Lesson 28: Combined Circuits

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Ashley Hodgeson

Review: Series Circuits

There is only one possible pathway for the electricity to travel around a series circuit. Consequently, if any of the three switches in the circuit below is open, electricity will not flow to any of the loads. If one of them is off, they will all be off. If your house was wired in series, when you turned on a light switch every light in the house would come on.

The following rules must be followed in all series connections:

I_T = I₁ = I₂ = I₃ = …

R_T = R₁ + R₂ + R₃ + …

V_T = V₁ + V₂ + V₃ + …

Review: Parallel Circuits

In a parallel circuit, there is more than one path for the electricity to flow along. If one of the switches is open, electricity can still flow through the other branch. Each switch controls only one load.

The following rules must be followed in all parallel connections:

V_T = V₁ = V₂ = V₃

I_T = I₁ + I₂ + I₃

Combined Circuits:

A combined circuit contains some series connections and some parallel connections. When you look at the circuit below, you should see that there are two pathways which the electrons can flow along to get around the circuit. The green pathway starts at the positive terminal, goes through R₁, then through R₄, and finally back to the negative terminal of the power source.

The orange pathway also starts at the positive terminal, but instead of passing through R₁, it passes through R₂ and R₃ instead. Then, like the green pathway, it passes through R₄ and finally back to the negative terminal of the power source. The current can flow through R_1, or through R₂ and R₃, however, if it goes through R₂ if must go through R₃. We say that R₂ and R₃ are connected together in series, which are then connected in parallel with R₁. R₄ is connected in series because regardless of the path taken around the circuit, all of the current must pass through R₄.

Learn more about combined circuits.

A Problem Solving Strategy…

Combined circuits are often too complicated to solve as they are presented. They must first be reduced into simpler circuits. Our goal is to replace the parallel components of the circuit with series components that have the same overall effect on the circuit.

For example, consider the component below:

As electricity flows from point A to point B, it must pass through either R₁ or R₂. Or you could replace R₁ and R₂ with a single resistor that has the same overall resistance as R₁ and R₂ combined.

R₁ and R₂ combined have a resistance of 3.6 ?. So does R₃. By replacing R₁ and R₂ with R₃, we have made the circuit simpler and easier to solve. To solve combined circuits, we need to simplify them first. To solve a combined circuit, you need to determine the current, potential difference, and resistance of all of the components of the circuit.

Example 1: Solving a Combined Circuit

When solving the circuit below for example, it is important to remember to use series equations for components that are connected in series and parallel equations for components that are connected in parallel.

The first step is to combine R₂ and R₃ into a new resistor that we will call R₅. R₅ doesn’t really exist in the circuit, it is just used to make the circuit easier to solve. Since the two resistors are connected in series we will use the following rule:

R₅ = R₂ + R₃ = 4? + 2? = 6?.

This is a simplified version of our original circuit. We can further simplify it by combining R₁ and R₅ into a new resistor that we will call R₆. Since the two resistors are connected in parallel we will use the following rule: 

Now that we have simplified the circuit to a series circuit, it should be relatively easy to solve. Be careful to ensure that you are using series rules for series connections and parallel rules for parallel connections.

Using our series rules:

R_T = R₄ + R₆ = 8 ? + 2 ? = 10 ?

And since the circuit above is a series circuit and currents in a series circuit are all equal, then I₄ and I₆ also equal 3 A.

Using Ohm’s Law, we can now find V₄ and V₆:

V₄ = I₄R₄ = (3A)(8?) = 24 V

V₆ = I₆R₆ = (3A)(2?) = 6 V

The potential difference will drop 6 V as the electricity passes through R₆. This means it must also drop 6 V as it passes through either R₁ or R₅ from our second diagram. Therefore, V₁ = V₅ = 6 V.

I₁ can be found using Ohm’s Law:

Since R₁ is connected in parallel with R₅, therefore:

I₆ = I₁ + I₅

I₅ = I₆ – I₁= 3 A – 2 A = 1A

Therefore the current through R₂ and R₃ will also equal 1A.

Ohm’s Law can now be used to calculate the final two potential differences.

V₂ = I₂R₂ = (1A)(4?) = 4 V

V₃ = I₃R₃ = (1A)(2?) = 2 V

The solutions to combined circuits often involve several simplification steps. It is a good idea to draw out each step as you go. With practice you will find that you can often do several simplifications in one step. Before going on, go back and try this example again on your own.