# The components of power

One of the first things people do when presented with power data is to plot them. However, since power is the product of pedal force and pedal speed (or equivalently, torque and angular velocity) it may be equally interesting to decompose power into its two separate components and to compare power with each of them.

I have plotted the components of power from Gary Gellin's 2003 San Bruno Hillclimb. The plots below show (by distance in kilometers) cadence (in rpm), average torque over a single crank revolution (in newton-meters), and power (in watts). I have superimposed a slightly smoothed line onto the plots to help show the overall pattern. In addition, because gear usage influences cadence and torque, I have also included that plot. Perhaps you will be able to discern that in these data power appears to track crank torque (and thus, pedal force) more closely than cadence. Note in particular the sections from about kilometer 1 to kilometer 2, and again from about kilometer 2.1 to about kilometer 3.0, during which the gears were constant. Over those two sections, Gary's cadence varied from a low of about 80rpm up to more than 100rpm.

Below are four additional plots. The upper left scatterplot shows the relationship between power and cadence; the upper right shows the relationship between power and crank torque; and the lower left shows the relationship between the two components. The lower right plot superimposes three "isopower" contours on the rpm-torque plot at the lower left and its signficance will be explained in the section below. Evidently, the linear correlation between power and torque is higher (r = 0.9) than that between power and cadence (r = 0.03 if we exclude zero power readings, which essentially means no linear relationship). Note also that the correlation between cadence and torque is moderately high and negative (r = -0.4, once again excluding zero power readings), so that in general the higher the cadence the lower is the torque.

Although the correlation between cadence and torque is obvious, each point in the bivariate panel plot above shows individual observations, and does not link the points in sequence (as the plots at the top of this page do). The plot below shows the time-sequence of the cadence-torque relationship, smoothed over 30-second intervals. If you follow the path of the line from 1 to 30, you can trace the cadence-torque combinations Gary used during the climb that day. Below that is a supplementary plot that shows the gear choice during the climb, with each 30-second interval keyed to the numbers in the cadence-torque plot. For example, you can see that intervals 6, 7, 8, and 9 were done in the same gear.

In other words, since we know that power is the product of cadence and torque (or pedal speed and pedal force), looking for a relationship between power and cadence without taking into account torque is like looking for a relationship between the area of a rectangle and its height without taking into account its length. Worse, suppose that although some rectangles are slightly shorter and some slightly taller, they all lie within a narrow range of height (just as most cadences lie within a narrow range of each other). You will be able to plot the rectangle area against the rectangle height, just as one is able to plot power against cadence, and you will find a particular height for which your particular sample of rectangles has the largest area. It does not follow that rectangles of that height will always have the largest area, nor that that height represents a "sweet spot" for rectangle area.