Two coils. connected as shown in Fig. 32-28, separately have...

Two coils. connected as shown in Fig. $32-28$, separately have inductances $L_{1}$ and $L_{2}$. Their mutual inductance is $M$. (a) Show that this combination can be replaced by a single coil of equivalent inductance given by $$ L_{\mathrm{eq}}=L_{1}+L_{2}+2 M $$ (b) How could the coils in Fig. $32-28$ be reconnected to yield an equivalent inductance of $$ L_{\mathrm{eq}}=L_{1}+L_{2}-2 M ? $$ (This problem is an extension of Problem 7, but the requirement that the coils be far apart has been removed.)

Two coils. connected as shown in Fig. $32-28$, separately have inductances $L_{1}$ and $L_{2}$. Their mutual inductance is $M$.
(a) Show that this combination can be replaced by a single coil of equivalent inductance given by
$$
L_{\mathrm{eq}}=L_{1}+L_{2}+2 M
$$
(b) How could the coils in Fig. $32-28$ be reconnected to yield an equivalent inductance of
$$
L_{\mathrm{eq}}=L_{1}+L_{2}-2 M ?
$$
(This problem is an extension of Problem 7, but the requirement that the coils be far apart has been removed.)

Key Concepts

Windings Polarity and Dot Convention

The dot convention is a method used to indicate the relative polarity of the windings in coupled coils. It helps to determine whether the mutual inductance will contribute positively or negatively to the total inductance when the coils are connected in series. Correct interpretation of the dot convention is essential for proper circuit design and analysis.

Series Connection of Inductors

When inductors are connected in series, the total or equivalent inductance is affected not only by the individual self inductances of the coils but also by any mutual inductance present between them. The contributions from mutual inductance can either add to or subtract from the total inductance depending on the relative orientation and connection polarity.

Equivalent Inductance of Coupled Coils

The equivalent inductance of coupled coils takes into account both the self inductances and the mutual inductance between them. Depending on the connection (aiding or opposing), the mutual inductance M is added to or subtracted from the sum of the individual inductances, resulting in L_eq = L1 + L2 ± 2M. This concept is critical for analyzing and designing circuits involving magnetically coupled coils.

Self Inductance

Self inductance is the inherent property of a coil by which it opposes changes in the current flowing through it. This property is quantified by the inductance value L, and it represents the ability of a coil to store magnetic energy in its own magnetic field when current flows through it.

Mutual Inductance

Mutual inductance refers to the phenomenon where a change in current in one coil induces an electromotive force (emf) in another nearby coil. The mutual inductance, M, quantifies the coupling between the coils and depends on factors such as the geometry, orientation, and distance between the coils.

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