 # Standing On the Shoulders of Mr. Ohm: Using Kirchhoff’s Law for Complex Circuits

Ohm’s Law is your golden ticket for calculating the voltage, current, or resistance in a simple series or parallel circuit, but what happens when your circuit is more complicated? You might be designing electronics that have both parallel and series resistance, and Ohm’s Law starts to fall down. Or what if you don’t have a constant current source? In these situations, when you can’t only use V = IR, then it’s time to stand on the shoulders of Ohm and use Kirchhoff’s Circuit Law. Here we’ll be looking at what Kirchhoff’s Circuit Law is, and how to use it to analyze the voltage and current of complex electrical circuits.

## What is Kirchhoff’s Circuit Law?

When you’re building a complex circuit that includes bridges or T networks, then you can’t solely rely on Ohm’s Law to find the voltage or current. This is where Kirchhoff’s Circuit Law comes in handy, which allows you to calculate both the current and voltage for complex circuits with a system of linear equations. There are two variations of Kirchhoff’s Law, including:

• Kirchhoff’s Current Law: To analyze the total current for a complex circuit
• Kirchhoff’s Voltage Law: To analyze the total voltage for a complex circuit
• When you combine these two laws, you get Kirchhoff’s Circuit Law

Like any other scientific or mathematical law named after their creator, Kirchhoff’s Circuit Law was invented by German Physicist Gustav Kirchhoff. Gustav was known for many achievements in his lifetime, including the theory of spectrum analysis which proved that elements give off a unique light pattern when heated. When Kirchhoff and chemist Robert Bunsen analyzed these light patterns through a prism, they discovered that each element in the periodic table has its own unique wavelength. The discovery of this pattern allowed the duo to uncover two new elements, cesium and rubidium. Gustav Kirchhoff (left) and Robert Bunsen (right)

Kirchhoff later went on to apply his spectrum analysis theory to study the composition of the sun, where he discovered many dark lines in the sun’s wavelength spectrum. This was caused by gas from the sun absorbing specific wavelengths of light, and this discovery marked the beginning of a new age of research and exploration in the field of astronomy.

A bit closer to home in the world of electronics, Kirchhoff announced his set of laws for analyzing the current and voltage for electrical circuits in 1845, known today as Kirchhoff’s Circuit Law. This work builds upon the foundation outlined in Ohm’s Law and has helped paved the way for the complex circuit analysis that we rely on today.

## First Law – Kirchhoff’s Current Law

Kirchhoff’s Current Law states that the amount of current that enters a node is equal to the amount of current leaving a node. Why? Because when current enters a node, it has no other place to go except to exit. What goes in must come out. You can identify a node where two or more paths are connected via a common point. In a schematic, this will be the junction dot connecting two intersecting net connections.

Take a look at the image below to understand this Law visually. Here we have two currents entering a node, and three currents leaving the node. According to Kirchhoff’s Current Law, the relationship between these currents entering and exiting the node can be represented as I1 + I2 = I3 + I4 + I5. Kirchhoff’s Current Law, current in must equal current out. (Image source)

When you balance this equation as an algebraic expression, then you conclude that current entering and exiting a node will always equal 0, or I1 + I2 + (-I3 + -I4 + -I5) = 0 Everything has to balance out, and Kirchhoff called this principle the Conservation of Charge.

Let’s look at an example circuit to see how this works. Below we have a circuit with four nodes: A, C, E, and F. Current first flows from its voltage source and separates at Node A, which then flows through resistors R1 and R2. From there, the current recombines at Node C and splits again to flow through resistors R3, R4, and R5 where it meets Node E and Node F. (Image source)

To validate Kirchoff’s Current Law in this circuit, we need to take the following steps:

1. Calculate the total current of the circuit
2. Calculate the current flowing through each node
3. Compare input and output currents at specific nodes to validate Kirchoff’s Current Law.

### 1. Calculate total current

Here we use Ohm’s Law to get the total current of our circuit with I = V/R. We already have our total voltage of 132V, and now we just need to find the total resistance in all of our nodes. This requires the simple method of calculating the total resistance of resistors wired in parallel, which is: Starting at Node AC, we get the following resistance for parallel resistors R1 and R2: And moving on to Node CEF, we get the following resistance for parallel resistors R3, R4, and R5: We now have our total resistance of 11 Ohms for the entire circuit, which we can then plug into Ohm’s Law I = V/R to get the total current in our circuit: ### 2. Calculate node currents

Now that we know we have 12 amps flowing out of our circuit, we can calculate the current at each set of nodes. We’ll again enlist the help of Ohm’s Law in the form of I = V/R to get the current for each node branch.

First, we need the voltages for node branches AC and CF: Then we can calculate the current for each node branch: ### 3. Validate Kirchhoff’s Current Law

With the current for each node branch calculated, we now have two distinct reference points that we can use to compare our input and output currents. This will allow us to analyze our circuit and validate Kirchhoff’s Current Law like so: ## Second Law – Kirchhoff’s Voltage Law

Kirchhoff’s Voltage Law states that in any closed loop circuit the total voltage will always equal the sum of all the voltage drops within the loop. You’ll find voltage drops occurring whenever current flows through a passive component like a resistor, and Kirchhoff referred to this law as the Conservation of Energy. Again, what goes in must come out.

Take a look at the image below to understand this visually. In this circuit, we have a voltage source, and four areas in the circuit where the voltage will encounter a passive component, which will cause a distinct voltage drop. Since these passive components are connected in series, you can simply add the total voltage drops together, and compare it to the total voltage to get a relationship that looks like this: Let’s start with a straightforward circuit to demonstrate how this works. In the example below, we have two known variables, the total voltage and the voltage drop across R1. (Image source)

What we need to figure out is the voltage drop across R2, and we can use Kirchoff’s Voltage Law to figure this out with the following relationship: Since the total voltage drop in the circuit has to equal the total voltage source, this provides an easy way to calculate our missing variable. If you wanted to express this relationship as a proper algebraic expression, you’d get the sum of all voltage drops and the total voltage equalling zero as shown here: Let’s look at another example. In the circuit below we have three resistors connected in series with a 12 volt battery. To validate Kirchoff’s Voltage Law in this circuit, we need to take the following steps:

1. Calculate the total resistance of the circuit
2. Calculate the total current of the circuit
3. Calculate the current through each resistor
4. Calculate the voltage drop across each resistor

Compare the voltage source to total voltage drop to validate Kirchoff’s Voltage Law

### 1. Calculate total resistance

Since all of our resistors are wired in series, we can easily find the total resistance by just adding all of the resistance values together as so: ### 2. Calculate the total current

Now that we know our total resistance, we can again use Ohm’s Law to get the total current of our circuit in the form of I = V/R, which looks like this: ### 3. Calculate current through each resistor

Since all of our resistors are wired in series they will all have the same amount of current flowing through them, which we can express as: ### 4. Calculate the voltage drop across each resistor

Our final calculation will again use Ohm’s Law to give us the total voltage drop for each resistor in the form of V = IR, which looks like this: ### 5. Validate Kirchoff’s Voltage Law

Now we have all of the data we need, including the total voltage of our circuit, along with each voltage drop across each of our resistors. When putting all of this together, we can easily validate Kirchhoff’s Voltage Law with this relationship: This can also be expressed as: As you can see, the total voltage equals the total voltage drop in our circuit. What goes in must come out, and Kirchhoff’s Law works yet again!

## Process for Using Kirchhoff’s Circuit Law

With an understanding of how Kirchhoff’s Circuit Law works, you now have a new tool in your toolbox for analyzing voltage and current in complete circuits. When using these Laws out in the wild consider using the following step-by-step process:

1. First, begin by labeling all of the known voltages and resistances on your circuit.
2. Then name each branch on your circuit with a current label, such as I1, I2, I3, etc. A branch is a single or group of components connected between two nodes.
3. Next, find Kirchhoff’s Current Law for each node in your circuit.
4. Then find Kirchhoff’s Voltage Law for each of the independent loops in your circuit.

Once you have Kirchoff’s Current and Voltage Laws calculated, ou can then use your equations to find any missing currents. Ready to try it on your own? Take a look at the circuit below and see if you can validate Kirchoff’s Current Law and Voltage Law with a little bit of help from Ohm! 