Basic Concepts

        There is a good algorithm that can be stated for how to write the KVL equations for a loop.  We'll need that when we examine this circuit again.  Here's the statement of the algorithm.

• Pick a starting point on the loop you want to write KVL for.
• Imagine walking around the loop - clockwise or counterclockwise.
• When you enter an element there will be a voltage defined across that element.  One end will be positive and the other negative.
• Pick the sign of the voltage definition on the end of the element that you enter.  Conversely, you could choose the sign of the end you leave, except that you have to be consistent all the way around the loop.
• Write down the voltage across the element using the sign you got in the previous step.
• Keep doing that until you have gone completely around the loop returning to your starting point.
• Set your result equal to zero.

        Before we write the KVL equations, we need to notice something.  The answer to Question #7 may not be correct.  Let's think a little deeper to be sure we have it right.  We took the correct answer to be two loops.  In a sense that's correct, but in another sense there are three loops.  In the picture below, each of the three buttons - when pressed - will show you one of the three loops.  There's a loop there that you might not have thought about.  Click the three buttons to see the three loops.

• For the first loop (Battery, Element #1, Element #2)
• -V_B + V₁ + V₂ = 0
• For the second loop (Element #2, Element #3, Element #4).  Note, you have to be careful with this one because you might not expect the voltage across Element #3 to be defined the way it is.
• -V₂ - V₃ + V₄ = 0
• For the third loop (Battery, Element #1, Element #3, Element #4)
• -V_B + V₁ - V₃ + V₄ = 0

        Actually, that's not right, because we do not get three independent equations. There are only two independent equations we can write.

        That's not immediately obvious, so write the three equations as shown below.  We'll put a horizontal line between the first two and the third equation.

-V_B + V₁ + V₂ = 0
-V₂ - V₃ + V₄ = 0

Can you see that you can add the first two equations to get the third?  (Actually, there is a -V₂ and a +V₂, and those are the only things that cancel out when you add.)  The third equation can be obtained from the first two equations, so it is not an independent equation.  When you have the first two equations you can get the third from them!

        What this means is that you have to be careful when you write KVL.  You can write too many equations, and in being careful you might not write enough.  Fortunately, if you look at a circuit you can almost always see how many independent loops there are by inspection.  Going back to our question about how many loops there are in this circuit, the answer is that there are three loops but only two independent loops.

        Now, let's see if you can apply your knowledge of KVL to solve a few simple problems.

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        You're on your way to being a KVL expert.  Try your hand on these problems, and you should be ready to move on to the next lesson.

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P1.   Here is a circuit.  Determine all of the loops in this circuit and write KVL for every loop.

P2.   Here is a circuit.  The voltage across element 1 is V1, etc.  Write the KVL equations for the following loops:

• V_in, and elements 1, 4, and 5.
• V_in, and elements 1, 2, 6, and 7.
• V_in, and elements 1, 2, 3, and 8.

P3.   In the circuit below, write the KVL equations for:

• The "center loop" with elements 5, 4, 2, 6, and 7;
• The loop with elements 5, 4, 2, 3, and 8.

        Show how these KVL equations can be derived from the KVL equations of Problem 3.

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What If You Need More Than The KVL Equations?

        KVL gives a lot of information about a circuit.  However, it doesn't give you everything you need to know all of the time.  Sometimes you will need information from KCL - Kirchhoff's Current Law.  You will also, always need the information that is contained in the descriptions of the elements themselves.  For example, resistors have a relationship between the current flowing through the resistor and the voltage across the resistor.  That information is crucial if there's a resistor in a circuit.

        Still, without KVL you won't get very far analyzing circuits, so you will definitely use the information and concepts you learned in this lesson.