Series and parallel circuits

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Left: Series / Right: Parallel
Arrows indicate direction of current. The red bars represent the voltage as it drops in the series circuit. The red bars in the parallel circuit do not drop because the voltage across each element in a parallel circuit is the same.

Series and parallel electrical circuits are two basic ways of wiring components. The naming describes the method of attaching components, i.e. 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 (i.e. a conductor) to the corresponding ends of the other. However, when 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 a demonstration, 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, on the other hand, each bulb is wired separately to the battery in two loops, the bulbs are said to be in parallel.

The measurable quantities used here are R, resistance, measured in ohms (Ω), I, current, measured in amperes (Amps)(A) (coulombs per second), and V, voltage, measured in volts (V) (joules per coulomb).

1 Series circuits
2 Parallel circuits
3 See also

[edit] Series circuits

Series circuits are sometimes called cascade-coupled or daisy chain-coupled.

The current that enters a series circuit has to flow through every element in the circuit. Therefore, all elements in a series connection have equal currents. Two ammeters placed anywhere in the circuit would prove this.

[edit] Resistors

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

$R_\mathrm{total} = R_1 + R_2 + \cdots + R_n$

for components in series, having resistances $\ R_1$ , $\ R_2$ , etc.

To find the current, $\ I$ use Ohm's law $I = \frac{V}{R_{total}}$

To find the voltage across any particular component with resistance $\ R_i$ , use Ohm's law again. $V_i = I \cdot R_i$

Where $\ I$ is the current, as calculated above.

Note that the components divide the voltage according to their resistances, so, in the case of two resistors:

$\frac{V_1}{V_2} = \frac{R_1}{R_2}$

[edit] Inductors

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:

$\ L_\mathrm{total} = (L_1 + M) + (L_2 + M)$

$\ L_\mathrm{total} = (L_1 - M) + (L_2 - M)$

Which formula is the correct one, depends how the magnetic fields of both inductors influence each other.

When there are more than two inductors, it gets more complicated, since you have to take into account the mutual inductance of each of them and how each coils influences the other.

So for three coils, there are three mutual inductances ( $M 12, M 13$ and $M 23$ ) and eight possible equations.

[edit] Capacitors

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:

${1\over{C_\mathrm{total}}} = {1\over{C_1}} + {1\over{C_2}} + \cdots + {1\over{C_n}}$

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.

[edit] 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.

To find the total current, I, use Ohm's Law on each loop, then sum. (See Kirchhoff's circuit laws for an explanation of why this works). Factoring out the voltage (which, again, is the same across parallel components) gives:

$I_\mathrm{total} = V * \left(\frac{1} {R_1} + \frac{1} {R_2} + \cdots + \frac{1} {R_n}\right)$

[edit] Notation

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

$R_\mathrm{total} = R_1 \| R_2 = {R_1 R_2 \over R_1 + R_2}$

[edit] Resistors

To find the total resistance of all the components, add together the individual reciprocal of each resistance of each component, and take the reciprocal of the sum:

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

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

for components in parallel, having resistances R₁, R₂, etc.

The above rule can be calculated by using Ohm's law for the whole circuit

R total = V / I total

and substituting for I_total

To find the current in any particular component with resistance R_i, use Ohm's law again.

I i = V / R i

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

I 1 / I 2 = R 2 / R 1

[edit] Inductors

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:

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

${1\over{L_\mathrm{total}}} = {1\over{L_1}} + {1\over{L_2}} + \cdots + {1\over{L_n}}$

Once again, if the inductors are situated in each others' magnetic fields, one has to take into account mutual inductance. If the mutual inductance between two coils in parallel is M then the equivalent inductor is:

${1 \over L_\mathrm{total}} = {1 \over (L_1 + M)} + {1 \over (L_2 + M)}$

${1 \over L_\mathrm{total}} = {1 \over (L_1 - M)} + {1 \over (L_2 - M)}$

${ L_\mathrm{total}} = {(L_1L_2 - M^2) \over L_1+L_2 + 2M}$

${ L_\mathrm{total}} = {(L_1L_2 - M^2) \over L_1+L_2 - 2M}$

And once again, which formula is the correct one, depends how the magnetic fields of both inductors influence each other.

The principle is the same for more than two inductors, but you now have to take into account the mutual inductance of each inductor on each other inductor and how they influence each other. So for three coils, there are three mutual inductances ( $M 12, M 13$ and $M 23$ ) and eight possible equations.