CVT- Continuously Variable Transmission

# How much does CVT benefit?

 It is advantageous to use a CVT, instead of a manual transmission. This is mainly because the engine will operate always on the optimum regimes and throttle-positions, adapted to the varying road conditions and power demands.
 But is this advantage enough to overcome the inherent limitations and dissipations of the common friction CVTs? In order to quantify the advantage of using a CVT, we will ignore the inherent limitations and dissipations of the common friction CVTs. This will be a simplified approach. To compare performances, we calculate how much time [seconds] is necessary to accelerate a car from rest to 100 km/h using Manual Transmission (MT). Then we will calculate it with the same car, but using a CVT. In both cases, we will neglect all energy losses such as clutch transitions, aerodynamics, etc, and we consider the road is horizontal.
1st case: Manual Transmission (MT):

Consider a utility vehicle:

 mass: M=1250 kg; power: Power=75cV @ 5700 rpm
 Note: 'mass' includes passengers and luggage (350kg)

The transmission ratios (output speed/input speed) include the differential ratio:

 i1=0.066 i2=0.095 i3=0.14 i4=0.19 i5=0.28
 Example: to calculate the car speed on first gear at 5700 rpm:  V=5700rpm * i1 * wheel_radius * k1 = 43 km/h            {Eq.1} Note: k1 is units conversion factor, k1=(2*PIrad/1rot)/(60s/1min)*(1km/1000m)/(1h/3600s)=0.3768min*km/m*h
 We are considering a real case: it's a small vehicle sold in Europe in 1991. This is the engine's power/torque diagram. To simplify calculations, we will consider the blue line as the torque values. The torque value used, is the torque corresponding to the maximum power (@5700rpm). According to the technical specifications, this car takes 12.1s to accelerate from rest to 100 km/h. If we include luggage and passengers, the value should rise to about 15.5s. Anyway, we will have to calculate this MT case theoretically in order to compare with the corresponding theoretical calculations for the CVT case.
During each gear, the torque will be almost constant, and so will be the car's acceleration. Force=Power/Velocity. Also, Acceleration=Force/Mass. Thus,

Acceleration = Power / ( Velocity * Mass ).

Consider the ultimate car speed during the 1st gear, that is 43km/h.(@5700rpm). The power is 55kW, so the constant acceleration value can be calculated by:

Acceleration1st=55kw / ( 43km/h * Mass ) * k2 = Acc1=3.7 m/s˛

 Note: k2 is units conversion factor, k2=(1km/1000m)/(1h/3600s)=3.6km.s/m.h

Similarly for the other gears: Acc2=2.6 m/s˛ ; Acc3=1.8 m/s˛ ; Acc4=1.3 m/s˛ .
The 5th gear acceleration isn't requires because 100km/h are attained in 4th gear.

The time required to attain 100km/h is the sum of the time spent on each gear.
The uniformly accelerated movement has the equation:

Final_Velocity = Initial_Velocity + Acceleration * Time     <=>

Time = (Final_Velocity - Initial_Velocity) / Acceleration

Let's calculate the sime during the 1st gear:

Time1 = ( 43km/h - 0Km/h ) / ( Acc1 * k2 ) = 3,2s

Similarly for the other gears and summing up, we conclude that the theoretical required time to accelerate from rest to 100km/h, using a manual transmission (MT), will be:

 MT_Time = 11.9s

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 Note:  The difference between the real value (15.5s) and the theoretical value (11.9) may be easily understood considering about 3.5s for gear shifting.

2nd case: Continuously Variable Transmission (CVT):

Now we will calculate the required time to accelerate from rest to 100km/h, using a Continuously Variable Transmission (CVT). To simplify calculations, we will consider the IVT case, because it allows continuous ratio variation from rest.

To maximize acceleration, power must be kept on it's greatest value:

While accelerating, Force = Power / Velocity ,

According to Newtons Law, Force = Mass * Acceleration ,

Equating, becomes, Power / Velocity = Mass * Acceleration ,

Acceleration is the derivate of Velocity in order to time: dv/dt.

Thus, separating variables:

Velocity * Mass * dv/dt / Power = 1 ,

Integrating (§) this differential equation on both sides, becomes,

§ ( Velocity * Mass / Power ) dv  = § (1) dt

Considering null constants, ( Velocity˛/2 ) * (Mass / Power) = Time

Substituting Velocity=100km/h, M=1250kg, Power=75cV, results on:

 CVT_Time = 8.8s

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You may compute the CVT advantage regarding other car specifications. Please use the CVT vs MT Calculator.

 Conclusion: The Continuously Variable Transmission (CVT) is 35% more performant than the Manual Transmission (MT). With same car and engine, the CVT takes only 75% of the time to accelerate to 100km/h, compared to the MT.

 This means that: -Although the known CVTs (ex: "V"belt, Toroidal) have greater inherent dissipations than MT, they may be still advantageous. -To take full advantage of the whole 35% improvement of CVT, it would be necessary to invent something as a "Geared CVT", (without the unavoidable limitations of friction CVTs). -If it is difficult to eliminate significantly the energy losses of some friction drives, then these CVT will hardly improve performances up to 35%. Power-split and similar techniques may help here.Perhaps, the most remarkable practical proof of the CVT performance, was the CVT use in the 800cV Formula One Canon-Williams-Renault, in 1993. Without so much development as the MT version, the experimental CVT Formula One was 1 second faster per lap.

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