Series and parallel circuits

$${\displaystyle I=I_{1}=I_{2}=\cdots =I_{n}}$$

In a series circuit, the current is the same for all of the elements.

In a series circuit, the voltage is the sum of the voltage drops of the individual components (resistance units).

$${\displaystyle V=V_{1}+V_{2}+\dots +V_{n}=I\left(R_{1}+R_{2}+\dots +R_{n}\right)}$$

The total resistance of two or more resistors connected in series is equal to the sum of their individual resistances:

$${\displaystyle R_{\text{total}}=R_{\text{s}}=R_{1}+R_{2}+\cdots +R_{n}.}$$

Here, the subscript s in R_s denotes "series", and R_s denotes resistance in a series.

Electrical conductance presents a reciprocal quantity to resistance. Total conductance of a series circuits of pure resistances, therefore, can be calculated from the following expression:

$${\displaystyle {\frac {1}{G_{\text{total}}}}={\frac {1}{G_{1}}}+{\frac {1}{G_{2}}}+\cdots +{\frac {1}{G_{n}}}.}$$

For a special case of two conductances in series, the total conductance is equal to:

$${\displaystyle G_{\text{total}}={\frac {G_{1}G_{2}}{G_{1}+G_{2}}}.}$$

Inductors follow the same law, in that the total inductance of non-coupled inductors in series is equal to the sum of their individual inductances:

$${\displaystyle L_{\mathrm {total} }=L_{1}+L_{2}+\cdots +L_{n}}$$

However, in some situations, it is difficult to prevent adjacent inductors from influencing each other as the magnetic field of one device couples with the windings of its neighbors. This influence is defined by the mutual inductance M. For example, if two inductors are in series, there are two possible equivalent inductances depending on how the magnetic fields of both inductors influence each other.

When there are more than two inductors, the mutual inductance between each of them and the way the coils influence each other complicates the calculation. For a larger number of coils the total combined inductance is given by the sum of all mutual inductances between the various coils including the mutual inductance of each given coil with itself, which we term self-inductance or simply inductance. For three coils, there are six mutual inductances ${\displaystyle M_{12}}$, ${\displaystyle M_{13}}$, ${\displaystyle M_{23}}$ and ${\displaystyle M_{21}}$, ${\displaystyle M_{31}}$ and ${\displaystyle M_{32}}$. There are also the three self-inductances of the three coils: ${\displaystyle M_{11}}$, ${\displaystyle M_{22}}$ and ${\displaystyle M_{33}}$.

Therefore

$${\displaystyle L_{\text{total}}=\left(M_{11}+M_{22}+M_{33}\right)+\left(M_{12}+M_{13}+M_{23}\right)+\left(M_{21}+M_{31}+M_{32}\right)}$$

By reciprocity, ${\displaystyle M_{ij}}$ = ${\displaystyle M_{ji}}$ so that the last two groups can be combined. The first three terms represent the sum of the self-inductances of the various coils. The formula is easily extended to any number of series coils with mutual coupling. The method can be used to find the self-inductance of large coils of wire of any cross-sectional shape by computing the sum of the mutual inductance of each turn of wire in the coil with every other turn since in such a coil all turns are in series.

Capacitors follow the same law using the reciprocals. The total capacitance of capacitors in series is equal to the reciprocal of the sum of the reciprocals of their individual capacitances:

$${\displaystyle {\frac {1}{C_{\text{total}}}}={\frac {1}{C_{1}}}+{\frac {1}{C_{2}}}+\cdots +{\frac {1}{C_{n}}}.}$$

Equivalently using elastance (the reciprocal of capacitance), the total series elastance equals the sum of each capacitor's elastance.

Two or more switches in series form a logical AND; the circuit only carries current if all switches are closed. See AND gate.

A battery is a collection of electrochemical cells. If the cells are connected in series, the voltage of the battery will be the sum of the cell voltages. For example, a 12 volt car battery contains six 2-volt cells connected in series. Some vehicles, such as trucks, have two 12 volt batteries in series to feed the 24-volt system.

In a parallel circuit, the voltage is the same for all elements.

$${\displaystyle V=V_{1}=V_{2}=\dots =V_{n}}$$

The current in each individual resistor is found by Ohm's law. Factoring out the voltage gives

$${\displaystyle I_{\text{total}}=I_{1}+I_{2}+\cdots +I_{n}=V\left({\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+\cdots +{\frac {1}{R_{n}}}\right).}$$

To find the total resistance of all components, add the reciprocals of the resistances ${\displaystyle R_{i}}$ of each component and take the reciprocal of the sum. Total resistance will always be less than the value of the smallest resistance:

$${\displaystyle {\frac {1}{R_{\text{total}}}}={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+\cdots +{\frac {1}{R_{n}}}.}$$

For only two resistances, the unreciprocated expression is reasonably simple:

$${\displaystyle R_{\text{total}}={\frac {R_{1}R_{2}}{R_{1}+R_{2}}}.}$$

This sometimes goes by the mnemonic product over sum.

For N equal resistances in parallel, the reciprocal sum expression simplifies to:

$${\displaystyle {\frac {1}{R_{\text{total}}}}=N{\frac {1}{R}}.}$$

and therefore to:

$${\displaystyle R_{\text{total}}={\frac {R}{N}}.}$$

To find the current in a component with resistance ${\displaystyle R_{i}}$, use Ohm's law again:

$${\displaystyle I_{i}={\frac {V}{R_{i}}}\,.}$$

The components divide the current according to their reciprocal resistances, so, in the case of two resistors,

$${\displaystyle {\frac {I_{1}}{I_{2}}}={\frac {R_{2}}{R_{1}}}.}$$

An old term for devices connected in parallel is multiple, such as multiple connections for arc lamps.

Since electrical conductance ${\displaystyle G}$ is reciprocal to resistance, the expression for total conductance of a parallel circuit of resistors reads:

$${\displaystyle G_{\text{total}}=G_{1}+G_{2}+\cdots +G_{n}.}$$

The relations for total conductance and resistance stand in a complementary relationship: the expression for a series connection of resistances is the same as for parallel connection of conductances, and vice versa.

Inductors follow the same law, in that the total inductance of non-coupled inductors in parallel is equal to the reciprocal of the sum of the reciprocals of their individual inductances:

$${\displaystyle {\frac {1}{L_{\text{total}}}}={\frac {1}{L_{1}}}+{\frac {1}{L_{2}}}+\cdots +{\frac {1}{L_{n}}}.}$$

If the inductors are situated in each other's magnetic fields, this approach is invalid due to mutual inductance. If the mutual inductance between two coils in parallel is M, the equivalent inductor is:

$${\displaystyle {\frac {1}{L_{\text{total}}}}={\frac {L_{1}+L_{2}-2M}{L_{1}L_{2}-M^{2}}}}$$

If ${\displaystyle L_{1}=L_{2}}$

$${\displaystyle L_{\text{total}}={\frac {L+M}{2}}}$$

The sign of ${\displaystyle M}$ depends on how the magnetic fields influence each other. For two equal tightly coupled coils the total inductance is close to that of every single coil. If the polarity of one coil is reversed so that M is negative, then the parallel inductance is nearly zero or the combination is almost non-inductive. It is assumed in the "tightly coupled" case M is very nearly equal to L. However, if the inductances are not equal and the coils are tightly coupled there can be near short circuit conditions and high circulating currents for both positive and negative values of M, which can cause problems.

More than three inductors become more complex and the mutual inductance of each inductor on each other inductor and their influence on each other must be considered. For three coils, there are three mutual inductances ${\displaystyle M_{12}}$, ${\displaystyle M_{13}}$ and ${\displaystyle M_{23}}$. This is best handled by matrix methods and summing the terms of the inverse of the ${\displaystyle L}$ matrix (3×3 in this case).

The pertinent equations are of the form:

$${\displaystyle v_{i}=\sum _{j}L_{i,j}{\frac {di_{j}}{dt}}}$$

The total capacitance of capacitors in parallel is equal to the sum of their individual capacitances:

$${\displaystyle C_{\text{total}}=C_{1}+C_{2}+\cdots +C_{n}.}$$

The working voltage of a parallel combination of capacitors is always limited by the smallest working voltage of an individual capacitor.

Two or more switches in parallel form a logical OR; the circuit carries current if at least one switch is closed. See OR gate.

If the cells of a battery are connected in parallel, the battery voltage will be the same as the cell voltage, but the current supplied by each cell will be a fraction of the total current. For example, if a battery comprises four identical cells connected in parallel and delivers a current of 1 ampere, the current supplied by each cell will be 0.25 ampere. If the cells are not identical, cells with higher voltages will attempt to charge those with lower ones, potentially damaging them.

Parallel-connected batteries were widely used to power the valve filaments in portable radios. Lithium-ion rechargeable batteries (particularly laptop batteries) are often connected in parallel to increase the ampere-hour rating. Some solar electric systems have batteries in parallel to increase the storage capacity; a close approximation of total amp-hours is the sum of all amp-hours of in-parallel batteries.