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Parallel circuit

In electrical circuits series and parallel are two basic ways of wiring components. The naming comes after the method of attaching components, i.e. one after the other, or next to each other. As a demonstration, consider a very simple circuit consisting of two lightbulbs and one 9V 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 (A) (coulombs per second), and V, voltage, measured in volts (V) (joules per coulomb).

Series circuits

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

The same current has to pass through all the components in the series. An ammeter placed anywhere in the circuit would measure the same amount.

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

R_total = R₁ + R₂ + ... + R_n

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

• To find the current, I, use Ohm's law.

I = V/R_total

• To find the voltage across any particular component with resistance R_i , use Ohm's law again.

V=IR_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, V₁/V₂ = R₁/R₂

Inductors follow the same law, in that the total inductance of inductors in series is equal to the sum of their individual inductances:

<math>L_{total}=L_1+L_2+ \cdots +L_n<math>

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:

<math>{1\over{C_{total}}}={1\over{C_1}}+{1\over{C_2}}+ \cdots +{1\over{C_n}}<math>

Parallel circuits

The voltage is the same across all the components in parallel.

• 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 all components) gives:

<math>I_{total} = V \cdot \left(\frac{1} {R_1} + \frac{1} {R_2} + ... + \frac{1} {R_n}\right)<math>

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

1 / R_total = 1 / R₁ + 1 / R₂ + ... + 1 / 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, I₁/I₂ = R₂/R₁

Inductors follow the same law, in that the total inductance of inductors in parallel is equal to the reciprocal of the sum of the reciprocals of their individual inductances:

<math>{1\over{L_{total}}}={1\over{L_1}}+{1\over{L_2}}+ \cdots +{1\over{L_n}}<math>

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

<math>C_{total}=C_1+C_2+ \cdots +C_n<math>

Notation

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

<math> R_{total} = R_1 \| R_2 = {R_1 R_2 \over R_1 + R_2} <math>