# Derivation: Total (Equivalent) Inductance of a Series and Parallel Circuit of Coils

In the following we want to derive the total inductance $$L$$ (also called equivalent inductance) of two coils connected in parallel and of two coils connected in series. One coil has the inductance $$L_1$$ and the other $$L_2$$.

Since the magnetic field generated by the coil is proportional to the current $$\class{red}{I}$$ flowing through the coil, the magnetic field and hence the magnetic flux $$\Phi_{\text m}$$ can be written as follows:

Magnetic flux is equal to inductance times current
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The inductance $$L$$ of the coil is the proportionality constant here. The applied AC voltage $$U$$ and the resulting current $$I$$ are related via the Faraday's law of induction:

Voltage is proportional to the negative time change of the current
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## Total inductance of a series circuit of coils

Consider two coils connected in series, to which an alternating voltage $$U$$ is applied. Thus, the time-dependent total voltage $$U$$ is between the both coils. The voltages $$U_1$$ and $$U_2$$ are between the ends of the individual coils:

Sum of the individual voltages in a series circuit
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For all three voltages $$U$$, $$U_1$$ and $$U_2$$ we insert the induction law 2 (the minus sign cancels out on both sides):

Sum of the individual voltages via induction law
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The total current $$\class{red}{I}$$ passes through the two coils according to the junction rule (1st Kirchhoff rule). So the currents are all equal: $$\class{red}{I} = \class{red}{I_1} = \class{red}{I_2}$$. Insert the current $$\class{red}{I}$$ into Eq. 4:

Sum of the individual voltages via induction law and equal current
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The time derivative of the current occurs on both sides of the equation, so it can be canceled out:

Total inductance for two coils
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The inductance in a series circuit add up to a total inductance. If we had $$n$$ instead of two coils connected in series, then we analogously sum up the individual inductances:

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## Total inductance of a parallel circuit of coils

Let us now consider two parallel connected coils. If an AC voltage $$U$$ is applied to the parallel circuit, an AC current $$I$$ flows. This current splits at the junction into the currents $$I_1$$ and $$I_2$$, which flow to the first and second coils, respectively. According to Kirchhoff's junction rule, the total current is given by the sum of the individual currents:

Total current through a parallel circuit
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Differentiate both sides of Eq. 8 with respect to time:

Time derivative of the total current is the sum of the time derivatives of the individual currents
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This way you can insert the induction law 2 into Eq. 9 for the time derivatives:

Ratio of voltage to inductance
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According to the mesh rule (2nd Kirchhoff rule), the total voltage $$U$$ also drops at the individual coils. This means: $$U = U_1 = U_2$$. Substitute this voltage into Eq. 10:

Total voltage per total inductance
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We can cancel out the voltage $$U$$ in Eq. 11 on both sides:

Reciprocal of the total inductance for a parallel circuit
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As you can see: the total inductance of a parallel circuit of two coils is not equal to the sum of the individual inductances. We can also rearrange the equation with respect to the total inductance:

Formula for total inductance of two coils connected in parallel
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If we have not 2 but $$n$$ coils connected in parallel, then analogously we sum the reciprocals of the individual inductances to get the reciprocal of the total inductance:

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