Kirchhoff’s Laws

Part 1: Fundamental Concepts: Kirchhoff’s Laws

With what you know so far, you can calculate what happens in a large variety of circuit  configurations.  As the complexity of the circuits being analyzed increases, the method of simplifying the circuit and repeatedly applying Ohm’s Law can become tedious.  With some additional information, larger and more complex circuits can easily dealt with.

The additional information needed is provided by Kirchhoff's first and second laws.  They were developed in 1845 by German physicist Gustav Robert Kirchhoff.  These laws describe the way that current flows through a circuit and how the voltages are distributed around it.

Kirchhoff’s First Law

Kirchhoff's First Law states that the sum of the current entering a point, or node, in a circuit must equal the sum of the current leaving that node.  In other words, current is not created or destroyed in a circuit, and all of the current that enters a point in a circuit must leave that point.  This should make intuitive sense given that the current flowing in a circuit has a physical basis.  The number of electrons that flow into a single point in a circuit must be the same as the number of electrons that leave that point.  A circuit cannot create or destroy electrons because a circuit cannot create and destroy matter.  A simple, partial circuit diagram is shown below in Figure 1 to illustrate the point.

The schematics shown on the left and right hand side of Figure 1 are equivalent.  A point, or node, in circuit does not need to have a physical representation that looks like a point.  The commonly connected center of the circuit is a point, as far as the circuit is concerned.  This is because no matter the representation physically or schematically, all of the resistors in the center of the circuit are connected to each other with a conductor.  As a consequence, they are all at the same potential (or voltage) and the current can move freely along the center conductor.

In this case, the currents I1, I2, I3, and I5 are flowing into the network, and currents I4 and I6 are flowing out of it.  Given all the current that enters must leave, we know that I1 + I2 + I3 + I5 = I4 + I6 even though we don’t know the value of those currents and the magnitude of the voltages that are causing them to flow.

Figure 1. The sum of the currents entering a node equals the sum of the currents leaving a node.

The direction and magnitude of the current flows are entirely based on what is forcing them, the voltages, in the circuit.  In this case, it is the voltage present on the ends of the resistors and the voltage at the center node.  When one of the voltages is increased, all of the current flows adjust and their magnitudes change.  This is because the node at the center of the circuit changes voltage when the current flow into one of the resistors changes causing a change in the voltage drops across all of the resistors.

Kirchhoff’s Second Law

Kirchhoff's Second Law deals with the voltages present around a loop in a circuit.  It states that the voltages around a loop in a circuit must sum to 0.  You saw this earlier when dealing with series circuits.  It is the same concept stated differently.  When dealing with a series circuit, it was noted that the sum of the voltages in a group of series connected resistors must equal the voltage across group. Figure 3 from the section on series circuits, is repeated below as Figure 2 for convenience.

Figure 2. Three resistors in series.

 For this to be true, that the sum voltages around a loop is 0, the polarity of the voltages being summed must be accounted for.  Looking at Figure 2, we will use the loop that the current is flowing in to keep things straight.  The goal is to add the values together as we work around the loop the current is flowing in.  Following the current flow, if you find a positive value on the first terminal of a component you encounter, we treat that as a positive value. If you encounter a negative terminal first, we treat that as a negative value.  This is essentially the same way that you treat voltage sources when they are combined in series.

In the figure above, starting out just before R1 and moving in the direction of the current flow, we first encounter a positive value of 2V.  We add this to the next value we encounter, also positive, for R2.  Continuing on, we encounter the third positive value for R3, so we add that to the total.  Leaving R3, we find the voltage source next in the loop and encounter the negative terminal first, which we also add to our total as a negative number.  If you were keeping track, the sum of the four values is 0V as shown below.

2V + 2V + 2V - 6V = 0V

Kirchhoff’s Second Law applies to every loop that current can flow in, even though there may be more than one loop in the circuit as a whole.  You can use Kirchhoff’s first and second laws to write equations that then can be solved to find the voltages and currents that are present in the circuit.  This will be explored in more depth in the next section.

Most circuits contain more than one branch.  Each branch of the circuit can form a loop that current can flow in.  The following figure illustrates the concept.

Figure 3. Summing voltages around loops.

The discussion from here to the end of the section is a bit dense.  Stick with it.  Everything that is being discussed you already know, it is simply being presented for a slightly more busy circuit, shown in Figure 3, with more loops.

There are three loops shown.  The loops are drawn around small series connected circuits that have either a voltage source (such as Vs) or something like a voltage source.  For instance, in loop 2 you can imagine one of the resistors, for example R2, acting like a voltage source.  Because there is a current flowing through R2, it will have a voltage drop across it.  That voltage drop can act like a source which induces current to flow through loop 2.

If we look at the first loop, while ignoring the second and third loops for the moment, we can see how Kirchhoff's Second Law applies.  The voltage source, Vs, is connected to two resistors, R1 and R2, whose voltage drops are V1 and V2.  Kirchhoff's Second Law tells us that the sum around this loop is 0, therefore, V1+V2-Vs = 0.  Remember to pay attention to the polarity of the voltages shown in the figure. To assign polarities to the voltages that appear across the resistors, follow the current flow.  The end of the resistor that the current enters will be more positive than the end it exits.  Determining how this is done is covered in more detail in the next section.

In reality, we aren’t ignoring the second and third loops when we look at loop 1. This is because the voltage V2 is really the voltage across the parallel combination of R2 with the equivalent series resistance of R3 and R4.  We don’t know what V2 is, but we don’t need to at this point.  The goal is to write the equations that describe the voltage drops in the three loops.

If we look at what is happening in the second loop, we can see that the voltage drop across the series connected resistors R3 and R4 must be the same as the voltage drop V2 because the resistors are connected across R2.  The positive side of V2 is connected to the left side of R3, and the negative side of V2 is connected to the bottom of R4.  Therefore, the voltage drop V3 + V4 must be equal to V2  which gives us V3 + V4 - V2 = 0. 

When looking at the second loop, you might reasonably ask how we know the polarity of the voltage V3 since it is connected to a node with both a positive and a negative symbol.  Consider the way in which current is flowing.  It always flows from positive to negative.  In this circuit, the current from Vs flows through R1 first.  It then branches and flows through both R2 and R3. The left hand side of R3 is positive because current is flowing into it.  Another way to determine the polarity is to start with R4.  It has a direct connection to the negative side of the voltage source, so its polarity is easy to see.  From that, the polarity of R3 becomes obvious.  In the next section we will see that even if we get this wrong, the math will tell us by producing a negative result.  The goal is to identify all of the loops — getting some of the polarities wrong will not lead to the wrong answer.

The same thought process can be applied to the third outer loop as well.  The equation for the third loop can also be found by combining the first and second loop equations.

It is convention to think about the negative side of the voltage source as return or 0V.  By definition then, all of the voltages in the circuit are positive.  If you placed the negative (black) lead of a meter on the currents common return path (the negative side of Vs) in the circuit shown in Figure 3, all of the voltages in this circuit will be positive as you would expect.  The polarities shown for the individual resistors are correct, but each one is shown relative to the other side of the that resistor.