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

## Table of contents

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**

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**

## 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**

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**

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**

The time derivative of the current occurs on both sides of the equation, so it can be canceled out:

**Total inductance for two coils**

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:

**Total inductance of coils connected in series**

## 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**

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**

This way you can insert the induction law 2

into Eq. 9

for the time derivatives:

**Ratio of voltage to inductance**

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**

We can cancel out the voltage \(U\) in Eq. 11

on both sides:

**Reciprocal of the total inductance for a parallel circuit**

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**

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:

**Total inductance of coils connected in parallel**