Equivalent Circuits - Understanding Series And Parallel Combination Of Resistance

Equivalent Circuits - Understanding Series and Parallel Combination of Resistance

In electrical circuits, resistors can be connected in different ways - series and parallel.
Series combination: Resistors are connected end to end, forming a single path for current.
Parallel combination: Resistors are connected side by side, providing multiple paths for current.
Series combination increases the total resistance, while parallel combination decreases it.
It is crucial to understand and analyze equivalent circuits for solving complex problems.

Series Combination

In series combination, resistors are connected one after another.
The total resistance is the sum of individual resistances.
Formula: Rt = R1 + R2 + R3 + …
Example: Three resistors of values 2Ω, 3Ω, and 4Ω in series.
- Rt = 2Ω + 3Ω + 4Ω = 9Ω
In series combination, the current remains the same throughout the circuit.
Voltage across each resistor is different and varies according to their respective resistances.

Series Combination - Voltage Distribution

In a series combination, the total voltage across the circuit is the sum of voltages across individual resistors.
Example: In a circuit with three resistors of 2Ω, 3Ω, and 4Ω in series, and a total voltage of 12V.
- Vt = V1 + V2 + V3 = 12V
Voltage across each resistor can be obtained using Ohm’s law: V = I * R.
The current remains the same throughout the series combination.

Series Combination - Current Distribution

In a series combination, the same current flows through all the resistors.
The total current entering the series combination is equal to the current flowing through each resistor.
Example: In a circuit with three resistors of 2Ω, 3Ω, and 4Ω in series, and a total current of 3A.
- It = I1 = I2 = I3 = 3A
The current remains constant throughout the series combination.

Parallel Combination

In a parallel combination, resistors are connected side by side.
The total resistance reduces as more paths are available for current to flow.
Formula: 1/Rt = 1/R1 + 1/R2 + 1/R3 + …
Example: Three resistors of values 2Ω, 3Ω, and 4Ω in parallel.
- 1/Rt = 1/2Ω + 1/3Ω + 1/4Ω
- 1/Rt = (6 + 4 + 3)/12
- 1/Rt = 13/12
- Rt = 12/13 Ω (~0.92 Ω)

Parallel Combination - Voltage Distribution

In a parallel combination, the voltage across each resistor is the same.
Example: In a circuit with three resistors of 2Ω, 3Ω, and 4Ω in parallel, and a total voltage of 10V.
- V1 = V2 = V3 = 10V
The voltage remains constant across all the resistors in the parallel combination.

Parallel Combination - Current Distribution

In a parallel combination, the total current entering the combination splits among the resistors.
The sum of currents through each resistor adds up to the total current.
Example: In a circuit with three resistors of 2Ω, 3Ω, and 4Ω in parallel, and a total current of 5A.
- It = I1 + I2 + I3 = 5A
The current splits among the parallel resistors based on their individual resistances.
The inverse of resistance determines the fraction of current flowing through each resistor.

Equivalent Resistance - Series Combination

In a series combination, the total resistance (Rt) is equal to the sum of individual resistances (R1, R2, R3…).
Example: In a circuit with three resistors of 4Ω, 5Ω, and 6Ω in series.
- Rt = 4Ω + 5Ω + 6Ω
- Rt = 15Ω
The equivalent resistance in series is always greater than the individual resistances.

Equivalent Resistance - Parallel Combination

In a parallel combination, the reciprocal of the total resistance (1/Rt) is equal to the sum of reciprocals of individual resistances (1/R1, 1/R2, 1/R3…).
Example: In a circuit with three resistors of 8Ω, 10Ω, and 12Ω in parallel.
- 1/Rt = 1/8Ω + 1/10Ω + 1/12Ω
- 1/Rt = (15 + 12 + 10)/120Ω
- 1/Rt = 37/120Ω
- Rt = 120/37Ω (~3.24Ω)

Superposition of Voltage and Current Sources

In some circuits, both voltage and current sources may be present.
Superposition theorem allows us to handle such circuits by considering only one source at a time.
To apply superposition, we turn off all sources except one, and calculate the corresponding response.
We repeat this process for each source separately and then sum the responses.

Superposition Example: Voltage Source

Consider a circuit with a voltage source of 10V and a current source of 2A.
To apply superposition, we turn off the current source and analyze the circuit with only the voltage source.
The voltage across each resistor is calculated using Ohm’s law: V = I * R.
The total voltage is obtained by summing up the individual voltages.

Superposition Example: Current Source

In the same circuit, we turn off the voltage source and analyze the circuit with only the current source.
The current flowing through each resistor is calculated using Ohm’s law: I = V / R.
The total current is obtained by summing up the individual currents.

Total Response from Superposition

After analyzing the circuit with each source separately, we obtain the individual voltages and currents.
The total voltage and current for the original circuit are obtained by summing up the corresponding values obtained from each source.

Equivalent Resistance for Complex Circuits

For more complex circuits, finding the equivalent resistance becomes essential.
We can simplify complex circuits into an equivalent circuit with a single resistor.
The equivalent resistance is the resistance value that, when connected to a voltage source, produces the same current flow as the original circuit.

Finding Equivalent Resistance - Series and Parallel Combination

In complex circuits, we can use the concepts of series and parallel combinations to find the equivalent resistance.
Identify regions of the circuit that can be simplified using series or parallel combinations.
Replace each identified region with the appropriate equivalent resistance.
Repeat until the entire circuit is reduced to a single equivalent resistance.

Equivalent Resistance Example - Complex Circuit

Consider a complex circuit with resistors connected in series and parallel combinations.
Identify the different regions that can be simplified using series and parallel combinations.
Replace each region with the equivalent resistance, reducing the circuit step by step.

Equivalent Resistance Example - Continued

Keep simplifying the circuit until it is reduced to a single equivalent resistance.
The equivalent resistance is found by replacing all the resistors with their equivalent values and computing the overall resistance.

Equivalent Resistance Example - Final Equivalent Circuit

After simplifying the entire circuit, we are left with a single equivalent resistance and the voltage source.
The original complex circuit is reduced to a simple circuit that can be easily analyzed using the known laws and principles.

Summary

Equivalent circuits allow us to simplify complex circuits by finding a single resistance.
Series and parallel combinations are used to determine the equivalent resistance.
Superposition theorem is utilized when both voltage and current sources are present.
Understanding and analyzing equivalent circuits is crucial in solving complex problems effectively.

Properties of Resistors in Series

The current flowing through each resistor in series is the same.
The total resistance in a series combination is equal to the sum of individual resistances.
The voltage across each resistor in series is proportional to its resistance.
The total voltage across the series combination is the sum of individual voltages.

Example of Resistors in Series

Consider three resistors connected in series: R1 = 4Ω, R2 = 6Ω, R3 = 8Ω.
The total resistance in series would be: Rt = R1 + R2 + R3 = 4Ω + 6Ω + 8Ω = 18Ω.
The current flowing through each resistor would be the same, let’s say I = 2A.
The voltage across each resistor in series can be calculated using Ohm’s law: V = I * R.
Example calculations: V1 = 2A * 4Ω = 8V, V2 = 2A * 6Ω = 12V, V3 = 2A * 8Ω = 16V.
The total voltage across the series combination would be: Vt = V1 + V2 + V3 = 8V + 12V + 16V = 36V.

Properties of Resistors in Parallel

The voltage across each resistor in parallel is the same.
The total resistance in a parallel combination is less than the resistance of the smallest resistor.
The current splits between the parallel resistors based on their individual resistances.
The sum of the currents flowing through each resistor in parallel is equal to the total current.

Example of Resistors in Parallel

Consider three resistors connected in parallel: R1 = 4Ω, R2 = 6Ω, R3 = 8Ω.
The total resistance in parallel can be calculated using the formula: 1/Rt = 1/R1 + 1/R2 + 1/R3.
Example calculations: 1/Rt = 1/4Ω + 1/6Ω + 1/8Ω = (6 + 4 + 3)/(24) = 13/24.
Inverse of the total resistance: Rt = 24/13Ω (~1.85Ω).
Let’s assume the total current flowing through the parallel combination is I = 3A.
The current flowing through each resistor can be calculated using Ohm’s law: I = V / R.
Example calculations: I1 = 3A * (1.85Ω/4Ω) = 1.3875A, I2 = 3A * (1.85Ω/6Ω) = 0.925A, I3 = 3A * (1.85Ω/8Ω) = 0.694A.
The sum of the currents flowing through each resistor: It = I1 + I2 + I3 = 1.3875A + 0.925A + 0.694A = 3A.

Analysis of Complex Circuits

Complex circuits may contain a combination of series and parallel resistors.
The circuit can be simplified step by step using series and parallel combinations.
Identify regions of the circuit that can be simplified using these combinations.
Replace each region with the corresponding equivalent resistance.
Repeat the process until the entire circuit is simplified.

Example Complex Circuit Analysis

Consider a complex circuit with multiple resistors connected in series and parallel.
Break down the circuit into simpler sections and simplify one section at a time.
Replace each section with its equivalent resistance and adjust the overall circuit accordingly.

Equivalent Resistance - Further Analysis

It is possible for a complex circuit to have multiple equivalent resistances.
Identify sections of the circuit that can be simplified individually.
Calculate the equivalent resistance for each section and replace it in the circuit.

Step-by-Step Simplification Example

Consider a complex circuit with resistors connected in various combinations.
Start by identifying simple series and parallel combinations.
Replace each combination with its equivalent resistance.
Gradually simplify the circuit until only one equivalent resistance remains.

Final Equivalent Circuit Example

After simplifying the entire complex circuit, we end up with a single equivalent resistance.
The original complex circuit is now reduced to a simple circuit with minimum components.
This equivalent circuit can be easily analyzed using known formulas and principles.

Importance of Equivalent Circuits

Equivalent circuits provide a way to simplify complex circuit configurations.
They allow us to focus on the important aspects of the circuit without getting overwhelmed.
The analysis of equivalent circuits helps in understanding the behavior and characteristics of electrical circuits.
Equivalent circuits are extensively used in various electrical engineering applications.