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Old Sep 23, 2010, 01:48 PM
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How to calculate flight time - guide

When I recently start to deal with multi-rotor designs, I noticed how those systems are really power hungry: batteries with capacity up to 4000 and more mAh for only 10 minutes or less flying time. As I'm used to fly EasyStar plane for 30 and more minutes with 2200 mAh LiPo, that seamed as quite a difference, but I fast realized that 4, 6 or 8 motors constantly spinning in order to keep that weird looking contraption airborne just need that big amount of power. The more power it needs, less battery run time we will have.

As I'm primarily interested in AP, non-aerobatic multi-rotor that can be airborne as much as possible, my first reaction was: I'll will add one more battery and double my flight time. Of course, very soon became obvious that this is not that simple . Inspired with many posts on this forum, I started to gather information that could help me understand relation between multi-rotor configuration and flying time. I have found a lot of useful information, so I tried to compile short overview. I would be thankful if you could help me by checking if this make sense (keep in mind that this is approximation only)

Let’s imagine that we have following multi-rotor setup for analysis:
  • hexacopter
  • airframe weight (without battery) 2000 g
  • battery 3S, 2200 mAh, 200g
  • APC 10x4.7 prop
  • motor similar to Hacker A20-22L

We want this thing to fly and we would like to know how much flying time we can expect. We know copter weight, but how can we connect it to flying time? Although relation between weight and battery run time may seam a bit complicated, it's in fact quite straightforward. Here is what connects those parameters:
  • weight defines required prop/motor thrust
  • prop/motor thrust defines needed power
  • power defines needed current
  • currents determines battery run time


It is basically set for 4 standard physical and electrical rules. Each rule (we refer it as step) will result with data that we will use as input for next step. As initial input we have our weight and as final output battery run time. By solving the data for each rule, we can easy connect weight to flight time. Let’s analyze each step…
Step 1 - Thrust

If we want hexacopter from our example to hover, we have to beat the gravity. Weight of system is 2200 g (airframe + battery). Its 6 motors/props have to produce at least 2200 g of thrust in order to beat the gravity. As we have 6 motors, each one has to produce 2000 / 6= 367 g of thrust (with assumption that all motors/props generate equal thrust). Easy, right . Now we have to find out power needed to generate that amount of thrust.

Step 2 - Power

In order to get required power, we need some kind of relationship between thrust and power. Best way for this is to use „thrust to power“ graph, which gives us relation between static thrust and power for given motor, propeller and battery. There are some such graphs on the net, but if you lack some, with some effort you can do them your self. This is done in a way that you run motor through several tests with given prop and fully charged battery, where you gradually increase motor power (throttle stick position) and measure achieved thrust together with consumed power. From measured data you can than plot „thrust to power“ graph which gives needed relation. There are lot ideas and hints in this forum how to measure static thrust, so I will not go into details.

Note: as power which motor will need is determined by propeller and battery (number of cells), certain graph is valid only for prop and battery that are used during testing. If you change prop or number of cells, you have to use graph for that prop and battery combination.

In our example will use one such graph, for motor similar to A20-22L, which I derived from Dr. Kiwi data. For simplicity reasons, I represent it in linear form (although it’s quite similar to real curve in given thrust range). In reality graph can be a complex curve. Still, regardless to curve shape, method is the same. We will use this graph only for illustration purposes, don’t take it as a real reference for A20-22L motor.




Going back to our example… We said that for hover we need 367 g of thrust per motor. If we look into chart, we can see that motor will take around 37W in order to produce that thrust with 10x4.7 prop on 3s LiPo. Now we know required power, let’s find out the current.

Step 3 - Current

By knowing required power and battery voltage, it is straightforward to calculate motor current:

I = P / U
I – motor current (A)
P – motor power (W)
U – battery voltage (V)


As we have 3s LiPo with nominal voltage of 11.1 V, required current is:

37/11.1 = 3.3 A

This is current that single motor will draw in order to hover our hexacopter. As we have total of 6 motors, overall current consumption will be 6 * 3.6 = 19.8 A. One step left…


Step 4 – Battery run time

By knowing battery capacity and current consumption, we can calculate how long we can draw that amount of current from battery. For that we will use following formula:

T = C / I * 60
T – time (min)
C – battery capacity (Ah)
I – current (A)


For our example we finally get (2.2 / 19.8) * 60 = 6.6 min. There it is

Of course, calculation assumes ideal battery. Real batteries have internal resistance, discharge rate, etc., so in real life this time would be a bit shorter. Still, this should give us good battery run-time indication.


So, our hexacopter with AUW of 2200 g, with 2100 mAh battery should be able to hover for around 6 minutes. Not that much impressive. Let’s see what will happen if we use additional battery, same as the first one. How much flight we could expect from that?

Well, AUW of hexacopter with two batteries will be 2000 + 200 + 200 = 2400 g. Required hover thrust per motor is then 2400/6 = 400 g. From graph we can see that required power is around 45W, which gives 45 / 11.1 = 4.1 A per motor or 24.6A for all 6 of them. As our battery capacity is now 4.4 Ah, we are getting following flying time:

(4.4 / 24.6) * 60 = 10.7 min

We can see that we have increased battery capacity by 100%, but extended hover time for 62%. People often think that doubling the batteries will in same way double the flight time. Of course that this is not completely true, because by adding additional batteries we are adding additional weight which increase current consumption, therefore making batteries last shorter.


Besides getting battery run-time, we can also get other useful information from calculated data. For instance, by knowing required motor power to hover, you can see in which power range will motor run or how much power is there in reserve for maneuvers. In our example, hover will require 40W, while motor is rated to lets say 180W. Big reserve, so maybe smaller motor could be better suited.

Also from motor current that we calculated in step 3 and motor parameters Rm and Io, we can calculate motor efficiency. By knowing motor efficiency, we can judge how efficient our system will be in hover. I will not go into details about efficiency, as this is out of scope of this topic, but I will give formula that picked up on the net, so you can try to play with data:

Efficiency (%) = Pout / Pin * 100

Pin = Vin x Iin
Pout = (Vin - Iin * Rm) * (Iin - Io)
Vin – input voltage (measured at ESC input)
Iin – input current (measured at ESC input)
Rm - motor resistance
Io - current without load


Well, that was it

To summarize, by knowing for given motor-prop-battery combo “thrust to power” relation, it is easy to predict how flying time will change with different AUW, battery, motor-prop combo, etc. Of course it will be still only approximation, as we are not taking into account things like power consumption changes during maneuvering, battery voltage drops with higher currents and other discharge properties, ESC influence, etc., but still it should give us quite good view on certain multi-rotor configuration.

What do you think? Does it make sense?

BR
Ivan
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Last edited by ikorman; Sep 24, 2010 at 03:57 PM.
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Old Sep 24, 2010, 04:56 AM
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thanks for that i started reading thinking "i will never make sense of this" but you explained it really well now im off to go borrow some scales
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Old Sep 24, 2010, 03:54 PM
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Quote:
Originally Posted by n3m1s1s View Post
thanks for that i started reading thinking "i will never make sense of this" but you explained it really well now im off to go borrow some scales
Thx. I did try to make it understandable, but still I'm not sure if I nailed it right - so every feedback is appreciated

For beginners it's not easy to battle through the information forest, so I hope that small "guides" like this one will help them.

BR
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Old Oct 17, 2010, 10:23 PM
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Very well explained! I'm building/designing mine from scratch and this will come in handy.
As for thrust/power curve, is there a method to derive this?
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Old Oct 20, 2010, 08:54 AM
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Originally Posted by bluehash View Post
Very well explained! I'm building/designing mine from scratch and this will come in handy.
As for thrust/power curve, is there a method to derive this?
Well, there are many different methods for this. If connected measured data points are similar to line, you can approximate them with linear equation by using "least-squares" method. Here is example how to do it in Excel:

http://www.csupomona.edu/~seskandari...illiam_Lee.pdf

It could be good starting point for further reading.

Ivan
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Old Oct 20, 2010, 09:01 AM
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Thanks!
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Old Oct 27, 2010, 05:31 PM
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Nice clear explanation!

Some questions. The power to thrust curve appears to be shaft power to thrust. The efficiency of the motor comes into play to determine motor input power as you indicate. Best case on a lot of motors is around 85% efficiency.

As well, there is a loss in the ESC (heat lost). Dont know but I would guess at least 5% loss.

As well, there is the 80% rule for lippo. So of the 2200 mah, you have 1760 of usable capacity.

I am getting a quad shortly and flying time is one of my concerns. 8 mins in a helicopter is a long flight. I think 10 mins of hovering in a quad is not going to be long enough.
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Old Oct 27, 2010, 09:45 PM
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Hover Time Estimator

Ivan, dont want to threadnap on you as it is your calculations that are used in this spreadsheet so kudos to you.

I have put together a small spreadsheet with a number of inputs. The thrust and motor shaft watts are from Ivans table above. If you put in the rest of the numbers it will calculate an estimate of hover time.

There are a lot of "quesses" such as motor efficiency and esc efficiency. The ratio of thrust to motor shaft watts is basically an efficiency rating of the prop as well.

Would be nice to have feedback from some real life machines to tweak the numbers a bit.
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Old Oct 28, 2010, 09:24 AM
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Quote:
Originally Posted by gpach View Post
Nice clear explanation!

Some questions. The power to thrust curve appears to be shaft power to thrust. The efficiency of the motor comes into play to determine motor input power as you indicate. Best case on a lot of motors is around 85% efficiency.

As well, there is a loss in the ESC (heat lost). Dont know but I would guess at least 5% loss.

As well, there is the 80% rule for lippo. So of the 2200 mah, you have 1760 of usable capacity.

I am getting a quad shortly and flying time is one of my concerns. 8 mins in a helicopter is a long flight. I think 10 mins of hovering in a quad is not going to be long enough.
Well, when I have done this I had i mind that power could be measured at ESC input, therefore you will take into account ESC losses and efficiency.

Regarding Excel calculator, your are welcome to create it I have something similar for my internal use, but didn't have enough time make it user friendly.

BR
Ivan
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Old Oct 28, 2010, 10:05 AM
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Ivan, trust me, mine is not much, just a spreadsheet.

After rereading your post, and comprehending it better. I noticed that your power/thrust graph includes the motor efficiency. I have modified my spreadsheet to take motor efficiency off the table.

As well, I found another thread
http://www.rcgroups.com/forums/showthread.php?t=768115
Where someone has actually posted a chart of hover watts versus weight over a wide range of AUW's.

I have backchecked his real life results and adjusted the numbers in my table.
It would appear that "complete system" watts per gram runs in the range of 0.105 to 0.133 w/g

For me its good info to be able to calculate an expected air time give changes from 3S to 4S or one to two batteries.
As in Heli's weight is your enemy.
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Old Dec 30, 2011, 07:33 AM
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Great Job explaining this! Awesome!
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Old Dec 30, 2011, 07:22 PM
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I've also been curious about equations that define flight time. From intuition we know that flight time will increase as you increase the battery size, but after some point it must start to decrease, right? I'm trying to derive an equation that shows this, and lets us calculate what is the battery size that maximizes flight time. However, I must be making an inadequate assumption, because my equation does not show such a thing.

I come up with an equ that shows an asymptotic approach to a limit. I guess this makes more sense. I suppose the rub lies in the fact that i assume the same frame/motors are used no matter what the battery size is. This of course can't happen in real life, as the power requirement would keep going up, and you'd have to change the propulsion system, increasing its mass. Has anyone else gone through such an exercise?
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