Resistors in Series and Parallel

Ever wondered how your smartphone, fan, or even fairy lights know exactly how much current to draw without exploding? The answer lies in small pieces of hardware known as resistors that control the flow of electric current behind the scenes. But here's where it gets even more interesting—resistors don't act alone; they collaborate in series, parallel, or even a mix of both to determine how a circuit will behave. In this lesson, we will learn about these collaborations and how they work in real-life situations. 

1.0Understanding the Basics

Before delving into what resistors are and how they work in arrangements, let’s just quickly recall three fundamental concepts, which are the core of electricity. 

• Potential Difference(Voltage): Potential difference is simply the energy per unit charge, provided by a power source, required to move electrons through a circuit. The SI unit is Volt and the symbol is V. 
• Electric Current: It is the flow of electric charge, caused by a potential difference, through a conductor. It is generated by the movement of electrons. The SI unit is the Ampere, and the Symbol is I. 
• Resistance: Resistance is the basic property of the conductor, which resists the flow of current through it. It has the symbol R with Ohm as the SI unit. 

These three quantities make up Ohm’s Law, which gives a mathematical relation between these quantities: $V=IR$

2.0Introduction to Resistors

A resistor is a passive electronic component of an electric circuit, specifically designed to resist the flow of electric current. The device converts electric energy into heat and stops the current when it reaches the desired level. The amount of opposition provided by each resistor is different and is called resistance, which is also measured in Ohms. Most of the time, in a circuit, multiple resistors are used and hence can be arranged in series, parallel, or a combination of both to achieve a specific resistance. 

3.0Resistors in Series

In an electric circuit, when multiple resistors are connected end-to-end and have a single path for current to flow, such an arrangement is known as a series arrangement. 

Circuit Diagram: 

Current in Series: In a series arrangement of resistors, as there is only one way for current to flow, the current (I) remains constant through all the resistors. 

Voltage Division: Unlike the current, the overall voltage (V) across the resistors is divided among each resistor according to their respective resistances. Hence, it can be said: 

$V_{Total}=V_1+V_2+V_3+...+V_n$

Here, according to Ohm’s law, each V_i = IR_i; therefore: 

$I{R}_{Total}=I{R}_1+I{R}_2+I{R}_3+...+I{R}_n$

Resistors in Series Formula for Equivalent Resistance

The equivalent resistance is the numerical value of a single resistance provided by a combination of multiple resistors. In the case of the series combination of resistors, the equivalent resistance is simply the sum of the individual resistance of each resistor. Hence, the resistors in series formula for equivalent resistance can be expressed as: 

$R_{eq}=R_1+R_2+R_3+...+R_n$

4.0Resistors in Parallel

When multiple resistors are connected to the same two points, which ultimately provides multiple paths for current to flow, such a combination of resistors is known as a Parallel combination. 

Circuit Diagram: 

Current Division: Unlike the series combination, in the parallel combination of resistors, the overall current splits among each resistor. Note that the lower the resistance, the higher the current through these resistors. Mathematically, the current division can be explained as: 

$I_{Total}=I_1+I_2+I_3+...+I_n$

Voltage in Parallel: As the resistors in a parallel combination are connected directly to the same two points in the circuit, each resistor experiences the same voltage across it. Hence, in the above equation, each value of current is I_i = VRn. Therefore, the equation becomes: 

$\frac{V}{R_{Total}}=\frac{V}{R_1}+\frac{V}{R_2}+\frac{V}{R_3}+...+\frac{V}{R_n}$

Resistors in Parallel Formula

In a parallel combination of resistors, the equivalent resistance always remains less than the smallest resistor among them. Mathematically, the resistance of resistors in parallel can be expressed as: 

$\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+...+\frac{1}{R_n}$

5.0Combination of Resistors (Series and Parallel)

In real-life scenarios, resistors are hardly found connected in only series or only parallel. Most often, the practical application of resistors involves a combination of both, which helps in balanced control over current and voltage. Some real-life combination of resistors examples include home wiring systems, electronic devices, fairy lights, etc. 

To numerically solve for the equivalent circuits of such combinations, simply reduce the parallel parts, then the series parts, or vice versa, until we are left with a single equivalent resistance. 

Example: Calculate the total equivalent resistance of the circuit, with two resistors, R₁ = 6 Ω and R₂ = 3 Ω, connected in parallel. And this parallel group is then connected in series with a resistor R₃ = 4 Ω.

Solution: First, let’s solve the parallel part that is R₁ and R₂: 

$\frac{1}{R_{Parallel}}=\frac{1}{R_1}+\frac{1}{R_2}$

$\frac{1}{R_{Parallel}}=\frac{1}{6}+\frac{1}{3}=\frac{3}{6}=\frac{1}{2}$

$R_{Parallel}=2\Omega$

Now, solve for the series resistor R₃ with R_Parallel: 

$R_{Series}=R_3+R_{Parallel}$

$R_{Series}=2+4=6\Omega$

Hence, the total equivalent resistance is $6\Omega$. 

6.0Applications of Series and Parallel Resistors 

Applications of Resistors in Series:

1. Voltage Dividers: Employed in sensors and analogue circuits to generate varying voltages from a common source.
2. Current Limiting: Shields delicate components such as LEDs from excessive current.
3. Increasing Resistance: Employed to decrease the current in power supply circuits.

Applications of Resistors in Parallel:

1. Redundancy: Guarantees that if one path fails, the current continues to flow.
2. Lowering Resistance: Employed to draw increased current in power circuits.
3. Load Sharing: Protects from overheating by distributing current over several paths