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### Course: Class 12 Physics (India)>Unit 3

Lesson 9: Series and parallel resistors

# 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.
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, $\text{R1}$, $\text{R2}$, and $\text{R3}$ 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 $\text{R1}$, $\text{R2}$, and $\text{R3}$ 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 ${\text{I}}_{\text{s}}$ is driving current $i$ towards $\text{R1}$, $\text{R2}$, and $\text{R3}$. 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:
$i={i}_{\text{R1}}+{i}_{\text{R2}}+{i}_{\text{R3}}$
$v={i}_{\text{R1}}\cdot \text{R1}\phantom{\rule{2em}{0ex}}v={i}_{\text{R2}}\cdot \text{R2}\phantom{\rule{2em}{0ex}}v={i}_{\text{R3}}\cdot \text{R3}$
This is enough to get going. Rearrange the three Ohm's Law expressions to solve for current in terms of voltage and resistance:
${i}_{\text{R1}}=\frac{v}{\text{R1}}\phantom{\rule{2em}{0ex}}{i}_{\text{R2}}=\frac{v}{\text{R2}}\phantom{\rule{2em}{0ex}}{i}_{\text{R3}}=\frac{v}{\text{R3}}$
Substitute these expressions into the sum of currents:
$i=\frac{v}{\text{R1}}+\frac{v}{\text{R2}}+\frac{v}{\text{R3}}$
Factor out the common term $v$,
$i=v\left(\frac{1}{\text{R1}}+\frac{1}{\text{R2}}+\frac{1}{\text{R3}}\right)$
Now remember that we already know $i$ (it is a property of the current source), so we can solve for $v$:
$v=i\phantom{\rule{0.167em}{0ex}}\frac{1}{\left(\frac{1}{\text{R1}}+\frac{1}{\text{R2}}+\frac{1}{\text{R3}}\right)}$
This expression looks just like Ohm's Law, $v=i\phantom{\rule{0.167em}{0ex}}\text{R}$, 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.
${\text{R}}_{\text{parallel}}=\frac{1}{\left(\frac{1}{\text{R1}}+\frac{1}{\text{R2}}+\frac{1}{\text{R3}}\right)}$
The equivalent parallel resistor is the reciprocal of the sum of reciprocals. We can write this equation another way by rearranging the giant reciprocal,
$\frac{1}{{\text{R}}_{\text{parallel}}}=\frac{1}{\text{R1}}+\frac{1}{\text{R2}}+\frac{1}{\text{R3}}$
Ohm's Law applied to parallel resistors,
$v=i\phantom{\rule{0.167em}{0ex}}{\text{R}}_{\text{parallel}}$
From the "viewpoint" of the current source, the equivalent resistor ${R}_{\text{parallel}}$ 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,
$\frac{1}{{\text{R}}_{\text{parallel}}}=\frac{1}{\text{R1}}+\frac{1}{\text{R2}}+\text{…}+\frac{1}{{\text{R}}_{\text{N}}}$

### 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.
$v={i}_{\text{R}1}\cdot {\text{R}}_{1}\phantom{\rule{2em}{0ex}}v={i}_{\text{R}2}\cdot {\text{R}}_{2}\phantom{\rule{2em}{0ex}}v={i}_{\text{R}3}\cdot {\text{R}}_{3}\phantom{\rule{2em}{0ex}}$
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:

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

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

problem 4
In the example, $\text{R1}$ and $\text{R3}$ happen to differ in resistance by a ratio of $1:10$ $\left(50\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }$ vs. $500\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }\right)$. What is the ratio of their currents?
Choose 1 answer:

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

## Special case - two resistors in parallel

Two resistors in parallel have an equivalent resistance of:
${\text{R}}_{\text{parallel}}=\frac{1}{\left(\frac{1}{\text{R1}}+\frac{1}{\text{R2}}\right)}$
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 ${R}_{\text{parallel}}$?
Let $\text{R1},\text{R2}=\text{R}$
${R}_{\text{parallel}}=\frac{\text{R}\cdot \text{R}}{\text{R}+\text{R}}=\frac{\text{R}\cdot \text{R}}{2\text{R}}$
${R}_{\text{parallel}}=\frac{1}{2}\text{R}$
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,
$\frac{1}{{\text{R}}_{\text{parallel}}}=\frac{1}{\text{R1}}+\frac{1}{\text{R2}}+\text{…}+\frac{1}{{\text{R}}_{\text{N}}}$
For two parallel resistors it is usually easier to combine them as the product over the sum:
${\text{R}}_{\text{parallel}}=\frac{\text{R1}\cdot \text{R2}}{\text{R}1+\text{R}2}$
${\text{R}}_{\text{parallel}}$ 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?

• 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
(13 votes)
• 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.
(44 votes)
• 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)
• 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.
(9 votes)
• 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 ?
(3 votes)
• When given a choice, current will always go through the path of least resistance, literally in this example.
(4 votes)
• If I have a more complicated circuit, do the resistors still have the same voltage?
(2 votes)
• If the ends of the resistors are connected to each other, then they share the same voltage and they are for sure in parallel. It does not matter that other complicated things are connected as well. The resistors are still in parallel.
(3 votes)
• 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)
• 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)
• 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)
• Bit of a simple question, but how did we convert
50Ω + 100Ω + 500Ω
to
0.02 + 0.01 + 0.002?
Thanks!
(2 votes)
• does an equal current pass through all branches at a node in a circuit?
(1 vote)
• Definitely not. The only thing you know for sure is that all the currents flowing into a node add up to zero (Kirchhoff's Current Law). Other than that one rule, the currents can have any value.
(2 votes)
• Is it counted as parallel is they are connected vertically on the diagram?
(1 vote)
• Resistors are in parallel if their terminals are connected to the same two nodes. It doesn't matter if the resistors are drawn vertical or horizontal.
(2 votes)
• Hi, this might have been answered already and I have just missed it, but what is the exact reason why current is constant when resistors are in series and voltage is constant with resistors in parallel? Thanks.
(1 vote)
• Current in a series circuit is the same everywhere in the circuit because there is only a single current path. Imagine garden hoses connected in series. All the water that goes into the first hose has to come out the end of the last hose. If you measure flow (in liters or gallons/minute) at any point along the series hoses the measurement has to be the same at every point. If it wasn't, water would be leaking out of the hose somewhere, or hiding in some secret chamber inside the hose.

Parallel connections have the same voltage because the components are connected to only two circuit nodes. If you want to measure voltage you only have two points to touch with your voltmeter probes.
(2 votes)