Resistors in Series and Parallel: Formula Derivation and Examples

© Eugene Brennan

Formulas for Resistors in Series and Parallel

Resistors are ubiquitous components in electronic circuitry both in industrial and domestic consumer products. Often, in circuit analysis, we need to work out the values when two or more resistors are combined. In this tutorial, we'll work out the formulas for resistors connected in series and parallel.

Some Revision: A Circuit With One Resistor

In an earlier tutorial, 'How to Understand Electricity: Volts, Amps and Watts Explained on Appliances', you learned that when a single resistor with resistance R ohms was connected in a circuit with a voltage source V, the current I through the circuit was given by Ohm's Law:

Ohms Law

I = V/R

Example: A 240 V mains supply is connected to a heater with a resistance of 60 ohms. What current will flow through the heater?

Current = V/R = 240/60 = 4 amps

Schematic of a simple circuit. A voltage source V drives a current I through the resistance R. © Eugene Brennan

Derivation of Formula for Resistance When Resistors Are in Series

Now, let's add a second resistor in series. Series means that the resistors are like links in a chain, one after another. We call the resistors R₁ and R₂.
Because the resistors are linked together, the voltage source V causes the same current I to flow through both of them.

Two resistors connected in series. The same current I flows through both resistors. © Eugene Brennan

There will be a voltage drop or potential difference across both resistors.

Let the voltage drop measured across R₁ be V₁ and let the voltage measured across R₂ be V_2, as shown in the diagram below.

Voltage drop across resistors connected in series. © Eugene Brennan

From Ohm's Law, we know that for a circuit with a resistance R and voltage V:

I = V/R

Therefore, rearranging the equation by multiplying both sides by R

IR = V

or switching around

V = IR

So for resistor R₁

V₁ = IR₁

and for resistor R₂

V₂ = IR₂

Kirchoff's Voltage Law

From Kirchoff's Voltage Law, we know that the sum of voltages around a closed loop in a circuit adds up to zero. We decide on a convention, so voltage sources with arrows pointing clockwise from negative to positive are considered positive and voltage drops across resistors are negative. So, in our example:

V - V₁ - V₂ = 0

Rearranging

V = V₁ + V_{2 ................} (i.e., the voltage V equals the sum of the drops across the resistors)

Substitute for V₁ and V₂ calculated earlier

V = IR₁ + IR₂ = I(R₁ + R₂)

Divide both sides by I

V/I = R₁ + R₂

But from Ohm's Law, we know I = V/R, so rearranging:

V/I = R = total resistance of the circuit. Let's call it R_total

Therefore

V/I = R_total = R₁ + R₂

In general, if we have n resistors:

R_total = R₁ + R₂ + ...... R_n

So, to get the total resistance of resistors connected in series, we just add all the values.

Formula for resistors connected in series

Example 1

Five 10k resistors and two 100k resistors are connected in series. What is the combined resistance?

Answer

Resistor values are often specified in kiloohm (abbreviated to "k") or megaohms (abbreviated to "M")

1 kiloohm or 1k = 1000 ohms

1 megaohm or 1M = 1000,000 ohms

So total resistance = sum of the resistances

= 5 x (10k) + 2 x (100k)

= 50k + 200k

= 250k or 250,000 ohms

Example 2

Three 47 ohm, five 1.2k, four 100k and two 3.3M resistors are connected in series. What is the total resistance?

Answer

We often replace the decimal point in resistor values with the multiplier to avoid misreading if, e.g., the "dot" gets erased from the value printed on a component or in documents. So 1.2k becomes 1k2.

So total resistance = sum of the resistances

= 3 x 47 + 5 x 1k2 + 4 x 100k + 2 x 3M3

= 3 x 47 + 5 x 1200 + 4 x 100,000 + 2 x 3,300,000

= 141 + 6000 + 400,000 + 6,600,000

= 7,006,141 ohms

Derivation of Formula for Resistance of Two Resistors in Parallel

Next, we'll derive the expression for resistors in parallel. Parallel means all the ends of the resistors are connected together at one point, and all the other ends of the resistors are connected at another point.

When resistors are connected in parallel, the current from the source is split between all the resistors instead of being the same as was the case with series connected resistors. However, the same voltage is now common to all resistors.

Two resistors connected in parallel. © Eugene Brennan

Let the current through resistor R₁ be I₁ and the current through R₂ be I₂.

The voltage drop across both R₁ and R₂ is equal to the supply voltage V.

Therefore, from Ohm's Law

I₁ = V/R₁

and

I₂ = V/R₂

Kirchoff's Current Law

From Kirchoff's current we know the current entering a node (connection point) is equal to the current leaving the node.

Therefore,

I = I₁ + I₂

Substituting the values derived for I₁ and I₂ gives us

I = V/R₁ + V/R₂

= V(1/R₁ + 1/R₂)

The lowest common denominator (LCD) of 1/R₁ and 1/R₂ is R₁R₂ so we can replace the expression (1/R₁ + 1/R₂) by

R₂/R₁R₂+ R₁/R₁R₂

Switching around the two fractions

= R₁/R₁R₂+ R₂/R₁R₂

and since the denominator of both fractions is the same

= (R₁ + R₂)/R₁R₂

Therefore,

I = V(1/R₁ + 1/R₂) = V(R₁ + R₂)/R₁R₂

Rearranging by dividing by dividing both sides of the equation by V and taking the reciprocal of both sides gives us:

V/I = R₁R₂/(R₁ + R₂)

But from Ohm's Law, we know V/I = total resistance of the circuit. Let's call it R_total.

Therefore,

V/I = R_total = R₁R₂ / (R₁ + R₂)

So, for two resistors in parallel, the combined resistance is the product of the individual resistances divided by the sum of the resistances.

Formula for two resistors connected in parallel.

Example

A 100 ohm resistor and a 220 ohm resistor are connected in parallel. What is the combined resistance?

Answer

For two resistors in parallel, we just divide the product of the resistances by their sum.

So total resistance = 100 x 220 / (100 + 220) = 22000/320 = 8.75 ohms

Derivation of Formula for Resistance of Multiple Resistors in Parallel

If we have more than two resistors connected in parallel, the current I equals the sum of all the currents flowing through the resistors.

Multiple resistors in parallel. © Eugene Brennan

So for n resistors

I = I₁+ I₂+ I₃. ........... + I_n

= V/R₁+ V/R₂+ V/R₃+ ............. V/R_n

= V(1/R₁+ 1/R₂ + V/R₃ ........... 1/R_n)

Rearranging

I/V = (1/R₁ + 1/R₂ + V/R₃ ........... 1/R_n)

If V/I = R_total then

I/V = 1/R_total = (1/R₁ + 1/R₂ + V/R₃ ........... 1/R_n)

So, our final formula is

1/R_total = (1/R₁ + 1/R₂ + V/R₃ ........... 1/R_n)

We could invert the right side of the formula to give an expression for R_total; however, it's easier to remember the equation for the reciprocal of resistance.
So, to calculate the total resistance, we calculate the reciprocals of all the resistances first and sum them together, giving us the reciprocal of the total resistance. Then, we take the reciprocal of this result, giving us R_total.

Formula for multiple resistors in parallel

Example

Calculate the combined resistance of three 100-ohm and four 200-ohm resistors in parallel.

Answer

Let's call the combined resistance R.

So

1/R = 1/100 + 1/100 + 1/100 + 1/200 + 1/200 + 1/200 + 1/200

We can use a calculator to work out the result for 1/R by summing all the fractions and then inverting to find R, but let's try and work it out "by hand".

So

1/R = 1/100 + 1/100 + 1/100 + 1/200 + 1/200 + 1/200 + 1/200

= 3/100 + 4/200

To simplify a sum or difference of fractions, we can use a lowest common denominator (LCD). The LCD of 100 and 200 in our example is 200

Therefore, multiply the top and bottom of the first fraction by 2 giving:

1/R = 3/100 + 4/200 = (2 x 3) / (2 x 100) + 4/200

= 6 / 200 + 4/200

= (6 + 4)/200 = 10/200

and inverting gives R = 200 / 10 = 20 ohms. No calculator needed!

Recommended Books

Introductory Circuit Analysis by Robert L Boylestad and available from Amazon covers the basics of electricity and circuit theory and also more advanced topics such as AC theory, magnetic circuits and electrostatics. It's well illustrated and suitable for high school students and also first and second-year electric or electronic engineering students. New and used versions of the hardcover 10th edition are available on Amazon. Later editions are also available.

References

Boylestad, Robert L. (1968) Introductory Circuit Analysis (6th ed. 1990) Merrill Publishing Company, London, England.

Disclaimer 

This article is accurate and true to the best of the author’s knowledge. Content is for informational or entertainment purposes only and does not substitute for personal counsel or professional advice in business, financial, legal, or technical matters.

© 2020 Eugene Brennan

The Cork-Dublin Transmission Pipeline and Gas Distribution to Kilcullen

Gas distribution network in central Kilcullen from the interactive Dial Before You Dig website. Image courtesy Gas Networks Ireland.

On 4th March last year, a press release was issued by Uisce Éireann to media and local public representatives to the effect that:

“Civil Engineers are carrying out pre-work activities on the 225mm pipeline, which involves a localised crossing over the High Transmisison (sic) Gas main under the supervision of Gas Networks.”

That line is the Cork-Dublin gas transmission pipeline, and the work was being carried out at Kennycourt, just after the right turn for Kilgowan when heading towards Dunlavin. The 225 mm pipeline was a water main which was being installed for the Dunlavin Water Supply Rationalisation Project.

The gas transmission network in Ireland. Image courtesy Gas Networks Ireland.

The Kinsale Head Gas Field was discovered in 1971 by the US Company Marathon Oil Corporation. Gas well heads were installed in 1977 and production began in 1978. Construction of an 18-inch (450 mm) diameter pipeline to Dublin by Bórd Gáis Éireann began in 1981 and was completed in 1983. The pipeline runs cross-country through the Kennycourt, Mooretown, Carnalway, Harristown area on its way to Dublin. It passes under the river near Ardenode. It operates at 70 bar, or approximately 1015 psi (pounds per square inch). For comparison, your car tyres are pumped up to about 2.2 bar. 

The Kinsale Head gas field ceased production in 2020.

Gas Networks Ireland doesn’t publish a fixed burial depth for transmission pipelines. High-pressure gas transmission pipelines in Europe are designed in accordance with standards such as EN 1594, which set out functional requirements for pipelines operating above 16 bar, including protection from external interference. Burial depth is determined by design and risk assessment and varies depending on location, with greater cover typically used at road crossings, railways, and other high-load areas.

You’ve probably seen the white poles along roads with what look like cowls on top, marking the line. I erroneously thought these were some type of safety vent, but when I reported a damaged one near Coghlanstown, which had probably been cut by a tractor during autumn hedge cutting, I learned they’re simply markers, and what looks like a cowl is in fact a disk which helps aerial survey teams, including helicopter patrols, identify the pipeline route. A Prior Information Notice (PIN) for Gas Networks Ireland’s aerial surveillance services was published in June 2024 on the Irish Government’s eTenders portal which stated that:

“Gas Networks Ireland requires the provision of an aerial surveillance patrol service of their gas pipeline system in the Republic of Ireland and Northern Ireland to ensure there are no works or digging being carried out on or near the pipeline and no excessive vegetation growth on or near the pipeline.”

Kilcullen’s gas supply comes from a hub connection on the Cork-Dublin pipeline located north of Ballymore Eustace. You can register and view an interactive map of the transmission and distribution network, with detail down to street level, on the Dial Before You Dig website here.

Immersion Heaters and Potential Fire Risks

Immersion heater terminal box. © Eugene Brennan

So I was doing some gardening for the sister today and she mentioned a smell that was coming from her immersion. This immediately rang alarm bells in my engineering head, knowing what was likely to have happened—So I investigated.

My Telemetry Software Didn't Control Voyager 1

Credit: NASA/JPL-Caltech

NASA's decision to switch off an instrument on Voyager 1 to keep the spacecraft operating reminded me of my much less glamorous work in a past life, when I wrote telemetry software to control pumps and other equipment in sewage and water treatment plants.. But they were only down the road from Dublin in Kildare, Laois and other places throughout the country. If the remote software didn't work, or crashed, often due to frustratingly annoying transient bugs, usually a watchdog circuit kicked in to reboot the SCADA system (Supervisory Control and Data Acquisition). If there was a big problem and frequent crashes or anomalies, technicians would reinstall software. Eventually, I added the facility to upload new versions of software. All this was much less sophisticated than the complex software that runs on computers nowadays. Voyager 1 is nearly a light day away meaning a radio signal takes nearly a day to reach the spacecraft. Not much hope of sending people out to do an upgrade. Perhaps they can upload new code.

 

Edit: Yes, they can upload new "patches" and code updates. Code is restricted to around 64k. Voyager 1 has six computers, three of which are backups. The data bus is 16 to 18 bits wide and the address bus is 12 bits wide.

How Can Insects Survive Falling From a Height?

Forces on a falling insect. © Eugene Brennan

One of the reasons insects don't hurt themselves when they fall is because they have a low terminal velocity and are travelling relatively slowly when they reach the ground. Terminal velocity is the maximum velocity attained by a falling body in a fluid such as air or water and subject to a force called drag, (resistance to motion). For humans, it's around 120 mph or 54 m/s lying flat. For an insect, it's typically around 2 m/s. Once the increasing drag force balances the weight (another force), there's no net force and an object stops accelerating (Newton's second law of motion). Also when an insect hits the ground, because it has such a small mass, the g-force on its body is small because it has to shed so little momentum, and that also helps prevents damage. So it's a combination of low terminal velocity, small mass and also having an outer protective layer or exoskeleton that gives them protection.

Learn How to Solder: A Complete Guide for Beginners

Image by permission Weller Tools

A Basic Skill When Building Electronic Circuits

Soldering is a fundamental skill essential to learn when building electronic circuits or fixing circuit boards.

Kippure TV Mast Visible From Kilcullen

Kippure in the Wicklow Mountains, photographed from Kilcullen. © Eugene Brennan

I took this with my 10-year-old Nikon D5300 SLR from St. Brigid's Well on Saturday evening. The second photo below is a crop of the 6000-pixel-wide image. If you have good eyesight, you can see the 127-metre-tall television transmission mast on Kippure on a clear day. It's approximately 7 m taller than the Spire in O'Connell Street in Dublin and located 17.5 miles away from Kilcullen as the crow flies. (I can just about make it out with the naked eye). At night of course it's very noticeable, and the beacons on the mast can be seen flashing in the distance. They used to be red, but they changed them to white at some stage. I've seen the flashing from as far away as Kilmeague.

I haven't figured out why this camera always takes such dull photos. I reckon smartphones must boost the colour saturation, highlights and contrast in images to make them more vivid. I have the setting for this (can't remember what it's called) set as "neutral" in the camera so that images aren't altered. I can always change the settings later during post processing.

Edit: An FB photography group I'm a member of confirms that this is the case and images are flat if neutral photo style is selected. Saving photos as JPGs compresses images and reduces filesize, so I'm going to retake the photo and save as raw to see whether it makes much of a difference as regards detail and artefacts.

The TV mast on Kippure (a crop from the image above) © Eugene Brennan

 

Why Is Most of an Iceberg Under Water? And How Much?

Original uncaptioned image Lurens, public domain image via Pixabay.com

Ninety Percent of an Iceberg Is Below the Waterline

Icebergs can be massive, but did you know that you're only seeing literally the tip of them above the water? Yes, up to 90% of an iceberg's volume can be under water. But why does that amount sink and not just float above sea level like the rest of it? In this article we explore why.

What's the Difference Between Analogue and Digital?

© Eugene Brennan

Analogue vs. Digital

We often hear about analogue and digital in the context of communications, sound recording, cameras, TV, radio, and electronic devices. But what exactly is the difference, and is digital better than analogue? Why has digital replaced analogue in audio, digital imaging, and electronic communication?

Vodafone Internet Service Down After Thunderstorm

 

The Internet was dropping intermittently since the thunderstorm the other night. A Vodafone technician replaced the modem yesterday afternoon, but still no go. Then the service disappeared completely when the technician was still working on it.

Why Is Binary Used in Electronics and Computers?

Why is binary used in electronics? Geralt via Pixabay.com

Why Do Computers Use Binary?

The binary numbering system is the basis for the storage, transfer and manipulation of data in computer systems and digital electronic devices. This system uses base 2 rather than base 10, which is what we are familiar with for counting in everyday life.

How to Convert Decimal to Binary and Binary to Decimal

Binary number and its decimal equivalent. © Eugene Brennan

Base 2, the Basis for Binary Code

The base 2, or binary numbering system is the basis for all binary code and data storage in computing systems and electronic devices. This guide shows you how to convert from binary to decimal and decimal to binary.

How to Convert Hex to Binary and Binary to Hexadecimal

© Eugene Brennan

The Hexadecimal Numbering System

The base 16, also known as hexadecimal (abbreviated to hex) numbering system is regularly used in computer coding for conveniently representing a byte or word of data. This guide shows you how to convert from hex to binary and binary to hexadecimal.

The Andromeda Galaxy: "It's Big, And Also Faaarrr Away"

The apparent size of the Andromeda Galaxy, compared to that of the Moon. AI image created by Grok.

At a distance of 2.5 million light-years, or 24 million trillion km, the Andromeda Galaxy is the most distant object visible to the naked eye. We can see it because space is mostly empty, and light from the stellar object reaches us relatively unchanged, unlike what happens when we view distant objects on the horizon on Earth.

Loading Data for the Kilcullen 38 kV Substation.

Loading data for the Kilcullen 38 kV Substation, courtesy ESB Networks.

From the ESB Networks document 38 kV & 110 kV Station Special Load Readings. 

The table shows summer and winter loads in megawatt (MW) for transformers T41, T42 and T424.

It Came From Outer Space....

Nostoc commune (Latin name), also known as "jelly fungus". © Eugene Brennan

 Well not quite. This is probably Nostoc commune (Latin name), also known as "jelly fungus", "star jelly" or "witch's butter". It's not actually a fungus, but a colony of bacteria. There's loads of it in a patch on my yard, but it sometimes grows on lawns. It's extremely slippery on a lawn. Apparently it's eaten as a salad in the Philippines when dried out. I'm a fussy eater so I won't be trying it anytime soon.

More information on jelly fungus here on Wikipedia.

Off-Road Lanes for Cycling in Kildare and Wicklow

Lane from Barretstown to Bishopshill Commons. © Eugene Brennan

I'm always on the lookout for new trails and farm lanes for cycling on. Some are marked with a cul-de-sac sign, and just end up at farmhouses. However, others actually link two roads. Usually, there’s no sign at the junction where they meet a main road, or even a road number, so it can be a mystery where they go.

Available Substation Capacity in Ireland

Information on substations. Image courtesy ESB Networks.

This interesting interactive map on the ESB Networks website shows transformer capacities. A "substation" indicated on the map could be an actual town substation like the one at Sunnyhill, with 1, 2 or more transformers and associated switchgear and circuit breakers, or a transformer in a cabinet or small building (common in the newer housing estates).

What Is the Ground Wire For?

© Eugene Brennan

What Is a Ground Wire?

The hot and neutral cores in a power cable are used to supply current to the appliances in your home. So what about the green wire? Why is it needed? This is known as the "ground" and is an additional wire that is included for the safety of you and your home.

Re-Discovering the Mill Stream Arch

One mill is marked on the OSI c.1837 first-edition six-inch map. Image courtesy the National Library of Scotland, licence CC-BY.

I'm still trying to make sense of the arrangement of buildings around the Mill Stream, as shown on old OSI maps, south of where the Mass path crosses the stream. These are visible on the OSI c. 1837 first-edition 6-inch map.  On the later 25-inch map, there's a distinct offset in the route of the stream near the location of the arch. The bed of the stream also appears raised at this point. Possibly a weir was located here to create a pressure head for a mill wheel.

 

There's a distinct offset in the stream course. Image © Tailte Éireann

I hadn't seen the arch since around 1981. I thought it was covered with scrub and brambles over the decades, and was submerged below the waterline, but not so. This was a good time of the year to investigate it before the brambles get going.  

My theory is that it could have been the opening to a culvert/mill race that diverted water to a mill in the adjoining field beside the cemetery car park or it may have been the return feed to the stream. One mill is marked on the 6-inch, first-edition OSI map and was located just over the wall from where the bins used to be kept in the cemetery. The ruins of the mill is still there. There's an arched culvert under the Mile Mill road, south of the bend at the entrance to the cemetery, that fed water to this mill.

I think the arched opening into the stream is unlikely to be a latrine outlet or drain as it's too wide. I thought most of the arch was submerged below the waterline, but possibly the bed of the stream has dropped over the last four decades due to erosion. 

An image from the Lawrence Collection shows a building in the field, adjacent to the cemetery car park.

A brick arch, located in a wall adjacent to the Mill Stream in Kilcullen. © Eugene Brennan

© Eugene Brennan

A building, now demolished, visible in a photo from the Lawrence Collection. © National Library of Ireland

 

The Mill Stream north of the Mass path bridge was also re-routed post-1939, the earlier route passing through the adjacent field, the latter being depicted as "liable to floods" on the map. 

The map above is a ChatGPT-sharpened version of a screenshot of the c. 1837 first-edition six-inch map available on the National Library of Scotland website. The scans appear to be a bit clearer than those available on Tailte Éireann's Irish Townland and Historical Map viewer.
Map reproduction licence CC-BY, courtesy the National Library of Scotland.