Unlike the previous series resistor circuit, in a parallel resistor network the circuit current can take more than one path as there are multiple paths for the current. Then parallel circuits are classed as current dividers.
Since there are multiple paths for the supply current to flow through, the current may not be the same through all the branches in the parallel network. However, the voltage drop across all of the resistors in a parallel resistive network IS the same. Then, Resistors in Parallel have a Common Voltage across them and this is true for all parallel connected elements.
So we can define a parallel resistive circuit as one where the resistors are connected to the same two points (or nodes) and is identified by the fact that it has more than one current path connected to a common voltage source. Then in our parallel resistor example below the voltage across resistor R1 equals the voltage across resistor R2 which equals the voltage across R3 and which equals the supply voltage. Therefore, for a parallel resistor network this is given as:
In the following resistors in parallel circuit the resistors R1, R2 and R3 are all connected together in parallel between the two points A and B as shown.
Parallel Resistor Circuit
In the previous series resistor network we saw that the total resistance, RT of the circuit was equal to the sum of all the individual resistors added together. For resistors in parallel the equivalent circuit resistance RT is calculated differently.
Here, the reciprocal ( 1/R ) value of the individual resistances are all added together instead of the resistances themselves with the inverse of the algebraic sum giving the equivalent resistance as shown.
Parallel Resistor Equation
Then the inverse of the equivalent resistance of two or more resistors connected in parallel is the algebraic sum of the inverses of the individual resistances.
If the two resistances or impedances in parallel are equal and of the same value, then the total or equivalent resistance, RT is equal to half the value of one resistor. That is equal to R/2 and for three equal resistors in parallel, R/3, etc.
Note that the equivalent resistance is always less than the smallest resistor in the parallel network so the total resistance, RT will always decrease as additional parallel resistors are added.
Parallel resistance gives us a value known as Conductance, symbol G with the units of conductance being the Siemens, symbol S. Conductance is the reciprocal or the inverse of resistance, ( G = 1/R ). To convert conductance back into a resistance value we need to take the reciprocal of the conductance giving us then the total resistance, RT of the resistors in parallel.
We now know that resistors that are connected between the same two points are said to be in parallel. But a parallel resistive circuit can take many forms other than the obvious one given above and here are a few examples of how resistors can be connected together in parallel.
Various Parallel Resistor Networks
The five resistive networks above may look different to each other, but they are all arranged as Resistors in Parallel and as such the same conditions and equations apply.
Resistors in Parallel Example No1
Find the total resistance, RT of the following resistors connected in a parallel network.
The total resistance RT across the two terminals A and B is calculated as:
This method of reciprocal 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 we can use a much simpler and quicker formula to find the total or equivalent resistance value, RT and help reduce the reciprocal maths a little.
This much quicker product-over-sum method of calculating two resistor in parallel, either having equal or unequal values is given as:
Resistors in Parallel Example No2
Consider the following circuit which has only two resistors in a parallel combination.
Using our formula above for two resistors connected together in parallel we can calculate the total circuit resistance, RT as:
One important point to remember about resistors in parallel, is that the total circuit resistance ( RT ) of any two resistors connected together in parallel will always be LESS than the value of the smallest resistor in that combination.
In our example above, the value of the combination was calculated as: RT = 15kΩ, where as the value of the smallest resistor is 22kΩ, much higher. In other words, the equivalent resistance of a parallel network will always be less than the smallest individual resistor in the combination.
Also, in the case of R1 being equal to the value of R2, that is R1 = R2, the total resistance of the network will be exactly half the value of one of the resistors, R/2.
Likewise, if three or more resistors each with the same value are connected in parallel, then the equivalent resistance will be equal to R/n where R is the value of the resistor and n is the number of individual resistances in the combination.
For example, six 100Ω resistors are connected together in a parallel combination. The equivalent resistance will therefore be: RT = R/n = 100/6 = 16.7Ω. But note that this ONLY works for equivalent resistors. That is resistors all having the same value.
Currents in a Parallel Resistor Circuit
The total current, IT entering a parallel resistive circuit is the sum of all the individual currents flowing in all the parallel branches. But the amount of current flowing through each parallel branch may not necessarily be the same, as the resistive value of each branch determines the amount of current flowing within that branch.
For example, although the parallel combination has the same voltage across it, the resistances could be different therefore the current flowing through each resistor would definitely be different as determined by Ohms Law.
Consider the two resistors in parallel above. The current that flows through each of the resistors ( IR1 and IR2 ) 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.
Kirchhoff’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:
IT = IR1 + IR2
By using Ohm’s Law, we can calculate the current flowing through each parallel resistor shown in Example No2 above as being:
The current flowing in resistor R1 is given as:
IR1 = VS ÷ R1 = 12V ÷ 22kΩ = 0.545mA or 545μA
The current flowing in resistor R2 is given as:
IR2 = VS ÷ R2 = 12V ÷ 47kΩ = 0.255mA or 255μA
thus giving us a total current IT flowing around the circuit as:
IT = 0.545mA + 0.255mA = 0.8mA or 800μA
and this can also be verified directly using Ohm’s Law as:
IT = VS ÷ RT = 12 ÷ 15kΩ = 0.8mA or 800μA (the same)
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:
Then parallel resistor networks can also be thought of as “current dividers” because the supply current splits or divides between the various parallel branches. So a parallel resistor circuit having N resistive networks will have N-different current paths while maintaining a common voltage across itself. Parallel resistors can also be interchanged with each other without changing the total resistance or the total circuit current.
Resistors in Parallel Example No3
Calculate the individual branch currents and total current drawn from the power supply for the following set of resistors connected together in a parallel combination.
As the supply voltage is common to all the resistors in a parallel circuit, we can use Ohms Law to calculate the individual branch current as follows.
Then the total circuit current, IT flowing into the parallel resistor combination will be:
This total circuit current value of 5 amperes can also be found and verified by finding the equivalent circuit resistance, RT of the parallel branch and dividing it into the supply voltage, VS as follows.
Equivalent circuit resistance:
Then the current flowing in the circuit will be:
Resistors in Parallel Summary
So to summarise. When two or more resistors are connected so that both of their terminals are respectively connected to each terminal of the other resistor or resistors, they are said to be connected together in parallel. The voltage across each resistor within a parallel combination is exactly the same but the currents flowing through them are not the same as this is determined by their resistance value and Ohms Law. Then parallel circuits are current dividers.
The equivalent or total resistance, RT of a parallel combination is found through reciprocal addition and the total resistance value will always be less than the smallest individual resistor in the combination. Parallel resistor networks can be interchanged within the same combination without changing the total resistance or total circuit current. Resistors connected together in a parallel circuit will continue to operate even though one resistor may be open-circuited.
Thus far we have seen resistor networks connected in either a series or a parallel combination. In the next tutorial about Resistors, we will look at connecting resistors together in both a series and parallel combination at the same time producing a mixed or combinational resistor circuit.