What is Series and Parallel Circuit - It's Introduction, Mathematical Expressions, Why to Use

Series Circuit:

In series circuits components are connected one after another, forming a single path for current. Series circuits are used when you want the same current to flow through all components, and you want to increase the total resistance.

Imagine series circuit as single pipe with multiple valves, if any valve closes, water stops completely. Water pressure drops at each valve. Same water flows through all valves.

Battery (+) -- Bulb 1 -- Bulb 2 -- Bulb 3 -- Battery (-)

Hence in series:

1. Current is the same through each component.

In a series circuit, components are connected one after the other, so there is only one path for current. If one component burns out. All components stop working. Like Christmas lights where one bad bulb kills the whole string.

Imagine a water pipe with two restrictions i.e. resistances. The same amount of water i.e. current, must flow through both because there's no alternative path. So what goes in at one end must come out at the other. No water or current can disappear or appear magically. So, current is the same.

In a series circuit, there is only one path for electrons to flow. The electrons that
leave the negative terminal of the battery must travel through each component in sequence to return to the positive terminal. Each electron pushes the one in front. When an electron enters R₁, it displaces an electron inside R₁. That electron displaces another, creating a chain reaction. The same displacement rate propagates through all resistors. Conservation of charge: Electrons don't vanish, so current is constant.

There is no branching, so the number of electrons per second i.e. current passing through each component is the same. At the atomic level, the same electrons or rather, the same rate of electron flow must go through each component because there is no alternative path. If one component resists the flow, it reduces the current for the entire circuit.

Current through Bulb 1 = Current through Bulb 2 = Current through Bulb 3

I(total) = I₁ = I₂ = I₃

2. Voltages add up. Total voltage is the sum of the voltages across each component.

In a series circuit, each component has a resistance. The battery provides a certain voltage i.e. energy per charge. As electrons pass through a resistor, they lose some of their energy as heat, light, etc. Inside a resistor, electrons collide with atoms, losing kinetic energy as heat. Each collision transfers energy from electron to atom.

This energy loss per charge is the voltage across that resistor. Since the electrons must pass through each resistor in turn, they lose energy in each one. The total energy lost per charge i.e. sum of voltage drops must equal the energy provided per charge by the battery (conservation of energy). So, the battery voltage is divided among the component. The total pressure drop across the series combination must equal the battery's pressure. So, voltages add up.

Battery gives each electron energy,
E = e × V
In R₁, Electron loses energy e × V₁
In R₂, Same electron loses e × V₂
Total energy lost = e × (V₁ + V₂ + V₃)

By energy conservation,
e × V_battery = e × (V₁ + V₂ + V₃)
V_battery = V₁ + V₂ + V₃

If battery is 12V and all bulbs identical: each gets 4V.
Battery voltage = Bulb1 voltage + Bulb2 voltage + Bulb3 voltage

V(total) = V₁ + V₂ + V₃

3. Resistances add up. Total resistance is the sum of the individual resistances.

Think of running through three different mud pits. Mud pit 1 slows you by factor R₁. Mud pit 2 slows you by additional factor R₂. Mud pit 3 slows you by additional factor R₃.

Total resistance = R₁ + R₂ + R₃

In a series circuit, the resistors are connected one after the other. The total resistance is the sum of the individual resistances because each resistor opposes the flow of electrons, and the electrons have to overcome each resistance in sequence. The more resistors in series, the more difficult it is for the electrons to flow like adding more sections to a narrow pipe. Mathematically, the total resistance is the sum of the individual resistances because the current is the same through each, and the total voltage is the sum of the individual voltages.

So the total resistance seen by the battery is R_total = R₁ + R₂ + R₃

More bulbs = more resistance = dimmer bulb

Why Use Series?

• When you need voltage division
e.g.12V battery to power 3V LED
• You have low voltage but need to power multiple low-voltage devices.
• When components must work/die together. e.g. safety circuits, fire alarm circuits, if wire cuts, alarm triggers
• When you want to measure current. e.g. ammeter in series
• Simple and cheap wiring. You're building decorative lights, where coordinated failure is acceptable
• You want a simple circuit with fewer wires
• You need current to be identical through components
• Used in battery packs to increase voltage.
e.g. 1.5V + 1.5V + 1.5V = 4.5V total
• Current limiting resistors with LEDs, Dimmer switches

Parallel circuit:

Imagine parallel circuit as main pipe with multiple branches with their own valve. If valve 1 closes, valves 2 & 3 still work. Each branch gets full pressure. Water splits between branches.

Components are connected across the same two points, providing multiple paths for current. Parallel circuits are used when you want the same voltage across all components, and you want to allow components to operate independently.

                   --- Bulb 1 ---
               /                         \
Battery (+) ---- Bulb 2 ---- Battery (-)
               \                          /
                   --- Bulb 3 ---


Hence in parallel:

1. Voltage is the same across all components.

In a parallel circuit, each component is connected directly to the same two points i.e. nodes of the circuit. Therefore, the potential difference or voltage between these two points is the same for each Component. At the atomic level, the electric field which is related to the voltage is the same across each branch because they are connected to the same nodes. The battery maintains a constant voltage across these two points, so each component gets the full push or voltage from the battery.

Battery gives each electron the same energy boost regardless of which path it takes. All electrons start at same low energy i.e. battery negative. All electrons end at same high energy i.e. battery positive. For every path electron takes,
Energy gain per electron = e × V

Consider two resistors R1 and R2 in parallel with a battery of voltage V. By the definition of parallel connection, the voltage across each resistor is the same equal to the battery voltage V.

Each bulb gets full battery voltage. Bulb1 gets 12V, bulb2 gets 12V, bulb3 gets 12V

V(total) = V₁ = V₂ = V₃

2. Currents add up. The total current is the sum of the currents through each component.

Think of a river splitting into multiple streams. Stream 1 carries 30% of water, stream 2 carries 50% of water and stream 3 carries 20% of water. Total water flow is equal to sum of all stream flows. Water takes multiple paths to same destination.

In a parallel circuit, the electrons from the battery split at the junction or node and take multiple paths. The amount of current in each branch is determined by the resistance of that branch (ohm's law i =V/R). Since the voltage is the same across each branch, the current in each branch is V divided by the resistance of that branch. The total current from the battery is the sum of the currents in each branch. At the atomic level, the electrons divide themselves among the available paths.
The more paths, the more electrons will flow through that path. The total current is the sum of the currents in all paths.

Current from the battery (I_total) splits into two paths: I₁ through R1 and I₂ through R2.

By KCL, Total current = I₁ + I₂ + I₃

If one components burns out. Other components stay on. Like house lights turning off kitchen light doesn't affect bedroom

3. The reciprocal of the total resistance is the sum of the reciprocals of the individual resistances, so total resistance is less than any individual resistance.

Adding more parallel paths is like adding more pipes for water to flow. More paths mean less overall resistance to flow.

When resistors are connected in parallel, the total resistance decreases because there are multiple paths for the electrons to flow. The electrons can choose more than one path, so it is easier for them to flow like adding more lanes to a highway. Mathematically, the reciprocal of the total resistance is the sum of the reciprocals of the individual resistances. This is because the total current is the sum of the currents in each branch, and the voltage is the same.
So, 1/R_total = 1/R1 + 1/R2 +... .
This means that the total resistance is less than the smallest individual resistance.

Each resistor provides an independent conduction channel. Total conductance is sum of individual conductances.
Conductance = 1/resistance

The total resistance R_total is given by,1/R(total) = 1/R₁ + 1/R₂ + 1/R₃

More bulbs = less total resistance = battery drains faster

More paths = more conductance = less total resistance

Why Use parallel?

• When devices need full voltage
e.g. appliances in your home
• Reliability is critical, one failure shouldn't stop others.
e.g. home wiring, car systems
• You want to add/remove devices without affecting others, so you can
turn off one appliance without affecting
others and car lighting systems.
• You need to power devices with different current requirements and when you want to increase current capacity.