# Series Circuits

**Series Circuits** are circuits which satisfy the following rules:

- The
**same current flows through each element**, the current doesn't branch off - They have
**one common terminal**which is not connected to another current carrying component

The key concept is that the **current is the same through series components.**

A series circuit with several resistors is shown below, we say that the resistors are **connected in series.**

**com1** and **com2** are common terminals between **R1, R2** and **R2 **and **R3** hence all three resistors are in **series** and **the same current flows through them all.**

To identify series connections in more complex circuits, we just need to ask ourselves if the **same current would flow through the 2 or more components** and if the answer is yes, then they are connected in series.

## Resistors in Series

When resistors are connected in series, the total resistance is **equal to the sum of the resistances.**

Each resistor **opposes** current flow hence the net effect is simply the total resistance added together through the current path.

Adding more resistors in series **increases** the total resistance of the circuit which **decreases **the current by Ohm's Law.

The above can also be proven mathematically from Ohm's Law:

Hence the above circuit is equivalent to the following circuit below where a **single equivalent resistor** replaces them all.

In short: **Resistance adds in series**.

## Voltage in Series

Series connections of **voltage sources** can be represented by a single source with a **voltage equal to the sum of the individual sources.**

Similarly, **voltage drops** add up when components are added in series.

Regarding the **voltage drops adding in series**, this is due to the total voltage supplied by the voltage sources being *dropped *across the circuit and so if we have say two voltage sources and two resistors it follows that the total voltage supplied by those sources will be dropped across the two resistors.

Shown below is a circuit with two voltage sources and two resistors connected in series:

To intuitively understand why **voltage sources add in series**, which is incredibly important in really understanding voltage, it helps to imagine performing **voltage readings **at various points in the circuit.

In the circuit above we have defined the **negative terminal** of **voltage source Vs1** to be **0V **and so it follows that on the **positive terminal** the voltage read would be **1V **as a coulomb of charge here would have 1V of energy higher than the other terminal.

Now, if we measure up to the **negative terminal** of the **other voltage source Vs2**, we would **still read 1V** as we treat a **wire **as having a **resistance of 0Ω** (even in reality it is so small you will read very close to the same result even with a very high precision instrument).

Looking at the **positive side **though, what would we read? Remember that fundamentally a voltage source provides its rated voltage across its terminals and that voltage is **always relative to another point**, so if we take the negative terminal of V1 to be our reference of 0V then the reading would be **2V**. The reason is that Vs2 provides a voltage *aka potential difference* of 1V across its terminals and so we must be **1V above **the the negative terminal which was 1V.

The above circuit is also equivalent to the circuit below after summing the series voltage sources and resistors together:

In short: **Voltage adds in series**.

### Examples:

**1. **What is the total resistance between terminals a and b in the circuit below?

**same current**flows through all resistors as R1 and R2 have one common terminal and R2 and R3 have one common terminal, the current cannot branch off.This means all three resistors are

**connected in series**, hence the total resistance is equal to the sum of the 3:

**2. a)** What is the current I in the circuit below:

**equivalent simple series circuit**with a

**single resistor**and

**voltage source**. We can find the equivalent total series resistance by

**adding**together the two series resistors,

**R1**and

**R2**: Next we can find the equivalent total voltage provided by the voltage sources by summing

**V1**and

**V2**: Now to calculate the current

**I**we can use Ohm's law using the equivalent resistance and voltage of the circuit we have just calculated:

**b)** What is the voltage drop across resistor R1?

**c)** What is the voltage drop across resistor R2?

Todo: Add example of designing circuit with certain voltage and resistance using bag of parts.

**3.** A junior engineering student has a bag of parts containing 1k, 2k, 5k and 4.7k resistors as well as 1.8V, 2.2V and 2.8V batteries. The lab demonstrator asks the student to create a circuit which supplies 5V to an 8k load, how would this circuit look?

**series resistors**and

**series voltage sources**to help us construct this circuit. Recall that resistance and voltage sources simply

**add in series**and hence the question becomes which

**combination**of resistors and batteries will achieve the

**target values**? How can we make 8k from 1k, 2k, 5k and 4.7k? Well 1k+2k+5k=8k computes so let's use them! How can we make 5V from 1.8V, 2.2V and 2.8V? 2.2V+2.8V=5V so that will also work. The circuit then becomes 1k,2k,5k all connected in series to a 2.2V and 2.8V battery as shown below:

### Questions:

** 1. a) **Determine the series resistance in the circuit below:

#### Answer

**b) **What is the current flowing through the circuit?

#### Answer

**c) **What is the voltage across R1?

#### Answer

**d)** What is the voltage across R2?

#### Answer

**2. a)** Determine the resistance in the circuit below:

#### Answer

**b) **What is the current flowing through the circuit?

#### Answer

**c) **What is the voltage across R1?

#### Answer

**d) **What is the voltage across R2?

#### Answer

e) What is the voltage across R3?

#### Answer

## Lab:

Let's build further intuition by constructing series circuits using LTSpice.

To begin with let's create a simple circuit with a voltage source and multiple resistors. We can start by adding a voltage source as shown below:

From here let's connect up a few resistors and add a ground connection:

Now we can run a **transient simulation** to measure all of the **voltages **and the **series current** with settings as below:

We can then left click just above the voltage source to see 5V shown as we expect given we have a 5V ideal voltage source:

Let's then probe after R1 and R2 to see all of the voltages in the same chart by performing two single left clicks at each of the nodes.

Finally let's **measure the current** by left clicking on any of the resistors or voltage source:

From the bottom left we can see that the current flowing through the circuit is 833.333uA.

We can then try the same circuit with a **current source** as below:

Now if we measure the current we expect to see 1mA flowing:

How about the voltage produced by the current source for this resistance? We can measure that by left clicking at the node connecting the current source and R1:

Turns out to be 6V. Which we expect as 1mA*6k=6V.

Finally, let's try adding multiple voltage sources in a circuit such as below:

We can then run the simulation again and confirm that we measure 8V (3V+5V) at the node connecting R1 and Vs2:

We can then measure the series current by clicking on either voltage source or resistor:

4mA which makes sense as 8V/4k = 4mA.

I encourage you to play around with other circuits to get a feel for how this all works!