Resistors in Parallel

Resistors in Parallel

Resistors are said to be connected together in "Parallel" when both of their terminals are respectively connected to each terminal of the other resistor or resistors. The voltage drop across all of the resistors in parallel is the same. Then, Resistors in Parallel have a Common Voltage across them and in our example below the voltage across the resistors is given as:

V_R1 = V_R2 = V_R3 = V_AB = 12V

In the following circuit the resistors R₁, R₂ and R₃ are all connected together in parallel between the two points A and B.

Parallel Resistor Circuit

In the previous series resistor circuit we saw that the total resistance, R_T of the circuit was equal to the sum of all the individual resistors added together. For resistors in parallel the equivalent circuit resistance R_T is calculated differently.

Parallel Resistor Equation

Here, the reciprocal ( 1/Rn ) value of the individual resistances are all added together instead of the resistances themselves. This gives us a value known as Conductance, symbol G with the units of conductance being the Siemens, symbol S. Conductance is therefore the reciprocal or the inverse of resistance, ( G = 1/R ). To convert this conductance sum back into a resistance value we need to take the reciprocal of the conductance giving us then the total resistance, R_T of the resistors in parallel.

Example No1

For example, find the total resistance of the following parallel network

Then the total resistance R_T across the two terminals A and B is calculated as:

This method of calculation can be used for calculating any number of individual resistances connected together within a single parallel network. If however, there are only two individual resistors in parallel then a much simpler and quicker formula can be used to find the total resistance value, and this is given as:

Example No2

Consider the following circuit with the two resistors in parallel combination.

Using our two resistor formula above we can calculate the total circuit resistance, R_T as:

One important point to remember about resistors in parallel, is that the total circuit resistance (R_T) of any two resistors connected together in parallel will always be LESS than the value of the smallest resistor and in our example above  R_T = 14.9kΩ were as the value of the smallest resistor is only 22kΩ. Also, in the case of R₁ being equal to the value of R₂, ( R₁ = R₂ ) the total resistance of the network will be exactly half the value of one of the resistors, R/2.

Consider the two resistors in parallel above. The current that flows through each of the resistors ( I_R1 and I_R2 ) connected together in parallel is not necessarily the same value as it depends upon the resistive value of the resistor. However, we do know that the current that enters the circuit at point A must also exit the circuit at point B. Kirchoff's Current Laws. states that "the total current leaving a circuit is equal to that entering the circuit - no current is lost". Thus, the total current flowing in the circuit is given as:

I_T = I_R1 + I_R2

Then by using Ohm's Law, the current flowing through each resistor can be calculated as:

Current flowing in R₁ = V/R₁ =  12V ÷ 22kΩ = 0.545mA

Current flowing in R₂ = V/R₂ =  12V ÷ 47kΩ = 0.255mA

giving us a total current I_T flowing around the circuit as:

I_T = 0.545mA + 0.255mA = 0.8mA or 800uA.

The equation given for calculating the total current flowing in a parallel resistor circuit which is the sum of all the individual currents added together is given as:

I_total = I₁ + I₂+ I₃  + ..... I_n

Then parallel resistor networks can also be thought of as "current dividers" because the current splits or divides between the various branches and a parallel resistor circuit having N resistive networks will have N-different current paths while maintaining a common voltage.