Understanding Kirchhoff’s Circuit Law

Kirchhoff’s Circuit Laws define voltage and current relationships and serve as the foundation for understanding complicated circuits.

In one of our earlier lesson, we looked about resistors and how one equivalent resistance (RT) can represent a series, parallel, or combination of resistor connections. Ohm’s Law is readily followed by these circuits when calculating voltage and current.

But if circuits get more complex, like bridge or T networks, Ohm’s Law might not be enough to fully understand them. We use Kirchhoff’s Circuit Laws to solve for currents and voltages in such complicated situations.

We are able to compute the unknown electrical values by using these fundamental rules to create the circuit equations in a methodical manner.

German scientist Gustav Kirchhoff developed two interlinked laws governing electrical circuits in 1845. These ideas, collectively referred to as Kirchhoff’s Circuit Laws, deal with the essential ideas of energy and current conservation.

Kirchhoff’s Current Law (KCL) explicitly addresses the flow of current in a closed loop, stating that the total current entering a junction must equal the total current leaving that junction.

On the other hand, the primary objective of Kirchhoff’s Voltage Law (KVL) is on voltage sources inside of closed loops. It states that all of the voltages within the loop must add up to zero algebraically.

Kirchhoffs First Law: The Current Law, (KCL)

Consider a circuit junction as the location where wires converge. According to Kirchhoff’s Current Law, the total current entering the junction (imagine incoming traffic) and the total current exiting the junction (imagine outgoing traffic) must be precisely equal. This makes reasonable since charge cannot vanish within the junction.

Consider current to be like water running through pipes. Water cannot accumulate or disappear at a branching point unless the total volume of water flowing in and out of the system equals zero. Electrical charge operates in a circuit in a similar way.

An equation representing KCL may be written as follows: I(entering) = current entering junction; I(exiting) = current leaving junction. These currents must add up to zero:

I_(entering) + I_(exiting) = 0

Kirchhoff’s theory is widely referred to as the Conservation of Charge.

Kirchhoff’s Current Law

Referring the above diagram, positive values are assigned to I1, I2, and I3, the three currents that enter the node. On the other hand, negative values are assigned to I4 and I5, the two currents that leave the node.

This standard represents the idea of current flow and is consistent with Kirchhoff’s Current Law (KCL). Using this method, the KCL equation for this particular node may be rewritten as follows:

I₁ + I₂ + I₃ – I₄ – I₅ = 0

To make the idea of a node clearer: A junction or connecting point where two or more current-carrying conductors (wires, cables) or circuit components (resistors, capacitors, etc.) converge is referred to as a node in an electrical circuit. Current flows via these components and conductors to create a closed-loop pathway.

Use of Kirchhoff’s Current Law in Parallel Circuit Analysis: When examining parallel circuits, Kirchhoff’s Current Law is especially helpful. This is due to the fact that in a parallel arrangement, the source’s total current splits and passes through each branch before merging again at a junction. With the help of KCL, we are able to join the currents in these branches at the node, or the point of reconnection.

Kirchhoffs Second Law: The Voltage Law, (KVL)

Kirchhoff’s Voltage Law (KVL) governs the relationship between voltage sources and voltage drops in a closed loop circuit. It stipulates that all voltages (also known as electromotive forces, or EMFs) around the loop must add up to zero algebraically.

To put it another way, the total voltage measured around a closed-loop network powered by a voltage source is precisely equal to the sum of all the voltage drops experienced within that same loop.

Understanding this with Conservation of Energy:

This law conforms with the fundamental idea of energy conservation. Energy can only be altered within a closed loop; it cannot be generated or destroyed.

This is reflected in KVL, which states that the algebraic total of the energy given by the voltage sources and the algebraic total of the energy spent by the voltage drops around the loop must equal one another. Because energy cannot be generated inside the loop, the net voltage around it must be zero.

Kirchhoff’s Voltage Law

Applying KVL in a Loop:

In order to apply Kirchhoff’s Voltage Law (KVL), a closed loop must be traversed consistently in either a clockwise or counterclockwise manner, with the voltage drops observed being added up.

Keeping the loop going in this fashion is vital. Based on the polarity of each voltage drop with respect to the selected traversal direction, a positive or negative sign is assigned to it.

By doing this, the energy flow inside the loop is guaranteed to be precisely reflected in the final voltage sum. A summation that deviates from a consistent path will be inaccurate.

KVL is very useful for evaluating series circuits. The same current passes successively via each component in a series setup. We are able to correlate the voltage drops across each element with the overall voltage provided by the source by applying KVL to a loop that includes the complete series circuit.

Terminology used in Circuit Analysis:

When doing circuit analysis with Kirchhoff’s Laws (KCL and KVL), the circuit components are characterized by several essential terms:

• Circuit: A closed loop that offers a constant path for electrical current to flow is referred to as a circuit.
• Path: A path in a circuit is a single connection made between sources or elements. Consider it as a designated path for current to flow.
• Node: A junction point connecting two or more circuit elements is known as a node. It is essentially the junction of the circuit’s various branches, and in schematic diagrams, it is frequently depicted as a dot.
• Branch: A branch is a segment of a circuit that consists of a single part (such as a resistor) or a collection of parts connected between two nodes. It’s a particular path that current in the circuit may travel.
• Loop: A loop is a straightforward, closed path in a circuit that avoids repeatedly passing through any element or node. Consider tracing the path of a single current flow through the circuit without going through any connections again.
• Mesh: A mesh is a unique kind of loop that does not consist of any other loops. There are no smaller closed paths present within this single, closed loop circuit.

The behavior of current and voltage in a circuit is dependent on how its components are connected.

• Series: Components are connected in series if they all experience the same current flow.
• Parallel: When the same voltage is applied across all components, they are connected in parallel.

A Fundamental DC Circuit is shown below with the above explained specifications:

Solving a Kirchhoff’s Circuit Law Problem

Calculate the current passing through R3, the 36Ω resistor.

The circuit has 3 branches, 2 nodes (A and B) and 2 independent loops.

Using Kirchhoff’s Current Law, KCL the equations are given as:

At node A :    I₁ + I₂ = I₃

At node B :    I₃ = I₁ + I₂

Using Kirchhoff’s Voltage Law, KVL the equations are given as:

• Loop 1 is given as :    9 = R₁ I₁ + R₃ I₃ = 9I₁ + 36I₃
• Loop 2 is given as :    18 = R₂ I₂ + R₃ I₃ = 18I₂ + 36I₃
• Loop 3 is given as :    9 – 18 = 9I₁ – 18I₂

• As I₃ is the sum of I₁ + I₂ we can rewrite the equations as;
• Eq. No 1 :    9 = 9I₁ + 36(I₁ + I₂)  =  45I₁ + 36I₂
• Eq. No 2 :    18 = 18I₂ + 36(I₁ + I₂)  =  36I₁ + 54I₂

The above two “Simultaneous Equations” may now be simplified by subtraction to provide the values of I1 and I2.

45I₁ + 36I₂ = 9
36l₂ = 9 – 45l₁
l₂ = 9/36 – 45l₁/36
l₂ = 0.25 – 1.25l₁

36I₁ + 54I₂ = 18
36l₁ + 54(0.25 – 1.25l₁) = 18
36l₁ + 13.5 – 67.5l₁ = 18
-31.5l₁ = 18 – 13.5
l₁ = – 0.142

45I₁ + 36I₂ = 9
45l₁ = 9 – 36l₂
l₁ = 9/45 – 36l₂/45
l₁ = 0.2 – 0.8l₂

36I₁ + 54I₂ = 18
36(0.2 – 0.8l₂) + 54l₂ = 18
7.2 – 28.8l₂ + 54l₂ = 18
7.2 + 25.2l₂ = 18
25.2l₂ = 18 – 7.2
l₂ = 0.42

Substitution of I₁ in terms of I₂ gives us the value of I₁ as -0.143 Amps

Substitution of I₂ in terms of I₁ gives us the value of I₂ as +0.429 Amps

As :    I₃ = I₁ + I₂

The current flowing in resistor R₃ is given as :    -0.143 + 0.429 = 0.286 Amps

and the voltage across the resistor R₃ is given as :    0.286 x 36 = 10.29 volts

The negative sign for I₁ means that the direction of current flow initially chosen was wrong, but never the less still valid. In fact, the 20v battery is charging the 10v battery.

References: Kirchhoff’s Circuit Law

https://www.khanacademy.org/science/physics/circuits-topic/circuits-resistance/a/ee-kirchhoffs-laws