductance $L_{eff}$ and replace the other inductors with short-circuits (wires).

For instance, considering the circuit in Fig. 1, inductors $L_1$ and $L_2$ are connected in series. Therefore, we can keep one the inductors, say $L_1$, replace its value with $L_{eff}=L_1+L_2$, and replace inductor $L_2$ with a wire.

L L1 L2 L3 L4 L5 R Leff L3 L4 L5 R Leff L3 L4 L5
Fig. 1. When combining multiple inductors that are connected in series, we keep one inductor and short-circuit the others. The 3 diagrams are equivalent with each other.
Parallel

If two or more inductors $L_1$, $L_2$, ... $L_n$ are connected in parallel they are equivalent with a inductor with $$L_{eff}=\frac{1}{\frac{1}{L_1}+\frac{1}{L_2}+...+\frac{1}{L_n}}$$ That means we can replace one of the $n$ inductors with an equivalent inductor with inductance $L_{eff}$ and remove all the other inductors. When we have only two inductors connected in parallel, we can replace them with a single inductors with inductance $$L_{eff}=\frac{L_1 L_2}{L_1+L_2}$$

For instance, considering the circuit in Fig. 2, inductors $L_3$, $L_4$, and $L_5$ are connected in parallel. Therefore, we can keep one the inductors, say $L_3$, replace its value with $L_{eff}=\frac{1}{\frac{1}{L_3}+\frac{1}{L_4}+\frac{1}{L_5}}$, and remove inductor $L_4$ from the circuit. Similarly, we could keep $L_4$ and remove $L_3$ and $L_5$, or keep $L_5$ and remove $L_3$ and $L_4$.

L L1 L2 L3 L4 L5 R L1 L2 Leff R L1 L2 Leff R L1 L2 Leff
Fig. 2. When combining multiple inductors that are connected in parallel, we keep one inductor and remove the others. The 4 diagrams are equivalent with each other.
See also