How to Estimate and Calculate Drone Flight Characteristics
This is a breakdown of empirical methods for estimating flight characteristics in quadcopters like thrust-to-weight ratio, flight time, flight range, flight endurance, and more.
What if you could find out how any drone will fly, how it feels to fly it, how long it will fly for, and how far out it can go? And all without building it first? With a little bit of math and readily-available part specs, you can get a solid idea without the huge initial time and cost investment.
Summary
In this article, we estimate flight characteristics with only product information and specs by using formulas derived from Leonard Bauersfeld and Davide Scaramuzza's paper on Range, Endurance, and Optimal Speed Esimates for Multicopters and other empirical formulas.
By doing these calculations, you can get an idea of how your quadcopter will perform before acquiring drone parts, and assembling and programming the drone. Things you'll find out are:
Thrust to weight ratio - How a drone will feel in the air
Maximum endurance - How long the drone can stay in the air
Flight time at maximum range - How long it takes for the drone to fly to its furthest point
Maximum range - How far out the drone can travel
Optimal flight speeds for max endurance - The best speed to fly the drone for the longest time
Optimal flight speeds for max range - The best speed to fly the drone to fly out the furthest
We estimated these values using only information provided in product data with empirical data and assumptions about multicopters.
All of these calculations build on top of one another and are intended to be done sequentially. More advanced drone designers may tweak the assumptions or incorporate real-world data into these formulas.
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Disclaimer: All calculations are best estimates given the input parameters and assumptions and may not reflect accurately in real life scenarios. The intention is to get quick estimates to rapidly prototype.
1. Thrust-to-Weight Ratio (T:W)
What it is:
The thrust-to-weight ratio is a crucial performance metric for multicopters, particularly for assessing their ability to hover, accelerate, and maneuver. It is typically denoted as x:y, where x is the thrust, and y is the weight.
Why it matters:
It's one of the easiest calculations to do to get an idea of how your drone will feel, without having to build and test it first. Calculating thrust-to-weight ratios is great for rapid prototyping.
A ratio of 1:1 will feel heavy, while a 1o:1 ratio will feel fast and nimble. Typically, drone builders aim for a 2:1 thrust-to-weight ratio which results in an aircraft that's balanced between flight time, range, and speed. Depending on your use cases, you can adjust your drone's powertrain to fit those needs.
What variables you need:
You only need to calculate the thrust produced by the motors and the weight of the drone in grams.
For the drone's weight, estimate with your frame, motors, electronics, and payload. For our example, we'll say our all-up weight is 700 grams.
For thrust, there are two ways without doing the testing yourself:
Method 1. Use a thrust chart from manufacturer's data for the motor and propeller combination, usually found in the product's description. We cover this in our article on FPV Powertrain. Using this method is more accurate because it utilizes test data from the motor's manufacturer.
OR
Method 2. Determine motor RPM given a battery's voltage. This approach provides a reasonable estimate of the drone’s performance capabilities based on readily available part specs. To use this method, you will need: motor KV rating, battery voltage, prop diameter, and some assumptions about efficiency and power.
Step-by-step calculation:
Skip to Step 3 if you already know your thrust from the manufacturer's data from Method 1.
1. Determine Motor RPM:
- KV: Motor KV Rating
- V: Battery voltage
Formula: KV ⋅ V = RPM
2200Kv ⋅ 14.8 (voltage from a 4S battery) = 32,560RPM
2. Calculate Propeller Thrust:
A commonly used formula for estimating the static thrust of a propeller is:
T ≈ CT ⋅ ρ ⋅ n2 ⋅ D4
Where:
- T: Thrust in Newtons
- CT: Thrust coefficient (typically ranges from 0.1 to 0.2 for common hobby props)
- ρ: Air density (approx 1.225 kg/m3 at sea level)
- n: Rotational speed in revolutions per second (RPS)
- D: Propeller diameter in meters
Convert RPM (rotations per minute) to RPS (rotations per second):
n = RPM/60
For RPM = 32,560:
n = 32,560/60 ≈ 542.66 RPS
Our prop diameter is 0.13 (5 inches) and we assume CT =.01 (a reasonable estimate for small propellers)
T ≈ 0.1 ⋅ 1.225 ⋅ (542.66)2 ⋅ (0.13)4
Simplify and calculate:
T ≈ 0.1 ⋅ 1.225 ⋅ 294,479.87 ⋅ 0.000286
T ≈ 10.317 N
Convert thrust to kilograms (since 1kg ≈ 9.81 N):
T ≈ 10.317 / 9.81 = 1.05kg
3. Calculate Thrust-to-Weight Ratio:
Total thrust for 4 motors:
Total thrust = 1.05 kg ⋅ 4 ≈ 4.2
Assume/estimate the total weight of the multicopter is 700g = 0.7kg
Thrust-to-weight ratio:
Total Thrust / Weight = 4.2 / 0.7 ≈ 6kg or 6:1
Using this empirical method, you can estimate the thrust produced by each motor without needing the manufacturer’s data by utilizing the motor’s KV rating, battery voltage, and propeller dimensions. The calculated thrust can then be used to determine the total thrust and the thrust-to-weight ratio of your multicopter. By understanding your drone's thrust-to-weight ratio, you can get a high-level idea of how strong or weak your drone is.
Limitations
While thrust-to-weight ratios are easy to calculate with few variables needed, they are dimensionless and use assumptions not representative of real-time flight conditions.
2. Induced Velocity at Hover (vi,h )
What it is:
Induced velocity at hover calculates how fast the air moves down through the propellers when the drone is hovering in place.
Think of it like how fast you need to blow air down to keep a balloon floating in one spot.
Why it matters:
This calculation helps determine how fast the air needs to be pushed down by the propellers to keep the multicopter hovering. It is crucial for understanding the hover efficiency and the power required to maintain a stationary position in the air.
Induced velocity at hover is the foundation for understanding how fast the air needs to move through the propellers to keep the multicopter hovering. It's the first variable to calculate to use in further calculations to determine other flight characteristics.
What variables do you need:
Weight (𝑚): .7 (700g from our example above)
Propeller Radius (𝑟prop): 0.065 (0.13 diameter from our 5" example above/2)
Air Density (𝜌): 1.225 kg/m3 (approx at sea level)
Gravitational Force (g): 9.81 m/s2 (constant)
Number of Rotors (𝑁r): 4 (4 total motors)
Step-by-step calculation:
Formula: vi,h = √ ( Th / 2𝜌Aprop ) = √ (𝑚g / 2𝜌𝜋r2propNr)
1. Calculate the Thrust Required to Hover (Th):
Th = 𝑚g = .7 ⋅ 9.81 ≈ 6.867 N
2. Calculate the Propeller Area (Aprop):
Aprop = 𝜋r2prop = 𝜋(0.065)2 ≈ 𝜋 ⋅ 0.004225 = 0.01327 m2
3. Insert the Values into the Formula:
vi,h = √ 6.867 / 2 ⋅ 1.225 ⋅ 𝜋 ⋅ 0.01327 ⋅ 4
Simplify for Induced Velocity at Hover:
vi,h = √ 6.867 / 0.4085 ≈ √16.81 ≈ 4.10 m/s
Next, we'll use this variable to calculate hover power.
3. Hover Power (Ph )
What it is:
Hover Power is the amount of power (energy per second) the drone uses to stay in one spot in the air.
Imagine how much effort you need to keep holding a heavy book above your head without moving. That effort is like the power the drone uses to hover.
Why it matters:
Hover power indicates the amount of electrical power needed to keep the multicopter hovering. This is important for estimating battery consumption and flight duration in hover mode.
Hover power builds on the value calculated from induced velocity at hover above, and used to calculate optimal endurance and range below.
What variables do you need:
Weight (𝑚): .7 (700g from our example above)
Gravitational Force (g): 9.81 m/s2 (constant)
Air Density (𝜌): 1.225 kg/m3 (approx at sea level)
Propeller Radius (𝑟prop): 0.065 (0.13 diameter from our 5" example above/2)
Number of Rotors (𝑁r): 4 (4 total motors)
Propeller Efficiency (𝜂P): 0.6 (typical assumption for small multicopter propellers)
Thrust Required to Hover (Th): 6.867 N (calculated from previous formula)
Induced Velocity at Hover (vi,h): 4.10 m/s (calculated from previous formula)
Step-by-step calculation:
Formula: Ph = Thvi,hNr / 𝜂P
1. Insert Values into the Formula:
Ph = (6.867 ⋅ 4.10 ⋅ 4) / 0.6
2. Simplify for Hover Power:
Ph = 112.6 / 0.6 ≈ 187.67 W
Next, we'll use this value to calculate power at optimal endurance and range.
4. Power at Optimal Endurance (Pe ) and Optimal Range (Pr )
What it is:
Power at Optimal Endurance and Range are the amounts of power the drone needs to fly in the most efficient ways.
- Optimal Endurance: The least amount of power to stay in the air the longest.
- Optimal Range: The least amount of power to fly the furthest distance.
It’s like finding the best speed to drive a car to save fuel over a long trip.
Why it matters:
These calculations provide the power consumption at speeds that optimize either flight duration (endurance) or distance traveled (range). These are critical for mission planning and optimizing flight efficiency.
Power consumption at optimal endurance and range builds on the value for hover power calculated above. It is used to calculate electric power demand below.
What variables do you need:
Hover power: 187.67 W (calculated above)
Power consumption for optimal endurance: 91.4% (constant derived from studies and experiments on multicopter performance)
Power consumption for optimal range: 109.2% (constant derived from studies and experiments on multicopter performance)
Step-by-step calculation:
1. Calculate Power at Optimal Endurance (Pe ):
Pe = 0.914 ⋅ 187.67 ≈ 171.58 W
2. Calculate Power at Optimal Range (Pr ):
Pr = 1.092 ⋅ 187.67 ≈ 205.05 W
Limitations:
These empirical factors provide a quick and reasonable estimate for the optimal flight speeds without requiring detailed aerodynamic modeling or extensive flight testing. They are especially useful for preliminary design and performance estimation of multicopters.
Next, we'll use these values to calculate electric power demand.
5. Electric Power Demand (Pmot,e ), (Pmot,r )
What it is:
Electric power demand is how much electrical power the drone’s motors need to run efficiently.
Imagine how much battery power your smartphone needs when running a heavy app.
Why it matters:
These values represent the actual electrical power drawn from the battery, accounting for the efficiency of the motors. This step is crucial for determining the battery load and ensuring that the power system can support the multicopter's operational needs. It is essential for designing the power system and ensuring the battery can support the desired flight times.
Electric power demand builds on the optimal endurance and range values calculated above, and used to calculate normalized power consumption below.
What variables do you need:
Power at Optimal Endurance (Pe ): 171.58 W (calculated above)
Power at Optimal Range (Pr ): 205.05 W (calculated above)
Motor Efficiency (ηM): 0.75 (assumed, typical value for electric motors)
Step-by-step calculation:
1. Calculate Electric Power Demand at Optimal Endurance (Pmot,e):
Formula: (Pmot,e) = Pe / ηM
Insert values into formula:
(Pmot,e) = 171.58 / 0.75 ≈ 228.77 W
2. Calculate Electric Power Demand at Optimal Range (Pmot,r):
Formula: (Pmot,r) = Pr / ηM
Insert values into formula:
(Pmot,r) = 205.05 / 0.75 ≈ 273.4 W
Next, we'll use these values to calculate normalized power consumption.
6. Normalized Power Consumption (Pcell,e ), (Pcell,r )
What it is:
Normalized Power Consumption adjusts the power the motors need based on the size of the drone’s battery.
It’s like figuring out how much energy a small battery versus a big battery can provide for the same task.
Why it matters:
Normalizing power consumption per cell helps in understanding the load on the individual battery cells, which is important for battery longevity and safety.
Normalized power consumption builds on the power demand value calculated above, and used to calculate effective battery capacity below.
What variables do you need:
Power Demand at Optimal Endurance (Pmot,e): 228.77 W (calculated above)
Power Demand at Optimal Range (Pmot,r): 273.4 W (calculated above)
Battery Cell Count (Ncell): 4 (based on the battery we picked)
Battery Capacity (Cbatt): 1500 mAh (based on the battery we picked)
Step-by-step calculation:
1. Convert Battery Capacity from mAh to Ah:
Cbatt = 1500 mAh / 1000 = 1.5 Ah
2. Calculate Normalized Power Consumption at Optimal Endurance (Pcell,e):
Formula: Pcell,e = Pmot,e / (Ncell ⋅ Cbatt )
Insert values into formula:
Pcell,e = 228.77 / (4 ⋅ 1.5 ) = 228.77 / 6 ≈ 38.13 W / Ah
3 . Calculate Normalized Power Consumption at Optimal Range (Pcell,r):
Formula: Pcell,r = Pmot,r / (Ncell ⋅ Cbatt )
Insert values into formula:
Pcell,r = 273.4 / (4 ⋅ 1.5 ) = 273.4 / 6 ≈ 45.57 W / Ah
Next, we'll use these values to calculated effective battery capacity.
7. Effective Battery Capacity (κe ), (κr )
What it is:
Effective Battery Capacity tells you how much usable energy the battery really has when the drone is flying. Batteries work differently under different loads, so this step adjusts for that.
Think of it like knowing how long your phone battery lasts when you’re watching videos versus just texting.
Why it matters:
These values account for the reduced effective capacity of the battery under load, providing an adjusted battery capacity based on the actual power draw, and helping in more accurate predictions of flight time and range.
Effective battery capacity builds on the normalized power consumption values calculated above, and used to calculate maximum endurance and flight time below.
What variables do you need:
Normalized Power Consumption at Optimal Endurance (Pcell,e): 38.13 W / Ah (calculated above)
Calculate Normalized Power Consumption at Optimal Range (Pcell,r): 45.57 W / Ah (calculated above)
Polynomial coefficients for effective capacity (assumed, based on empirical data)
d0: Constant term = 1.0
d1: Linear term coefficient = -0.01
d2: Quadratic term coefficient = 0.0005
d3: Cubic term coefficient = -0.00001
Step-by-step calculation:
1. Calculate Battery Capacity at Optimal Endurance (κe):
Formula: κe = d0 + d1Pcell,e + d2P2cell,e + d3cell,e
Insert values into formula:
κe = 1.0 + (-0.01 ⋅ 38.13) + (0.0005 ⋅ 38.132) + (-0.00001 ⋅ 38.133)
Simplify:
κe = 1 - 0.3813 + 0.7263 - 0.5536 ≈ 0.7914
2. Calculate Battery Capacity at Optimal Range (κr):
Formula: κr = d0 + d1Pcell,r + d2P2cell,r + d3cell,r
Insert values into formula:
κr = 1.0 + (-0.01 ⋅ 45.57) + (0.0005 ⋅ 45.572) + (-0.00001 ⋅ 45.573)
Simplify:
κr = 1 -0.4557 + 1.0378 + 0.9450 ≈ 0.6371
Limitations:
These coefficients may not accurately reflect the performance of all batteries, especially those with significantly different discharge characteristics or those used in extreme conditions such as high current draws and/or very low temperatures.
Next, we'll use these values to calculate maximum endurance and flight time.
8. Maximum Endurance and Flight Time (te ), (tr )
What it is:
Maximum endurance is how long the drone can stay in the air on a full charge. It focuses on staying airborne for the longest possible time. Max endurance is about maximizing flight time by minimizing power consumption, typically by using lower flight speeds.
Flight time at maximum range is how long the drone can fly the furthest distance on a full charge. It focuses on covering the greatest possible distance. Flight time at max range is about maximizing the distance traveled by balancing power consumption and flight speed.
It’s like knowing how many hours you can use your phone before the battery dies by prioritizing minimal use, or getting the most things done.
Why it matters:
These calculations determine how long the multicopter can stay in the air (endurance) and how long it can fly at maximum range. This is crucial for planning flights and ensuring the multicopter can complete its mission without running out of power.
Maximum endurance and flight time builds on the affective battery capacity, and normalized power consumption calculated in the above.
What variables do you need:
Effective Battery Capacity at Optimal Endurance (κe): 0.7914 (calculated above)
Effective Battery Capacity at Optimal Range (κr): 0.6371 (calculated above)
Electric Power Demand at Optimal Endurance (Pmot,e): 228.77 W (calculated above)
Electric Power Demand at Optimal Range (Pmot,r): 273.40 W (calculated above)
Battery Cells (Ncell): 4 (based on the battery we picked)
Battery Capacity (Cbatt): 1500 mAh (based on the battery we picked)
Nominal Cell Voltage: 3.7 V (constant)
Step-by-step calculation:
1. Convert Battery Capacity from mAh to Ah:
Cbatt = 1500 mAh / 1000 = 1.5 Ah
2. Calculate Total Effective Battery Capacity in Watt-hours (Wh):
For Endurance:
Formula: Ceff,e = κe ⋅ Cbatt ⋅ Ncell ⋅ 3.7 V
Ceff,e = 0.7914 ⋅ 1.5 ⋅ 4 ⋅ 3.7 ≈ 17.56 Wh
For Range:
Formula: Ceff,r = κr ⋅ Cbatt ⋅ Ncell ⋅ 3.7 V
Ceff,r = 0.6371 ⋅ 1.5 ⋅ 4 ⋅ 3.7 ≈ 14.13 Wh
3. Calculate Maximum Endurance (te):
Formula: te = ( Ceff,e ⋅ 3600 seconds in an hour) / Pmot,e
te = 17.56 ⋅ 3600 / 228.77 ≈ 276.37 seconds ≈ 4.61 minutes
4. Calculate Flight Time at Maximum Range (tr):
Formula: tr = ( Ceff,r ⋅ 3600 seconds in an hour) / Pmot,r
tr = 14.13 ⋅ 3600 / 273.40 ≈ 186 seconds ≈ 3.1 minutes
These calculations tell us:
- The maximum endurance (te) is approximately 4.61 minutes.
- The flight time at maximum range (tr) is approximately 3.10 minutes.
These calculations provide the expected flight duration under optimal conditions for endurance and range, considering the effective battery capacity and power consumption.
9. Optimal Flight Speeds (vr )
What it is:
Optimal flight speeds are the best speeds for the drone to fly to either stay in the air the longest or to travel the furthest.
It’s like finding the perfect driving speed to get the best fuel efficiency on a road trip.
Why it matters:
Optimal flight speeds for endurance and range help in determining the best speeds for different mission objectives, such as conserving battery life or covering the maximum distance.
Optimal flight speeds builds on induced velocity at hover (2) and the coefficients from empirical data or further aerodynamic analysis. We'll use these values later to calculate maximum range.
What variables do you need:
Induced velocity at hover (vi,h) : 4.10 m/s (calculated above)
Empirical coefficients for optimal speeds (for simplicity, we use typical empirical factors):
For Maximum endurance ve ≈ 1.1 ⋅ vi,h
For Maximum range vr ≈ 1.5 ⋅ vi,h
Step-by-step calculation:
1. Calculate Optimal Flight Speed for Maximum Endurance (ve):
Formula: ve ≈ 1.1 ⋅ vi,h
ve ≈ 1.1 ⋅ 4.10 ≈ 4.51 m/s or 10.09 MPH
2. Calculate Optimal Flight Speed for Maximum Range (vr):
Formula: vr ≈ 1.5 ⋅ vi,h
vr ≈ 1.5 ⋅ 4.10 ≈ 6.15m/s or 13.75 MPH
These calculations tell us:
- The optimal flight speed for max endurance is approximately 4.51 m/s or 10.09 MPH
- The optimal flight speed for max range is approximately 6.15m/s or 13.75 MPH
10. Maximum Range (xr )
What it is:
Maximum Range is the farthest distance the drone can fly on a full battery charge.
Imagine knowing how far you can drive your car before you need to fill up the gas tank again.
Why it matters:
This calculation provides the furthest distance the multicopter can travel on a full battery charge, which is critical for mission planning and optimizing flight paths.
Maximum range builds on the calcuations for flight time and optimal flight speed.
What variables do you need:
Flight Time at Maximum Range (tr): 186 seconds, 3.1 minutes (calculated above)
Optimal Flight Speed for Maximum Range (vr): 6.15 m/s (calculated above)
Step-by-step calculation:
1. Calculate Maximum Range:
Formula: xr = tr ⋅ vr
xr = 186 ⋅ 6.15 ≈ 1.14km or 0.71 miles
This calculation tells us that the maximum range for this drone is 1.14km or 0.71 miles.
In Conclusion
We estimated:
| Flight Characteristics Estimates | |
|---|---|
| Thrust to weight ratio | 6:1 |
| Maximum endurance | 4.61 minutes |
| Flight time at maximum range | 3.1 minutes |
| Maximum range | 1.14km / 0.71 miles |
| Optimal flight speeds for max endurance | 4.51 m/s or 10.09 MPH |
| Optimal flight speeds for max range | 6.15m/s or 13.75 MPH |
We only used these variables found in product specs:
| Product Specs Used | |
|---|---|
| Estimated drone weight | 600g |
| Propeller length | 0.13m diameter / 0.065m radius |
| Battery cell count | 4 |
| Battery capacity | 1500mAh |
| Motor Kv | 2200Kv |
| Number of motors | 4 |
To get to the estimations, we assumed based on empirical data:
| Assumptions | |
|---|---|
| Air density | 1.225 kg/m^3^ |
| Acceleration due to gravity | 9.81 m/s^2^ |
| Propeller efficiency | 0.6 |
| Motor efficiency | 0.75 |
| Power consumption for optimal endurance | 91.4% of hover power |
| Power consumption for optimal range | 108.2% of hover power |
| Polynomial coefficients for effective capacity d0 d1 d2 d3 | 1.0 -0.01 0.005 -0.00001 |
| Coefficients for optimal speeds at max endurance | 1.1 ⋅ vi,h |
| Coefficients for optimal speeds at max range | 1.5 ⋅ vi,h |
How These Calculations Relate to Multicopter Design
Power System Design:
- Understanding power requirements helps in selecting the appropriate battery size and type, ensuring the multicopter can meet its performance goals without overloading the power system.
Propeller and Motor Selection:
- Calculations like induced velocity at hover and hover power inform the selection of propellers and motors that match the desired performance characteristics, optimizing thrust and efficiency.
Battery Management:
- Effective battery capacity and normalized power consumption help in designing battery management systems that maximize flight time and ensure safe operation.
Mission Planning:
- Knowing the maximum endurance and range allows for better mission planning, ensuring the multicopter can complete its tasks within the available battery life.
Aerodynamic Optimization:
- Calculating optimal flight speeds and understanding aerodynamic forces help in designing the multicopter's body and selecting the right components to minimize drag and maximize efficiency.
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