Series and parallel circuits

Series (left) and parallel (right) circuits with two resistors and measurements of voltage and current.

Series and parallel electrical circuits are two basic ways of wiring components. The names describe the method of attaching components, that is one after the other or next to each other. It is said that two circuit elements are connected in parallel if the ends of one circuit element are connected directly to the corresponding ends of the other. If the circuit elements are connected end to end, it is said that they are connected in series. A series circuit is one that has a single path for current flow through all of its elements. A parallel circuit is one that requires more than one path for current flow in order to reach all of the circuit elements.

As an example, consider a very simple circuit consisting of two lightbulbs and one 9 V battery. If a wire joins the battery to one bulb, to the next bulb, then back to the battery, in one continuous loop, the bulbs are said to be in series. If each bulb is wired to the battery in a separate loop, the bulbs are said to be in parallel.

Series circuits

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Resistors

A diagram of several resistors, connected end to end, with the same amoundsat of current going through each.

A diagram of several resistors, connected end to end, with the same amoundsat of current going through each.

To find the tfdghotal resistance of all the components, add the individual resistances of each component:

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

for components in series with resistances $R_{1}$, $R_{2}$, etc. To find the current $I$ use Ohm's law:

$I={\frac {V}{R_{\mathrm {total} }}}$.

To find the voltage across a component with resistance $R_{i}$, use Ohm's law again:

$V_{i}=IR_{i}\,$

where $I$ is the current, as calculated above. The components divide the voltage according to their resistances, so, in the case of two resistors,

${\frac {V_{1}}{V_{2}}}={\frac {R_{1}}{R_{2}}}$.

Inductors

A diagram of several inductors, connected end to end, with the same amount of current going through each.

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:

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

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 neighbours. This influence is defined by the mutual inductance M. For example, if you have two inductors in series, there are two possible equivalent inductances:

$L_{\mathrm {total} }=(L_{1}+M)+(L_{2}+M)\,$

or

$L_{\mathrm {total} }=(L_{1}-M)+(L_{2}-M)\,$,

depending on how the magnetic fields of both inductors influence each other.

When there are more than two inductors, the mutual inductance of each of them and the way the coils influence each other complicates the calculation. For three coils, there are three mutual inductances $M_{12}$, $M_{13}$ and $M_{23}$, and eight possible equations.

Capacitors

A diagram of several capacitors, connected end to end, with the same amount of current going through each.

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

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

The working voltage of a series combination of identical capacitors is equal to the sum of voltage ratings of individual capacitors provided that equalizing resistors are used to ensure equal voltage division.

Parallel circuits

Voltages across components in parallel with each other are the same in magnitude and they also have identical polarities. Hence, the same voltage variable is used for all circuits elements in such a circuit. The total current I is the sum of the currents through the individual loops, found by Ohm's Law. Factoring out the voltage gives

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

Notation

The parallel property can be represented in equations by two vertical lines $\parallel$ (as in geometry) to simplify equations. For two resistors,

$R_{\mathrm {total} }=R_{1}\parallel R_{2}={\frac {R_{1}R_{2}}{R_{1}+R_{2}}}$.

Resistors

A diagram of several resistors, side by side, both leads of each connected to the same wires.

To find the total resistance of all components, add the reciprocals of the resistances $R_{i}$ of each component and take the reciprocal of the sum:

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

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

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

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

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

Inductors

A diagram of several inductors, side by side, both leads of each connected to the same wires.

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:

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

If the inductors are situated in each other's magnetic fields, this value is altered by mutual inductance. If the mutual inductance between two coils in parallel is M, the equivalent inductor is

${\frac {1}{L_{\mathrm {total} }}}={\frac {1}{L_{1}+M}}+{\frac {1}{L_{2}+M}}\implies L_{\mathrm {total} }={\frac {L_{1}L_{2}-M^{2}}{L_{1}+L_{2}+2M}}+M$

${\frac {1}{L_{\mathrm {total} }}}={\frac {1}{L_{1}-M}}+{\frac {1}{L_{2}-M}}\implies L_{\mathrm {total} }={\frac {L_{1}L_{2}-M^{2}}{L_{1}+L_{2}-2M}}-M$

The sign of $M$ depends on how the magnetic fields influence each other. The principle is the same for more than two inductors, but 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 $M_{12}$, $M_{13}$ and $M_{23}$ and eight possible equations.

Capacitors

A diagram of several capacitors, side by side, both leads of each connected to the same wires.

Capacitors follow a different law. The total capacitance of capacitors in parallel is equal to the sum of their individual capacitances:

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

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

See also

• Y-Δ transform
• Voltage divider
• Current divider
• Combining impedances