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The oldest patent for an electric bike I've been able to find at the US Patent and Trademark Office is this one, by Ogden Bolton, Jr. of Canton Ohio, which was filed in September 1895 and granted three months later. You can see from these original diagrams that it bears an amazingly close resemblance to modern electric bikes. In the general picture on the left, you can see there's a hub motor on the rear wheel (blue), a battery suspended from the frame (red), and a simple handlebar control to make the thing stop and go. In the more detailed cutaway of the hub motor on the right, you can see there's a six-pole magnet in the center (orange) bolted to the frame and an armature (made from coiled wire, yellow) that rotates around it when the current is switched on. It's quite a hefty motor even by modern standards; Ogdon mentions "a heavy current at low voltage—for instance, to carry one hundred amperes at ten volts." So that's 1000 watts, which is about twice the power of a typical modern bike hub motor.
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An e-bike is identified as a "motor assisted cycle" (MAC) in British Columbia, which differs from electric mopeds and scooters, which are "limited-speed motorcycles." Motor assisted cycles must: have an electric motor of no more than 500 W; have fully operable pedals; not be capable of propelling the device at a speed greater than 32 km/hr [19.9 mph]. The engine must disengage when (a) the operator stops pedaling, (b) an accelerator controller is released, OR (c) a brake is applied. A driver's license, vehicle registration, and insurance are all not required. A bike helmet must be worn.
Both bikes have passable cadence sensor pedal assist that kicks in a little late and lets go a little early. Torque sensors are too expensive for this price level so you are just not going to get the same responsiveness as a bike store e-bike. Rattan has 5 levels of pedal assist while the Ancheer has 3. Over long periods of pedaling the PAS evens out or you can just use the throttle.
Vehicles with an electric power and power of less than 300W are classified as "not a motor vehicle". Such electric bicycles must comply with the same rules as bicycles. You must wear a helmet even on a scooter or bike under 300W. If the power is over 300W or a combustion engine is used it is a "low powered vehicle" and the moped rules apply. Specifically, a drivers license and registration are required.
Arkansas does not define E-bikes. The following definition describes a combustion engine. E-bikes being electric do not have a cylinder capacity and thus this law is not technically applicable. The state defines a "Motorized bicycle" as "a bicycle with an automatic transmission and a motor of less than 50cc." Riders require either a certificate to operate a motorized bicycle, a motorcycle license, a motor-driven cycle license, or a license of class A, B, C or D. Certificates cannot be issued to riders under 10 years of age.
Pedelecs are much like conventional bicycles in use and function — the electric motor only provides assistance, for example, when the rider is climbing or struggling against a headwind. Pedelecs are therefore especially useful for people in hilly areas where riding a bike would prove too strenuous for many to consider taking up cycling as a daily means of transport. They are also useful for riders who more generally need some assistance, e.g. for people with heart, leg muscle or knee joint issues.
Some electric bikes claim to use a neat trick called regenerative braking. If you start pedaling the bicycle or going downhill, the spinning wheels turn the electric motor in the hub in reverse and start charging up the batteries. In practice, regenerative braking is nowhere near as useful on an electric bicycle as it is on an electric train or car. An electric bike has much less mass and velocity than either a train or car, so it never gains (or loses) anything like as much kinetic energy when it starts and stops. You'd have to go down an awful lot of hills to charge up the batteries completely and that's usually not practical. And what's the point in pedaling the wheels simply to charge the battery? You might as well have bought an ordinary bicycle to start with!
Power assisted bicycles are classified in two categories in Saskatchewan. An electric assist bicycle is a 2 or 3 wheeled bicycle (sic.: 3 wheeled bicycle) that uses pedals and a motor at the same time only. A power cycle uses either pedals and motor or motor only. Both must have engines with 500 watt power or less, and must not be able exceed 32 km/h (20 mph), i.e., electric motor cuts out at this speed or cycle is unable to go this fast on a level surface. The power cycle has to meet the Federal Motor Vehicle Safety Standards for a power-assisted bicycle. The power cycle requires at least a learner's driving licence (class 7), and all of the other classes 1-5 may operate these also. The electric assist bicycle does not require a licence. Helmets are required for both. Both are treated as bicycles regarding rules of the road. Gas powered or assisted bicycles are classified as motorcycles regardless of engine size or if using pedals plus motor. Stickers identifying the bicycle's compliance with the Federal classification may be required for power cycles by some cities or municipalities. 
Available products (motors, senors, batteries, control systems. etc.) change fast since the e-bike market shows a strong growth. In principle, get a rear motor if you want a lot of raw power and acceleration (e.g. for a 500W motor). Middle motors are more energy efficient and less subject to problems (e.g. overheating). Middle motors usually are combined with integrated rear hub gears, and rear motors with derailleurs. Both have advantages and disadvantages. E.g. the former require less maintenance, are easy to use in go-and-stop traffic, but require taking off weight on pedals when shifting. Therefore the latter (derailleurs) are better for speedy overland and hills cycling.
In Manitoba Electric Bikes can be classified as a scooter or a moped/mobility vehicle depending on the power of the engine used and its top speed. If the engine on the Electric Bike is less than 50cc and it cannot exceed 50 km per hour the rider is not required to have a motorcycle licence or any specific training. Electric Bikes can be driven by anyone with a Class 5 driver's licence in any stage of the graduated licensing process. A Class 5 Learners Licence requires one to be age 16 (parents consent if under 18) and a visit to a licensing office to pass a vision test and a written knowledge test about the rules of the road and traffic signs.
"Medical Exemptions" are also a standard right in the State of Texas for motorcycles & even bicyclists. Through Texas's motorcycle helmet law (bicycle helmet laws from city ordinances), it is only required for those 21 years old or younger to wear a helmet. However, a medical exemption, written by a certified licensed medical physician or licensed chiropractor, which exempts one from wearing a helmet, can be used for bicyclists if helmets are required.
The Ohio Revised Code 4511.01  distinguishes motorized bicycles and mopeds from motorcycles or scooters by describing them as "...any vehicle having either two tandem wheels or one wheel in the front and two wheels in the rear, that is capable of being pedaled and is equipped with a helper motor of not more than fifty cubic centimeters piston displacement that produces no more than one brake horsepower and is capable of propelling the vehicle at a speed of no greater than twenty miles per hour on a level surface." One brake horsepower converts to 0.75 kW, or (rounded) 750W. Thus, a bicycle with an electric helper motor operating under 750W, and not propelling the bicycle over 20 mph, does not qualify to be registered under Ohio state law. Local jurisdictions may have other regulations.
On February 4, 2014, SB997 was introduced by Senator Matt Smith, which seeks to amend PA Vehicle Code to include "Pedalcycle with Electric Assist". In a memo addressed to all senate members, Smith said the definition shall include "bicycles equipped with an electric motor not exceeding 750 watts, weighing not more than 100 pounds, are capable of a maximum speed of not more than 20 mph, and have operable pedals."
(15.5) "Electric assisted bicycle" means a device with two or three wheels which has a saddle and fully operative pedals for human propulsion and also has an electric motor. For such a device to be considered an electric assisted bicycle, it shall meet the requirements of the Federal Motor Vehicle Safety Standards, as set forth in 49 C.F.R. Section 571, et seq., and shall operate in such a manner that the electric motor disengages or ceases to function when the brakes are applied. The electric motor in an electric assisted bicycle shall: