Day 7: Basic Circuit Analysis – Step-by-Step Solved Examples
Welcome to Day 7 of the Electrical Fundamentals – Zero to Advanced course at Electrical Designer World!
In the previous lessons, we studied voltage, current, resistance, Ohm’s law, series and parallel circuits, electrical power, and Kirchhoff’s Current Law (KCL) and Voltage Law (KVL).
Now it is time to combine these concepts through basic circuit analysis.
Circuit analysis is the process of calculating unknown voltages, currents, and resistances in an electrical circuit. It helps us understand how a circuit behaves and how electrical components interact with one another.
For beginners, the best way to learn circuit analysis is to start with simple series circuits, progress to parallel circuits, and then solve mixed circuits using Ohm’s law, KCL, and KVL.
In this lesson, you will learn the complete step-by-step process with circuit examples, formulas, solved calculations, and practice questions.
Table of Contents
1. What Is Basic Circuit Analysis?
Basic circuit analysis is the process of determining electrical quantities such as voltage, current, and resistance using circuit laws and component relationships.
An electrical circuit may contain:
- Voltage sources such as batteries or DC power supplies.
- Resistors and other electrical components.
- Conducting wires and connection nodes.
- Series and parallel branches.
- Unknown currents and voltages that must be calculated.
The main objective is to determine how voltage and current are distributed throughout the circuit.
For example, if a 24 V supply is connected to three resistors, circuit analysis helps us calculate the total resistance, source current, and voltage drop across each resistor.
Why is circuit analysis important?
Circuit analysis is useful for understanding electrical networks, checking circuit behaviour, troubleshooting faults, and developing the foundation for advanced electrical engineering.
It also prepares you for studying AC circuits, electrical machines, and power systems.
2. Important Formulas for Circuit Analysis
Before solving a circuit, remember these fundamental formulas.
Ohm’s law
V = I × R
I = V / R
R = V / I
Series resistance
Rₜ = R₁ + R₂ + R₃ + …
Parallel resistance
1/Rₜ = 1/R₁ + 1/R₂ + 1/R₃ + …
For two resistors in parallel:
Rₜ = (R₁ × R₂) / (R₁ + R₂)
Kirchhoff’s Current Law
The algebraic sum of currents at a node is zero.
ΣI = 0
Kirchhoff’s Voltage Law
The algebraic sum of voltages around a closed loop is zero.
ΣV = 0
Electrical power
For a DC circuit:
P = V × I
These formulas provide the foundation for solving many basic electrical circuits.
3. Steps to Analyse an Electrical Circuit
Follow this systematic method whenever you solve a circuit problem.
Step 1: Identify the components
Read the circuit diagram carefully and identify the voltage source, resistors, nodes, branches, and any given current or voltage values.
Step 2: Identify series and parallel connections
Check which resistors carry the same current and which resistors are connected across the same two nodes.
Do not assume two components are in series or parallel merely because they appear next to each other in a drawing. Their electrical connections determine the relationship.
Step 3: Simplify the circuit
Combine clearly identifiable series resistors or parallel resistor groups to obtain an equivalent resistance.
Repeat the process where possible.
Step 4: Calculate the total current
Once the equivalent resistance is known, use Ohm’s law:
Iₜ = V / Rₜ
Step 5: Calculate individual voltages and currents
Work backwards through the simplified circuit.
Use Ohm’s law to calculate voltage drops and branch currents. Apply KCL at nodes and KVL around loops when required.
Step 6: Verify the answer
Check that:
- Series voltage drops add to the appropriate source voltage.
- Branch currents add to the total current.
- Calculated values satisfy Ohm’s law.
- The units and signs are consistent.
This method will help you solve circuit problems accurately and systematically.
4. Example 1: Series Circuit Analysis
A 24 V DC source is connected to three resistors in series:
- R₁ = 4 Ω
- R₂ = 6 Ω
- R₃ = 10 Ω
Calculate the total resistance, circuit current, and voltage across each resistor.
Step 1: Calculate total resistance
For a series circuit:
Rₜ = R₁ + R₂ + R₃
Rₜ = 4 + 6 + 10
Rₜ = 20 Ω
Step 2: Calculate total current
Using Ohm’s law:
Iₜ = V / Rₜ
Iₜ = 24 / 20
Iₜ = 1.2 A
The same current flows through every resistor in this series circuit.
Step 3: Calculate voltage across each resistor
For R₁:
V₁ = I × R₁
V₁ = 1.2 × 4
V₁ = 4.8 V
For R₂:
V₂ = 1.2 × 6
V₂ = 7.2 V
For R₃:
V₃ = 1.2 × 10
V₃ = 12 V
Step 4: Verify using KVL
V₁ + V₂ + V₃ = 24 V
4.8 + 7.2 + 12 = 24 V
The voltage drops equal the source voltage.
Final answers:
- Total resistance = 20 Ω
- Circuit current = 1.2 A
- Voltage drops = 4.8 V, 7.2 V, and 12 V
5. Example 2: Parallel Circuit Analysis
A 12 V DC supply is connected across three parallel resistors:
- R₁ = 6 Ω
- R₂ = 12 Ω
- R₃ = 24 Ω
Calculate the equivalent resistance, branch currents, and total current.
Step 1: Calculate equivalent resistance
1/Rₜ = 1/6 + 1/12 + 1/24
Using a common denominator:
1/Rₜ = 4/24 + 2/24 + 1/24
1/Rₜ = 7/24
Therefore:
Rₜ = 24/7 ≈ 3.43 Ω
Step 2: Calculate each branch current
All three resistors have the same 12 V supply across them.
For R₁:
I₁ = 12 / 6 = 2 A
For R₂:
I₂ = 12 / 12 = 1 A
For R₃:
I₃ = 12 / 24 = 0.5 A
Step 3: Calculate total current
Using KCL:
Iₜ = I₁ + I₂ + I₃
Iₜ = 2 + 1 + 0.5
Iₜ = 3.5 A
Step 4: Verify using Ohm’s law
Iₜ = V / Rₜ
Iₜ = 12 / (24/7)
Iₜ = 3.5 A
The two methods give the same result.
Final answers:
- Equivalent resistance ≈ 3.43 Ω
- Branch currents = 2 A, 1 A, and 0.5 A
- Total current = 3.5 A
6. Example 3: Series-Parallel Circuit Analysis
A 24 V DC supply is connected to a 4 Ω resistor in series with a parallel combination of 6 Ω and 12 Ω resistors.
Calculate the total resistance, total current, voltage drops, and branch currents.
Step 1: Identify the circuit arrangement
The 6 Ω and 12 Ω resistors are connected in parallel. This parallel group is in series with the 4 Ω resistor.
Step 2: Calculate the parallel resistance
Rₚ = (R₁ × R₂) / (R₁ + R₂)
Rₚ = (6 × 12) / (6 + 12)
Rₚ = 72 / 18
Rₚ = 4 Ω
Step 3: Calculate total resistance
Rₜ = 4 + 4
Rₜ = 8 Ω
Step 4: Calculate total current
Iₜ = V / Rₜ
Iₜ = 24 / 8
Iₜ = 3 A
Step 5: Calculate the voltage drop across the series resistor
V₁ = Iₜ × R₁
V₁ = 3 × 4
V₁ = 12 V
Step 6: Calculate voltage across the parallel group
Vₚ = 24 − 12
Vₚ = 12 V
Each parallel resistor therefore has 12 V across it.
Step 7: Calculate branch currents
For the 6 Ω resistor:
I₁ = 12 / 6 = 2 A
For the 12 Ω resistor:
I₂ = 12 / 12 = 1 A
Step 8: Verify using KCL
Iₜ = I₁ + I₂
3 = 2 + 1
The current balance is correct.
Final answers:
- Total resistance = 8 Ω
- Total current = 3 A
- Voltage across the series resistor = 12 V
- Voltage across the parallel group = 12 V
- Branch currents = 2 A and 1 A
7. Example 4: Circuit Analysis Using KVL
Kirchhoff’s Voltage Law is particularly useful when a circuit contains a closed loop and the voltage drops or source voltages must be related.
Consider a 12 V DC source connected in series with two resistors:
- R₁ = 2 Ω
- R₂ = 4 Ω
Find the circuit current and voltage across each resistor using KVL and Ohm’s law.
Step 1: Assign a current
Let the clockwise current be I.
Step 2: Write the KVL equation
Starting at the negative terminal of the source and travelling through the source from negative to positive gives a voltage rise of 12 V.
Traversing the resistors in the direction of current gives voltage drops of 2I and 4I.
Therefore:
12 − 2I − 4I = 0
12 − 6I = 0
6I = 12
I = 2 A
Step 3: Calculate voltage drops
V₁ = I × R₁
V₁ = 2 × 2
V₁ = 4 V
V₂ = I × R₂
V₂ = 2 × 4
V₂ = 8 V
Step 4: Verify KVL
12 − 4 − 8 = 0
The equation is satisfied.
This example demonstrates how KVL and Ohm’s law can be used together to calculate unknown circuit quantities.
8. Example 5: Circuit Analysis Using KCL
Suppose a total current of 5 A reaches a node. Two branches carry currents of 2 A and I₃ away from the node.
Find I₃ using KCL.
Step 1: Identify entering and leaving currents
Entering current = 5 A
Leaving currents = 2 A and I₃
Step 2: Apply KCL
Sum of entering currents = Sum of leaving currents.
5 = 2 + I₃
I₃ = 5 − 2
I₃ = 3 A
Step 3: Verify
2 + 3 = 5 A
The total current leaving the node equals the current entering it.
This is the basic principle used to analyse current distribution at circuit junctions.
9. Introduction to Nodal and Mesh Analysis
When circuits become more complex, simplifying series and parallel resistors may not be enough. Two widely used systematic methods are nodal analysis and mesh analysis.
Nodal analysis
Nodal analysis uses Kirchhoff’s Current Law to calculate unknown node voltages relative to a selected reference node.
The general process is:
- Select a reference node, usually called ground.
- Assign voltage variables to the other essential nodes.
- Express branch currents using node voltages and component relationships.
- Apply KCL at the unknown-voltage nodes.
- Solve the resulting equations.
For a resistor connected between nodes at voltages V₁ and V₂, the current from node 1 to node 2 is:
I = (V₁ − V₂) / R
This equation follows from Ohm’s law.
Mesh analysis
Mesh analysis uses Kirchhoff’s Voltage Law to solve planar circuits by assigning currents to independent meshes.
The general process is:
- Identify the independent meshes.
- Assign a mesh current to each mesh.
- Write a KVL equation around each mesh.
- Express resistor voltage drops using the mesh currents.
- Solve the simultaneous equations.
For a resistor shared by two meshes, the current through the shared resistor is determined by the difference between the mesh currents, using the chosen reference directions.
These methods are useful for larger networks where direct circuit simplification is not possible.
10. Common Mistakes in Circuit Analysis
Avoid these common errors when solving circuit problems.
Mistake 1: Confusing series and parallel connections.
Check the actual electrical nodes. Components are parallel only when they connect across the same two nodes.
Mistake 2: Adding parallel resistances directly.
For parallel resistors, use the reciprocal formula or the two-resistor product-over-sum formula.
Mistake 3: Assuming the current is the same in every branch.
Current is the same through components in a simple series path, but it divides between parallel branches.
Mistake 4: Forgetting voltage polarity.
When writing KVL equations, keep track of voltage rises and drops.
Mistake 5: Using inconsistent units.
Convert values to compatible units before performing calculations.
Mistake 6: Skipping verification.
Always check current balance, voltage balance, and the reasonableness of your answers.
11. Practical Applications of Basic Circuit Analysis
Electrical troubleshooting
Circuit analysis helps engineers compare measured voltages and currents with expected values to identify possible faults.
Electronic circuits
Resistor networks, voltage dividers, and many basic electronic circuits can be analysed using Ohm’s law, KCL, and KVL.
Electrical power distribution
Understanding voltage drops and current distribution provides a foundation for analysing electrical distribution circuits.
Electrical engineering design
These fundamentals prepare engineers to understand more advanced calculations involving electrical loads, cable voltage drop, transformers, and power systems.
For practical EPC design, remember that basic circuit calculations are only one part of the design process. Real installations also require appropriate standards, equipment ratings, protection coordination, installation conditions, and engineering checks.
Frequently Asked Questions
1. What is basic circuit analysis?
Basic circuit analysis is the process of calculating unknown electrical quantities, such as voltage and current, using circuit laws and component relationships.
2. Which formulas are most important for circuit analysis?
Ohm’s law, series and parallel resistance formulas, KCL, and KVL are the essential starting points.
3. How do you solve a series circuit?
Add all series resistances, calculate the current using Ohm’s law, and determine individual voltage drops.
4. How do you solve a parallel circuit?
Calculate equivalent resistance, find the total current, calculate branch currents, and verify that branch currents add to the total current.
5. When should KCL be used?
Use KCL at circuit nodes to calculate unknown currents and verify current balance.
6. When should KVL be used?
Use KVL around closed loops to relate voltage rises and drops.
7. What is nodal analysis?
Nodal analysis is a method that uses KCL equations to calculate unknown node voltages.
8. What is mesh analysis?
Mesh analysis is a method that uses KVL equations to calculate currents in independent meshes of a planar circuit.
Day 7 Practice Questions
Test your understanding of basic circuit analysis.
- Three resistors of 2 Ω, 4 Ω, and 6 Ω are connected in series across a 24 V source. Calculate total resistance and current.
- Two resistors of 6 Ω and 12 Ω are connected in parallel across 12 V. Calculate the branch currents and total current.
- A 24 V source is connected to a 4 Ω resistor in series with a parallel group of 6 Ω and 12 Ω resistors. Calculate total resistance and total current.
- A 5 A current enters a node. Two currents of 1.5 A and I₃ leave it. Find I₃.
- A 20 V source supplies two series resistors of 4 Ω and 6 Ω. Calculate the current and voltage drops.
- What is the main principle behind nodal analysis?
- What is the main principle behind mesh analysis?
- State two common mistakes made when solving series-parallel circuits.
Answer Key
- Rₜ = 2 + 4 + 6 = 12 Ω; I = 24 / 12 = 2 A.
- I₁ = 12 / 6 = 2 A; I₂ = 12 / 12 = 1 A; total current = 3 A.
- Parallel resistance = 4 Ω; total resistance = 8 Ω; total current = 3 A.
- 5 = 1.5 + I₃; I₃ = 3.5 A.
- Rₜ = 10 Ω; I = 2 A; voltage drops = 8 V and 12 V.
- Kirchhoff’s Current Law.
- Kirchhoff’s Voltage Law.
- Confusing series and parallel connections, and adding parallel resistances directly.
Further Reading: Basic Circuit Analysis
Explore these resources to improve your understanding of electrical circuit analysis:
- All About Circuits – Online Textbook — learn circuit theory, Ohm’s law, network analysis, and electrical calculations.
- OpenStax College Physics — review current, resistance, and fundamental circuit relationships.
- Schneider Electric Electrical Installation Guide — explore practical electrical installation and distribution principles.
Conclusion
Basic circuit analysis brings together the most important concepts from the first six days of this course.
By combining Ohm’s law, series and parallel resistance formulas, Kirchhoff’s Current Law, and Kirchhoff’s Voltage Law, you can calculate unknown voltages, currents, and resistances in many simple electrical circuits.
The most important habit is to solve each circuit systematically: identify the connections, simplify the network where possible, calculate unknown quantities, and verify your results.
Continue learning:
- Explore the Electrical Fundamentals – 30-Day Course
- Read Day 3: Ohm’s Law
- Read Day 4: Series and Parallel Circuits
- Read Day 5: Electrical Power and Energy
- Read Day 6: Kirchhoff’s Laws
- Day 8: Electrical Resistance and Resistivity — add the link after publishing the next lesson.
Keep learning step by step with Electrical Designer World!