Making sense of Maxwell’s Equations

Reading Time: < 1 minute

I’m currently learning about modelling of electromagnetic systems, and in that process, I’m trying to wrap my head around Maxwell’s Equations. Maxwell’s Equations can be used to determine electric and magnetic fields, which are important for machines using them for work, such as electrical motors.

There are some aspects to those that are similar to fluid dynamics, with electrical fields having their source in electrical charges, while magnetic fields are free of sources. Much of that is expressed using curl and divergence. These are differential operators used to characterise vector fields.

I came across a pretty good video by 3Blue1Brown of Khan Academy fame introducing the concepts of curl and divergence, and also showing the connection to fluid and electrodynamics:

Aktivieren Sie JavaScript um das Video zu sehen.
https://www.youtube.com/watch?v=rB83DpBJQsE

The other video, which goes over Maxwell’s Equations and explains them pretty well is by DrPhysicsA:

Aktivieren Sie JavaScript um das Video zu sehen.
https://www.youtube.com/watch?v=AWI70HXrbG0

Be careful, though: At or around minute 18:13, DrPhysicsA explains how Maxwell’s Second Law (or Gauss’ Law for Magnetism) breaks down when you place your control surface such that only one pole of the magnet is inside and the other is outside. According to DrPhysicsA, that would lead to non-zero flow through the control surface, violating Gauss’ Law for Magnetism.

But actually, that is not the case. One has to consider that the field also flows inside the magnet. Namely, there is flow between the north- and south-pole inside the magnet in opposite direction to that outside the magnet. Thus, the flow inside the magnet and the flow outside that magnet through the control surface still cancel out. Gauss’ Law for Magnetism therefore very elegantly states that magnetic poles always come in pairs.

Lift, Drag and Thrust — Aerodynamics Primer

Reading Time: 12 minutes

Ever wondered how lift is generated on the wing of an aircraft or how a propeller generates thrust? There’s a legend that goes like this: The particles of the air want to stay together, but as the particle that goes along the upper part of the wing needs to go a longer way, it needs to be faster, and according to Bernoulli’s Principle, faster air means lower pressure, leading to lift. Nice explanation, isn’t it? So very romantic, the particles that just won’t let themselves be separated.

It’s just that…it’s completely wrong! Air particles aren’t married to each other, and it may even be that they are completely mixed up — in case of turbulent flow — with the wing still generating lift! In addition, Bernoulli’s Principle does not account for friction in the air, and thus by itself cannot explain drag.

For our further analysis of our quadrocopter model, we need to model the behaviour of the propeller and the engine that drives it. Modelling the behaviour of the propeller involves the use of the formulae for thrust F_T and power P

(1)   \begin{eqnarray*} F_T &= c_T \rho D^4 n^2 \\ P &= c_P \rho D^5 n^3 \end{eqnarray*}

and working with these. Specifically, we need to understand the nature of the thrust and power coefficients c_T and c_P.

So, let’s have a look how lift and drag on an airfoil come to be, how we can use dimensional analysis to get the typical formulae for describing them, and where to find what’s missing. Here’s the route for today’s trekking tour:

Reaction Forces: When a Ball meets a Wall

Let’s first recap a bit of physics basics. Remember Newton’s Third Law of Motion?

For every action, there is an equal and opposite reaction.

Let’s imagine we are sitting in a car at rest, with the motor running. If we now press the accelerator, the engine will provide power that is transferred to the wheels, which in turn will exert a force on the street. Now, the wheel is obviously accelerated to turn so that its bottom moves towards the back of our car — the opposite direction of where we want to go!

So it cannot be the wheel that provides our forward acceleration, because the wheel enacts a force in the wrong direction. Instead, it’s the street surface which makes us move: The force accelerating the wheel makes the wheel enact that force on the street surface, and the street surface will enact an opposite force of equal size on the wheel, accelerating us forward.

In this example, the action is the force acting from the wheel on the street surface, and the reaction is the force acting from the street surface on the wheel and — in consequence — on the whole car. And essentially, that also explains lift and drag. So, thanks for reading, have a good night! …

A ball being thrown against a wall at an angle (viewed from above).

Of course, it’s a bit more involved. For a better understanding, let’s imagine a ball being thrown at a wall, as indicated in the figure to the right. When the ball hits the wall, it cannot proceed further in the same direction, so its velocity vector and with it its momentum must change. The change of velocity is indicated in the figure by \Delta v.

As we know from Newton’s Second Law of Motion, a change of velocity requires a force acting on the ball. It is reasonable to assume that this force is enacted at the contact area of the ball to the wall.

Again, applying Newton’s Third Law of Motion, there must be an opposite force with the same magnitude in the system. But where is it?

Well, there is only one other object touching the contact area, and that is the wall. Thus, the reverse force must be acting on the wall. If we hadn’t fixed the wall in our reference system, we would see it being propelled backwards.

But, how big is this force? Well, in our example, we cannot tell exactly. However, we know the change of the force over time in a qualitative manner:

  1. When the ball first touches the wall, the force is practically non-existent.
  2. Then, when the ball is compressed more and more, it acts against being compressed with increasing force, until it is maximally compressed and at rest.
  3. From then on, it is accelerated again in its new direction, and as it decompresses in the process, the force decreases again.

The total change in momentum until the time t is the integral of this force over time (assuming, we are in a vacuum and there is no internal friction in the ball):

(2)   \begin{equation*} \Delta \vec{p}\left(t\right) = \int_0^t \vec{F}\left(\tau\right) d\tau \end{equation*}

If you want to check the validity of this equation, just consider that \Delta \vec{p} = m \Delta \vec{v} and F = m \vec{a} = m \frac{d}{dt} \vec{v}. If you look closely, you can see that Conservation of Momentum follows from Newton’s Third Law of Motion, according to which for the total forces we have \vec{F} = 0, and thus for the total momentum in the system, \frac{d}{dt} \vec{p}=0 must hold.

We call \vec{F}\left(t\right) the reaction force, as it is a reaction to the change of momentum of part of the system (the ball).

However, although we cannot exactly determine the force, we can derive something about the change of momentum. Looking at the velocity triangle in the figure above, we see that \Delta v is proportional to the magnitudes of v_1 and v_2 (just try scaling the triangle in your head). We know that the latter are equal, so the change in velocity is proportional to the magnitude v_1. It is also proportional to the mass of the ball, so we have

(3)   \begin{equation*} \Delta \vec{p} \sim m v_1 \end{equation*}

Now, if we were to constantly throw balls onto the wall, all having the same mass and initial velocity, we could determine the mean force acting on the wall. Let’s assume that over a time interval of length \Delta t we would throw a mass of \Delta m onto the wall. Then the change in momentum over that time would be

(4)   \begin{equation*} \Delta \vec{p}_{tot} \sim \Delta m v_1 \end{equation*}

Let’s further assume that we throw the balls at a constant frequency, so that we get a mean mass-flow rate of \dot{m} \approx \frac{\Delta m}{\Delta t}. The mean force over the same time would then be

(5)   \begin{equation*} \vec{F} = \frac{\Delta \vec{p}_{tot}}{\Delta t} \sim \frac{\Delta m}{\Delta t} v_1 = \dot{m} v_1 \end{equation*}

If we now think of the balls as being a gas of density \rho moving at velocity v_1 through a tube with cross-section A, the mass flow rate is given by

(6)   \begin{equation*} \dot{m} = \rho A v_1 \end{equation*}

If we plug all of this together, we get the force to be

(7)   \begin{equation*} \vec{F} \sim \rho A \left(v_1\right)^2 \end{equation*}

which already looks eerily like the usual formula for lift and drag:

(8)   \begin{equation*} F = c \rho A v^2 \end{equation*}

Reaction Forces on an Wing

Let’s have a look at the cross-section of a wing and how the air flows around it:

Schematic Airflow around an asymmetric airfoil

The drawing above is only a schematic, but there are some important aspects of the airflow visualised there. What we see is that the air flows in a laminar fashion around the airfoil (that’s what we call the cross-section of a wing). An airflow is laminar if all the sheets of air are nicely separated and there is no turbulence.

We also see that the air on the left side is quite parallel to the horizontal plane, while on the right side it briefly flows downward, before assimilating to the free-flow again. As we have now learned, this indicates a change in momentum, which means that there is a force in play. As there should be: We would expect the wing to generate lift, an upwards force, so the resulting change in momentum should be downwards.

But where does this change in momentum come from? Think about what would happen if there was no change in momentum: The air would simply flow through the wing. The air cannot do that, so it must change direction to avoid the wing. It will again change direction when the wing “moves away” from the flow again, as the pressure of the air flowing beside it will push it that way. That change of direction implies a change of momentum, and that change of momentum must come with a force acting on the air.

So finally, we know where lift and drag come from: They represent the sum of all these forces along the wing, or the net force, and these forces result from the air having to follow the slope of the airfoil. There is a general convention to define lift and drag according to the direction of the free-stream:

  • Lift is the part of the force perpendicular to the direction of the free-stream, with positive lift pointing upwards, and
  • drag is the part of the force parallel to the direction of the free-stream, with positive drag pointing in the direction of the free-stream flow.

Again, we ask ourselves: How big are these forces? And again, we cannot give that number exactly just from this basic model — specifically, if we consider friction –, but we can characterise it quite well using dimensional analysis.

Dimensional Analysis of Lift and Drag

Now it’s time to use what we learned in a previous article about dimensional analysis — but this time we apply it to our wing. From our previous considerations, we already have identified a few parameters that may influence lift and drag:

  • the density of the air in the free-stream \rho_\infty,
  • the size and form of the wing, represented by its projected surface area A, and
  • the free-stream velocity of the air v_\infty.
Angle of Attack on an Airfoil

Geometrically, the so-called angle of attack \alpha will also play a role, as shown in the figure above. This is the angle between the direction of the free stream and the chord line of the airfoil — the theoretical line from the leading to the trailing edge.

If we are working at velocities close to the speed of sound, we need to consider compressibility. Thus, another parameter is the speed of sound in the free stream a_\infty. The speed of sound may differ between the free stream and around the wing, as it depends on the density of the air, and this may well be influenced by the flow around the wing. Therefore, we only consider the free-stream speed of sound — the speeds of sound elsewhere in the stream would be expressible based on this.

Further, we also need to consider internal friction in the air — otherwise there will be no drag at all, according to a finding that is known as D’Alembert’s Paradox. The friction will be represented by the free-stream viscosity \mu_\infty. The friction between two sheets of fluid is proportional to the size of the contact surface and the gradient of velocity between both. The viscosity is the proportionality constant.

So, we finally have the following parameters influencing lift and drag, with their dimensions (M for mass, L for length, and T for time):

QuantitySymbolDimension
Lift/DragF\frac{ML}{T^2}
Angle of Attack\alpha1
Area of the WingAL^2
Free-Stream Velocityv_\infty\frac{L}{T}
Density of the Air\rho_\infty\frac{M}{L^3}
Viscosity\mu\frac{M}{LT}
Speed of Sounda_\infty\frac{L}{T}

Obviously, the lift and drag forces are functions of the other parameters:

(9)   \begin{equation*} F = f\left(\alpha,A,v_\infty,\rho_\infty,\mu,a_\infty\right) \end{equation*}

The dimensions used in these parameters are mass M, length L and time T. We’ll use the density \rho_\infty, the free-stream velocity v_\infty and the wing area A to represent them:

DimensionSymbolRepresentation
MassM\rho \sqrt{A^3}
LengthL\sqrt{A}
TimeT\frac{\sqrt{A}}{v_\infty}

Using these, we find the following expressions with equivalent dimensions:

QuantityDimensionEquivalent Expression
Lift/Drag\frac{ML}{T^2}\rho A {v_\infty}^2
Angle of Attack11
Viscosity \frac{M}{LT}\rho \sqrt{A} v_\infty
Speed of Sound \frac{L}{T}v_\infty

We could now use \frac{\mu}{\rho \sqrt{A} v_\infty} as the dimensionless quantity to represent friction. If you are interested in aerodynamics, that expression may look familiar to you. The Reynolds Number is often used to represent the influence of friction in fluid flows, but it looks slightly different:

(10)   \begin{equation*}  {Re}_\infty := \frac{\rho_\infty l v_\infty}{\mu_\infty} \end{equation*}

So, first of all, nominator and denominator are reversed — larger numbers indicate less friction –, and instead of \sqrt{A} we have the so-called characteristic length l in there. In aerodynamics, the chord length c is usually used for l, so we might not want to deviate from that. Thus, we will use the Reynolds-Number as a dimensionless representation of viscosity.

For the speed of sound, we’d get \frac{a_\infty}{v_\infty}, which is the inverse of the Mach Number. So, we’d rather use the Mach Number M_\infty:=\frac{v_\infty}{a_\infty} directly. Thus, our relationship has the following form:

(11)   \begin{equation*} \frac{F}{\rho {v_\infty}^2 A} = C\left(\alpha,{Re}_\infty,M_\infty\right) \end{equation*}

with

(12)   \begin{eqnarray*} {Re}_\infty &:=& \frac{\rho c v_\infty}{\mu_\infty} \\ M_\infty &:=& \frac{v_\infty}{a_\infty} \end{eqnarray*}

This is the formula in the form that you’ll find in most aerodynamics literature. The value of C\left(\alpha,{Re}_\infty,M_\infty\right) is called the coefficient of lift when calculating lift and the coefficient of drag when calculating drag.

So, where we originally had six parameters to vary in an experiment (\alpha, c resp. A, v_\infty, a_\infty, \rho_\infty and \mu_\infty), we are now left with three parameters to vary (\alpha, {Re}_\infty and M_\infty), and we have a formula that will help us deal with the rest of the variation without additional experiments.

Propeller Aerodynamics

On our multicopter, we don’t have wings. We have propellers, which could be described as rotating wings. Lift turns to thrust, drag turns to torque, torque turns to required power.

The performance of a propeller is determined by a relationship between these quantities:

  • Thrust force F,
  • propeller torque Q,
  • propeller power (due to torque) P,
  • rotational frequency of the propeller n,
  • propeller diameter D,
  • velocity of the air in free-stream v_\infty,
  • density of the air in free-stream \rho_\infty,
  • viscosity of the air in free-stream \mu_\infty,
  • speed of sound in free-stream a_\infty.

Now, here is an excercise for you: See if you can find the usual descriptions of F, Q and P based on dimension analysis:

(13)   \begin{eqnarray*} F &:= c_t\left(J,Re,M\right) \rho D^4 n^2 \\ Q &:= c_q\left(J,Re,M\right) \rho D^5 n^2 \\ P &:= c_p\left(J,Re,M\right) \rho D^5 n^3 \end{eqnarray*}

You will again come across the Reynolds Number Re and the Mach Number M. I’ll give you a few hints for these:

  • The Reynolds Number for propellers is usually based on the chord and tangential velocity at 75% of propeller radius for the characteristic length and velocity. The symbol used for the former is typically c_{\frac{3}{4}}. Yes, you’ll have to add that to the list of dimensional parameters.
  • The Mach Number is typically based on the velocity of the propeller tip.

In addition, you’ll find the advance ratio relevant:

(14)   \begin{equation*} J := \frac{v_\infty}{n D} \end{equation*}

With some basic geometry knowledge you should be able to figure these out. If you get stuck, I recommend looking at this paper by Robert Deters, Gavin Ananda and Michael Selig of the University of Illinois – Urbana-Champaign (UIUC) [1].

Where to find out more

Now that we found out in general what the aerodynamic coefficients can do for us, we’d of course like to know how to get them. Well, there’s always measurement. If you don’t want to do it yourself, the University of Illinois – Urbana-Champaign (UIUC) provides a pretty extensive source of data for airfoils and propellers:

Or perhaps you also want to go back to the origins of flight, to the systematic experiments with large numbers of airfoils at NACA, the National Advisory Committee for Aeronautics, a predecessor of today’s NASA [2]?

However, we can also try to model and simulate the behaviour of airflow around objects. If we assume friction to be negligible (non-viscous or inviscid flow), there’s a whole theory of fluid dynamics modelled around that assumption: potential flow. (Actually, that also requires that the flow is free of rotation.) This is quite a powerful approach as it allows to model fluid flow as a potential field (similar to gravity), and many complicated flows can be described by superposition of basic flows such as uniform flow, flow sources/sinks or circular vortices.

Also, thin airfoil theory makes heavy use of potential flow. It gives us some pretty good first-order estimates of the lift coefficient and the coefficient of the so-called pitch moment for thin airfoils. An airfoil is considered thin if its thickness is small compared to its chord length.

One of the most well-known results from that is the estimated gradient of the lift coefficient of a thin 2D airfoil for small angles of attack (with the angle of attack \alpha being given in radians):

(15)   \begin{equation*} \frac{\partial}{\partial \alpha} c_l = 2 \pi \end{equation*}

However, modelling flow around airfoils using potential flow has one important drawback: there is no drag at all! As drag due to friction is not modelled, it cannot be considered. But even if we sum up the forces due to change in momentum over the airfoil, there is no component along the direction of the free-stream flow.

That means: In potential flow, airfoils generate lift, but no drag. This realisation is known as D’Alembert’s Paradox, after the 18th-century French mathematician Jean le Rond d’Alembert.

Still, potential flow gets us pretty far towards a solution for viscous flow: We can approximate a first solution, and then add corrections, e.g., using boundary layer models. The boundary layer is the area around an object within which the velocity gradient is large enough for friction to matter. By first determining a potential flow solution, the thickness of the boundary layer can be estimated. Then, another potential flow solution is found, increasing the thickness of the airfoil by that of the boundary layer. This is an iterative process which goes on until a good, stable solution is found. Using that solution, it is then possible to derive the amount of drag produced inside the boundary layer.

Then, there’s induced drag, which comes from the fact that our wings are not infinitely wide — although we can approximate infinitely wide wings by making them long and slender. This is what is done with sail planes, who have wings with pretty high aspect ratios. To actually estimate the lift and drag of a real wing, we can use lifting line theory, which again uses a simple model based on potential flow to transform lift and drag data on 2D-airfoils into lift and drag data for a finite wing.

And for the really complicated cases, we can try to explicitly solve the Navier-Stokes equations numerically using computational fluid dynamics (CFD). With these, we can create a virtual wind tunnel, and approximate the air flow quite well. However, there is a lot of computational power involved, and the setup requires much more intricate models of our objects than the other approaches.

Of course there are also tools available for the simpler methods. Modelling the performance of 2D-airfoils in inviscid and viscous, subsonic flow is supported by the well-known XFOIL software. The tool XFLR5 extends this to wing design using lifting-line and vortex sheet methods.

Conclusion

So, now we know where lift, drag, propeller thrust, torque and power come from, and how we can characterise them. These formulae allow us to do a good bit of work, and at least get some pretty good estimates for the performance of a wing — if we know the coefficients. They do not allow us to directly derive these coefficients, but of course there are different methods to approximate them for individual forms of wings: measurements, thin-airfoil theory, potential flow, boundary-layer methods, lifting-line theory or CFD.

Now we are well-equipped for modelling the aerodynamics part of our multicopter engines. Next time, we’ll look into modelling our engine with a DC motor.

[1] [doi] R. W. Deters, G. K. A. Krishnan, and M. S. Selig, “Reynolds number effects on the performance of small-scale propellers,” in 32nd AIAA applied aerodynamics conference, 2014.
[Bibtex]
@InProceedings{Deters.Krishnan.ea2014,
author = {Robert W. Deters and Gavin Kumar Ananda Krishnan and Michael S. Selig},
title = {Reynolds Number Effects on the Performance of Small-Scale Propellers},
booktitle = {32nd {AIAA} Applied Aerodynamics Conference},
year = {2014},
month = {jun},
publisher = {American Institute of Aeronautics and Astronautics},
doi = {10.2514/6.2014-2151},
url = {https://m-selig.ae.illinois.edu/pubs/DetersAnandaSelig-2014-AIAA-2014-2151.pdf},
}
[2] E. N. Jacobs, K. E. Ward, and R. M. Pinkerton, “The characteristics of 78 related airfoil sections from tests in the variable-density wind tunnel,” National Advisory Committee for Aeronautics, Washington, DC, techreport 460, 1933.
[Bibtex]
@TechReport{Jacobs.Ward.ea1933,
author = {Jacobs, Eastman N. and Ward, Kenneth E. and Pinkerton, Robert M.},
title = {The characteristics of 78 related airfoil sections from tests in the variable-density wind tunnel},
institution = {National Advisory Committee for Aeronautics},
year = {1933},
type = {techreport},
number = {460},
address = {Washington, DC},
month = nov,
abstract = {An investigation of a large group of related airfoils was made in the NACA variable-density wind tunnel at a large value of the Reynolds number. The tests were made to provide data that may be directly employed for a rational choice of the most suitable airfoil section for a given application. The variation of the aerodynamic characteristics with variations in thickness and mean-line form were systematically studied.},
file = {:Jacobs.Ward.ea1933 - The characteristics of 78 related airfoil sections from tests in the variable-density wind tunnel.pdf:PDF},
keywords = {AIRFOILS; AERODYNAMIC CHARACTERISTICS; WIND TUNNEL TESTS; THICKNESS; AIRFOIL PROFILES; AERODYNAMIC COEFFICIENTS; LIFT DRAG RATIO; ASPECT RATIO; ANGLE OF ATTACK; HIGH REYNOLDS NUMBER; TABLES (DATA); GRAPHS (CHARTS)},
owner = {ralfg},
timestamp = {2019-11-02},
url = {https://ntrs.nasa.gov/search.jsp?R=19930091108},
}

Dimensional Analysis of a DC-Motor

Reading Time: 9 minutes

For the construction of my first quadrocopter, I bought some brushless DC motors, some propellers, a set of drivers for brushless motors, a battery, built a geometrically matching frame and started working with that. This approach quite probably has not led to a good design: I might get more flight time with the same payload by some better design.

However, there’s so many parameters to tweak: Battery capacity, voltage, size and mass of the engines, size and pitch of the propellers, frame geometry, etc. All of these are interrelated by complicated relationships, most of which are not exactly known to us. Varying all the parameters to find some optimum may prove both time-consuming and costly: We’d have to order different parts with different parameters, measure them under a whole lot of different situations, and then find the optimal solution.

However, there’s a pretty clever tool to reduce the set of variables to consider: Dimensional Analysis. The very short version is: Any relationship between a set of physical quantities can be represented by a smaller set of physical quantities and their dimensionless relationship to the remaining quantities. This allows us to massively simplify experiment design.

In this article, we’ll have a look at the basics of dimensional analysis using the so-called Buckingham \Pi Theorem [1], and try to use a combination of physical insight and dimensional analysis to characterise the performance of a DC motor in comparison to some of its basic parameters such as size or the strength of the magnetic material used. So, this is the plan for today’s tour:

Dimensional Analysis: An example

Schematic view of a mathematical pendulum

Let’s have a look at a mathematical pendulum. We have a mass m fixed to an arm or rope of length l, and the strength of local gravity shall be given by the gravitational acceleration g. We want to determine T, the period of the pendulum. Some basic physical insight tells us, that T may depend on the mass, the length of the arm, local gravity and the angle of initial deflection \alpha_0:

(1)   \begin{equation*} T = f\left(m,l,g,\alpha_0) \end{equation*}

Dimensional analysis allows us to get a general picture of these influences. Let’s try to express T as a dimensionless multiple of some product of powers of m, l and g. Note that we do not consider \alpha_0 in this product, as it has no dimension itself and thus no power of \alpha_0 can contribute to making the product dimensionless:

(2)   \begin{equation*} k_T = T m^{c_m} l^{c_l} g^{c_g} \end{equation*}

Solving this equation so that k_T is dimensionless leads us to

(3)   \begin{equation*} k_T = T \sqrt{\frac{g}{l}} \end{equation*}

Now, we can write our relationship from Eq. 1 in the following way:

(4)   \begin{equation*} T \sqrt{\frac{g}{l}} = \bar{f}\left(m,l,g,\alpha_0) \end{equation*}

However, on the left-hand side, we have a dimensionless number, so the function \bar{f} must also map its parameters to dimensionless numbers. \bar{f} cannot depend on any product of powers of m, l and g, as such a product would not be dimensionless: No power of l and g can cancel the mass dimension present in any non-zero power of m, and the same is true for any other combination. There also cannot be a function that only depends on the value of the dimensioned parameter, as then the value of the function would change if we used feet as unit for the length instead of meters.

Thus, the function on the right can actually only depend on dimensionless quantities, which leaves \alpha_0. So, we finally get

(5)   \begin{equation*} T \sqrt{\frac{g}{l}} = T\left(\alpha_0) \end{equation*}

So, dimensional analysis tells us, that the period of the pendulum only depends on \sqrt{\frac{g}{l}} and the initial deflection angle \alpha_0. We arrived at this conclusion only by analysing the dimensions of the quantities involved.

We might not know T\left(\alpha) in general, but we could measure it for some set of values of g, l and \alpha_0, and could then determine it for any other set of values of g and l if only we keep \alpha_0 constant. We say that \alpha_0 determines the similitude of two problems with different parameters.

If we were to look into the actual equations of motion, we’d get the same result — and we’d also get the form of T\left(\alpha_0\right). Depending on the complexity of the equations involved, this could prove very tedious.

The Buckingham Pi Theorem

What we just applied is known as the Buckingham \Pi Theorem, named for Edgar Buckingham, an American physicist living from 1867 to 1940. Buckingham was not the first to notice and apply the principle, but still today it is named for him.

The theorem essentially states that, if there is a physically meaningful relationship between physical quantities, then there must be a function describing this relationship which only depends on dimensionless ratios of these quantities. He also described a procedure to find these ratios.

Let’s have a look at the proof idea: Assume that we have n physical quantities, which we shall name q_1, \ldots, q_n, and there is some — possibly unknown — relationship between them, which we express in the form

(6)   \begin{equation*} F\left(q_1,\ldots,q_n\right) = 0 \end{equation*}

Obviously, the value of F does not have a dimension. If it did, we could simply divide by that dimension and still have a similar equation without dimension. Thus, F also cannot depend on dimensioned quantities. If it did, its value would differ depending on the units used, even if the quantities were the same.

As a consequence, we can select k of the quantities, and express the others as dimensionless multiples of these. Let’s say, that q_1,\ldots,q_k were the selected quantities. Then we can write Eq. 6 as follows:

(7)   \begin{equation*} F\left(q_1,\ldots,q_k,\frac{q_{k+1}}{{q_1}^{r_{k+1,1}}\ldots{q_k}^{r_{k+1,k}}},\ldots\right) = 0 \end{equation*}

Now, as argued before, F cannot depend on dimensioned quantities, so F must actually be independent of the q_1,\ldots,q_k and can only depend on the dimensionless ratios:

(8)   \begin{equation*} F\left(\frac{q_{k+1}}{{q_1}^{r_{k+1,1}}\ldots{q_k}^{r_{k+1,k}}},\ldots,\frac{q_{n}}{{q_1}^{r_{n,1}}\ldots{q_k}^{r_{n,k}}}\right) = 0 \end{equation*}

Finding the Basis Quantities

Now, how do we find which quantities to use as our basis q_1,\ldots,q_k? It may help to note that term “basis” is used intentionally here, as these quantities must form a basis for the vector space of dimensions used within q_1,\ldots,q_n. In our pendulum example, we had quantities of time, length, mass and acceleration, which is length per time squared. Thus, we have three dimensions and four variables (or “vectors” in linear algebra parlance), and we can select three of these four to represent all the units.

There’s some useful rules for selecting them, but they do not uniquely determine the selection:

  • If one of your variables is a dependent variable — such as the period in our pendulum example — you should not select it as basis variable. Otherwise it will also occur in the ratios for the independent variables, and that would not help us in characterising the relationship properly.
  • Independent variables should be independent of each other. For example, for a propeller you should not include the diameter, the mean chord and the aspect ratio, as the latter is defined by the former two. Instead, use only two of these.
  • Each dimension that occurs in at least one variable must also occur in at least one other variable. Otherwise we cannot create a fraction of variables in which the dimension does not occur.

Specifically the third rule also may give us important insight into the problem, such as when our problem only contains a single variable referencing a specific dimension. In that case our relationship either does not actually depend on that quantity, or we are missing another quantity that influences the relationship. The missing dimension may guide us as to where to look.

How does a DC-Motor work?

Now, let’s use our new knowledge to analyse the performance of a DC motor and its relationship with some of its basic parameters. This will allow us to make better decisions on selecting motors in the future.

So, how does a DC motor work? Essentially, it turns electrical energy into mechanical energy via a magnetic field. For that, it uses two magnetic fields, leading to a magnetic force. That force results in a moment around the axis of the motor. When the motor turns around its axis, one of the fields needs to change its orientation so that the motor keeps turning. This is achieved either mechanically in brushed motors or electronically in brushless motors.

Configuration of a Brushless DC-Motor with coils in the center and permanent magnets on the shell.

The figure above shows the basic mechanical configuration of a Brushless DC (BLDC) motor in outrunner configuration with permanent magnets for one part of the field, and the other field being generated by the coils in the middle. By switching the coils on and off, the inner field can be rotated. In an inrunner configuration, the fixed coils would be outside, and the static magnets would be on the axis in the middle.

For BLDC motors special brushless controllers are required that electronically determine the angular position of the magnets and switch the coils on and off as required. There are different setups for these controllers, some using hall sensors to determine the position of the magnets, others monitoring the induction voltage on the unconnected coil and using that to determine the proper timing for switching.

Schematic of the Electrical Configuration of a DC Motor

The figure to the right shows the basic electrical configuration of a DC motor, indicating the parasitic inductance and resistance of the coils in the motor. In addition, the turning motor also acts as a generator, inducing a counter-voltage, often called counter electromagnetic force (EMF) or back-EMF.

The torque acting on the axis is approximately proportional to the current flowing through the coils, and the reverse voltage generated is proportional to the rotational velocity. The proportionality factors are usually named k_t:=\frac{Q}{I} for the torque coefficient and k_v:=\frac{U}{\omega} for the voltage coefficient.

Efficiency Considerations for the DC-Motor

On the electrical side, we have losses on the resistor. On the magnetic-mechanical side, losses may come from constantly changing the magnetic field inside the metal of the stator. In addition, there may be aerodynamic losses from the turning motor.

The efficiency factor \eta is found as the quotient \frac{P_{out}}{P_{in}}, where P_{in} is the power injected into the system, and P_{out} is the usable power provided by the system.

On the electrical side, we have

(9)   \begin{equation*} \eta_{el} = \frac{k_v \omega I}{k_v \omega I + R I^2} = \frac{1}{1 + \frac{R I}{k_v \omega}} \end{equation*}

On the mechanical side, we have

(10)   \begin{equation*} \eta_{mech} = \frac{k_t I \omega}{k_v \omega I} = \frac{k_t}{k_v} \end{equation*}

Dimensional Analysis of the DC-Motor

We are reaching the final goal of this article: Determining the influences of dimensions on the performance on a DC motor. Specifically, we want to check the influence of some relevant dimensional parameters on the voltage and torque coefficients.

From our previous insight, we assume that k_v and k_t depend on

  • the geometric size of the motor, represented by the motor diameter D and its height h,
  • the number of poles p, and
  • the number of windings n.

Let’s have a look at the dimensions of our quantities. We denote the dimension of a quantity by square brackets. The dimension of D and h is a length: \left[D\right]=\left[h\right]=L. The dimension of k_v is voltage per rotational frequency, which is voltage times time. Voltage is energy per charge, or energy times time per current. So finally, we arrive at \left[k_v\right] = \frac{M L^2}{I T^2}, with M denoting mass, I denoting current and T denoting time.

It is quite obvious that we have a dimension \frac{M}{I T^2} which occurs in only one quantity, notably in k_v. Obviously, the relationship must also include additional quantities. One reasonable quantity to consider is the strength of the of the magnets. The strength of the magnets is given by their remanence B_r, which happens to have the dimension \frac{M}{I T^2}, just what we need.

Now we have all the dimensions we need — L and \frac{M}{I T^2}. Let’s recapitulate the dimensions of all the quantities:

QuantitySymbolDimension
Back-EMF Coefficientk_v\frac{M L^2}{I T^2}
Motor SizeDL
Motor HeighthL
RemanenceB_r\frac{M}{I T^2}

The back-EMF coefficient k_v is our dependent variable, so we cannot use that for expressing the dimensions. Thus, we use D and B_r:

DimensionExpression
LD
\frac{M}{I T^2}B_r

Now, we can express the unit of k_v using these variables:

(11)   \begin{equation*} \left[ B_r D^2 \right] = \frac{M L^2}{I T^2} \end{equation*}

So, as a consequence \frac{k_v}{B_r D^2} is dimensionless. Similarly, the units of D and h are the same, so that \lambda:=\frac{h}{D} is a dimensionless quantity. We call \lambda the aspect ratio of the motor.

Using the Buckingham \Pi Theorem, the following relationship must hold:

(12)   \begin{equation*} \frac{k_v}{B_r D^2} = f_v\left(n,p,\lambda\right) \end{equation*}

We know that the dimensions of k_v and k_t must be the same — otherwise \eta = \frac{k_t}{k_v} cannot be dimensionless. Also, it is reasonable to assume that k_t also depends on the same parameters as k_v. In consequence, the following relationship must also hold:

(13)   \begin{equation*} \frac{k_t}{B_r D^2} = f_t\left(n,p,\lambda\right) \end{equation*}

And as a final finding, we can thus also express the efficiency in a similar manner:

(14)   \begin{equation*} \eta_{mech} = \frac{k_t}{k_v} = \frac{f_t\left(n,p,\lambda\right)}{f_v\left(n,p,\lambda\right)} =: f_\eta\left(n,p,\lambda\right) \end{equation*}

These relationships enable us to compare motors with different sizes and magnet strengths, as long as the number of windings and the aspect ratio are the same. To determine the relationships expressed by f_t, f_v and f_\eta, we only need to determine \frac{k_v}{B_r D^2} and \frac{k_t}{B_r D^2} for motors with different values of n, p and \lambda. We do not need to explicitly vary D and B_r.

Conclusions

In this article, we learned about dimensional analysis, a very powerful tool which allows us to reduce the complexity of the analysis of physical relationships, just by looking at the dimensions. This method is widely used in aerodynamics and is the basis, for example, for the common formulae for lift and drag of wings and airfoils, or for thrust, torque and power for propellers.

Thus, to get a rough estimate on the performance of one system, we can scale up the performance data we have on another system, as long as the similarity parameters — which are the non-dimensional numbers to our coefficient parameters — are close enough. This strongly simplifies the dimensioning of a system, and allows us to get at least a rough ballpark setup for our design. We may later on refine the design and get some actual measurements, but we save a lot of up-front effort by getting to an

[1] [doi] E. Buckingham, “On physically similar systems; illustrations of the use of dimensional equations,” Phys. rev., vol. 4, p. 345–376, 1914.
[Bibtex]
@Article{Buckingham1914,
author = {Buckingham, E.},
title = {On Physically Similar Systems; Illustrations of the Use of Dimensional Equations},
journal = {Phys. Rev.},
year = {1914},
volume = {4},
pages = {345--376},
month = oct,
doi = {10.1103/PhysRev.4.345},
issue = {4},
numpages = {0},
publisher = {American Physical Society},
url = {https://babel.hathitrust.org/cgi/pt?id=uc1.31210014450082&view=1up&seq=905},
}