Prius nomogram

After finding a nomogram of how the Prius’s continuously variable transmission (or Power Split Device, in Toyota-speak) works, I thought it would be a fun exercise to use PyNomo to create my own Prius PSD nomogram.

The PSD is a planetary gearset, with the sun gear attached to Motor-Generator 1 (MG1), the planet carrier attached to the Internal Combustion Engine (ICE), and the ring gear attached to Motor-Generator 2 (MG2). MG2 drives the front wheels directly.

In the nomogram, the car is travelling 65 mph. Because this is a continuously variable transmission, there are infinite ways of achieving this road speed. MG2 must turn at about 3850 RPM because it drives the front wheels directly. However, the engine can be running at anywhere between idle speed and flat-out. MG1 will then turn at a corresponding speed according to the relationship shown in the nomogram.

The nomogram also illustrates how the Prius can run in “EV mode,” where the car drives under electric power alone. With the ICE off, MG1 rotates backwards and MG2 rotates forward. As the car speeds up and slows down in EV mode, the isopleth line see-saws up and down the MG1 and MG2 scales, anchored at the ICE’s 0 RPM point.

Part of the elegance of the PSD is that there is no reverse gear. MG2 simply rotates backwards with the engine off. Cool!

There is an interactive Flash-based version on

Watching for nomograms

Cleaning out drawers last weekend, I took a minute to fiddle with two slide rules that I inherited, one from each of my grandfathers. I’ve always found them to be a mysterious link to the past, where clever tricks made all kinds of things possible that we take for granted today. I’d like to be able to make more use of them, but they’re just not that practical any more.

Yesterday I stumbled on a link to a gallery of similarly outdated but intriguing tools called nomograms, which are graphical solutions to equations. Ron Doerfler assembled a calendar for 2010 featuring a number of intricate nomograms. He updates his site infrequently, but I certainly hope he is able to put together a similar calendar for 2011. I will definitely check back later in the year! It will be good for engineer-cred.

As an electrical engineer, I still have nightmares about trying to understand Smith charts in college. The Smith chart is another type of nomogram used for a number of purposes, including matching components, transmission line characteristics, and other high-frequency uses. Again, the chart is fascinating to look at, but it’s really a product of the black arts. Stay away.

Thinking more about where I’ve seen nomograms, I recall on highway maps there used to be a fuel economy calculator. There were two horizontal lines and a diagonal line between them. On one horizontal scale, you would select how many gallons it took to fill the tank, and on the other, you would choose how far you went on that tank. The center scale showed the car’s fuel economy for that tank. For the life of me, I can’t find any examples online, so I tried out PyNomo to create one of my own.

In the example above, I went 650 miles on a tank of fuel, and it took 14.5 gallons to refill the tank. The line between those two points intersects the MPG scale at about 45 MPG.

The PyNomo site has some additional examples that fly way beyond my comprehension, but they are neat to look at.

I also found a nomogram for homebrewers to calculate IBUs, and one illustrating the operation of planetary gear set in a Prius.