1. Formula for equivalent inductance of two coupled coils in series

When two coils of self-inductances $L_1$ and $L_2$ have mutual inductance $M$, the equivalent inductance in series depends on whether the fluxes aid or oppose each other:

• Series aiding: $L_{eq}=L_1+L_2+2M$
• Series opposing: $L_{eq}=L_1+L_2-2M$

2. Check the given options

• Option A: $\frac{1}{L_1}+\frac{1}{L_2}+\frac{1}{M}$ This is dimensionally incorrect for inductance.

• Option B: $L_1+L_2+M$ Incorrect, because mutual inductance contributes as $2M$ in series combination.

• Option C: $L_1+L_2+2M$ Correct for series aiding.

• Option D: $L_1+L_2-2M$ Correct for series opposing.

3. Interpretation of the question

Since the question simply asks for the equivalent self-inductance of two coils in series with mutual inductance $M$, the standard result depends on the relative connection:

$L_{eq}=L_1+L_2\pm 2M$

So, without specifying whether the coils are aiding or opposing, both C and D are possible depending on the connection.

4. Comparison with stored answer

The stored correct answer is D, which is only valid for the series opposing case. But the question does not specify that condition. Hence the stored answer is incomplete/incorrect for the question as written.

Therefore, the proper answer should be:

$L_{eq}=L_1+L_2\pm 2M$

and among the given single options, the question is ambiguous.