 # Series Elements

## Introduction

Electronic components are said to be in series when the current that flows through one of the components must also flow entirely though the other components. Keep in mind that charge is neither created nor destroyed in a circuit, nor does charge accumulate anywhere in a circuit. If there is only a single path through which current (and thus charge) can flow, then all the components along that path are in series. Using an analogy involving water, this would be like connecting multiple pipes together, end to end. The water flows from one pipe straight into the next pipe. Figure 1. Example of a circuit where all the elements are in series.

Figure 1 shows a circuit in which two resistors, $R_1$ and $R_2$, are in series. In fact, these resistors are also in series with a 3.3V voltage source. The current that flows from the voltage source must flow through $R_1$, and that current must, in turn, flow through $R_2$. Figure 2. Example of a circuit where some of the elements are in series are some are not.

On the other hand, the circuit shown in Fig. 2 consists of some elements that are in series and some that are not. In this circuit, the current that flows through the voltage source must also flow through the resistors $R_1$ and $R_2$. However, below the resistor $R_2$ there is an “intersection” that allows the current to flow along more than one path. The current can, and will, split between going through $R_3$ and $R_4$. $R_3$ and $R_4$ are said to be in parallel. We won't explore that topic further on this page, but you can find out more about parallel components by following the link available via the box on the right.

## Series Resistors

When two resistors are in series, they can be thought of as behaving as a single equivalent resistor that has a resistance equal to the sum of the individual resistances. This fact is illustrated in Fig. 3 where, on the left, two discrete resistors are shown, $R_1$ and $R_2$. $R_1$ has a resistance of 10 kΩ (10,000 Ω) and $R_2$ has a resistance of 5 kΩ (5,000 Ω). Figure 3. Series resistors can be replaced with a single equivalent resistor whose resistance is the sum of the series resistors.

Note that in this depiction of the circuit we have seemingly split apart the voltage source and there does not appear to be a path for charge to flow from ground (at the bottom) to the point that is labeled 3.3V at the top. However, in this type of representation of a circuit where the voltage produced by a voltage source is shown rather than the voltage source itself, although it is not explicitly drawn, it is understood that there is a path for current to flow from one side of the source to the other. Thus, other than explicitly giving the resistances of the resistors, the circuit on the left in Fig. 3 is identical to the circuit shown in Fig. 1.

In the circuit shown on the right of Fig. 2, the two individual resistors have been replaced by a single equivalent resistor with a resistance equal to the sum of the resistances of the two individual resistors. In this case we have:

${R_{{\rm{eq}}}} = {R_1} + {R_2} = 15\;{\rm{k}}\Omega$

If you are interested in why we can add series resistances, here are a few of the details. When it comes to circuit components, we typically are concerned with the current through them and the voltage across them. When elements are in series, they have the same current through them. Let's call this current $I$. Assuming this current is flowing in the circuit shown on the left of Fig. 2, using Ohm's Law, we know the voltage across $R_1$ is $IR_1$. Let's write $V_{R1}$ for this voltage. Similarly, let's write $V_{R2}$ for the voltage across $R_2$ which is given by $IR_2$. The voltage across the two resistors is simply the sum of the voltages across each individual resistor. Let's call this combined voltage $V_{\mathrm{eq}}$. Putting things together, we can express the combined voltage as:

\begin{eqnarray} V_{\mathrm{eq}} &=& V_{R1} + V_{R2} \\ &=& IR_1 + IR_2 \\ &=& I(R_1 + R_2) \\ &=& I R_{\mathrm{eq}} \end{eqnarray}

In the last two expressions we have effectively defined the equivalent resistance $R_{\mathrm{eq}}$ as being the sum $R_1 + R_2$. In general, for any number of resistors that are in series, the individual resistors can be replaced with a single equivalent resistance whose resistance is the sum of the individual resistances. This combining of resistors to form an equivalent resistance is independent of the type of source that is present. All that matters is that the resistors are in series.

## Important Points:

• The definition of “in series” pertains to any type of circuit component, not just resistors. If the current that passes through one component must also pass through a second, the two components are in series.
• Any number of components can be in series.
• In the schematic representation of series components, the wire that connects the components may have “turns” or “bends.” However, the wire cannot have “branches” or “intersections” with other wires that provide more than one path for the current to flow.
• Series resistors can be replaced with a single equivalent resistor whose resistance is the sum of the resistances of the series resistors.
• A circuit may consists of both series and non-series components.

1. What is the equivalent resistance that could be used to replace the two series resistors shown below? 2. What is the equivalent resistance that could be used to replace the series resistors shown below? 3. Which resistors in the following figure are in series, if any? 4. Which resistors in the following figure are in series, if any? 5. Here is a rather tricky question! Which resistors, if any, in the following figure are in series? 1. The total equivalent resistance is $1100$ Ω.
2. The total equivalent resistance is $10$ kΩ.
3. Resistors $R_1$ and $R_2$ are in series because the same current must pass through them. The current can split between $R_3$ and $R_4$ and hence those resistors are not in series with each other nor with $R_1$ and $R_2$.
5. $R_1$ and $R_4$ are considered to be in series because any current that flows through $R_1$ must ultimately flow through $R_4$. The individual resistors $R_2$ and $R_3$ are not in series with any other resistors, but ultimately the current that flows into the combination of $R_2$ and $R_3$ via $R_1$ must flow out through $R_2$.