# How to calculate theorectical engine max. torque from max. power and RPM?

I'm trying to come up with a formula to estimate an engine's maximum torque (and at which RPM) given other values and some assumptions.

I'm approximating an engine's torque curve as 2 linear functions $t_1(r)$ and $t_2(r)$, with $r$ being the engine's RPM, that connects to each other at a specific RPM. $t_1(r)$ has a positive slope and $t_2(r)$ has a negative slope, making the RPM that connects the both functions the point of maximum torque, namely $r_{t_{max}}$.

From Wikipedia, I found out that the engine's power relates to the torque, approximately, through the formula: $Power = (Torque · RPM)/5252$. From this, I assumed that the power curve in the example could be written as $p_i(r) = (t_i(r) · r)/5252$.

Given the engine's maximum power $P_{max}$ (and corresponding RPM, $r_{p_{max}})$, maximum RPM (redline) $r_{max}$ and corresponding torque at redline RPM $T_f$, is it possible to obtain a formula to calculate theoretical maximum torque $T_{max}$ (and corresponding RPM $r_{t_{max}}$)?

For example, for a car with max. power 375hp at 5500rpm, with 6500rpm redline and 175lb-ft of torque at that redline, what would be the maximum torque (and at which RPM) if one would use the said torque curve assumption?

I attempted to reach a formula through the following reasoning:

Since $t_2(r)$ starts at the maximum torque point ($[r_{t_{max}},r_{max} ]$ range) and the maximum power is between $r_{t_{max}}$ and $r_{max}$, $r_{p_{max}}$ is within the range of $t_2(r)$.

Calling $t_2(r)$ as $t(r)$:

$t(r) = a·r+b$

$t(r_{t_{max}}) = a·r_{t_{max}}+b = T_{max}$

$t(r_{max}) = a·r_{max}+b = T_f$

$a = (T_{max} -T_f)/(r_{t_{max}} - r_{max})$

$b = (r_{t_{max}}·T_f - r_{max}·T_{max})/(r_{t_{max}} - r_{max})$

Since $p_2(r) = t_2(r)·r/5252$ (assuming $p(r)$ as $p_2(r)$):

$p(r) = a·r² + b·r$

Which would be a quadratic function. Looking up at Wikipedia, the following formula gives the vertex point of a quadratic function:

$x_{max} = -b/(2·a)$

So the point of maximum of $p(r)$ would be:

$r_{p_{max}} = - ((r_{t_{max}}·T_f - r_{max}·T_{max})/(r_{t_{max}} - >r_{max}))/(2·((T_{max} -T_f)/(r_{t_{max}} - r_{max})))$

Which I reduced to:

$r_{t_{max}} = (T_{max}/T_f)·(r_{max}-2r_{p_{max}})+2r_{p_{max}}$

Which, initially, looked fine, but soon I realized that this formula doesn't use the engine maximum power and appears to be only sensitive to the $(T_{max}/T_f)$ ratio.

EDIT:

Here is a graph with an example:

Within the ranges of $t_1(x)$ and $t_2(x)$, $y$ varies between 0 and 1, which here represents the fraction of the maximum torque (like a normalized function). $p_1(x)$ and $p_2(x)$ are power curves resulting from the given torque curves. If maximum RPM $r_{max}$ was at $x=4.0$ in this graph, the redline torque $T_f$ would be $y=0.5$. Maximum power $P_{max}$ would then be $y=2.25$ at $x=3.0$ RPM ($r_{p_{max}}$). Consequently, maximum torque $T_{max}$ would be at $x=2.0$ RPM ($r_{t_{max}}$).

The real question I'm posing here is to find $T_{max}$ and $r_{t_{max}}$, given $r_{max}$, $P_{max}$, $r_{p_{max}}$ and $T_f$, and given the assumption about the linearity of the torque curve. If the problem has more than one solution - and thus a indeterminate system - then at least a formula with the relation between $T_{max}$ and $r_{t_{max}}$ would be enough.

• Your problem is not well posed in that you miss one piece of information to get your answer. Your 2 data point at max power and max RPM allow you to fit a straight line, but max torque can be anywhere on that line. To know where it is, or equivalent where it crosses the other straight line, you need more information. – Manuel Fortin Oct 20 '17 at 0:51
• agree with manuel. better approach would be to obtain the actual torque curve for the engine and identify the maximum torque point, rather than trying to do curve fits in the face of missing data. You might also find it useful to review the literature dealing with engine performance curves and the relationship between torque, power, and rpm. – niels nielsen Oct 20 '17 at 4:15
• @user40292, that is exactly the scenario that I'm trying to deal with: in the face of missing data. Sometimes such missing data cannot be found on the Internet and trips to a dyno would be expensive. – Hydren Oct 20 '17 at 13:50
• @ManuelFortin, which data is needed? Could you shed some light? – Hydren Oct 20 '17 at 13:51
• Anything that gives you some information about where the two lines of the torque curve intersect. It could be the RPM at max torque, or two points that define the other line segment (for low RPM). Currently, you have more unknown that equations. – Manuel Fortin Oct 20 '17 at 14:30