In this article we are going to understand how the **engine torque** is produced, how **engine power** is calculated and what is a **torque and power curve**. Also, we are going to have a look at the engine torque and power maps (surfaces).

By the end of the article, the reader will be able to understand the difference between torque and power, how they affect the longitudinal dynamics of the vehicle and how to interpret torque and power curves at full load.

### Definition of torque

**Torque** can be regarded as a **turning force** applied on an object. Torque (vector) is the cross product between a force (vector) and a distance (scalar). The distance, also called the **lever arm**, is measured between the force and the turning point. Similar to a force, torque is a vector and is defined by an amplitude and a direction of rotation.

Imagine that you want to tighten/loosen the bolts of a wheel. Pushing or pulling the handle of the wrench connected to a nut or bolt, produces a torque (turning force) that loosens or tightens the nut or bolt.

The torque *T [Nm]* is the product of the force *F [N]* and the length of the lever arm *a [m]*.

In order to increase the magnitude of the torque we can either increase the force, the length of the lever arm or both.

**Example**: Calculate the torque obtained on the bolt if the arm of the wrench has *0.25 m* and the applied force is *100 N* (which is approx. equivalent with a pushing force of *10 kg*)

The same torque could be obtained if the lever arm was *1 m* and the force only *25 N*.

The same principle applies to internal combustion engines. The torque at the crankshaft is produced by the force applied on the conrod journal through the connecting rod.

The torque *T* will be produce at the crankshaft on each conrod journal, every time the piston is in the power stroke. The lever arm *a* in this case is the **crank radius (offset)**.

The magnitude of the force *F* depends on the combustion pressure within the cylinder. The higher the pressure in the cylinder, the higher the force on the crankshaft, the higher the output torque.

The length of the lever arm has impact on the overall **engine balance**. Increasing it too much can lead to engine imbalance, which results in higher forces in the crankshaft journals.

**Example**: Calculate the torque at the crankshaft for an engine with the following parameters:

Cylinder bore, B [mm] | 85 |

Cylinder pressure, p [bar] | 12 |

Crank offset, a [mm] | 62 |

First, we calculate the area of the piston (assuming the the piston head is flat and its diameter is equal with the bore of the cylinder):

\[A_p = \frac{\pi B^2}{4}=\frac{\pi \cdot 0.085^2}{4}=0.0056745 \text{ m}^2\]Second, we’ll calculate the force applied to the piston. To get the force in *N* (Newton), we’ll use the pressure converted in *Pa* (Pascal).

Assuming that all the force in the piston goes into the connecting rod, the torque is calculated as:

\[T = F \cdot a = 680.94021 \cdot 0.062 = 42.218293 \text{ Nm}\]The standard unit of measurement for torque is *N·m* (Newton meter). Especially in the USA, the unit of measurement for engine torque is *lbf·ft* (foot-pounds). The conversion between *N·m* and *lbf·ft* is:

1 \text{ lbf} \cdot \text{ft} &= 1.355818 \text{ N} \cdot \text{m}\\

1 \text{ N} \cdot \text{m} &= 0.7375621 \text{ lbf} \cdot \text{ft}

\end{split} \]

For our particular example, the torque in imperial units (USA) is:

\[T = 42.218293 \cdot 0.7375621 = 31.138615 \text{ lbf} \cdot \text{ft}\]Torque *T [N]* can also be expressed as a function of the mean effective pressure of the engine.

where:

*p _{me} [Pa]* – mean effective pressure

*V*– engine displacement (volume)

_{d}[m^{3}]*n*– number of crankshaft rotations for a complete engine cycle (for a 4-stroke engine

_{r}[-]*n*)

_{r}= 2### Definition of power

In physics, **power** is the work done in time or, with other words, is **the rate of doing work**. In rotational systems, power *P [W]* is the product of the torque *T [Nm]* and angular velocity *ω [rad/s]*.

The standard unit of measurement for power is *W* (Watt) and for rotational speed is *rad/s* (radian per second). Most of the vehicle manufacturers are providing the power of the engine in *bhp* (brake horse power) and the rotational speed in *rpm* (rotations per minute). Therefore, we are going to use conversion formulas for both rotational speed and power.

To convert from *rpm* to *rad/s*, we use:

To convert from *rad/s* to *rpm*, we use:

The engine power can also be measured in *kW* instead of *W* for a more compact value. To convert from *kW* to *bhp* and reverse, we use:

P \text{ [bhp]} &= 1.36 \cdot P \text{ [kW]}\\

P \text{ [kW]} &= \frac{P \text{ [bhp]}}{1.36}

\end{split} \]

In some cases you might find *HP* (Horse Power) instead of *bhp* as unit of measurement for power.

Having rotational speed measured in *rpm* and torque in *Nm*, the formula to calculate **power** is:

P \text{ [kW]} &= \frac{\pi \cdot N \text{ [rpm]} \cdot T \text{ [Nm]}}{30 \cdot 1000}\\

P \text{ [HP]} &= \frac{1.36 \cdot \pi \cdot N \text{ [rpm]} \cdot T \text{ [Nm]}}{30 \cdot 1000}

\end{split} \]

**Example**. Calculate the engine power in both *kW* and *HP*, if the engine torque is *150 Nm* and engine speed is *2800 rpm*.

P &= \frac{\pi \cdot 2800 \cdot 150}{30 \cdot 1000} = 44 \text{ kW}\\

P &= \frac{1.36 \cdot \pi \cdot 2800 \cdot 150}{30 \cdot 1000} = 59.8 \text{ HP}

\end{split} \]

### Engine dynamometer

Engine speed is measured using a sensor on the crankshaft (flywheel). Ideally, to calculate power, we should also measure the torque at the crankshaft with a sensor. Technically, this is possible but not applied in the automotive industry. Because of the operating conditions of the crankshaft (temperatures, vibrations), measuring engine torque with a sensor is not a reliable technique. Also, the cost of a torque sensor is quite high. Therefore, engine torque is measured on the full range of speed and load, using a **dynamometer** (test bench), and mapped (stored) into the engine control unit.

The dynamometer is basically a brake (mechanical, hydraulic or electrical) which absorbs the power produced by the engine. The most used and best type of dynamometer is the **electric dynamometer**. This is actually an **electric machine** that can be operated as a **generator** or **motor**. By varying the generator’s load torque, the engine can be put in any operating point (speed and torque). Also, with the engine at fuel cut (no fuel injection), the generator can be run as an electric motor to spin the engine. This way engine friction and pumping torque losses can be measured.

For an electric dynamometer, the rotor is connected to the crankshaft. The link between rotor and stator is electromagnetic. The stator is fixed through a lever arm to a **load cell**. To balance the rotor, the stator will push against the load cell. The torque *T* is calculated by multiplying the force *F* measured in the load cell with the length of the lever arm *a*.

The engine parameters: brake torque, brake horse power (bhp) or brake specific fuel consumption (BSFC) contain the keyword “brake” because a dynamometer (brake) is used to measure them.

What comes out from a dynamometer engine test are **torque maps** (surfaces) which give the value of the engine torque at a specific engine speed and load (stationary operating points). The load of the engine is equivalent to the position of the accelerator pedal.

Example of **torque map for a gasoline, spark ignition (SI) engine**:

Enginetorque [Nm] | Accelerator pedal position [%] | ||||||||

5 | 10 | 20 | 30 | 40 | 50 | 60 | 100 | ||

Engine speed [rpm] | 800 | 45 | 90 | 107 | 109 | 110 | 111 | 114 | 116 |

1300 | 60 | 105 | 132 | 133 | 134 | 136 | 138 | 141 | |

1800 | 35 | 89 | 133 | 141 | 142 | 144 | 145 | 149 | |

2300 | 19 | 70 | 133 | 147 | 148 | 150 | 151 | 155 | |

2800 | 3 | 55 | 133 | 153 | 159 | 161 | 163 | 165 | |

3300 | 0 | 41 | 126 | 152 | 161 | 165 | 167 | 171 | |

3800 | 0 | 33 | 116 | 150 | 160 | 167 | 170 | 175 | |

4300 | 0 | 26 | 110 | 155 | 169 | 176 | 180 | 184 | |

4800 | 0 | 18 | 106 | 155 | 174 | 179 | 185 | 190 | |

5300 | 0 | 12 | 96 | 147 | 167 | 175 | 181 | 187 | |

5800 | 0 | 4 | 84 | 136 | 161 | 170 | 175 | 183 | |

6300 | 0 | 0 | 72 | 120 | 145 | 153 | 159 | 171 |

Example of **power map for a gasoline, spark ignition (SI) engine**:

Enginepower [HP] | Accelerator pedal position [%] | ||||||||

5 | 10 | 20 | 30 | 40 | 50 | 60 | 100 | ||

Engine speed [rpm] | 800 | 5 | 10 | 12 | 12 | 13 | 13 | 13 | 13 |

1300 | 11 | 19 | 24 | 25 | 25 | 25 | 26 | 26 | |

1800 | 9 | 23 | 34 | 36 | 36 | 37 | 37 | 38 | |

2300 | 6 | 23 | 44 | 48 | 48 | 49 | 49 | 51 | |

2800 | 1 | 22 | 53 | 61 | 63 | 64 | 65 | 66 | |

3300 | 0 | 19 | 59 | 71 | 76 | 78 | 78 | 80 | |

3800 | 0 | 18 | 63 | 81 | 87 | 90 | 92 | 95 | |

4300 | 0 | 16 | 67 | 95 | 103 | 108 | 110 | 113 | |

4800 | 0 | 12 | 72 | 106 | 119 | 122 | 126 | 130 | |

5300 | 0 | 9 | 72 | 111 | 126 | 132 | 137 | 141 | |

5800 | 0 | 3 | 69 | 112 | 133 | 140 | 145 | 151 | |

6300 | 0 | 0 | 65 | 108 | 130 | 137 | 143 | 153 |

The electronic control module (ECM) of an ICE has the torque map stored in the memory. It calculates (interpolates) the engine torque function of the current engine speed and load. In the ECM, the load is expressed as intake manifold pressure for gasoline (spark ignition, SI) engines and injection time or fuel mass for diesel (compression ignition, CI) engines. The engine torque calculation strategy has corrections based on temperature and intake air pressure.

Plotting the torque and power data, function of engine speed and load, gives the following surfaces:

For a better interpretation of the torque and power maps, a 2-D torque line can be plotted for a fixed value of the accelerator pedal position.

### Engine torque and power at full load

As you have seen, the torque and power of an internal combustion engine depend on both engine speed and load. Usually, engine manufacturers are publishing the torque and curve characteristics (curves) at **full load** (100% accelerator pedal position). Full load torque and power curves highlight the maximum torque and power distribution through the whole range of engine speed.

The shape of the above torque and power curves are not from a real engine, the scope being to explain the main parameters. Nevertheless, the shapes are similar to the real characteristics of a spark ignited (gasoline), port injection, atmospheric engine.

Engine speed *N _{e} [rpm]* is characterized by four main points:

*N _{min}* – is the minimum stable engine speed at full load

*N*– is the engine speed at maximum engine torque

_{Tmax}*N*– is the engine speed at maximum engine power; also called

_{Pmax}**rated engine speed**

*N*– is the maximum stable engine speed

_{max}At minimum speed, the engine should run smoothly, without oscillations or stalling. The engine should also allow operation at the maximum speed without any structural damage.

The **full load engine torque** curve *T _{e} [Nm]* is characterized by four points:

*T _{0}* – engine torque at minimum engine speed

*T*– maximum engine torque (peak torque or

_{max}**rated torque**)

*T*– engine torque at maximum engine power

_{P}*T*– engine torque at maximum engine speed

_{M}Depending on the type of intake air (atmospheric or turbocharged) the peak torque can be either a point or a line. For turbocharged or supercharged engines, maximum torque can be kept constant between two engine speed values.

The **full load engine power** curve *P _{e} [HP]* is characterized by four points:

*P _{0}* – engine power at minimum engine speed

*P*– maximum engine power (peak power or

_{max}**rated power**)

*P*– engine power at maximum engine torque

_{T}*P*– engine power at maximum engine speed

_{M}The area between minimum engine speed *N _{min}* and maximum torque engine speed

*N*is called

_{Tmax}**low end**torque zone. The higher the torque in this area, the better the launch/acceleration capabilities of the vehicle. When the engine is operating in this area, at full load, if the road resistance increase, the engine speed will decrease, which will result in a drop of engine torque and an

**engine stall**. For this reason, this area is also called

**unstable torque region**.

The area between maximum torque engine speed *N _{Tmax}* and maximum power engine speed

*N*is called the

_{Pmax}**power band**. During vehicle acceleration, for best performance, the gearshift (up) should be performed at maximum engine power. Depending on the gear ratios of the gearbox, after the gearshift, the selected gear will drop the engine speed at maximum torque, which will give optimum acceleration. Shifting the gears at maximum engine power will keep the engine speed within the power band.

The area between maximum power engine speed *N _{Pmax}* and maximum engine speed

*N*is called

_{max}**high end**torque zone. Higher torque results in higher output power, which translates in higher maximum vehicle speed and better acceleration at high speed.

When the engine speed is kept between maximum torque engine speed *N _{Tmax}* and maximum engine speed

*N*, if the vehicle road resistance increases, the engine speed will drop and the output torque will increase, thus compensating for the road load increase. For this reason, this area is called the

_{max}**stable torque region**.

Below you can find some full load torque and power curves examples for different types of engines. Notice the shape of the curves function of the type of the engine (spark ignited or compression ignited) and type of air intake (atmospheric or turbo(super)charged).

### Honda 2.0 engine torque and power at full load

Cylinders architecture | 4 in-line | |

Fuel | gasoline (SI) | |

Engine capacity [cm^{3}] | 1998 | |

Fuel injection | valve port | |

Air intake | atmospheric | |

Valve timing | variable | |

T_{max} [Nm] | 190 | |

N_{Tmax} [rpm] | 4500 | |

P_{max} [HP] | 155 | |

N_{Pmax} [rpm] | 6000 | |

N_{max} [rpm] | 6800 |

### Saab 2.0T engine torque and power at full load

Cylinders architecture | 4 in-line | |

Fuel | gasoline (SI) | |

Engine capacity [cm^{3}] | 1998 | |

Fuel injection | valve port | |

Air intake | turbocharged | |

Valve timing | fixed | |

T_{max} [Nm] | 265 | |

N_{Tmax} [rpm] | 2500 | |

P_{max} [HP] | 175 | |

N_{Pmax} [rpm] | 5500 | |

N_{max} [rpm] | 6300 |

### Audi 2.0 TFSI engine torque and power at full load

Cylinders architecture | 4 in-line | |

Fuel | gasoline (SI) | |

Engine capacity [cm^{3}] | 1994 | |

Fuel injection | direct | |

Air intake | turbocharged | |

Valve timing | fixed | |

T_{max} [Nm] | 280 | |

N_{Tmax} [rpm] | 1800 – 5000 | |

P_{max} [HP] | 200 | |

N_{Pmax} [rpm] | 5100 – 6000 | |

N_{max} [rpm] | 6500 |

### Toyota 2.0 D-4D engine torque and power at full load

Cylinders architecture | 4 in-line | |

Fuel | diesel (CI) | |

Engine capacity [cm^{3}] | 1998 | |

Fuel injection | direct | |

Air intake | turbocharged | |

Valve timing | fixed | |

T_{max} [Nm] | 300 | |

N_{Tmax} [rpm] | 2000 – 2800 | |

P_{max} [HP] | 126 | |

N_{Pmax} [rpm] | 3600 | |

N_{max} [rpm] | 5200 |

### Mercedes-Benz 1.8 Kompressor engine torque and power at full load

Cylinders architecture | 4 in-line | |

Fuel | gasoline | |

Engine capacity [cm^{3}] | 1796 | |

Fuel injection | valve port | |

Air intake | supercharged | |

Valve timing | fixed | |

T_{max} [Nm] | 230 | |

N_{Tmax} [rpm] | 2800 – 4600 | |

P_{max} [HP] | 156 | |

N_{Pmax} [rpm] | 5200 | |

N_{max} [rpm] | 6250 |

### BMW 3.0 TwinTurbo engine torque and power at full load

Cylinders architecture | 6 in-line | |

Fuel | gasoline | |

Engine capacity [cm^{3}] | 2979 | |

Fuel injection | direct | |

Air intake | dual-stage turbocharged | |

Valve timing | variable | |

T_{max} [Nm] | 400 | |

N_{Tmax} [rpm] | 1300 – 5000 | |

P_{max} [HP] | 306 | |

N_{Pmax} [rpm] | 5800 | |

N_{max} [rpm] | 7000 |

### Mazda 2.6 rotary engine torque and power at full load

Cylinders architecture | 2 Wankel | |

Fuel | gasoline | |

Engine capacity [cm^{3}] | 1308 (2616) | |

Fuel injection | valve port | |

Air intake | atmospheric | |

Valve timing | fixed | |

T_{max} [Nm] | 211 | |

N_{Tmax} [rpm] | 5500 | |

P_{max} [HP] | 231 | |

N_{Pmax} [rpm] | 8200 | |

N_{max} [rpm] | 9500 |

### Porsche 3.6 engine torque and power at full load

Cylinders architecture | 6 flat | |

Fuel | gasoline | |

Engine capacity [cm^{3}] | 3600 | |

Fuel injection | valve port | |

Air intake | atmospheric | |

Valve timing | variable | |

T_{max} [Nm] | 405 | |

N_{Tmax} [rpm] | 5500 | |

P_{max} [HP] | 415 | |

N_{Pmax} [rpm] | 7600 | |

N_{max} [rpm] | 8400 |

Key statements to keep in mind regarding engine power and torque:

**Torque**

- torque is a component of power
- torque can be increased by increasing the mean effective pressure of the engine or by lowering the torque losses (friction, pumping)
- having a lower maximum torque distributed on a range of engine speeds its better from the traction point of view than having a higher maximum torque point
- low end torque is very important for the launch capabilities of the vehicles
- high torque is beneficial in off-road situation, when the vehicle is operated at high road gradients but low speed

**Power**

- engine power depends on both torque and speed
- power can be increased by increasing the torque or the engine speed
- high power is important for high vehicle speeds, the higher the maximum power the higher the maximum speed of the vehicle
- engine power distribution at full load, through the engine speed range, affects the acceleration capability of the vehicle at high speeds
- for best acceleration performance, a vehicle should be operated in the power band, between maximum engine torque and power

For any questions or observations regarding this tutorial please use the comment form below.

Don’t forget to Like, Share and Subscribe!

## Kevin J Kelleher

Anthony, the two links I provided include an explanation of the logic, which is intuitive once the graphs of wheel torque vs mph in each gear are observed. No need for another article. You go all the way down to bmep, to your credit, then you go thin on the stable and unstable powerbands !

## Kevin J Kelleher

This conclusion of yours, under “Power”, is not correct:

“for best acceleration performance, a vehicle should be operated in the power band, between maximum engine torque and power”

Shifting at the rpm for max hp is always short shifting. See my links for optimum shift points, that are always above the max hp rpm.

## Kevin J Kelleher

Anthony,

Everything looks great, except when the optimum power band for shifting is defined by two rpm points.

Based on my spreadsheet work, the optimum shift points must include gearbox ratios for the axle torque in the next gear. It is an iterative process, and graphical solutions are the best approach. I have 2 links, this one shows the result of this spread sheet approach to find shift points for 3 different engines:

https://www.evodynamics.com/blogs/news/114447365-calculating-shift-points

And this link allows you to plug in your engine’s torque vs rpm curve incrementally. You also input your gear ratios. You have tables with this data.

http://glennmessersmith.com/shiftpt.html

As I noted before, for steep hp curves with little reduction near redline, that will be your shift point. For your 2L turbo example, the rpm band related to the shift point will be about centered on the peak hp rpm. A bit like an area under the hp curve.

Kevin

## Kevin J Kelleher

I made my spreadsheet to determine optimum shift points about 15+ years ago, when it was just a web argument with nobody providing calculations. Evodynamics, in 2016, published their method, which is exactly the same as what I had done. I encourage you to apply that method (2nd link) to the Toyota 2L diesel, using your 100% throttle torque values.

## Aravindh Babu

So on the torque surface that you have, how does an ECM actually move through that surface when driving a vehicle? Let’s say you stay in one gear, and you press down on the pedal so that the position goes from 5% to 20%. Does it try and hold speed constant while increasing the torque and then increase speed? Or does it hold torque constant while increasing the speed, and then increase the torque? Or is it some, more complicated, pathway? If so, what considerations might go into deciding that pathway?

Thanks

## Muhamad Samsul Bin Abd Samad

12bar of pressure is equal to 1,200,000p.a for your example…not 120,000…

## Alok

Dear Anthony,

This is a remarkably simple to understand stuff. Great job.

In real life an engine will always will have a set of transmission ratios (certainly more than 1) thus the torque/ power curve for engine speeds appear certainly different. I made a test rig to measure the torque / power for different gear ratios and realised that this way the engine will reveal better application details for a vehicle. I would like that to be published sometime!

Excellent initiative………

## Kevin J Kelleher

Hi,

Looks like a great presentation. As a Mechanical Engineering (PE) Consultant, that reviewed many papers destined for the ASME publications, I naturally focused on an apparent error. I left the comment below. I took data off dyno curves, at constant rpm increments, like 50 or 100 rpms, and created tables for calculated axle torque and related miles per hour, for each rpm on the curve. I repeated this for each gear ratio. I can provide more info, or perhaps send copies of some examples.

Regards,

Kevin Kelleher

Good job, except your “power band” is wrong. For most NA gas engines, shift at redline for max acc’n. If turbocharged and hp is about same magnitude as torque, you will shift beyond hp peak, but not at redline. Using excel, I created axle torque vs car mph curves for each gear, based on dyno curves, gear ratios, tire size, etc, and confirmed this. Also did the same with hp curves vs mph, with same results.

## Anthony Stark

Hi Kevin,

If you want you can write an article, send it together with some pictures and I can publish it as you being the author.

I also plan to write an article on how the gear ratios impact the output torque.

Thanks for your feedback.

## Kevin Kelleher

Good job, except your “power band” is wrong. For most NA gas engines, shift at redline for max acc’n. If turbocharged and hp is about same magnitude as torque, you will shift beyond hp peak, but not at redline. Using excel, I created axle torque vs car mph curves for each gear, based on dyno curves, gear ratios, ire size, etc, and confirmed this. Also did the same with hp curves vs mph, with same results.

## David Trudeau

This would be better and easier to understand without the imperial conversion. They could all be put at the end.

## Paurnima T

useful info.

Thank you so much!

## Uvanesh

Great site to learn. Detailed information on every topic. Thanks a lot ..!