Kirchhoff's Voltage Law

## Kirchhoff’s voltage law

Kirchhoff's Voltage Law is one of the three fundamental laws of electrical circuits (along with Ohm's Law and Kirchhoff's Current Law). It states that the algebraic sum of the voltages in a series circuit will equal zero. That probably doesn't sound very useful. Maybe even a little confusing. Here's a more practical way to put it. Each resistor or other component in a series circuit will have a certain voltage difference across it (assuming there is resistance and current flow). If you add all those voltages together they will add up to the battery voltage.

If there is a single resistor in a circuit, then all of the battery voltage will be seen across that one resistor. The following diagram shows the probes of a volt meter. The voltage that the meter will read is shown between the probes. If the probes are placed across the single resistor, the voltage will equal the battery voltage.

 With one resistor the entire battery voltage apperars across that resistor

If there are multiple resistors in a series circuit, the voltage is shared proportionally across each resistor. For example if you have 30 volts across two 20 ohm resistors in series, each resistor will have 15 volts across it. This would be true for two 10 ohm resistors, two 150 ohm resistors or any two equal resistors. The battery voltage would be evenly split across the two resistors.

In the following circuit there are three resistors, 20, 30 and 10 ohms (60 ohms, total).

 The total voltage across the components in a series circuit adds up to the battery voltage.

The 20 ohm resistor is 1/3 of the total resistance so it has 1/3 of the total voltage (10 volts). The 30 ohm resistor is 1/2 of the total resistance so it has 1/2 of the total voltage (15 volts). The 10 ohm resistor is 1/6 of the total resistance so it has 1/6 of the total voltage (5 volts). The circuit has a 30 volt battery. Therefore the voltages across the resistors are 10 volts, 15 volts and 5 volts respectively (1/3, 1/2 and 1/6 of the battery voltage). These add up to 30 volts.

Let's change the resistors to a 50 ohm resistor, a 75 ohm resistor and a 25 ohm resistor.

 The resistors are different but the proportions are the same. Therefore the voltages are the same.

The resistors are different but the proportions are the same. 1/3 of the voltage is across the 50 ohm resistor (10 volts), 1/2 of the voltage is across the 75 ohm resistor (15 volts) and 1/6 of the voltage is across the 25 ohm resistor (5 volts). The voltages would still add-up to the battery voltage of 30 volts.

In the next circuit, the black probe of the volt meter is connected to the negative side of the battery and remains there. The red probe is moved to different points around the circuit to read the voltages at these points in relation to the negative side of the battery.

 Voltages referenced to the negative side of the battery

Notice that when the red probe is connected to the positive side of the battery, the meter reads 30 volts, which is the battery voltage. This makes perfect sense because the meter is simply connected across the battery. When the red probe is moved to the other side of the 20 ohm resistor it reads 20 volts. There is a loss of 10 volts from the side of the resistor closest to the battery compared to the side farthest from the battery. This voltage drop, as it is called is the same 10 volts read when the meter probes were placed across the resistor.

When the red probe is moved to the other side of the 30 ohm resistor the meter will read 5 volts. There is a loss of another 15 volts. This 15 volt drop is the same 15 volts read when the meter was placed directly across the 30 ohm resistor. Finally, there is a 5 volt loss across the 10 ohm resistor, leaving zero volts on the negative side of the battery. All of the voltage was lost in the process of going around the complete circuit. This is like walking downstream along a river. As you walk in the direction of flow you go lower and lower. Finally you reach the ocean which is as low as you can go. Remember that height, like voltage is potential energy. As you walk downstream you go lower and lower losing potential energy. As you go around the circuit in the direction of conventional current flow you will have less and less voltage, finally reaching zero volts. This is a practical demonstration of Kirchhoff's Voltage Law. We start with the battery voltage, then subtract the voltage across each resistor, ending up with zero volts.

Notice that at the at the end of the circuit the red and black probes are both connected to the same place—the negative terminal of the battery. Remember that the voltmeter measures the voltage difference between the probes. If the probes are connected to the same point in the circuit there is no difference between them. The meter will read 0 volts telling you that there is no voltage difference between the probes. The meter does not read 0 volts because there is no voltage there. It is only telling you that both probes are at the same voltage.

### Unsticking some sticking points

Let's pause for a moment and make sure we understand this. This is a sticking point for many people. Repeating for emphasis, when you anchor the black probe at the negative side of a battery, then measure the voltages around the circuit with the red probe, you will get lower and lower voltages as you go from the positive terminal of the battery around the circuit. When the red probe reaches the negative side of the battery it will read 0 volts. This is not because there is no voltage at this point. It is only because the red and black probes are at the same point in the circuit. The meter is telling you that there is no voltage difference between the probes, not that there is no voltage at that point. Don't forget that the voltmeter tells you the difference between two voltages, not an absolute voltage. Zero volts is not the absence of voltage. A voltmeter reads zero volts when both probes are at the same voltage.

Another sticking point for some people is where current goes once it reaches the negative side of the battery. Some people have no problem with following the current around the loop, but wonder where it goes in the end. Remember that the battery acts like a pump that circulates electrical current around the circuit. It blows conventional current out the positive terminal and sucks it back in the negative side. The current just goes round and round the circuit.[1]

Series Circuits and Kirchhoff's Voltage Law

—————————