Voltage Divider Calculations Involving Three or More Resistors

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What is a voltage divider?

The voltage divider is a mathematical shortcut used to determine the voltage across an individual resistor in a series-connected circuit. From Ohm’s Law, we know that voltage is directly proportional to resistance. It follows that the ratio of the voltage across a resistor, relative to the supply voltage, is equal to the ratio of resistor relative to the total series resistance. We can express this as the equation:

\frac{R_X}{R_{total}} = \frac{V_{R_X}}{V_{supply}}

where X identifies the resistor of interest.

For a two-resistor circuit, we can rearrange the equation into the familiar form solving for R_2:

\frac{R_2}{R_1 + R_2} = \frac{V_{R_2}}{V_{supply}}

Solving for V_{R_2}:

V_{R_2} = V_{supply} \left( \frac{R_2}{R_1 + R_2} \right)

Figure 1: A voltage divider featuring three resistors. The meter is configured to measure the voltage across R2. In this example, the blue V_R2 icon represents the meter’s red probe and REF1 icon represents the black probe.
Figure 1: A voltage divider featuring three resistors. The meter is configured to measure the voltage across R2. In this example, the blue V_R2 icon represents the meter’s red probe and REF1 icon represents the black probe.975×731 57.6 KB

Figure 1: A voltage divider featuring three resistors. The meter is configured to measure the voltage across R2. In this example, the blue V_R2 icon represents the meter’s red probe and REF1 icon represents the black probe.

Focus on multiple resistor circuits

Many textbooks introduce the voltage divider using the two resistor formula as derived in this article. This is certainly a good place to start, but the utility of the voltage divider is enhanced when generalized and applied to circuits with three or more series-connected resistors. In its most useful form:

V_{R_X} = V_{supply} \left( \frac{R_X}{R_{total}} \right)

Tech Tip: Our ability to absorb new concepts is limited. This limitation is reflected in the textbook two-resistor voltage divider equation formatted to solve for the voltage across the lower resistance. This is done to sidestep a conversation about voltage references. In practical terms the voltage divider simplifies analysis because the black multimeter probe is always connected to ground.

As we gain additional experience, it becomes second nature to think about voltage as a relative measurement where we move the probes to measure the voltage across an element.

Ground reference vs floating reference

Resistor voltage drop is measured in a closed circuit using two probes where the probes are connected to two different nodes within the circuit. This is a fancy way of saying you can’t measure battery voltage with a single probe.

The traditional textbook two-resistor voltage divider is careful to ground reference the measurement. In the lab, this implies that the black multimeter probe is connected to ground. This allows us to explore a ground referenced input and output relationship as shown in this DigiKey Voltage Divider Calculator.

There is nothing that prevents us from calculating the voltage across R_1 using:

V_{R_1} = V_{supply} \left( \frac{R_1}{R_1 + R_2} \right)

However, we can no longer use the terms input and output as the calculated voltage across R1 is no longer the ground reference “output” of the circuit. Instead, we must recognize that we are measuring the “voltage drop” across R1.

Tech Tip: **For the simple two resistor voltage divider, the voltage across R2 is often identified as the “output” because it is ground referenced. For other resistor(s), we must use appropriate language describing the “voltage drop” across the resistor of interest.
This floating reference is shown in Figure 1. Recognize that the meter’s probes have been moved to measure the voltage directly across R2. The blue V_R2 icon represents the meter’s red probe and REF1 icon represents the black probe.

Using the generalized voltage divider equation, we can calculate the voltage for each resistor.

V_{R_1} = 10 \left( \frac{1500}{1500 + 1000 + 500} \right) = 5 \ VDC

V_{R_2} = 10 \left( \frac{1000}{1500 + 1000 + 500} \right) = 3.33 \ VDC

V_{R_3} = 10 \left(\frac{500}{1500 + 1000 + 500} \right) = 1.66\ VDC

In each case, the resistor of interest’s voltage is calculated as the ratio of the select resistor divided by the total series resistance. Note that Kirchhoff’s voltage law is supported in these calculations, as the summation of the individual resistor voltage drops is equal to the supply voltage.

Tech Tip: From a practical lab perspective, both of the voltmeter’s probes must be moved to measure the voltage drop associated with the resistor of interest.

Parting thoughts

This article presents a generalized equation for the voltage divider. Instead of a two-resistor solution, it is applicable to a series-connected circuit with many resistors. We must be very careful with terminology as the textbook two-resistor circuit is often accompanied with a ground referenced assumption. Instead of referring to the “output” of a voltage divider, we should think in terms of “voltage drop across the resistor of interest.” This is a subtle change in thinking that can have a profound influence as you continue your study of electronics.

TL;DR: Pay attention to your assumptions about circuit grounds.

A series of questions and critical thinking questions have been added to the end of this note. Please try to answer them. Give a thumbs up if you learned something. Finally, leave your questions and comments in the space below.

Best wishes,

APDahlen

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About this author

Aaron Dahlen, LCDR USCG (Ret.), serves as an application engineer at DigiKey. He has a unique electronics and automation foundation built over a 27-year military career as a technician and engineer which was further enhanced by 12 years of teaching (partially interwoven with military experience). With an MSEE degree from Minnesota State University, Mankato, Dahlen has taught in an ABET-accredited EE program, served as the program coordinator for an EET program, and taught component-level repair to military electronics technicians. Dahlen has returned to his Northern Minnesota home and thoroughly enjoys researching and writing educational articles about electronics and automation.

Highlighted Experience

Dahlen is an active contributor to the DigiKey TechForum. At the time of this writing, he has created over 160 unique posts and provided an additional 500 forum posts. Dahlen shares his insights on a wide variety of topics including microcontrollers, FPGA programming in Verilog, and a large body of work on industrial controls.

Questions

The following questions will help reinforce the content of the article.

  1. What is the equation for a multi-resistor voltage divider?

  2. Why is the generalized voltage divider shortcut formula more useful than two-resistor formula?

  3. With regards to Figure 1, solve for the R2 voltage if the supply is increased to 20 VDC and R3 is increased to 2 kΩ.

  4. Why do we stress the difference between “output” and “voltage drop”?

  5. Construct and then solve five unique voltage divider problems featuring multiple resistors. Strive for variety by solving for supply voltage as well as voltage across resistors in different positions.

  6. How is the voltage divider equation related to Ohm’s Law?

  7. True/False: When we talk about the “output” of a voltage divider, we are assuming a ground referenced system.

  8. ____________________ law is supported by the observation that the sum of the resistor voltage drops equals the supply voltage.

  9. Why do we include the “closed circuit” stipulation for a measuring the voltage drop in a resistive circuit?

  10. Describe multimeter loading? Hint: See the previous article.

Critical thinking questions

These critical thinking questions expand the article’s content allowing you to develop a big picture understanding the material and its relationship to adjacent topics. They are often open ended, require research, and are best answered in essay form.

  1. With regards to Figure 1, a 500 Ω resistor is added in parallel with R1. What is the first step required to solve the R1 voltage drop?

  2. A student is using an oscilloscope to measure the R_2 voltage in Figure 1. The probe’s ground clip is connected to the junction of R2 and R3. Why does the R2 voltage measure 4 VDC? Hint: What devices are ground referenced?

  3. What complications arise when we use the terms “input” and “output” to describe a voltage divider?

  4. An instrumentation amplifier may be used to measure floating voltages. With regards to Figure 1, add a three op-amp-based instrumentation amplifier to measure the R2 voltage.