Kirchhoff’s Current & Voltage Law (KCL & KVL) | Solved Example

KCL & KVL – Kirchhoff’s First & Second Laws with Solved Example

A German Physicist “Robert Kirchhoff” introduced two important electrical laws in 1847 by which, we can easily find the equivalent resistance of a complex network and flowing currents in different conductors. Both AC and DC circuits can be solved and simplified by using these simple laws which are known as Kirchhoff’s Current Law (KCL) and Kirchhoff’s Voltage Law (KVL).

Also note that KCL is derived from the charge continuity equation in electromagnetism while KVL is derived from Maxwell – Faraday equation for static magnetic field (the derivative of B with respect to time is 0).

Kirchhoff’s Current Law (KCL):

According to KCL:

At any moment, the algebraic sum of flowing currents through a point (or junction) in a network is Zero (0) or in any electrical network, the algebraic sum of the currents meeting at a point (or junction) is Zero (0). This law is also known as Point Law or Current law.

In any electrical network, the algebraic sum of incoming currents to a point and outgoing currents from that point is Zero. Or the entering currents to a point are equal to the leaving currents of that point.

In other words, the sum of the currents flowing towards a point is equal to the sum of those flowing away from it. Or the algebraic sum of the currents entering a node equals the algebraic sum of the currents leaving it.

Explanation of KCL:

Suppose some conductors are meeting at a point “A” as shown in fig 1.a. In some conductors, currents are incoming to the point “A” while in other conductors, Currents are leaving or outgoing from point “A”.

Consider the incoming or entering currents as “Positive (+) towards point “A” while the leaving or outgoing currents from point “A” is “Negative (-)”.
then:

I₁ + (–I₂) + (–I₃) + (–I₄) + I₅ = 0

OR

I₁ + I₅ – I₂ – I₃ – I₄ = 0

OR

I₁ + I₅ = I₂ + I₃ + I₄ = 0

i.e.

Incoming or Entering Currents = Leaving or Outgoing Currents

Or

ΣI Entering = ΣI Leaving

For instance, 8A is coming towards a point and 5A plus 3A are leaving that point in fig 1.b, therefore,

8A = 5A + 3A

8A = 8A.

Kirchhoff’s Voltage Law (KVL):

The Kirchhoff’s second law or KVL stated that;

In any closed path (or circuit) in a network, the algebraic sum of the IR product is equal to the EMF in that path.

In other words, in any closed loop (which is also known as Mesh), the algebraic sum of the EMF applied is equal to the algebraic sum of the voltage drops in the elements. Kirchhoff’s second law is also known as Voltage Law or Mesh law.

ΣIR= ΣE

Explanation of KVL:

A closed circuit is shown in fig which contains two connections of batteries E₁ and E₂. The overall sum of E.M.F’s of the batteries is indicated by E₁-E₂. The imaginary direction of current is also shown in the fig.

E₁ drives the current in such a direction which is supposed to be positive while E₂ interferes in the direction of current (i.e. it is in the opposite direction of the supposed direction of current) hence, it is taken as negative. The voltage drop in this closed circuit depends on the product of Voltage and Current.

The voltage drop occurs in the supposed direction of current is known as Positive voltage drop while the other one is negative voltage drop.

In the above fig, I₁R₁ and I₂R₂ are positive voltage drops and I₃R₃ and I₄R₄ are negative V.D.

If we go around the closed circuit (or each mesh), and multiply the resistance of the conductor and the flowing current in it, then the sum of the IR is equal to the sum of the applied EMF sources connected to the circuit.

The overall equation for the above circuit is:

E₁ – E₂ = i₁R₁ + i₂R₂ – i₃R₃ – i₄R₄

If we go in the supposed direction of the current as shown in the fig, then the product of the IR is taken as positive otherwise negative.

Good to Know:

Direction of the Current:

It is very important to determine the direction of current whenever solving circuits via Kirchhoff’s laws. Same like the case of election current and conventional current.

The direction of current can be assumed through clockwise or anticlockwise direction. Once you select the custom direction of the current, you will have to apply and maintain the same direction for the overall circuit until the final solution of the circuit.

If we got the final value as positive it means, the supposed direction of the current was correct. In case of negative values, the current of the direction is reversed as compared to the supposed one then.

Circuit Analysis by Kirchhoff’s Laws

Solved Example on KCL and KVL (Kirchhoff’s Laws)

Example:

Resistors of R₁= 10Ω, R₂ = 4Ω and  R₃ = 8Ω are connected up to two batteries (of negligible resistance) as shown. Find the current through each resistor.

Solution:

Assume currents to flow in directions indicated by arrows.

Apply KCL on Junctions C and A.

Therefore, current in mesh ABC = i₁

Current in Mesh CA = i₂

Then current in Mesh CDA = i₁ – i₂

Now, Apply KVL on Mesh ABC, 20V are acting in clockwise direction. Equating the sum of IR products, we get;

10i₁ + 4i₂ = 20   …   (1)

In mesh ACD, 12 volts are acting in clockwise direction, then:

8(i₁ – i₂) – 4i₂ = 12

8i₁ – 8i₂ – 4i₂ = 12

8i₁ – 12i₂ = 12   …   (2)

Multiplying equation (1) by 3;

30i₁ + 12i₂ = 60

Solving for i₁

30i₁ + 12i₂ = 60

8i₁ – 12i₂ = 12

___________
38i₁ = 72

The above equation can be also simplified by Elimination or Cramer’s Rule.

i₁ = 72 ÷ 38 = 1.895 Amperes = Current in 10 Ohms resistor

Substituting this value in (1), we get:

10 (1.895) + 4i₂ = 20

4i₂ = 20 – 18.95

i₂ = 0.263 Amperes = Current in 4 Ohms Resistors.

Now,

i₁ – i₂= 1.895 – 0.263 = 1.632 Amperes

Applications of Kirchhoff’s Laws

• Kirchhoff’s laws can be used to determine the values of unknown values like current and Voltage as well as the direction of the flowing values of these quintets in the circuit.
• These laws can be applied on any circuit* (See the limitation of Kirchhoff’s Laws at the end of the article), but useful to find the unknown values in complex circuits and networks.
• Also used in Nodal and Mesh analysis to find the values of current and voltage.
• Current through each independent loop is carried by applying KVL (each loop) and current in any element of a circuit by counting all the current (Applicable in Loop Current Method).
• Current through each branch is carried by applying KCL (each junction) KVL in each loop of a circuit (Applicable in Loop Current Method).
• Kirchhoff’s Laws are useful in understanding the transfer of energy through an electric circuit.

Limitations of Kirchhoff’s laws:

• KCL is applicable on the assumption that current flows only in conductors and wires. While in High Frequency circuits where, parasitic capacitance can no longer be ignored. In such cases, Current can flow in an open circuit because in these cases, conductors or wires are acting as transmission lines.
• KVL is applicable on the assumption that there is no fluctuating magnetic field linking the closed loop. While, in presence of changing magnetic field in a High Frequency but short wave length AC circuits, the electric field is not a conservative vector field. So, the electric field cannot be the gradient of any potential and the line integral of the electric field around the loop is not zero, directly contradicting KVL. That’s why KVL is not applicable in such a condition.
• During the transfer of energy from the magnetic field to the electric field where fudge has to be introduced to KVL to make the P.d (potential differences) around the circuit equal to 0.

Related Posts about electric circuits analysis theorems:

• Thevenin’s Theorem. Step by Step Procedure with Solved Example
• Norton’s Theorem. Easy Step by Step Procedure with Example (Pictorial Views)
• SUPERNODE Circuit Analysis | Step by Step with Solved Example
• SUPERMESH Circuit Analysis | Step by Step with Solved Example
• Maximum Power Transfer Theorem for AC & DC Circuits
• Compensation Theorem – Proof, Explanation and Solved Examples
• Substitution Theorem – Step by Step Guide with Solved Example
• Millman’s Theorem – Analyzing AC & DC Circuits – Examples
• Superposition Theorem – Circuit Analysis with Solved Example
• Tellegen’s Theorem – Solved Examples & MATLAB Simulation
• Voltage Divider Rule (VDR) – Solved Examples for R, L and C Circuits
• Current Divider Rule (CDR) – Solved Examples for AC and DC Circuits
• Ohm’s Law: Simple Explanation with Statement and Formulas
• Star to Delta & Delta to Star Conversion. Y-Δ Transformation