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Parallel resistors

Resistors are in parallel if their terminals are connected to the same two nodes. The equivalent overall resistance is smaller than the smallest parallel resistor. Written by Willy McAllister.
Components are in parallel if they share two nodes, like this:
In this article we will work with resistors in parallel, to reveal the properties of the parallel connection. Later articles will cover capacitors and inductors in series and parallel.

Resistors in parallel

Resistors are in parallel when their two terminals connect to the same nodes.
In the following image, start text, R, 1, end text, start text, R, 2, end text, and start text, R, 3, end text are in parallel. The two distributed nodes are represented by the two horizontal lines.
Resistors in parallel share the same voltage on their terminals.
The resistors in the following image are not in parallel. There are extra components (orange boxes) breaking up the common nodes between resistors. This circuit has four separate nodes, so start text, R, 1, end text, start text, R, 2, end text, and start text, R, 3, end text do not share the same voltage.

Properties of resistors in parallel

Figuring out parallel resistors is a little trickier than series resistors. Here is a circuit with resistors in parallel. (This circuit has a current source. We don't get to use those very often, so this should be fun.)
Current source start text, I, end text, start subscript, start text, s, end text, end subscript is driving current i towards start text, R, 1, end text, start text, R, 2, end text, and start text, R, 3, end text. We know the value of current i is some given constant, but we don't yet know the voltage v or how i splits up into the three resistor currents.
Two things we do know are:
  • The three resistor currents have to add up to i.
  • Voltage v appears across all three resistors.
With just this little bit of knowledge, and Ohm's Law, we can write these expressions:
i, equals, i, start subscript, start text, R, 1, end text, end subscript, plus, i, start subscript, start text, R, 2, end text, end subscript, plus, i, start subscript, start text, R, 3, end text, end subscript
v, equals, i, start subscript, start text, R, 1, end text, end subscript, dot, start text, R, 1, end text, v, equals, i, start subscript, start text, R, 2, end text, end subscript, dot, start text, R, 2, end text, v, equals, i, start subscript, start text, R, 3, end text, end subscript, dot, start text, R, 3, end text
This is enough to get going. Rearrange the three Ohm's Law expressions to solve for current in terms of voltage and resistance:
i, start subscript, start text, R, 1, end text, end subscript, equals, start fraction, v, divided by, start text, R, 1, end text, end fraction, i, start subscript, start text, R, 2, end text, end subscript, equals, start fraction, v, divided by, start text, R, 2, end text, end fraction, i, start subscript, start text, R, 3, end text, end subscript, equals, start fraction, v, divided by, start text, R, 3, end text, end fraction
Substitute these expressions into the sum of currents:
i, equals, start fraction, v, divided by, start text, R, 1, end text, end fraction, plus, start fraction, v, divided by, start text, R, 2, end text, end fraction, plus, start fraction, v, divided by, start text, R, 3, end text, end fraction
Factor out the common term v,
i, equals, v, left parenthesis, start fraction, 1, divided by, start text, R, 1, end text, end fraction, plus, start fraction, 1, divided by, start text, R, 2, end text, end fraction, plus, start fraction, 1, divided by, start text, R, 3, end text, end fraction, right parenthesis
Now remember that we already know i (it is a property of the current source), so we can solve for v:
v, equals, i, start fraction, 1, divided by, left parenthesis, start fraction, 1, divided by, start text, R, 1, end text, end fraction, plus, start fraction, 1, divided by, start text, R, 2, end text, end fraction, plus, start fraction, 1, divided by, start text, R, 3, end text, end fraction, right parenthesis, end fraction
This expression looks just like Ohm's Law, v, equals, i, start text, R, end text, but with the parallel resistors appearing in a double-reciprocal in place of of a single resistor.
We conclude:
For resistors in parallel, the overall resistance is the reciprocal of the sum of reciprocals of the individual resistors.
(This sounds complicated, but we will derive something simpler before we are done.)

Equivalent parallel resistor

The previous equation suggests we can define a new resistor, equivalent to the parallel resistors. The new resistor is equivalent in the sense that, for a given i, the same voltage v appears.
start text, R, end text, start subscript, start text, p, a, r, a, l, l, e, l, end text, end subscript, equals, start fraction, 1, divided by, left parenthesis, start fraction, 1, divided by, start text, R, 1, end text, end fraction, plus, start fraction, 1, divided by, start text, R, 2, end text, end fraction, plus, start fraction, 1, divided by, start text, R, 3, end text, end fraction, right parenthesis, end fraction
The equivalent parallel resistor is the reciprocal of the sum of reciprocals. We can write this equation another way by rearranging the giant reciprocal,
start fraction, 1, divided by, start text, R, end text, start subscript, start text, p, a, r, a, l, l, e, l, end text, end subscript, end fraction, equals, start fraction, 1, divided by, start text, R, 1, end text, end fraction, plus, start fraction, 1, divided by, start text, R, 2, end text, end fraction, plus, start fraction, 1, divided by, start text, R, 3, end text, end fraction
Ohm's Law applied to parallel resistors,
v, equals, i, start text, R, end text, start subscript, start text, p, a, r, a, l, l, e, l, end text, end subscript
From the "viewpoint" of the current source, the equivalent resistor R, start subscript, start text, p, a, r, a, l, l, e, l, end text, end subscript is indistinguishable from the three parallel resistors, because in both circuits, v is the same.
If you have multiple resistors in parallel, the general form of the equivalent parallel resistance is,
start fraction, 1, divided by, start text, R, end text, start subscript, start text, p, a, r, a, l, l, e, l, end text, end subscript, end fraction, equals, start fraction, 1, divided by, start text, R, 1, end text, end fraction, plus, start fraction, 1, divided by, start text, R, 2, end text, end fraction, plus, point, point, point, plus, start fraction, 1, divided by, start text, R, end text, start subscript, start text, N, end text, end subscript, end fraction

Current distributes between resistors in parallel

We worked out the voltage v across the parallel connection, so what's left to figure out is the currents through the individual resistors.
Do this by applying Ohm's Law to the individual resistors.
v, equals, i, start subscript, start text, R, end text, 1, end subscript, dot, start text, R, end text, start subscript, 1, end subscript, v, equals, i, start subscript, start text, R, end text, 2, end subscript, dot, start text, R, end text, start subscript, 2, end subscript, v, equals, i, start subscript, start text, R, end text, 3, end subscript, dot, start text, R, end text, start subscript, 3, end subscript
This becomes more informative if you do an example with real numbers.
Find the voltage v and the currents through the three resistors.
Show that the individual resistor currents add up to i.

Reflection

Based on the resistor currents you just solved:
Problem 1
The largest share of current goes through which resistor?
Choose 1 answer:
Choose 1 answer:

Problem 2
The smallest share of current goes through which resistor?
Choose 1 answer:
Choose 1 answer:

problem 3
Relative to the three parallel resistors, what value is the equivalent resistor?
Choose 1 answer:
Choose 1 answer:

problem 4
In the example, start text, R, 1, end text and start text, R, 3, end text happen to differ in resistance by a ratio of 1, colon, 10 left parenthesis, 50, \Omega vs. 500, \Omega, right parenthesis. What is the ratio of their currents?
Choose 1 answer:
Choose 1 answer:

problem 5
Which resistor has the highest voltage?
Choose 1 answer:
Choose 1 answer:

Special case - two resistors in parallel

Two resistors in parallel have an equivalent resistance of:
start text, R, end text, start subscript, start text, p, a, r, a, l, l, e, l, end text, end subscript, equals, start fraction, 1, divided by, left parenthesis, start fraction, 1, divided by, start text, R, 1, end text, end fraction, plus, start fraction, 1, divided by, start text, R, 2, end text, end fraction, right parenthesis, end fraction
It's possible do a bit of manipulation to eliminate the reciprocals and come up with another expression with just one fraction. Rather than just telling you the answer, it is a rite of passage to work through the algebra the first time. The answer is tucked away so you can try this on your own before peeking.

Special special case - two equal resistors in parallel

If two resistors in parallel have the same value, what is the equivalent R, start subscript, start text, p, a, r, a, l, l, e, l, end text, end subscript?
Let start text, R, 1, end text, comma, start text, R, 2, end text, equals, start text, R, end text
R, start subscript, start text, p, a, r, a, l, l, e, l, end text, end subscript, equals, start fraction, start text, R, end text, dot, start text, R, end text, divided by, start text, R, end text, plus, start text, R, end text, end fraction, equals, start fraction, start text, R, end text, dot, start text, R, end text, divided by, 2, start text, R, end text, end fraction
R, start subscript, start text, p, a, r, a, l, l, e, l, end text, end subscript, equals, start fraction, 1, divided by, 2, end fraction, start text, R, end text
Two identical resistors in parallel have an equivalent resistance half the value of either resistor. The current splits equally between the two.

Summary

Resistors in parallel share the same voltage.
The general form for three or more resistors in parallel is,
start fraction, 1, divided by, start text, R, end text, start subscript, start text, p, a, r, a, l, l, e, l, end text, end subscript, end fraction, equals, start fraction, 1, divided by, start text, R, 1, end text, end fraction, plus, start fraction, 1, divided by, start text, R, 2, end text, end fraction, plus, point, point, point, plus, start fraction, 1, divided by, start text, R, end text, start subscript, start text, N, end text, end subscript, end fraction
For two parallel resistors it is usually easier to combine them as the product over the sum:
start text, R, end text, start subscript, start text, p, a, r, a, l, l, e, l, end text, end subscript, equals, start fraction, start text, R, 1, end text, dot, start text, R, 2, end text, divided by, start text, R, end text, 1, plus, start text, R, end text, 2, end fraction
start text, R, end text, start subscript, start text, p, a, r, a, l, l, e, l, end text, end subscript is always smaller than the smallest parallel resistor.
Current distributes amongst parallel resistors, with the largest current flowing through the smallest resistor.

Want to join the conversation?

  • aqualine tree style avatar for user NJK-Home
    Physically, what is the difference between a voltage source and a current source? Don't they both supply a current and voltage in a circuit? Why use one over the other?
    Thanks
    NK
    (10 votes)
    Default Khan Academy avatar avatar for user
    • spunky sam orange style avatar for user Willy McAllister
      Voltage and current sources generate both voltage and current. The difference between them lies in which parameter (voltage or current) is being controlled.

      A constant voltage source (like a battery) is designed to generate a controlled voltage. When you put a constant voltage source in a circuit, the voltage across its terminals is always a constant value. Depending on what it is connected to, a voltage source provides (generates) whatever current is needed to keep the voltage on its terminals constant. Example: a 1.5 V battery connected to a 100 ohm resistor will generate a current of 1.5/100 = 15 mA. If you change the resistor to 10 ohms, the voltage will still be 1.5 V but the voltage source will now generate a current of 1.5/10 = 150 mA.

      Current sources may seem a little strange, but they behave exactly like a voltage source, but with current being controlled. A constant current source is designed to generate a controlled current. When you put a current source in a circuit, the current through the source is always a constant value. Depending on what it is connected to, a current source provides whatever voltage is needed to keep the current on its terminals constant. Example: suppose you have a constant current source set to current = 1 mA. If you connect a 100 ohm resistor across the current source, the voltage will be V = 1 mA x 100 ohms = 0.1 V. If you change the resistor to 1000 ohms, the current will still be 1 mA and the voltage generated by the current source will rise to V = 1 mA x 1000 ohms = 1 V.

      Most transistors (MOSFET, Bipolar) and the old vacuum tubes have a region of operation where they act just like a current source. As beginning engineers, current sources are not familiar because they are buried inside integrated circuits.
      (31 votes)
  • old spice man green style avatar for user Snehit Kashyap
    If I make a circuit with 5 resistors of 30 ohms each in series and a battery of 24 volts and if I connect a 23W led bulb will it glow ??
    (3 votes)
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    • spunky sam orange style avatar for user Willy McAllister
      If I guess that the 23 watt LED lightbulb you have is designed for home lighting, then the circuit you describe may not light the bulb. Home LED lightbulbs are designed to work with 110 VAC or 220 VAC connected to them. There is a lot of circuitry inside an LED lightbulb to make it work with high voltage. It's packed with a lot more stuff than just an individual LED semiconductor device.
      (8 votes)
  • blobby green style avatar for user Roshan Ajith
    What is the reason behind the largest share of current going through the smallest resistor and the smallest share of current going through the largest resistor ?
    (2 votes)
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  • aqualine ultimate style avatar for user Denis
    If I have a more complicated circuit, do the resistors still have the same voltage?
    (2 votes)
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  • primosaur sapling style avatar for user Joseph Mullan
    i don't understand how you use the equations to analyse a circuit
    (3 votes)
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  • piceratops seed style avatar for user pranathisrao
    Is it the resistance of a resistor that determines the current flowing through it, or is that as more resistors are added, there is less current to go through to the next resistor? (By Kirchhoff's Laws, the current flowing into a system equals the current flowing out. So if the resistors in the "do it yourself" questions were switched (500 ohms first, and 50 ohms last) how would it be different?)
    (1 vote)
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    • piceratops seed style avatar for user jboling5
      Current wants to "get away" from resistance, so the lower your resistance, the more current will pass through. If you switched those two resistors absolutely nothing would be different. This is one of the most fundamentally important concepts in Electrical Engineering, and there are a few things to be learned from this:

      1. Things that are in parallel have the same voltage. Think of voltage as the height of a cliff. Think of the wires as a water slide down the cliff. One slide may be steeper than the other (so water flows down more easily, it has less resistance) but they start at the same height on the cliff. Same voltage, but could be different current depending on the resistances.

      2. There are only two nodes in that diagram, see his previous lesson on this. Name the top node "A" and the bottom node "B". Mark each side of your circuit elements "A" and "B" as well. As long as the "A" side of the element and the "B" side of the element touch the "A" and "B" node respectively, you can redraw this diagram ANYWAY you want. This is because this is an ideal circuit where the wiring has absolutely no resistance. All of the circuit elements are practically touching each other.

      3. You asked if more resistors are added, would there be less current to go around. The answer is maybe. Let's say all the resistors in his example were 1 Ohm. They would each get 1/3 of the total current. Let's say he adds another 1 Ohm resistor in parallel-- now each resistor gets 1/4 the total current. You could add a resistor with a really high resistance in certain places of the circuit to make the current want to go somewhere else. You could add a resistor with a really low resistance in certain places of the circuit to make the current want to go there instead.
      (4 votes)
  • leaf blue style avatar for user Ayman Munassar
    An idea to calculate three parallel resistors: [R1R2R3/R3(R1+R2)+(R1R2)]
    I tried it in the above example and it does work.
    (1 vote)
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  • male robot hal style avatar for user Jesse Arias
    Bit of a simple question, but how did we convert
    50Ω + 100Ω + 500Ω
    to
    0.02 + 0.01 + 0.002?
    Thanks!
    (2 votes)
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  • duskpin ultimate style avatar for user Kinjal
    does an equal current pass through all branches at a node in a circuit?
    (1 vote)
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  • blobby green style avatar for user deranakin2
    Is it counted as parallel is they are connected vertically on the diagram?
    (1 vote)
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