Parallel circuits
If two or more components are connected in parallel they have the same potential difference (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 applicable to all circuit components connected in parallel. The total current I is the sum of the currents through the individual components, in accordance with Kirchhoff's circuit laws.
Resistors
The current in each individual resistor is found by Ohm's law. Factoring out the voltage gives
For only two resistors, the unreciprocated expression is reasonably simple:
This sometimes goes by the mnemonic "product over sum".
To find the current in a component with resistance Ri, use Ohm's law again:
The components divide the current according to their reciprocal resistances, so, in the case of two resistors,
An old term for devices connected in parallel is multiple, such as a multiple connection for arc lamps.
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:
If L1 = L2
The sign of 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. We are assuming 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 3 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 M12, M13 and M23. This is best handled by matrix methods and summing the terms of the inverse of the L matrix (3 by 3 in this case).
The pertinent equations are of the form:
Capacitors
Capacitors follow the same law using the reciprocals. The total capacitance of capacitors in parallel is equal to the sum of their individual capacitances:
Switches
Two or more switches in parallel, form a logical OR; the circuit carries current if at least one switch is 'on'. 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 contains four cells connected in parallel and delivers a current of 1 ampere, the current supplied by each cell will be 0.25 ampere. Parallel-connected batteries were widely used to power the valve filaments in portable radios but they are now rare.
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