Kirchhoff’s Voltage Law (KVL): Definition, Formula, Sign Convention, and Examples

20.5.3 Kirchhoff’s Voltage Law

1. Definition

The algebraic sum of all the potential rise, potential fall and emf round a closed loop in a circuit is zero.

$\boxed{\sum_{i=1}^{n}V_i + \sum_{i=1}^{n}E_i = 0}$

where V and E represent p.d. and emf respectively.

2. Sign Convention for Potential Rise and Potential Fall

(a) Potential rise

We may assign a positive (or a negative) sign to the numerical value of a p.d. V or an emf E to indicate the fact that the voltage is a potential rise.

(b) Potential fall

If we follow the sign convention stated, then we will have to assign a negative (or a positive) sign to the numerical value of a p.d. V or an emf E to indicate the fact that the voltage is a potential fall.

Example 20.4

Figure 20.15 An electrical circuit with five resistors connected in two loops, used to analyze current and voltage distribution.

Refer to the circuit shown in Figure 20.15. Apply:

(a) Kirchhoff’s current law to currents at node B
(b) Kirchhoff’s voltage law to the closed loops: (i) ABEFA and (ii) BCDEB

Answer

Step 1

(i) Draw the direction of the current (if this has not yet been done in the circuit given) that flows through every component found in the circuit.

(ii) Label each current, like I₁, I₂, I₃, and so on, as shown in Figure 20.16.

Figure 20.16 Current directions and labeled branch currents (I₁, I₂, I₃, …) in the circuit.

Step 2

Draw a “voltage arrow” beside each component. The arrow must point towards the direction of a potential rise. Figure 20.17 shows all these arrows round the closed loops ABEFA and BCBEB.

Figure 20.17 Voltage arrows indicating potential rise directions around closed loops ABEFA and BCDEB.

(a) Applying Kirchhoff’s Current Law at Node B

$I_1 − I_2 − I_3 = 0$

$I_1 = I_2 + I_3$

(b)(i) Closed Loop ABEFA

Figure 20.18 Traversal of loop ABEFA starting from point A through R₁, R₃, E₂, R₄, and E₁, showing the voltages encountered along the closed loop.

Starting from point A, move around the loop passing through R₁, R₃, E₂, R₄, E₁, and and finally come back to A as shown in Figure 20.18. We will encounter the following voltages:

Components	Potential Rise	Potential Fall	Direction of Movement
R₁		-I₁R₁	Opposite to arrow direction
R₃		-I₃R₃	Opposite to arrow direction
E₂		-E₂	'+' to '-'
R₄		-I₁R₄	Opposite to arrow direction
E₁	+E₁		'-' to '+'

Applying Kirchhoff' voltage law (KVL):

$(-I_1R_1) + (-I_3R_3) + (-E_2) + (-I_1R_4) + (+E_1) = 0$

$-I_1R_1 - I_3R_3 - I_1R_4 = -E_1 + E_2$

(ii) Closed Loop BCDEB

Figure 20.19 Traversal of loop BCDEB starting from point B through R₂, E₃, R₅, E₂, and R₃, showing the voltages encountered along the closed loop.

Starting from, say, point B, move through R₂, E₃, R₅, E₂, R₃, and finally come back to B so as to complete the loop, as shown in Figure 20.19. We will encounter the following voltages:

Components	Potential Rise	Potential Fall	Direction of Movement
R₂		-I₂R₂	Opposite to arrow direction
E₃	+E₃		'-' to '+'
R₅		-I₂R₅	Opposite to arrow direction
E₂	+E₂		'-' to '+'
R₃	+I₃R₃		Same as arrow direction

Applying Kirchhoff' voltage law (KVL):

$(-I_2R_2) + (+E_3) + (-I_2R_5) + (+E_2) + (+I_3R_3) = 0$

$-I_2R_2 - I_2R_5 + I_3R_3 = -E_2 - E_3$

Example 20.5: Kirchhoff’s Laws in a Complex Circuit

Figure 20.20 shows a circuit consisting of several resistors and batteries.

Problem:
Refer to the circuit shown in Figure 20.20. Determine the current flowing through each resistor. The internal resistances of the batteries may be neglected.

Solution

Step 1

Draw the direction of the current at each resistor and label each current. Let the currents be I₁, I₂, and I₃.

Step 2

Draw a voltage arrow beside each component indicating the direction of potential rise.

Step 3: Apply Kirchhoff’s Current Law (KCL)

At node B (or E):

−I₁ + I₂ + I₃ = 0   ...(1)

Step 4: Loop ABEFA (Apply KVL)

Starting from point A and moving around the loop:

(−I₂ × 4) + (−8) + (−I₂ × 3) + (−6) = 0

−7I₂ = 14

I₂ = −2.0 A

The negative sign indicates that the actual direction of I₂ is opposite to the assumed direction.

Step 5: Loop BCDEB (Apply KVL)

Starting from point B and moving around the loop:

(−I₁ × 3) + (4) + (−I₁ × 2) + (8) = 0

5I₁ = 12

I₁ = +2.4 A

The positive sign indicates that the actual direction of I₁ is the same as assumed.

Step 6: Substitute into Equation (1)

−(2.4) + (−2.0) + I₃ = 0

I₃ = +4.4 A

Applications of Kirchhoff’s Voltage Law (KVL)

Kirchhoff’s Voltage Law (KVL) is widely used in analyzing electrical circuits, especially those involving multiple loops and components.

• Analyzing Complex Circuits
  KVL is used to determine unknown voltages and currents in circuits with multiple loops.
• Loop (Mesh) Analysis
  KVL forms the basis of mesh analysis to solve for unknown currents in each loop.
• Circuit Design
  Engineers use KVL to design reliable electrical and electronic circuits such as power supplies and amplifiers.
• Energy Conservation Verification
  KVL ensures that the total energy supplied in a loop equals the total energy consumed.
• Troubleshooting Circuits
  KVL helps identify errors or faults by checking whether the sum of voltages in a loop equals zero.

Conclusion of Kirchhoff’s Voltage Law (KVL)

Kirchhoff’s Voltage Law (KVL) states that the total sum of all potential rises and potential drops in a closed loop is equal to zero. This principle is based on the law of conservation of energy, ensuring that energy supplied by sources is completely used by the components in the circuit.

KVL is essential for analyzing complex electrical circuits, especially those with multiple loops. By applying proper sign conventions and loop analysis, it becomes a powerful tool to determine unknown voltages and currents accurately.

Overall, Kirchhoff’s Voltage Law plays a crucial role in electrical engineering, circuit design, and problem-solving in physics.

Frequently Asked Questions (FAQ) – Kirchhoff’s Voltage Law (KVL)

1. What is Kirchhoff’s Voltage Law (KVL)?

Kirchhoff’s Voltage Law states that the algebraic sum of all voltages in a closed loop is equal to zero.

2. Why is KVL important?

KVL is important because it helps analyze complex circuits and is based on the principle of energy conservation.

3. What is meant by a closed loop?

A closed loop is a complete path in a circuit where current can start and return to the same point.

4. What is the sign convention in KVL?

Voltage rise is usually taken as positive, while voltage drop is taken as negative, depending on the direction of traversal.

5. Can KVL be applied to any circuit?

Yes, KVL can be applied to all electrical circuits, including simple and complex multi-loop circuits.

6. What is the difference between KVL and KCL?

KVL deals with voltages in a loop, while Kirchhoff’s Current Law (KCL) deals with currents at a node.

7. Is KVL based on any physical law?

Yes, KVL is based on the law of conservation of energy.

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