Kirchoff's Voltage Law
Why Do You Need To Know About KVL?
Using KVL To Write Equations For Circuits
What If?
Problems
You are at:  Basic Concepts - Kirchoff's Laws - KVL
Return to Table of Contents
Kirchhoff's Voltage Law  - Introduction

Kirchhoff's Voltage Law - KVL - is one of two fundamental laws in electrical engineering, the other being Kirchhoff's Current Law (KCL).

• KVL is a fundamental law, as fundamental as Conservation of Energy in mechanics, for example, because KVL is really conservation of electrical energy.
• KVL and KCL are the starting point for analysis of any circuit.
• KCL and KVL always hold and are usually the most useful piece of information you will have about a circuit after the circuit itself.

Goals For This Lesson

What should you be able to do after this lesson?  Here's the basic objective. Given an electrical circuit: Be able to write KVL for every loop in the circuit. Be able to solve the KVL equations, especially for simple circuits.
These goals are very important.  If you can't write KVL equations and solve them, you may well be lost when you take a course in electronics in a few years.  It will be much harder to learn that later, so be sure to learn it well now.  (And, if you need to review the concept of voltage, here is a link to that lesson.

Kirchhoff's Voltage Law

Here's a simple circuit.  It has three components - a battery and two other components.  Each of the three components will have a current going through it and a voltage across it.  Here we want to focus on the voltage across each element, and how those three voltages are related. We could measure voltage:
• Anywhere along the wire shown in purple
• Anywhere along the wire shown in green
• Anywhere along the wire shown in blue.
Note:  Any point along the green wire is at the same voltage, and the same situation pertains for the blue wire and the purple wire.

Here's the same circuit.  Here, with the button, you can move the dot representing charge around the circuit.

Answer these questions about what happens as that charge moves. Problems

Q1.  As the charge moves from the top of the battery to the top of Element #1 (along the wire shown in purple), how much energy does the charge lose? Q2.  As the charge moves from the top of Element #1 through Element #1 to the bottom of element #1, how much energy does the charge lose? Q3.  As the charge moves from the bottom of Element #1 to the top of Element #2, how much energy does the charge lose? Q4.  As the charge moves from the top of Element #2 through Element #2 to the bottom of element #2, how much energy does the charge lose? Q5.  As the charge moves from the bottom of Element #2 to the bottom of the battery, how much energy does the charge lose? Q6.  As the charge moves from the bottom of the battery through the battery to the top of the battery, how much energy does the charge lose?  The last question is tricky because the charge actually gains energy as it goes through the battery.  Now, we can track the energy acquired and given up by the charge as it traverses the circuit.  And, as the charge completes one round trip around the circuit - returning to its starting point - there can be no net gain of energy or no net loss.  That's really a statement of conservation of energy.  What you put in is what you get out.  TANSTAAFL!  (There Ain't No Such Thing As A Free Lunch.  It's not good English, but it says something that can't be said easily otherwise!)  Let's formalize that.
• The energy put into the charge as it goes through the battery is Vb * Q.
• The charge loses V1 * Q. as it goes through Element #1.
• The charge loses V2 * Q. as it goes through Element #2.
• The net energy put into the charge (What's put in minus what it loses!) is:
• Vb * Q - V1 * Q - V2 * Q = (Vb - V1 - V2)Q = 0
• We can note that the amount of charge is irrelevant in what we have learned here.  What we have learned is:
• Vb - V1 - V2 = 0, or
• Vb = V1 + V2 = 0
This can be paraphrased several ways:
• Voltage across the battery = Voltage across Element #1 + Voltage across Element #2.
• The algebraic sum of the voltages around a closed loop is zero.
There are some things to note about this conclusion - either way it is phrased.  Note the following:
• The conclusion does not depend on what the elements in the loop are.  They can be anything at all but still, the algebraic sum of the voltages around a closed loop will be zero.
• If you have a circuit with many loops, the algebraic sum of the voltages around any loop in the circuit is zero.
That last note will need a little explanation and work to be sure you understand it.  Consider this circuit. Q7.  How many loops are there in the circuit above? Writing The KVL Equations

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.
Now let's do that for the loops in the circuit above.

A Comment

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. Now, let's write KVL for each of the three loops.
• For the first loop (Battery, Element #1, Element #2)
• -VB + 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)
• -VB + V1 - V3 + V4 = 0
So, we get three equations - right?

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.

-VB + V1 + V2 = 0
-V2 - V3 + V4 = 0

-VB + V1 - 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. Problems

P1.  Using the same circuit you have been examining, answer the following questions.  For the first problem if the battery is a 9v battery, and you know that V1 = 3.7v, what is the value of V2 (in volts)?

Enter your answer in the box below, then click the button to submit your answer.  You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate answer) scale.

Your grade is:

P2.   You also know that V4 = 1.3v.  What is the value of V3 (in volts)?

Enter your answer in the box below, then click the button to submit your answer.  You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate answer) scale.

Your grade is:

P3.   Now, you have the voltages across all of the elements and the battery.  Check your knowledge of KVL by putting your numbers into each of the three loop (KVL) equations for the circuit.

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. Problems

P4.   Consider the case of two flashlight batteries.  Inside a flashlight they are usually stacked something like the way they are shown below.  There is also a voltmeter shown.  It is connected to measure the voltage of the two batteries in series.  Draw an equivalent circuit diagram that can be used to represent this situation.  Compute the voltage that is measured by the voltmeter using KVL if each battery has 1.54v.

Enter your answer in the box below, then click the button to submit your answer.  You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate answer) scale.

Your grade is:

P5.   Here is a circuit.  Determine all of the loops in this circuit and write KVL for every loop. P6.   Here is a circuit.  The voltage across element 1 is V1, etc.  Write the KVL equations for the following loops: • Vin, and elements 1, 4, and 5.
• Vin, and elements 1, 2, 6, and 7.
• Vin, and elements 1, 2, 3, and 8.
P7.   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.

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.

Problems

Links to Other Lessons on Basic Electrical Engineering Topics Send your comments on these lessons.