Voltage Divider Rule for Series Resistors

Hello! Welcome to your seventh lesson in our study of DC Circuit Fundamentals.

In the last lesson, we focused on simplifying resistor networks by calculating their series and parallel equivalent resistances. This skill is a prerequisite for much of what we will do next, as it allows us to methodically reduce complex circuits into simpler ones.

Today, we will build directly on the concept of series resistors to learn an essential circuit analysis shortcut: the Voltage Divider Rule. Your goal is to learn how to apply this rule to quickly find the voltage at any point in a series resistor network. This is one of the most frequently used tools in an electronics designer's toolkit, as it's fundamental to setting up specific voltage levels required by different parts of a circuit—a core task in building any analog or mixed-signal hardware.

1. The Concept of Voltage Division

Imagine a voltage source connected to a chain of resistors in series. As we learned from Kirchhoff's Voltage Law (KVL), the total voltage supplied by the source is "dropped" across the resistors. The sum of all the individual voltage drops must equal the source voltage.

But how is that total voltage distributed or divided among the resistors? The division is proportional to their resistance. A resistor with a larger resistance value will take a larger share of the total voltage, and a smaller resistor will take a smaller share.

Let's see how we would calculate the voltage at a point between two resistors using the methods we already know.

L4: Series and Parallel Resistors

The following article demonstrates how to find the voltage at an intermediate node in a series circuit by first calculating the total current and then using Ohm's Law. This is the 'long way' of doing it, and it provides a solid foundation for understanding the shortcut we're about to learn.

In this resource from the Makeability Lab, please read the subsection titled 'Example 1: Solve for voltage at VB'. Follow the three steps shown: (1) Solve for current, (2) Calculate voltage drop, and (3) Calculate the voltage at node B. This process reinforces concepts from our previous lessons.

As you saw in the example, calculating the intermediate voltage $V_{B}$ involved multiple steps. The Voltage Divider Rule is a formula that combines all those steps into one, allowing you to find that voltage directly.

2. The Voltage Divider Formula

For a simple circuit with two series resistors, $R_{1}$ and $R_{2}$ , connected to an input voltage $V_{in}$ , the output voltage $V_{o u t}$ taken across $R_{2}$ is given by the formula:

$V_{o u t} = V_{in} \cdot \frac{R _{2}}{R _{1} + R _{2}}$

Let's break this down:

$V_{in}$ is the total voltage applied across the entire series combination.
$R_{1} + R_{2}$ is the total equivalent resistance, which you learned to calculate in the last lesson.
$\frac{R _{2}}{R _{1} + R _{2}}$ is the ratio of the resistance you're measuring across to the total resistance. This fraction represents the proportion of the total voltage that will be dropped across $R_{2}$ .

This elegant formula is derived directly from Ohm's Law and KVL. Understanding this derivation is key to owning the concept, not just memorizing a formula.

L4: Series and Parallel Resistors

The same article we just looked at also provides a clear, step-by-step derivation of the voltage divider equation. Working through this will solidify your understanding of how it's connected to the fundamental laws.

Now, read the subsection titled 'Deriving the voltage divider equation'. Notice how it starts with basic Ohm's Law expressions and uses algebraic substitution to arrive at the final formula.

Generalizing the Rule

The rule can be generalized for a series circuit with any number of resistors. The voltage $V_{x}$ across any specific resistor $R_{x}$ in the chain is:

$V_{x} = V_{T o t a l} \cdot \frac{R _{x}}{R _{T o t a l}}$

Where $R_{T o t a l}$ is the sum of all resistances in the series chain.

Voltage Divider Rule: Example Circuit Analysis — This image provides a clear example of applying the voltage divider rule in a circuit with three series resistors. It shows how to calculate the voltage drop across each individual resistor (V1, V2, V3) and also across a combination of resistors (V12) by treating them as a single equivalent resistance in the numerator.

Test your understanding!

Using the circuit from the image above ( $V_{S} = 12 V, R_{1} = 3Ω, R_{2} = 12Ω, R_{3} = 9Ω$ ), what is the voltage at the node between $R_{1}$ and $R_{2}$ ? This voltage is equivalent to the voltage drop across $R_{2}$ and $R_{3}$ combined. (Hint: you can treat $R_{2} + R_{3}$ as the "bottom resistor" of the divider).

Show answer

First, find the total resistance: $R_{T o t a l} = 3Ω + 12Ω + 9Ω = 24Ω$ .
The resistance of the "bottom" part of the divider is $R_{23} = R_{2} + R_{3} = 12Ω + 9Ω = 21Ω$ .
Now, apply the voltage divider formula:
$V_{o u t} = V_{in} \cdot \frac{R _{23}}{R _{T o t a l}} = 12 V \cdot \frac{21Ω}{24Ω} = 12 V \cdot 0.875 = 10.5 V$
This is the same result shown for $V_{12}$ in the image, but approached from the opposite side ( $V_{12}$ is the voltage across R1 and R2, while we calculated voltage across R2 and R3). Let's re-read the question. The voltage at the node between R1 and R2 means we are measuring from that node to ground. So we are indeed finding the voltage across R2 and R3 combined. The image's calculation for V12 is for the voltage across R1 and R2, which isn't the same. Let's calculate the voltage drop across just R1: $V_{1} = 12 V \cdot \frac{3Ω}{24Ω} = 1.5 V$ . So the voltage at the node after R1 is $12 V - 1.5 V = 10.5 V$ . Our calculation was correct.

3. A Practical Consideration: The Loading Effect

So far, we've analyzed voltage dividers in isolation. In a real application, you build a voltage divider to provide a specific voltage to another part of the circuit, which we call the load. This load could be a sensor, an LED, or the input to another circuit block like an amplifier.

Crucially, any load you connect has its own internal resistance ( $R_{L o a d}$ ). When you connect this load across $R_{2}$ , it is now electrically in parallel with $R_{2}$ .

What does this do?

It creates a new equivalent resistance for the bottom part of the divider: $R_{2}^{'} = R_{2} ∣∣ R_{L o a d} = \frac{R _{2} \cdot R _{L o a d}}{R _{2} + R _{L o a d}}$ .
From our last lesson, we know that the result of a parallel combination is always less than the smallest individual resistance. So, $R_{2}^{'}$ will be less than $R_{2}$ .
Because the bottom resistance of our divider has decreased, the output voltage $V_{o u t}$ will also decrease.

This is called the loading effect, and it's a critical concept in practical electronics. A voltage divider is "stiff" if the output voltage doesn't change much when a load is connected, and "weak" or "soft" if it changes significantly.

Voltage Divider Circuit Explained!

This video provides an excellent, practical explanation of the loading effect. It demonstrates with clear calculations how connecting a load resistor changes the output voltage and establishes an important rule of thumb for designing reliable voltage dividers.

Watch the segment from 07:43 to 16:39. Pay close attention to how the instructor first calculates the 'unloaded' output voltage and then recalculates it after connecting a load. Notice the dramatic difference in output voltage when the load resistance is small compared to when it is large.

The key takeaway from the video is a practical design rule: For a voltage divider to provide a relatively stable output voltage, the load's resistance should be much larger (at least 10x, but ideally 100x or more) than the resistance of the divider's bottom leg ( $R_{2}$ ). This ensures the load doesn't draw significant current and "drags down" the voltage you so carefully set.

Conclusion

You have now added another powerful shortcut to your circuit analysis toolkit. The voltage divider is more than just a formula; it's a fundamental circuit pattern used everywhere to create reference voltages and interface between different parts of a system.

Key Takeaways:

The Voltage Divider Rule is a shortcut to find the voltage across a resistor in a series chain: $V_{x} = V_{T o t a l} \cdot \frac{R _{x}}{R _{T o t a l}}$ .
The voltage across a series resistor is directly proportional to its resistance relative to the total resistance.
The loading effect is a critical practical issue. Connecting a load to a voltage divider puts the load's resistance in parallel with the divider's bottom resistor.
This parallel combination lowers the equivalent bottom resistance, causing the output voltage to drop.
To minimize the loading effect, ensure your load's resistance is significantly higher than the divider's resistance.

In our next and final lesson for this module, we will explore the dual of this concept: the Current Divider Rule, which is a shortcut for finding how current splits between parallel branches in a circuit.

Recommended resources

1. The Concept of Voltage Division

2. The Voltage Divider Formula

Generalizing the Rule

3. A Practical Consideration: The Loading Effect

Conclusion