Panasonic System Power Delivery Explanation
This explanation refers to the latest 2008 standard unit, but the general principles are common to all.  However, from 2010 the phase down no longer starts at 15 kph (9.4 mph), Japanese law no longer applied to export models so power assist remains at maximum until approximately 22 kph (14 mph).  In addition, the chainwheel size and rear sprocket sizes have sometimes been changed to match export preferences.  It's also possible to change the pedal cadence on standard units by altering the size of the motor chain drive sprocket and for details on that see power cutoff cadence in the modifications section.

I've often advised of the need to pedal slowly in order to get full power on these motor units, There's been a number of queries about this and I've seen comments posted that the hill climbing on the Kalkhoff Agattu was best around 9 mph, another that top gear only was needed and yet another comment that it's better on hills than the flat.

The reason for all those on the earlier model of that bike with the standard rear sprocket is that 9.4 mph is the last speed in top gear at which full motor power is contributed, from then on up it reduces in four phased steps until it's zero at 15 mph.

EU law requires power to phase down as it approaches the 25 kph limit, but without the point specified and it certainly doesn't intend as early as 9 mph (14.4 kph).  Also the zero point in EU law is 25 kph (15.62 mph), not 15 mph, so why those two discrepancies?

The reasons are both due to national laws.

First the easy one to explain.  Where most EU countries use the electric pedal assist 25 kph limit, Germany has it's own 24 kph limit, so the German made Kalkhoff has it's 22 tooth rear sprocket chosen with that in mind so that power disappears by 24 kph (15 mph).  To exactly match general EU law a 21 tooth sprocket is needed to give the slight increase to 25 kph (15.62 mph).

The explanation for the very early phase down is more complex, and it results from Japanese law on electric assist pedal cycles.  The Japanese have determined that the maximum power allowable must not exceed the power the rider puts in (1 to 1), and that this should only be fully available within normal cycling speeds.  Unfortunately for us, the Japanese have a very different view on cycling speeds to that of UK riders,  They regard their common cycling speed as being 15 kph (9.4 mph), and for "sports" riders like our lycra brigade, 24 kph (15 mph), which would raise a laugh from UK club riders who can often average 20 mph for an hour in moderately hilly areas.

Since sports riders don't use electric assist bikes, only the "common" bike's speed is legislated for.   Therefore in accordance with that, the Panasonic design has integral to it the phase down above full assist at 15 kph (9.4 mph) when the bike is in top gear.

But that's not the end of it, for Japanese law prescribes the power phase down slope as well.  Since an analogue slope doesn't readily match digital electronic systems working, Panasonic have chosen to have four step downs giving a rough tail off of power corresponding with the legal requirement, which is expressed in Japanese law by this equation:

1 - ( [kph - 15] / 9 ) = assist factor

where kph is the road speed.

Quite simply what this means is the road speed in kph, minus 15 (the 1 to 1 power assist limit), then divided by 9 produces a result which is subtracted from 1.

In a practical example, at 20 kph (12.5 mph), taking 15 from that 20 kph gives us 5.  Then that 5 divided by 9 gives 0.55 recurring.  To complete the equation we take that 0.55 from 1 to give 0.45 or 45%, and that's the ratio of rider power that the motor is allowed to assist with at 12.5 mph.

At the lower speed of 11 mph (17.6 kph) using the same calculation, 17.6 minus 15 then divided by 9 and taken away from 1 gives 0.71 or 71% of the riders input given as motor assistance.

At the higher speed of 13 mph (20.8 kph), only 36% of rider input is given by the motor.

For the export markets Panasonic now have the High Power mode which gives 1.3 times assist, but this is not a blanket 1.3 times, it's 1.3 times full 1 to 1 assistance up to 9.4 mph, and then above that speed at 1.3 times the above equation percentage results.  So the three examples given in declining order of assist with speed become 92% at 11 mph, 59% at 12.5 mph and 47% at 13 mph.   You can see from that how speeding up from 11 mph to 13 mph by pedalling faster halves the help the motor gives you, whatever mode you are in, so making work for yourself.

Of course all these are for top gear, and since the motor drives through the gears, when you change down, the power phase down point and the power assist levels drop too.  On the Agattu with the Shimano Nexus Inter-7 hub, dropping from 7th to 6th gear is a 13% drop, so that's what you take from the above.  It means a maximum 1 to 1 power assist speed of 8.2 mph, down from 9.4 mph, but of course that is what makes these bikes such superb hill climbers, the ability to shift the maximum power down to a speed that copes with any hill, however steep.

For example, our 7 speed Agattu's bottom gear is 60% down on the top gear, so the maximum assist power is available from zero to 3.8 mph, about the slowest speed one could keep balance at.  This means a moderately fit 75 kilo rider providing a reasonable 200 watts could climb a 1 in 3.7 (27% hill) at that speed in highpower mode, and one would have to search far and wide for a hill that steep, there being only a handful of them in Britain.

For those with lesser hills, the rear hub sprocket can be reduced to shift the phase down point and end point of power assistance upwards, the limit set by the steepest hill a rider needs to climb, and ultimately the limits of the motor's power to maintain a high road speed against wind resistance.

To adjust the above calculations after the rear sprocket is changed, divide the equation answer by the new sprockets number of teeth and multiply by the old number of teeth, which is 22.

Of course all this is on a "nice to know" basis, so there's no need to carry around a mathmetician with calculator on the carrier.   In practice it's just a matter of slowing the pedal speed when more assistance is wanted, either by going slower or by changing up a gear  The fastest cadence (pedalling rotations) for the maximum assist at 9.4 mph in top gear is a very leisurely 39 per minute, or 1.5 seconds per rotation.