Resistors are one of the most fundamental passive components used in electronics. They restrict current flow to limit power, divide voltages, terminate transmission lines, and much more. Resistors are often connected in series configurations on circuit boards to combine their properties. But what is the exact rule governing resistors in series?

The basic principle for series resistors is that their resistance values simply sum. This adds up to a total resistance that relates voltages and currents across the overall combination. However, there are important caveats and secondary effects that derive from this rule.

This article provides a comprehensive look at resistors in series including:

- The fundamental series resistance rule
- Calculations using the rule
- Combining different resistor values
- Secondary effects of series resistors
- Troubleshooting faults
- Example scenarios
- Frequently asked questions

By building a deep understanding of series resistor circuits, electrical engineers can confidently analyze and design complex resistive networks.

## The Fundamental Rule

The foundational rule governing resistors in series is:

**The total resistance of resistors in series is equal to the sum of their individual resistance values.**

This means that the overall resistance of a chain of series resistors is simply the algebraic total of each resistor:

Rtotal = R1 + R2 + R3 + … + Rn

Where R1, R2, etc. are the resistance values of each individual resistor and Rtotal is the equivalent total resistance. This rule applies to any number of resistors connected end-to-end in series.

This simple but powerful concept enables analysis of complex resistor networks by reducing multiple resistors down to a single equivalent resistance. We can then relate voltage and current through the combination using Ohm’s Law.

## Calculating Series Resistance

Let’s look at some example resistor circuits to see the series rule in action.

Consider two resistors connected in series:

`R1 = 560 Ω R2 = 330 Ω`

[Diagram of two resistors, R1 and R2, connected end-to-end in series.]

Using the series rule, the total resistance is:

Copy code

`Rtotal = R1 + R2 = 560 Ω + 330 Ω = 890 Ω`

For three resistors in series:

`R1 = 1 kΩ R2 = 10 kΩ R3 = 5.1 kΩ`

[Diagram of three resistors, R1, R2, and R3 connected end-to-end in series.]

The total resistance is:

`Rtotal = R1 + R2 + R3 = 1 kΩ + 10 kΩ + 5.1 kΩ = 16.1 kΩ`

The series rule applies equally for any number of resistors chained together in this manner.

## Combining Different Resistor Values

One key advantage of series resistor circuits is the ability to combine different individual resistor values to achieve a desired total resistance.

For example, given the resistor values:

`R1 = 470 Ω R2 = 1.5 kΩ R3 = 2.2 kΩ`

These can be connected in series to produce a total resistance of:

`Rtotal = R1 + R2 + R3 = 470 Ω + 1.5 kΩ + 2.2 kΩ = 4.17 kΩ`

The freedom to mix and match values provides flexibility in arriving at the needed total resistance through different combinations.

Both fixed standard value resistors and variable types like potentiometers can be utilized in series chains.

## Secondary Effects of Series Resistors

While the series rule provides total resistance, some secondary effects should also be noted:

**Voltage Divider**

The voltage drops proportionally across each resistor, dividing the source voltage.

**Power Dissipation**

Total power dissipated is equal to the sum of power in each resistor.

**Physical Size**

End-to-end resistors take up more space on a circuit board.

**Parasitic Inductance**

Stray inductance increases slightly compared to a single resistor.

**Failure Risk**

More resistors equates to higher chance of failure.

So in addition to simple resistance summation, parasitics, power handling, board space, reliability, and other factors should be assessed when utilizing series resistor combinations.

## Troubleshooting Faulty Series Resistors

If resistors are connected incorrectly in a series circuit, the total resistance will deviate from the expected value. Some common faults are:

**Shorted Resistor**

A shorted resistor will drastically lower the total resistance since it provides a zero ohm path.

**Open Resistor**

An open resistor increases total resistance closer to infinity ohms as it blocks current flow.

**Incorrect Value**

A wrongly specified resistor value will make the calculated total resistance incorrect.

**Reversed Polarity**

If a resistor is backwards with reversed polarity, the total resistance stays the same but voltage drop is affected.

**Damaged Resistor**

A damaged resistor may increase in resistance affecting the total resistance.

Using basic volt-ohm meter measurements combined with visual inspection, these kinds of issues can be identified and debugged in series resistor circuits.

## Example Series Resistor Scenarios

Let’s look at some practical examples using series resistors:

**Voltage Divider**

Three resistors are used to divide down a 5V supply in the ratios 25%, 25% and 50% to generate reference voltages.

**EMI Filter**

Resistors are inserted in series with a power line to help attenuate electromagnetic interference coupled onto the cable by forming an R/C filter with shunt capacitors.

**Pulse Shaper**

A resistor and capacitor in series temporarily holds a voltage to convert a fast digital pulse into a slower analog waveform.

**Impedance Matching**

Carefully sized series resistors match a transmission line impedance to a mismatched source impedance over a limited frequency range.

**Attenuator**

Fixed and variable resistors in series proportionally attenuate an analog signal voltage.

**Pull-up Resistor**

A resistor pulls a digital signal high when no device is actively driving the line low.

These examples demonstrate the flexibility of series resistors for a wide variety of circuit functions.

### Frequently Asked Questions

Some common questions that arise about the rules for resistors in series:

**Does series resistance increase at high frequencies?**

At high frequencies, parasitic inductance causes impedance to increase above the static DC resistance.

**What happens if different resistor power ratings are used?**

Each resistor should be sized to withstand the voltage drop it will experience without exceeding its power rating.

**Can any value resistors be combined in series?**

Yes, any values can be used together. Standard values may require smaller resistors to be paralleled to achieve intermediate resistances.

**What is the disadvantage of series vs. parallel resistors?**

Series resistors take up more space and have higher failure rates. But series suits high voltage applications better.

**How does temperature affect series resistors?**

Temperature coefficients cause each resistor value to drift slightly with temperature. Self-heating also becomes more significant.

In summary, the fundamental series resistance rule provides a starting point for analysis while secondary effects must also be incorporated in practical circuit designs.

## Conclusion

The basic rule governing resistors in series dictates that the total resistance is the simple sum of the individual resistor values. This allows series resistor combinations to be reduced down to a single equivalent resistance for calculations. Engineers leveraging this rule gain flexibility in achieving needed resistances from available standard values.

However, practical aspects like power handling, voltage division, parasitics, and reliability should also be considered when utilizing series resistor configurations. Attention to these secondary effects ensures the circuit performs as intended. But fundamentally, the series resistance rule provides the foundation for efficient analysis and design of these ubiquitous circuits.