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Applying Kirchhoff's Rules: Examples & Problems

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  • 0:02 Complex Circuits
  • 0:54 Kirchoff's Rules
  • 2:08 The Current Rule
  • 4:05 The Voltage Rule
  • 6:29 Lesson Summary
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Lesson Transcript
Instructor: Sarah Friedl

Sarah has two Master's, one in Zoology and one in GIS, a Bachelor's in Biology, and has taught college level Physical Science and Biology.

Ohm's Law can be useful for analyzing simple circuits, but when circuits are more complex, as they often are, we can instead analyze the circuits using Kirchhoff's rules. Learn about Kirchhoff's rules in this lesson.

Complex Circuits

Up to this point in our lessons on series and parallel circuits, we've covered basic direct current circuits in a pretty good amount of detail. But, the types of circuits we've discussed so far have been fairly simple. They consisted of one loop, one power source, and only a handful of resistors.

But, we can't always make our circuit be just one single, easy to analyze loop. Sometimes our circuits are more complicated, which may lead us to the work of a man named Gustav Kirchhoff. What? You've never heard of him? That's unfortunate because he was a very smart man and a talented physicist. For example, he calculated that an electrical signal travels along a resistance-less wire at the speed of light! He also worked with Robert Bunsen, the inventor of, that's right, the Bunsen burner!

Kirchhoff's Rules

Kirchhoff developed two rules, or laws, regarding circuits. The first of Kirchhoff's rules states that the sum of the currents entering a junction must equal the sum of the currents leaving the junction. Because this law deals with the current at junctions in the circuit, it's also sometimes called Kirchhoff's current law or Kirchhoff's junction rule. All this law is saying is that we have a conservation of charge - the current may split and go in different directions at the junction, but if you added up the total amount of current in each branch they would equal the amount of current that originally came into that junction.

The second of Kirchhoff's rules states that the algebraic sum of the voltage differences across all elements of a closed loop must equal zero. This rule is based on the law of conservation of energy, and reminds us that as current encounters a resistor there's a 'voltage drop' and that the sum of all voltage drops is equal to the total supplied voltage. Since this rule deals with voltage across loops in the circuit, it's also sometimes called Kirchhoff's voltage law or Kirchhoff's loop rule.

Using the Current Rule

So, now that you know what Kirchhoff's rules are, let's look at how you might use them to help you analyze a more complicated circuit. Say, for example, that you have a complex circuit like this:

Complex circuit
diagram of complex circuit

Here, we use the letter I to represent current, the letter R to represent resistance, and the letter E to represent the power supply. You can tell by the arrows which way the current is flowing and, in some cases, how much current is flowing in that direction.

But, let's say that some part of the current flow is unknown. We can use Kirchhoff's first rule, the current law, to help us. In this circuit, we can see that 15A enters junction A, and that 7A goes through the right branch toward Resistor 2. How much current flows toward Resistor 3? We simply find the difference between the left branch and the total incoming current. Therefore, I1 - I2 = I3, or 15A - 7A = 8A.

It works the same no matter how many incoming sources of power or outgoing branches you have. If, for example, you have the type of junction where there are two sources of incoming current and three exit paths for the current to flow through, your equation would look like this: I1 + I2 = I3 + I4 + I5.

Junction for above example
diagram of single circular source with five currents

And, if you know how much current is leaving the junction in each branch, but only know how much is coming in from one of the branches, say I2, simply rearrange your equation so that you get that unknown value by itself. Then, solve for I3 + I4 + I5 - I2 = I1, and you'll get your answer!

Using the Voltage Rule

The voltage rule is a little bit more complicated because we have to pay close attention to which direction we're heading through the circuit as well as through the power source. Current flows from the negative terminal of our power supply, through the circuit, and back to the positive terminal of the power supply as it completes the loop. As current passes through a resistor along the circuit there's a 'voltage drop,' which means there is a loss in voltage.

The total loss of voltage around a circuit loop will equal the total voltage of the power supply. So, if you have a 12V battery powering the circuit and you have three resistors along that circuit, you'll have a total loss of 12V across the resistors. In contrast, as current flows through the power supply, it passes from the positive terminal to the negative terminal, and there's a 'voltage rise,' or increase in voltage. Again, this will be the same voltage as the power supply.

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