Power-to-weight ratio

In classical mechanics, instantaneous power is the limiting value of the average work done per unit time as the time interval Δt approaches zero (i.e. the derivative with respect to time of the work done).

${\displaystyle P=\lim _{\Delta t\rightarrow 0}{\tfrac {\Delta W(t)}{\Delta t}}=\lim _{\Delta t\rightarrow 0}P_{\mathrm {avg} }={\frac {d}{dt}}W(t)\,}$

The typically used metric unit of the power-to-weight ratio is ${\displaystyle {\tfrac {\text{W}}{\text{kg}}}\;}$ which equals ${\displaystyle {\tfrac {{\text{m}}^{2}}{{\text{s}}^{3}}}\;}$. This fact allows one to express the power-to-weight ratio purely by SI base units. A vehicle's power-to-weight ratio equals its acceleration times its velocity; so at twice the velocity, it experiences half the acceleration, all else being equal.

If the work to be done is rectilinear motion of a body with constant mass ${\displaystyle m\;}$, whose center of mass is to be accelerated along a (possibly non-straight) to a speed ${\displaystyle |\mathbf {v} (t)|\;}$ and angle ${\displaystyle \phi \;}$ with respect to the centre and radial of a gravitational field by an onboard powerplant, then the associated kinetic energy is

${\displaystyle E_{K}={\tfrac {1}{2}}m|\mathbf {v} (t)|^{2}}$

where:

${\displaystyle m\;}$ is mass of the body

${\displaystyle |\mathbf {v} (t)|\;}$ is speed of the center of mass of the body, changing with time.

The work–energy principle states that the work done to the object over a period of time is equal to the difference in its total energy over that period of time, so the rate at which work is done is equal to the rate of change of the kinetic energy (in the absence of potential energy changes).

The work done from time t to time t + Δt along the path C is defined as the line integral ${\displaystyle \int _{C}\mathbf {F} \cdot d\mathbf {x} =\int _{t}^{t+\Delta t}\mathbf {F} \cdot \mathbf {v} (t)dt}$, so the fundamental theorem of calculus has that power is given by ${\displaystyle \mathbf {F} (t)\cdot \mathbf {v} (t)=m\mathbf {a} (t)\cdot \mathbf {v} (t)=\mathbf {\tau } (t)\cdot \mathbf {\omega } (t)}$.

where:

${\displaystyle \mathbf {a} (t)={\frac {d}{dt}}\mathbf {v} (t)\;}$ is acceleration of the center of mass of the body, changing with time.

${\displaystyle \mathbf {F} (t)\;}$ is linear force – or thrust – applied upon the center of mass of the body, changing with time.

${\displaystyle \mathbf {v} (t)\;}$ is velocity of the center of mass of the body, changing with time.

${\displaystyle \mathbf {\tau } (t)\;}$ is torque applied upon the center of mass of the body, changing with time.

${\displaystyle \mathbf {\omega } (t)\;}$ is angular velocity of the center of mass of the body, changing with time.

In propulsion, power is only delivered if the powerplant is in motion, and is transmitted to cause the body to be in motion. It is typically assumed here that mechanical transmission allows the powerplant to operate at peak output power. This assumption allows engine tuning to trade power band width and engine mass for transmission complexity and mass. Electric motors do not suffer from this tradeoff, instead trading their high torque for traction at low speed. The power advantage or power-to-weight ratio is then

${\displaystyle {\mbox{P-to-W}}=|\mathbf {a} (t)||\mathbf {v} (t)|\;}$

where:

${\displaystyle |\mathbf {v} (t)|\;}$ is linear speed of the center of mass of the body.

The useful power of an engine with shaft power output can be calculated using a dynamometer to measure torque and rotational speed, with maximum power reached when torque multiplied by rotational speed is a maximum. For jet engines the useful power is equal to the flight speed of the aircraft multiplied by the force, known as net thrust, required to make it go at that speed. It is used when calculating propulsive efficiency.

Thermal energy is made up from molecular kinetic energy and latent phase energy. Heat engines are able to convert thermal energy in the form of a temperature gradient between a hot source and a cold sink into other desirable mechanical work. Heat pumps take mechanical work to regenerate thermal energy in a temperature gradient. Standard definitions should be used when interpreting how the propulsive power of a jet or rocket engine is transferred to its vehicle.

An electric motor uses electrical energy to provide mechanical work, usually through the interaction of a magnetic field and current-carrying conductors. By the interaction of mechanical work on an electrical conductor in a magnetic field, electrical energy can be generated.

Fluids (liquid and gas) can be used to transmit and/or store energy using pressure and other fluid properties. Hydraulic (liquid) and pneumatic (gas) engines convert fluid pressure into other desirable mechanical or electrical work. Fluid pumps convert mechanical or electrical work into movement or pressure changes of a fluid, or storage in a pressure vessel.

A variety of effects can be harnessed to produce thermoelectricity, thermionic emission, pyroelectricity and piezoelectricity. Electrical resistance and ferromagnetism of materials can be harnessed to generate thermoacoustic energy from an electric current.

All electrochemical cell batteries deliver a changing voltage as their chemistry changes from "charged" to "discharged". A nominal output voltage and a cutoff voltage are typically specified for a battery by its manufacturer. The output voltage falls to the cutoff voltage when the battery becomes "discharged". The nominal output voltage is always less than the open-circuit voltage produced when the battery is "charged". The temperature of a battery can affect the power it can deliver, where lower temperatures reduce power. Total energy delivered from a single charge cycle is affected by both the battery temperature and the power it delivers. If the temperature lowers or the power demand increases, the total energy delivered at the point of "discharge" is also reduced.

Battery discharge profiles are often described in terms of a factor of battery capacity. For example, a battery with a nominal capacity quoted in ampere-hours (Ah) at a C/10 rated discharge current (derived in amperes) may safely provide a higher discharge current – and therefore higher power-to-weight ratio – but only with a lower energy capacity. Power-to-weight ratio for batteries is therefore less meaningful without reference to corresponding energy-to-weight ratio and cell temperature. This relationship is known as Peukert's law.[53]

Capacitors store electric charge onto two electrodes separated by an electric field semi-insulating (dielectric) medium. Electrostatic capacitors feature planar electrodes onto which electric charge accumulates. Electrolytic capacitors use a liquid electrolyte as one of the electrodes and the electric double layer effect upon the surface of the dielectric-electrolyte boundary to increase the amount of charge stored per unit volume. Electric double-layer capacitors extend both electrodes with a nanoporous material such as activated carbon to significantly increase the surface area upon which electric charge can accumulate, reducing the dielectric medium to nanopores and a very thin high permittivity separator.

While capacitors tend not to be as temperature sensitive as batteries, they are significantly capacity constrained and without the strength of chemical bonds suffer from self-discharge. Power-to-weight ratio of capacitors is usually higher than batteries because charge transport units within the cell are smaller (electrons rather than ions), however energy-to-weight ratio is conversely usually lower.

Fuel cells and flow cells, although perhaps using similar chemistry to batteries, do not contain the energy storage medium or fuel. With a continuous flow of fuel and oxidant, available fuel cells and flow cells continue to convert the energy storage medium into electric energy and waste products. Fuel cells distinctly contain a fixed electrolyte whereas flow cells also require a continuous flow of electrolyte. Flow cells typically have the fuel dissolved in the electrolyte.

Power-to-weight ratios for vehicles are usually calculated using curb weight (for cars) or wet weight (for motorcycles), that is, excluding weight of the driver and any cargo. This could be slightly misleading, especially with regard to motorcycles, where the driver might weigh 1/3 to 1/2 as much as the vehicle itself. In the sport of competitive cycling athlete's performance is increasingly being expressed in VAMs and thus as a power-to-weight ratio in W/kg. This can be measured through the use of a bicycle powermeter or calculated from measuring incline of a road climb and the rider's time to ascend it.[103]

A locomotive generally must be heavy in order to develop enough adhesion on the rails to start a train. As the coefficient of friction between steel wheels and rails seldom exceeds 0.25 in most cases, improving a locomotive's power-to-weight ratio is often counterproductive. However, the choice of power transmission system, such as variable-frequency drive versus direct current drive, may support a higher power-to-weight ratio by better managing propulsion power.

Most vehicles are designed to meet passenger comfort and cargo carrying requirements. Vehicle designs trade off power-to-weight ratio to increase comfort, cargo space, fuel economy, emissions control, energy security and endurance. Reduced drag and lower rolling resistance in a vehicle design can facilitate increased cargo space without increase in the (zero cargo) power-to-weight ratio. This increases the role flexibility of the vehicle. Energy security considerations can trade off power (typically decreased) and weight (typically increased), and therefore power-to-weight ratio, for fuel flexibility or drive-train hybridisation. Some utility and practical vehicle variants such as hot hatches and sports-utility vehicles reconfigure power (typically increased) and weight to provide the perception of sports car like performance or for other psychological benefit.

Increased engine performance is a consideration, but also other features associated with luxury vehicles. Longitudinal engines are common. Bodies vary from hot hatches, sedans (saloons), coupés, convertibles and roadsters. Mid-range dual-sport and cruiser motorcycles tend to have similar power-to-weight ratios.

Power-to-weight ratio is an important vehicle characteristic that affects the acceleration of sports vehicles.