Editing Series and parallel circuits (section)

==Parallel circuits<span class="anchor" id="Zparallel"></span><span class="anchor" id="Xparallel"></span><span class="anchor" id="Yparallel"></span><span class="anchor" id="Bparallel"></span><span class="anchor" id="Parallel_circuit_anchor"></span>==
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{{resistors inductors capacitors in series and parallel.svg}}

If two or more components are connected in parallel, they have the same difference of potential (voltage) across their ends.  The potential differences across the components are the same in magnitude, and they also have identical polarities. The same voltage is applied to all circuit components connected in parallel. The total current is the sum of the currents through the individual components, in accordance with [[Kirchhoff's circuit laws#Kirchhoff's current law|Kirchhoff's current law]].

===Voltage<span class="anchor" id="Rparallel"></span>===
In a '''parallel circuit''', the voltage is the same for all elements.
<math display="block">V = V_1 = V_2 = \dots = V_n</math>

===Current===
The current in each individual resistor is found by [[Ohm's law]]. Factoring out the voltage gives
<math display="block">I = \sum_{i=1}^n I_i = V\sum_{i=1}^n{1\over R_i}.</math>

===Resistance units===
To find the total [[Electrical resistance|resistance]] of all components, add the [[Multiplicative inverse|reciprocals]] of the resistances <math>R_i</math> of each component and take the reciprocal of the sum. Total resistance will always be less than the value of the smallest resistance:
[[File:Resistors_in_parallel.svg|alt=A diagram of several resistors, side by side, both leads of each connected to the same wires.|border|center|x120px]]
<math display="block">R = \left(\sum_{i=1}^n{1\over R_i}\right)^{-1} = \left({1\over R_1} + {1\over R_2} + {1\over R_3} + \dots + {1\over R_n}\right)^{-1}</math>

For only two resistances, the unreciprocated expression is reasonably simple:
<math display="block">R = \frac{R_1 R_2}{R_1 + R_2} .</math>

This sometimes goes by the mnemonic ''product over sum''.

For ''N'' equal resistances in parallel, the reciprocal sum expression simplifies to:
<math display="block">\frac{1}{R} = N \frac{1}{R}.</math>
and therefore to:
<math display="block">R = \frac{R}{N}.</math>

To find the [[current (electricity)|current]] in a component with resistance <math>R_i</math>, use Ohm's law again:
<math display="block">I_i = \frac{V}{R_i}\,.</math>

The components divide the current according to their reciprocal resistances, so, in the case of two resistors,
<math display="block">\frac{I_1}{I_2} = \frac{R_2}{R_1}.</math>

An old term for devices connected in parallel is ''multiple'', such as multiple connections for [[arc lamp]]s.

==== Conductance ====
Since electrical conductance <math>G</math> is reciprocal to resistance, the expression for total conductance of a parallel circuit of resistors is simply:
<math display="block">G = \sum_{i=1}^n G_i = G_1 + G_2 + G_3 \cdots + G_n.</math>

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<span class="anchor" id="Lparallel"></span>===
[[Inductor]]s 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:
[[File:Inductors_in_parallel.svg|alt=A diagram of several inductors, side by side, both leads of each connected to the same wires.|border|center|x120px]]
<math display="block">L = \left(\sum_{i=1}^n{1\over L_i}\right)^{-1} = \left({1\over L_1} + {1\over L_2} + {1\over L_3} + \dots + {1\over L_n}\right)^{-1}.</math>

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 {{mvar|M}}, the equivalent inductor is:
<math display="block">L = \frac{L_1L_2 - M^2}{L_1 + L_2 - 2M}</math>

If <math>L_1 = L_2</math>
<math display="block"> L = \frac{L + M}{2}</math>

The sign of <math>M</math> 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 {{mvar|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 {{mvar|M}} is very nearly equal to {{mvar|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 {{mvar|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 <math>M_{12}</math>, <math>M_{13}</math> and <math>M_{23}</math>. This is best handled by matrix methods and summing the terms of the inverse of the <math>L</math> matrix (3×3 in this case).

The pertinent equations are of the form:
<math display="block">v_{i} = \sum_{j} L_{i,j} \frac{di_j}{dt} </math>

===Capacitors<span class="anchor" id="Cparallel"></span>===
The total [[capacitance]] of [[capacitors]] in parallel is equal to the sum of their individual capacitances:
[[File:Capacitors_in_parallel.svg|alt=A diagram of several capacitors, side by side, both leads of each connected to the same wires.|border|center|x120px]]
<math display="block">C = \sum_{i=1}^n C_i = C_1 + C_2 + C_3 \cdots + C_n.</math>

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

===Switches===
Two or more [[switch]]es in parallel form a [[logical disjunction|logical OR]]; the circuit carries current if at least one switch is closed. See [[OR gate]].

===Cells and batteries===
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 in voltage, cells with higher voltages will attempt to charge those with lower ones, potentially damaging them.

Parallel-connected batteries were widely used to power the [[Vacuum tube|valve]] filaments in [[portable radio]]s. 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.