A parallel circuit is a way of joining two or more electricity-producing devices such as PV cells or modules, or batteries by connecting positive leads together and negative leads together; such a configuration increases the current but the voltage is constant. The potential differences across the components are the same in magnitude, and they also have identical polarities. The same voltage is applicable 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 current law. Potential difference is measured in amps .

Voltage[edit]

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

{\displaystyle V=V_{1}=V_{2}=\ldots =V_{n}} V = V_1 = V_2 = \ldots = V_n

Current[edit]

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

{\displaystyle I_{\mathrm {total} }=V\left({\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+\cdots +{\frac {1}{R_{n}}}\right)}I_\mathrm{total} = V\left(\frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}\right).

Resistors[edit]

To find the total resistance of all components, add the reciprocals of the resistances {\displaystyle R_{i}}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:

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

{\displaystyle {\frac {1}{R_{\mathrm {total} }}}={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+\cdots +{\frac {1}{R_{n}}}}\frac{1}{R_\mathrm{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}.

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

{\displaystyle R_{\mathrm {total} }={\frac {R_{1}R_{2}}{R_{1}+R_{2}}}.}R_\mathrm{total} = \frac{R_1R_2}{R_1+R_2} .

This sometimes goes by the mnemonic “product over sum”.

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

{\displaystyle {\frac {1}{R_{\mathrm {total} }}}={\frac {1}{R}}\times N}\frac{1}{R_\mathrm{total}} = \frac{1}{R} \times N.

and therefore to:

{\displaystyle {R_{\mathrm {total} }}={\frac {R}{N}}}{R_\mathrm{total}} = \frac{R}{N}.

To find the current in a component with resistance {\displaystyle R_{i}}R_{i}

, use Ohm’s law again:
{\displaystyle I_{i}={\frac {V}{R_{i}}}\,}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}}}}\frac{I_1}{I_2} = \frac{R_2}{R_1}.

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

Since electrical conductance {\displaystyle G}G

is reciprocal to resistance, the expression for total conductance of a parallel circuit of resistors reads:
{\displaystyle {G_{\mathrm {total} }}={G_{1}}+{G_{2}}+\cdots +{G_{n}}}{G_\mathrm{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[edit]

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.

{\displaystyle {\frac {1}{L_{\mathrm {total} }}}={\frac {1}{L_{1}}}+{\frac {1}{L_{2}}}+\cdots +{\frac {1}{L_{n}}}}\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 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_{\mathrm {total} }}}={\frac {L_{1}+L_{2}-2M}{L_{1}L_{2}-M^{2}}}}{\displaystyle {\frac {1}{L_{\mathrm {total} }}}={\frac {L_{1}+L_{2}-2M}{L_{1}L_{2}-M^{2}}}}

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

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

The sign of {\displaystyle M}M

depends on how the magnetic fields influence each other. For two equal tightly coupled coils the total inductance is close to that of each 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 becomes 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}}M_{12}

, {\displaystyle M_{13}}M_{13} and {\displaystyle M_{23}}M_{23}. This is best handled by matrix methods and summing the terms of the inverse of the {\displaystyle L}L matrix (3 by 3 in this case).

The pertinent equations are of the form: {\displaystyle v_{i}=\sum _{j}L_{i,j}{\frac {di_{j}}{dt}}}v_{i}=\sum_{j} L_{i,j}\frac{di_{j}}{dt}

Capacitors[edit]

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

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

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

Switches[edit]

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

Cells and batteries[edit]

If the cells of a battery are connected in parallel, the