Have you ever heard of circuit simulation?
It is a way of calculating how fast a car can theoretically lap a circuit, based on the car’s performance and the racing line.
The circuit simulation I’m introducing here was created by Tatsu.
According to him, it was “based on the F-1 Circuit Simulation published in the Honda R&D Technical Review, and converted into an Excel spreadsheet.”
When I first heard about it, I thought, “What the heck is that?!”
Someone like me, who can’t even do two-digit addition, has no business being taught by Tatsu. But thanks to his kindness, he taught me all kinds of things about it.
Tatsu has made the entire process of creating the simulation publicly available because he wants this kind of method to become more widely used.
I studied the process myself by reading through the articles he published:
How to Create an Excel-Based Circuit Simulation (Part 1)
How to Create an Excel-Based Circuit Simulation (Part 2)
How to Create an Excel-Based Circuit Simulation (Part 3)
How to Create an Excel-Based Circuit Simulation (Part 4)
How to Create an Excel-Based Circuit Simulation (Part 5)
How to Create an Excel-Based Circuit Simulation (Course Diagram)
The basic idea is explained in Part 1, but it took me an absurd amount of time just to understand that one article.
That said, everything is broken down into individual sections, so if you take your time and work through them one by one, you should eventually be able to understand it.
Once you get it, you realize just how carefully and clearly the article is written.
If it hadn’t been explained this clearly, I probably never would have understood it… I mean that.
As a side note, when I was in high school, I got called into a teacher’s office and was told:
“Your test scores are good in every subject except math. Why are your math scores so terrible?! Your name even came up in a teachers’ meeting! So, along with a few other students, you’re going to take part in a special math program.”
And so I was forced to stay after school for several months, doing math I couldn’t make any sense of.
But what was I supposed to do? Japanese and English were easy because all I had to do was remember things and write them down. But when I looked at a math problem, I had absolutely no idea what I was supposed to do.
Even someone like me was able to understand it with enough effort, so I’m sure you can too.
Some wise person once said, “If you’ve got the energy, you can do anything.”
But even if you’re full of energy, you still can’t fire a Kamehameha, so I don’t think you can do *everything*.
But I think you can understand this one.
From here on, rather than explaining the specific steps for actually building a circuit simulation in Excel, I’ll try to explain the basic idea as I understand it.
If I break down the basic concept of this simulation, it comes down to four things.
First, you decide on the racing line, then you set the limits of the forces the tires can generate along that line.
From there, you calculate the possible speed at each point, and finally calculate the lap time for one complete lap.
Basic Idea #1: “The Racing Line Comes First”
First of all, the racing line is not calculated.
You enter the radius of curvature hundreds of times yourself and determine the racing line.
Basically, you enter it so that it follows the actual racing line.
Then, the calculations tell you, “If you drive this line, what is the fastest lap time you can achieve?”
If you think, “Maybe I could go faster with a line like this,” you can run the calculation again with that line.
If the calculated result is faster, great. If it’s slower, it’s no good. So evaluating the result is simple.
You keep drawing different lines and running the calculations until you find the fastest one.
Basic Idea #2: “What Speed Can the Car Reach at Maximum Lateral G?”
First, use a GPS data logger to measure the car’s maximum lateral G.
With GPS-based data loggers such as LAP+, the displayed lateral G can be higher than the actual value, so you take the two values at the point where maximum lateral G was recorded: the radius of curvature of the line and the speed. Then you recalculate the actual lateral G.
The actual calculation is not important here, so I’ll leave it out. If you’re interested, check out Tatsu’s articles explaining how to create the circuit simulation.
For now, all you really need to remember is that if you know the radius of curvature and the speed, you can calculate the lateral G.
Conversely, if you want to know the speed, you can calculate it if you know the other two.
The smaller the radius of curvature, the greater the lateral G at the same speed. Conversely, if you want to maintain the same lateral G, you have to slow down as the radius of curvature gets smaller.
In my case, based on my actual driving data, I set the maximum lateral acceleration the car can generate at 1.15 G.
Since the racing line has already been determined, we know the radius of curvature at every point of each corner. So we can calculate things like:
What speed produces 1.15 G of lateral acceleration at Turn 1?
What about Turn 2? Turn 3?
(Next, we also need to determine the maximum acceleration and braking G, but that’s a little more complicated, so I’ll skip it here. You don’t need to understand it for now. After all the calculations and whatnot, I set my car’s maximum acceleration at 0.7 G and maximum braking at 0.8 G.)
Basic Idea #3: “The Tire’s Grip Cannot Exceed the Friction Circle”
Here’s the friction circle, which I’m sure you’re all familiar with.
The forces a tire can generate in the longitudinal and lateral directions are limited, and this is generally represented by the “friction circle.”
In the circuit simulation I’m introducing here, this is handled as “tire utilization.”
Here, tire utilization is defined as:
{ (longitudinal G ÷ maximum longitudinal G)^2 + (lateral G ÷ maximum lateral G)^2 }^0.5
Again, the actual formula isn’t important. All you really need to remember is that if you know the “longitudinal and lateral G currently being generated” and the “maximum longitudinal and lateral G the tires can generate,” you can calculate tire utilization.
When you’re right at the limit of the friction circle, tire utilization is 1 (100%).
Basic Idea #4: “Calculate Speed by Working Backward from the Minimum-Speed Point”
This is the main point.
For example, let’s say the minimum speed at Turn 2 at Takasu Circuit is 45 km/h.
In that case, the car must be traveling at more than 45 km/h both before and after the point where the minimum speed occurs.
After all, it’s called the “minimum speed” for a reason.
With a racing line, the radius of curvature generally gets smaller as you approach the point where the minimum speed occurs, then gets larger again as you move away from it.
For example, when entering a corner, you start with the steering wheel straight, gradually turn it more and more, and reach maximum steering angle at the minimum-speed point. Naturally, the radius of curvature of the line gradually becomes smaller to match this steering input.
Now, let’s make an initial assumption about the “speed at each point along the racing line.”
Let’s assume that at every point along the racing line, the car is traveling at the speed required to generate 1.15 G of lateral acceleration at that point.
As I explained in Basic Idea #2, the speed corresponding to a given lateral G is determined once the radius of curvature is known.
And since we’re assuming that lateral G is always 1.15 G, the car’s speed changes according to the radius of curvature.
Obviously, the smaller the radius of curvature, the lower the speed. The larger the radius, the higher the speed.
Let’s look at Turn 2 at Takasu Circuit.
First, there’s a straight leading into the corner.
A straight line has zero curvature, so if you tried to generate 1.15 G of lateral acceleration there, the calculated speed would be infinite.
Of course, this does not mean that you can actually drive at infinite speed. Mathematically, it simply means that if you calculate the speed required to generate 1.15 G of lateral acceleration from curvature alone, the speed approaches infinity as the curvature approaches zero.
In reality, the line isn’t perfectly straight, so the curvature will be slightly greater than zero, but the calculated speed will still be extremely high.
(Again, we’re only talking about the speed required to generate 1.15 G of lateral acceleration at that radius of curvature.)
Conversely, at the point where the minimum speed occurs in Turn 2, the radius of curvature of the line is at its smallest, around 15 meters.
LAP+ can also display the turning radius, so if you look at actual driving data, you’ll probably find that most people are around there.
And based on the assumption I made at the very beginning, let’s say the speed at this point is about 45 km/h.
As the radius of curvature changes from infinity to 15 meters, there is a smooth progression through values such as 500 m, 100 m, 50 m, 30 m, 20 m, 17 m…
If you calculate the speed required to generate 1.15 G of lateral acceleration at each of those points, you might get something like 270 km/h, 120 km/h, 85 km/h, 65 km/h, 55 km/h, 50 km/h…
But here’s the bad news: my actual car cannot drive through Turn 2 with a speed progression like this… 85 km/h, 65 km/h, 55 km/h, 50 km/h…
The reason is that these speeds were calculated as the speeds required to generate 1.15 G of lateral acceleration at each radius of curvature, so the longitudinal tire force required for braking has not been taken into account.
A tire cannot generate more grip than the friction circle allows, and my car can generate a maximum of only 1.15 G of lateral acceleration.
But if you look at the speed progression above, the speed is decreasing.
The speed is decreasing… which means the car is braking.
Ignoring external forces such as aerodynamic drag, the car has to use tire forces to slow down. So this would mean that “the tires are continuously using all of their available grip laterally to generate 1.15 G, while somehow also using additional grip in the braking direction.”
That simply isn’t possible. It’s a contradiction.
So how can we calculate the actual speed progression?
The answer is simple: just make the speed at each point lower to begin with.
If the speed is lower, the car generates less lateral G at the same radius of curvature, which means the remaining tire grip can be used for braking.
Yay! Problem solved. End of story.
…Nope. How much do we need to reduce the speed?
At 85 km/h, should we drop all the way down to 10 km/h? Or maybe 2 km/h?
That would make the lap time ridiculously slow.
The whole point of a circuit simulation is to calculate the fastest possible lap time, so that wouldn’t be useful.
So, we want to minimize the reduction in speed. Let’s start by reducing the speed by just 1 km/h and run the calculation.
This is where the “tire utilization” from Basic Idea #3 comes in.
With all the necessary elements already determined, we can simply enter the numbers and calculate the tire utilization.
The result is that even after reducing the speed by just 1 km/h, from 85 km/h to 84 km/h, tire utilization is still far above 100%.
That won’t work.
Next, we immediately recalculate it after reducing the speed by 2 km/h.
The result… still over 100%.
So we recalculate it after reducing the speed by 3 km/h.
No good.
Then 4 km/h.
Still no good.
Then 5 km/h.
Still not enough.
And if that doesn’t work, we try 6 km/h… 7 km/h…
…and keep repeating the process until we get:
Exactly 100%!!!
A spreadsheet program’s “macro” function allows you to perform this kind of repeated process.
In other words, you can have the computer automatically repeat the calculation over and over until it reaches the value you’re looking for.
Once the speed at which tire utilization reaches 100% has been determined, you use the same process to calculate the speed at the previous point, working backward from there.
This is how the braking-side speeds are determined.
The same process is used for the acceleration side.
However, while the braking side only requires us to consider tire utilization, the acceleration side also requires us to take into account the “driving force per tire” and the “number of driven tires (drivetrain layout),” so the calculation is a little more complicated.
But basically, we’re doing the same thing.
Once the speeds on the acceleration and braking sides have been calculated, the lower of the two is selected as “the speed at that point.”
This determines the speed at every point along the racing line for one complete lap, which then allows us to calculate the lap time.
This time, I have skipped most of the actual calculation methods and explained only the basic concepts, within the limits of what I was able to understand.
To actually build a circuit simulation in Excel, you still have to solve several practical issues, so if you want to try it yourself, I recommend referring to Tatsu’s published instructions and giving it a go.
This may have been a complicated explanation, but I think this is basically how it works.
Until now, I hadn’t really understood what is actually happening physically when a car drives around a circuit as fast as possible, but I’m starting to get a better idea of it, little by little.
By the way, when calculating the speed on the acceleration side, the assumed acceleration performance differs between 2WD and 4WD, or between low-powered and high-powered cars. Because of this, the optimal racing line can also differ slightly (depending on the shape of the corner).
Also, yaw moment and similar factors are not used in this calculation at all, so how easy or difficult the car is to turn does not directly affect the lap time calculated by the simulation.
The simulation only calculates the movement of the car’s center of gravity.
In a real car, however, weight distribution and other factors do affect how the car performs.
So how do they affect it?
They affect how far the real car is from the ideal state.
In other words, some factor prevents the car from driving the ideal line at the ideal speed.
Factors such as drivetrain layout, tire response, how quickly the car responds to steering input, the magnitude of the gain, damping adjustment, alignment-related factors, the amount of body roll, the way load transfer occurs, and so on all affect either acceleration, braking, or cornering force.
Ultimately, they can all be expressed as the longitudinal and lateral G at a given point.
In other words, what this simulation calculates is the theoretical lap time when the car can drive the ideal line at the ideal speed.
How far the actual car moves away from that ideal state because of the setup or the driver’s inputs.
I think understanding that is also important when trying to understand what makes a real car fast.
So, that’s my explanation of how to think about circuit simulation.
So, everyone, can you fire a Kamehameha now… I mean, did you understand the basic idea?
This way of thinking is very important for understanding how a car behaves when trying to drive a fixed circuit as quickly as possible, so I hope you find it useful.