# Calculations: Speed loss from battery vs dynamo, dynamo vs. no dynamo

## 1. Speed loss from dynamo vs. no dynamo

Power = k1 * v + k2 * v3 (when there's no wind)

where v = the speed, k1 = rolling resistance constant, k2 = wind resistance constant. k1 and k2 depend on tyres, bicycle shape, position of the cyclist, etc.

Assuming a rolling resistance of 30 Watt at 30 km/h, we can easily calculate the required power to pedal at another speed, or calculate the new speed when given a certain power. I will apply this to show how much influence the efficiency of a dynamo has, on cycling speed. Here I assume the use of 3 Watt lighting, so no use of e.g. 3 high power LEDs in series powered by the dynamo (which is possible)...

1. Take someone who rides at a brisk pace (as I usually do): Suppose I ride at 30 km/h, position a bit bent forward. This takes about 200 Watt (Note: This is on a standard bike with ca. 37mm tyres, fenders, racks etc. Not a road bike and especially not a time trial bike! Only then is it possible to go 30 km/h with about 120W. That is a figure I read in some test about bicycle lighting but which is irrelevant in case of lighting!).

• Now suppose I switch on a regular dynamo with low efficiency that uses about 18 Watt at that speed. What will be the new speed given I've no more than 200 Watt to give? This can be solved with a 3rd degree equation (also assuming the dynamo drag is linear with speed), but I will approximate so you can get a feel for these things:

We are left with 182 W, the rolling resistance (+dynamo drag!) becomes just a little smaller (because the speed won't change much), so I will leave it constant. This means 182-30 Watt = 152 Watt is left to spend on air resistance instead of 170 W. Speed is linear with the 3rd degree root of the power available to overcome the air resistance, so v2/v1 = (152/170)^(1/3) = 0.963. This means the new speed is 30 * 0.963 km/h = 28.9 km/h (this is an approximation, with decreased rolling resistance you would get to about 29.0 km/h, which doesn't differ much, as was clear from the start).

• Now use a hub dynamo with high efficiency, which takes up 8 W instead of 18 W (all at 30 km/h). New speed (same approximation) is then ca. 29.5 km/h. This differs just by 0.6 km/h from the regular dynamo... A tiny breeze will have more influence!

• Finally, a switched off lamp on a bike with SON 28 dynamo hub, which takes 1.5 W at 30 km/h. New speed compared to a bike with standard front hub (assuming this has nearly zero resistance) is (168.5/170)^(1/3) * 30.00 km/h = 29.90 km/h. So, you lose at most 0.10 km/h by using the dynamo hub.

2. Now we take someone who rides slowly: Suppose I ride at 20 km/h without a dynamo (+standard 3 W lamp), then the required power is approximately: 20 W + (20/30)^3 * 170 = 70 Watt. N.B. This is the power for the same somewhat bent-forward position, but people who cycle slowly at about 20 km/h usually sit up fairly straight. I estimate that a total power of around 100 W is a more accurate amount in that case, which means the influence on speed by the hub dynamo is a little less than calculated here.

• To cycle with dynamo on, only 58 W is left over (dynamo takes 12 W to power the 3W lamp at this speed), which gives a new speed of ca. 18.3 km/h

• Now take a hub dynamo with high efficiency (5 W at 20 km/h). The new speed then becomes ca. 19.3 km/h. The difference is just 1.0 km/h compared to a regular dynamo...

• Finally, a switched off lamp on a bike with SON 28 dynamo hub, which takes 1.0 W at 20 km/h. New speed compared to a bike with standard front hub (assuming that has nearly zero resistance) is (49/50)^(1/3) * 20.00 km/h = 19.87 km/h. So, you lose at most 0.13 km/h by using the dynamo hub.

As you can see, the faster you ride, the less the influence of the dynamo. The question is: Is a high efficiency dynamo of interest more to those who ride slowly or those who ride quickly? (answer: Those who ride quickly, as those who ride slowly don't ride slowly because it's hard to cycle faster but because they just ride at a slow pace; The fast riders want every bit of speed, but in my view it's not worth a lot of money as the speed gain is minimal at speeds of say 30 km/h).

## 2. Speed loss from battery vs dynamo

To come

What must be weighed here is the drag from the additional weight of the batteries (if the batteries + lamp are heavier than the dynamo + dynamo lamp), which gives additional rolling resistance, vs. the power needed to run the dynamo lamps from the dynamo.

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