Derive relation of resistors connected in parallel
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1/R=(1/r1)+(1/r2)+......+(1/rn)
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The total current, I 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 ( I and I ) 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 = I + I
Then by using Ohm’s Law , the current flowing through each resistor of Example No2 above can be calculated as:
Current flowing in
R = V ÷ R = 12V ÷ 22kΩ = 0.545mA or 545μA
Current flowing in
R = V ÷ R = 12V ÷ 47kΩ = 0.255mA or 255μA
thus giving us a total current I flowing around the circuit as:
I = 0.545mA + 0.255mA = 0.8mA or 800μA
and this can also be verified directly using Ohm’s Law as:
I = V ÷ R = 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:
I = I + I + I ….. + I
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, I 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, R of the parallel branch and dividing it into the supply voltage, V 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, R 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.
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 ( I and I ) 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 = I + I
Then by using Ohm’s Law , the current flowing through each resistor of Example No2 above can be calculated as:
Current flowing in
R = V ÷ R = 12V ÷ 22kΩ = 0.545mA or 545μA
Current flowing in
R = V ÷ R = 12V ÷ 47kΩ = 0.255mA or 255μA
thus giving us a total current I flowing around the circuit as:
I = 0.545mA + 0.255mA = 0.8mA or 800μA
and this can also be verified directly using Ohm’s Law as:
I = V ÷ R = 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:
I = I + I + I ….. + I
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, I 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, R of the parallel branch and dividing it into the supply voltage, V 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, R 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.
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