Solving Circuits Using KCL and KVL | DC Circuits

Welcome to the third lesson of our electric circuits series, where we dive deeper into circuit analysis and problem-solving techniques. In our previous lessons, we established a strong understanding of fundamental electrical concepts such as voltage, current, resistance, and circuit structures. This included exploring complex circuit configurations like nodes, loops, branches, series, and parallel setups, as well as applying Ohm's Law and understanding the functions of voltage and current sources. With this, we are equipped to tackle more complex circuit analysis problems and explore advanced circuit laws.

If you missed our previous blogs on circuit basics and structures, such as learning about the basics of voltage and current and circuits structures and components, I recommend checking them out to ensure a solid understanding of fundamental concepts. We want to acknowledge Digilent’s invaluable support and collaboration in previous lessons and their continued support as we go through this lesson!

In this tutorial, we will focus on Kirchhoff's laws, which play an important role in analyzing electrical circuits. By understanding Kirchhoff's laws conceptually and mathematically, we aim to enhance our ability to solve practical circuit problems and gain a deeper insight into how electrical circuits work.

As we have seen previously, the series-parallel techniques and Ohm's Law are valuable tools for circuit analysis. Here's a quick recap on these essential concepts:

Series Circuits: Components are connected end-to-end, and the total resistance is the sum of individual resistances. Current remains the same across all components.

Parallel Circuits: Components are connected across common points, and the total resistance is calculated using the reciprocal of individual resistances. Voltage remains the same across all components.

Ohm's Law: The relationship between voltage, current, and resistance, expressed as V = I * R or I = V / R. It helps calculate voltage, current, or resistance when two of these values are known, forming the basis of circuit analysis.

Combining these techniques allows for the analysis of circuits with resistors in different configurations, simplifying complex circuits. However, while they are effective for simple circuits, they fall short when analyzing intricate circuits with multiple nodes and loops. This is where Kirchhoff's laws come into play. There are two main laws: Kirchhoff's Current Law (KCL) and Kirchhoff's Voltage Law (KVL).

Kirchhoff's Current Law (KCL) is a fundamental law that is based on the principle of conservation of charge. It states that the total current flowing into a junction (or node) in a circuit is equal to the total current flowing out of that junction.

Let’s think about this intuitively. Imagine a water pipe junction where water can flow in and out through various branches. KCL asserts that the total water flow into the junction must equal the total flow out of the junction for water conservation.

Since current is the flow of charges, it makes intuitive sense that currents flowing into the node from one branch find a way to flow out through another branch.

Mathematically, you can say that when multiple branches of a circuit meet at a single point (junction), the sum of all currents entering that junction is equal to the sum of all currents leaving it. This can be represented as:

Here, represents the summation of all currents entering the junction, and represents the summation of all currents leaving the junction. The Greek letter (sigma) denotes the summation operation in mathematics, indicating that we are adding up multiple related quantities.

Kirchhoff's Current Law (KCL) can also be stated as “the sum of all currents entering or leaving a node in any electrical network is always equal to zero”.

Let’s look at this with an example. In the below figure, we have a node A, with currents i1, i2 and i5 entering the node and i3 and i4 exiting the node.

If we assume passive sign convention such that the currents entering the node are positive, and the currents leaving the node are negative, according to KCL, the sum of currents entering a node is equal to the sum of currents exiting it.

Hence, we can see that the sum of all currents entering or exiting a node is zero.

Let's dive into a few more examples to better understand KCL in action:

Example 1:
Consider the following circuit with nodes A and B. Currents i1 and i5 enter node A, while i4 enters node B, and i3 and i2 exit node B. We are given that i1 = 4 A, i2 = 3 A, i3 = 2 A, i5 = 1 A, where A represents amperes. Determine the value of i4.

Solution
To find the value of i4 in the given circuit, we can apply Kirchhoff's Current Law (KCL). Since nodes A and B are electrically the same point, we can consider them as the same node.

According to KCL, the sum of currents entering a node is equal to the sum of currents leaving the node.

Hence, the currents entering the node are i1 and i4, while the currents leaving are i2, i3 and i5.

Applying KCL, we have: i1 + i4 = i2 + i3 + i5

Substitute the given values: 4 A + i4 = 3 A + 2 A + 1 A

Combine like terms: 4 A + i4 = 6 A

Now, isolate i4 by subtracting 4 A from both sides:

i4 = 6 A - 4 A
i4 = 2 A

So, the value of i4 in the circuit is 2 amperes (A).

Example 2
In a more complex circuit, consider the following setup: there are two voltage sources, v2 = 30 V and v3 = 20 V, along with three resistances: r1 = 100 ohms, r2 = 1000 ohms, r3 = 150 ohms. Current i1 passes through r1, i3 flows through r3 and i2 flows through r2. Calculate the branch currents i1, i2 and i3.

R1 = 100 ohms
R2 = 1000 ohms
R3 = 150 ohms

Solution
As you may observe, current i1 flows through r1, current i3 flows through r3 and current i2 flows through r2, meeting at node V1. We can assume that currents i1 and i3 flow into node V1 from their voltage sources v2 and v3 respectively. We can also assume that current i2 flows out of node V1 towards the ground.

Hence, applying KCL at the node,

According to Ohm’s Law, I = V/R. Hence, we can substitute the currents in an equation with their V/R equivalents. To find the potential difference across each resistor, we calculate the difference between the voltages at each node on either end. This potential difference, divided by the resistance, gives us the branch current.

Substituting the values in the KCL equation, i1+i3 = i2 can be written as:

Let’s simplify the equation further as follows:

Substituting the value of v1 in the equations for i1, i2 and i3, we get:

As you may observe, the value of i3 is a negative value. This simply means that the actual direction of current flow is opposite to our initial assumed value (according to passive sign convention) and that current i3 flows out of/exits node v1.

NOTE:
It is important to note that negative values of currents or voltages in circuit analysis indicate a direction opposite to the assumed direction (based on the passive sign convention). The passive sign convention assumes that current flows into the positive terminal of an element and that voltage drops occur in the direction of current flow. Therefore, a negative value signifies a reversal in the assumed direction, indicating that the actual current flow or the voltage drop is in the opposite direction.

Kirchhoff's Voltage Law, on the other hand, states that the sum of the voltage rises and drops around any closed loop in a circuit is equal to zero. In simpler terms, it emphasizes energy conservation within a circuit.

Mathematically, KVL can be expressed as:

Here, V stands for voltage, which is the electrical potential difference across a component in the circuit.

Rises refer to voltage increases across components like voltage sources (e.g., batteries or generators) as you traverse the loop in a specific direction. These voltage increases are typically positive in the equation. Drops indicate voltage decreases across components like resistors or other loads as you move along the loop. These voltage decreases are usually considered negative in the equation.

It is important to note that when encountering a new circuit element, pay attention to the voltage sign as you enter the element. If the sign is positive (+), it indicates a voltage drop across the element, and the element voltage must be subtracted. Conversely, if the sign is negative (-), it signifies a voltage rise across the element, and the element voltage must be added. This approach helps in correctly determining the voltage changes across elements in a circuit.

To apply KVL, the voltage rises (positive contributions) and voltage drops (negative contributions) across these components are considered as we traverse a closed loop. The sum of these voltages must equal zero for a closed loop.

Let's demonstrate KVL with an example circuit. Here, we have a simple example circuit with one loop. There is a voltage source V = 20 V and three resistors in series, r1, r2 and r3.

We can find out the current flowing through the circuit using the series/parallel techniques and Ohm’s law. Since the resistors are in series, we can find the total resistance by summing up the individual resistances.

We can find the current flowing through the circuit through Ohm’s law.

Using the value of I, we can find the voltage drop or difference across each resistor.

Now let us validate Kirchoff’s voltage law using the results we have obtained from our previous calculations.

In this circuit, we have one loop. As we traverse the loop clockwise, we account for voltage rises (from sources) and voltage drops (across resistors).

Applying KVL to the loop, we write:

This equation states that the sum of the voltage rise from Vs minus the voltage drops across R1, R2 and R3 equals zero in the loop.

This equation illustrates the energy balance within the loop, hence validating Kirchhoff's Voltage Law

Let's look into a few more examples to better understand KVL in action:

Example 1:
Consider the following circuit with resistors R1 = 5 ohms and R2 = 3 ohms and a current source I1 = 5A. Determine the voltage drop across the current source using KVL.

Solution:
Since the resistors are in series, the current passing through them is the same.

Hence, according to KVL,

Example 2:
Consider the following circuit with two loops. There are two voltage sources V1 = 5V and V2 = 2V. Currents I1 and I2 pass through loops 1 and 2 respectively with resistances R1 = 2 ohms, R2 = 3 ohms and R3 = 1 ohms. Using KVL, find currents I1 and I2.

Solution:
The given circuit has two loops with loop currents I1 and I2 passing through loop 1 and 2 respectively.

When dealing with circuits containing two touching loops, we encounter situations where shared resistors are part of both loops. In such cases, to determine the current through the shared resistor, we subtract one loop's current from the other loop's current. This approach ensures that we account for the flow of current through the resistor from both directions in the circuit. By considering the difference of currents between the two loops, we can address complex circuit configurations.

Resistor R3 is a shared resistor between loops 1 and 2. As we can observe, current I1 passes through R3 from b to g while current I2 passes through R3 from g to b. Hence, we can consider that the current through the middle resistor, R3, is the difference of currents I1 and I2.

Using KVL to find the loop equations, we get:

As previously mentioned, observing the sign of a new circuit element while traversing a closed loop is important. A positive sign implies a voltage drop, requiring subtraction of the element voltage, whereas a negative sign indicates a voltage rise, requiring addition of the element voltage. Hence, in loop 2, the negative sign upon entering the circuit element signifies the addition of the voltage source (2V).

Solving the simultaneous equations for loop 1 and loop 2, we get:

Now that we have understood the concepts of KCL and KVL, let's apply them together to solve electrical circuit problems. Here’s a step-by-step guide:

1. Identify Nodes: Begin by identifying nodes where currents converge or diverge. Label these nodes for clarity.

2. Define Branches and Loops: Outline the branches (components) and loops (closed pathways) within the circuit.

3. Apply KCL to nodes: At each node, apply Kirchhoff’s Current Law to write equations representing the current balance.

Solve these simultaneous equations at each node to find the unknown currents.

4. Apply KVL to loops: Choose loops within the circuit and apply Kirchhoff’s Voltage Law to write equations representing voltage rises and drops around each loop.

Solve these equations to determine unknown voltages or currents.

5. Verify and Analyze: Once you've solved the equations, verify the results by checking for consistency with Ohm's Law and other circuit principles. Analyze the circuit behavior based on the calculated values.

Let’s illustrate these steps with an example circuit and walk through the solution using Kirchhoff’s Laws.

Consider the following circuit. Use KCL and KVL to find the currents through each branch of the circuit as well as the voltages across each element.

1. Identify Nodes: Label the nodes in the circuit—let's call them Node A, Node B, Node C, Node D, Node E and Node F.

Here, we can also assume the current directions based on the voltage sources in the circuit (current normally flows from the positive terminal to the negative terminal of the source) or based on passive sign convention. We can also take this time to note that Nodes D, E, and F are, in reality, actually one single node.

2. Define Branches and Loops: Outline the branches (resistors, sources) and loops (closed pathways)—we have Loop 1 and Loop 2.

3. Apply KCL: At each node, write KCL equations based on current conservation.

4. Apply KVL: Choose loops (Loop 1 and Loop 2) and write KVL equations representing voltage rises and drops.

Substitute the current equations obtained using KCL into the loop equations and solve to find currents.

Using this value, we can find I1.

The negative value of I1 simply means that the actual direction of current flow is opposite to the direction of current that was initially assumed. Hence, our circuit would look like this:

Using the results of these equations, we can determine the voltages VR1, VR2 and VR3.

5. Verify and Analyze: Check the calculated values against Ohm's Law and circuit principles to ensure consistency.

Loop 1:

Loop 2:

This example demonstrates the practical application of Kirchhoff’s Laws in analyzing complex electrical circuits. By systematically applying KCL and KVL, we can solve circuit problems efficiently.

In summary, the exploration of Kirchhoff's laws has enhanced our understanding of circuit analysis and problem-solving techniques. Kirchhoff's current law and voltage law have provided us with powerful tools to understand and analyze complex circuits as well as conceptually and mathematically represent current and voltage distribution in a circuit.

Again, we are grateful to Digilent for their continued support and collaboration in bringing these educational resources to students and enthusiasts. Their commitment to academic development and affordable solutions has been invaluable to creating this learning journey.

In the upcoming tutorials, we'll continue exploring advanced circuit analysis techniques and practical applications, building upon the knowledge and skills we've acquired so far. Stay tuned for more exciting content!