Parallel Circuits

To determine whether or not you are working with a parallel circuit, trace the path of current flow from the battery. If you encounter a junction where the current can take two or more paths, you are dealing with a parallel circuit or different branches.

The equations used to solve parallel circuits are as follows:

V_battery = V₁ = V₂= V₃

(the voltage of the battery is the same in every branch)

I_total = I₁ + I₂+ I₃

(the total current from the battery is divided up into the branches, each branch has equal current only if the resistance of each branch is the same, higher resistance in a branch means a lower current in that branch)

1/R_eq = 1/R₁ + 1/R₂+ 1/R₃

(the equivalent resistance of parallel resistors is smaller than the smallest resistor)

Example Problem:

Solve for the voltage and current across each resistor if three resistors of values 1Ω, 2Ω, and 3Ω are connected in parallel to a battery of voltage 12 V.

Solution: In solving circuits containing branches, first reduce the resistors into an equivalent resistor.

1/R_eq = 1/R₁ + 1/R₂+ 1/R₃

1/R_eq = 1/1Ω + 1/2Ω + 1/3Ω

R_eq = 0.55Ω

(Taking the three resistors out of the circuit and replacing them with one resistor of value 0.55Ω will yield the same current from the battery.)

Secondly, solve for the total current in the circuit using Ohm’s Law, I = V/R, where V is the battery voltage and R is the equivalent resistance.

I_tot = V_tot/R_eq

I_tot = 12V/0.55Ω = 21.8 A

Next, solve for the current in each individual branch. We know that the voltage of each branch is the same as the battery, so all voltages are 12 V.

I₁ = V₁/R₁= 12 V/1Ω = 12 A

I₂ = V₂/R₂= 12 V/2Ω = 6 A

I₃ = V₃/R₃= 12 V/3Ω = 4 A

Check the answers by seeing if the total of all currents equals the total current coming from the battery.

Check: 12A + 6A + 4A = 22A (close enough to 21.8 A with rounding)

Combination Circuits

Combination circuits contain resistors that are configured in series and in parallel. To solve these problems, it is important to slowly reduce the circuit by combining parallel branches or series resistors when appropriate. Eventually, you should be able to reduce the circuit down to only one resistor. This is called the equivalent resistor and would give the same current from the battery as the combination circuit. The following circuit shows three branches, each with two resistors. To break the circuit down, combine the two resistors in each branch by adding them (they are in series). You would now have three branches, each with one resistor. To find the equivalent resistance, add the reciprocals of the three resistors in parallel.

(source)