In his 1926 landmark text, “Aerofoil and Airscrew Theory,” the great British
aerodynamicist Hermann Glauert suggested we “consider the case of a
windmill on an aeroplane.” Although Glauert offered no specific application
thereof, he knew the airborne wind turbine would one day find important
applications.
In 1998, American engineer Paul MacCready introduced “with caution”
regenerative electric-powered flight, where an aircraft would incorporate
energy storage, a propeller, and a wind turbine, or dual-role machine thereof,
to propel the aircraft and regenerate stored energy in updrafts.
In this multi-disciplinary study, we shall evaluate component and system
performance with the intent of showing that regenerative electric flight is both
feasible and useful. Although to some our charts may often appear technical,
while to others they may often appear rudimentary, in either case it's ok to fall
asleep during the presentation. But remember at least this:
We can fly our electric-powered aircraft for an hour or so and largely deplete
the battery ~ good enough. But if our aircraft incorporates the regeneration
feature, and if we take advantage of that, our flight can be longer, farther, and
safer. And, depending on geography and weather conditions near the end of
the flight, we can regenerate a few minutes before descent, and then further
regenerate during descent, potentially landing on a full charge.
1
2
The Great Frigate Bird is our role model for a "regen" aircraft. Prevalent in tropical areas
such as Baja California and the Galapagos Islands, the Frigate Bird has many capabilities
which our regen shall attempt to emulate. These include self-contained takeoff, emergency
thrust on demand, and flight sustained often or overall by vertical motion of the
atmosphere. We must also follow the example of the Frigate Bird by incorporating an
"energy rate sensor" so that we can recognize and capitalize on lift while minimizing
exposure to sink. Perhaps the Frigate Bird senses the slight differences in air temperature
which mark rising or sinking air. Perhaps it can sense changes in barometric pressure. But
whatever its energy-rate-sensing method, the Frigate Bird soars day and night on thermals*
at altitudes up to 2.8 km, largely on shoulder-locked wings, traveling great distances in a
given sortie which may last several days. Thus, like the Wandering Albatross, the Frigate
Bird soars for long periods over water as far as the eye can see. But of the two, only the
Frigate Bird can never land on the water, due to its permeable plumage. Thus the Frigate
Bird feeds mostly by following waterborne predators and plucking scraps from the surface.
* H. Weimerskirch, et.al., Frigate Birds Ride High on Thermals, Nature, Jan 2003
M. Allen, Autonomous Soaring for Improved Endurance of a Small Uninhabited Air Vehicle , AiAA 2005-1025
3
The powertrain of a regenerative electric aircraft begins with an energy-storage unit,
connected with electrical cables to a speed control which bi-directionally conditions the
motor-generator power. A gearbox accommodates the ratio of speeds, with the motorgenerator spinning perhaps ten times as fast as the windprop. The system always rotates in
the same direction, but when the power mode changes from propeller to turbine, the thrust,
torque, power, and current change sign.
For the system operating in cruise or in high-efficiency regeneration, if we assume 84%
efficiency for the powertrain (excluding the windprop) and 85% “isolated” windprop
efficiency, we obtain 71% “system efficiency.” System efficiency is lower during climb or
max-capacity regeneration.
We show above an optional solar panel package to support solar-augmented regenerativeelectric flight. This option also may prevent loss of charge on the ground. However, solar
power is not included in the performance analysis herein.
4
The representative thermal shown above, including contours of local updraft velocity
(u,m/s), has a diameter of 200-m at its base and 1-km diameter at its top where the
elevation is perhaps 4-km. A thermal is aptly named, since its air temperature is
typically 1oC higher than that of the surrounding air.
The peak-updraft core of 5-m/s resides at an elevation of 1000-m. In the core, a
regen aircraft with a still-air sink rate of 1 m/s, including the effects of regeneration,
would be carried aloft at 4 m/s. At the top of the thermal, updraft velocity falls to
zero.
We will study the performance of the regen operating in the thermal. However,
“regen time” is also well spent in ridge lift, wave lift, or final descent. But in the
thermal, the optimal trajectory yields the maximum total specific energy (zt),
representing the total kinetic, potential, and stored energy per unit weight, whereby
(zt) has the units of height in meters.
5
6
As illustrated in this figure for one blade, each blade sheds a helical wake.
As with a wing, the wake induces a “downwash” normal to the wake leading
edge (taken near 25% chord, as in the sketch above). Also as with a wing,
we can calculate the wake-induced velocities and aerodynamic loading with
a vector integration using the horseshoe vortices arranged along each blade.
This method, documented in our SAE technical paper “Math Modeling of
Propeller Geometry and Aerodynamics,” was used to compute the fixedgeometry windprop performance described next.
7
The "windprop" blade section shown above has its chord line at the angle (b) from the plane of
rotation. The blade-relative wind (w) is the vector combination of the airspeed (v) and
rotational velocity (w r). For the diagram representing pinwheeling, the blade section has zero
angle of attack(a) since the relative wind vector (w) is aligned with the chord. With the
specified uniform pitch, this condition applies from root to tip.
If we now increase the rotational speed while holding constant airspeed, the blade will develop
lift, thrust, and torque as a propeller. Conversely, reducing rotational speed, the blade will
develop negative values thereof, thus acting as a turbine. Alternatively, we can imagine
holding fixed rotational speed as flight velocity varies.
We are thus led to define a new term, the “speed ratio” (s), which applies to both propeller and
turbine operation, and which also highlights the pinwheeling regime separating the two powerexchange modes. We define (s) as the ratio of flight velocity to the “pinwheeling” flight velocity
where, for the stated pitch and rotational speed, windprop thrust in propeller mode falls to
zero. Any subsequent increase of airspeed (s>1) yields turbine operation. A speed ratio of
zero (s=0) represents ground static propeller-mode operation, where thrust and torque
coefficients must include the effects of stalled blades. Although the speed ratio (s) enjoys
some similarity to the more familiar “advance ratio” (J), only (s) describes at once the
relationship of the three conditions represented by propeller, pinwheel, and turbine operation.
Note that the relative wind vector (w) is shorter for the turbine mode. Local forces vary with
(w2), while shaft power varies roughly with the cube of rotational speed (w). Thus, turbine
operation is significantly “power limited” in relation to propeller operation.
8
Here we plot windprop efficiency versus the “speed ratio” (s) for two fixed-geometry, uniform-pitch
windprop designs with the same diameter, climb thrust, and symmetrical sections. The high-RPM
option has two blades with 14-deg blade tip angle, and the low-RPM design has eight blades with
30-deg blade tip angle. In either case, propeller efficiency has the traditional definition with shaft
power in the denominator, whereas turbine efficiency follows Glauert’s definition for an airborne
turbine, with shaft power in the numerator. Since for turbine operation both torque and force
change sign, turbine efficiency remains positive. Note that airborne turbine efficiency is not subject
to the “Betz Limit” (59%) of a ground-based wind turbine having a different definition of efficiency.
As noted earlier, the speed ratio (s) is defined as the ratio of flight velocity to “pinwheel flight
velocity,” where thrust and torque fall to zero with the windprop operating as a propeller at a stated
rotational speed. Windprop efficiency is zero in the pinwheel regime (s1). At speed ratios above
unity, the windprop operates as a turbine. For both propeller and turbine operation, the curves
terminate at the first appearance, anywhere along the blade, of blade section maximum lift
coefficient (cl_max).
Finally, we plot force coefficient (F), versus speed ratio (s). The force (f) is non-dimensionalized as
a group with windprop disk area and flight dynamic pressure (q). This characterization, with the
formula in the blue box, allows us to relate the installed thrust-to-drag ratio (T/D), aircraft drag
coefficient (cD), wing area (s), windprop radius (R), number of windprops (Nwp), and still-air climb
rate (dz/dt). Regardless of operational mode, installed thrust (T) includes the normalized change in
drag (DD/D) due to windprop system addition.
9
10
Let's now characterize the performance of electrical machines with fixed magnetic-field
strength (B). We show a "classical" brushed machine which, if fitted with several armature
loops at different clock angles, will largely exhibit the performance of the one loop shown
at the moment of peak torque development. The moving charge has two velocity
components, one for machine rotation and the other for current parallel to the machine
axis. These velocities crossing the magnetic field give rise to forces on the charge, one
along the path and the other normal to the path, yielding torque. Along the current path,
work on the charge increases its potential energy, giving rise to "electromotive force, or
EMF" designated (e) with units of Voltage (representing energy per charge, N-m per
Coulomb). For the classical machine shown, (e) is proportional to (B), as well as to the
number (N) of armature-loop turns, armature length (L), diameter (D), and rotational speed
(w). Furthermore, we find that the torque (t) is proportional to current (i), with the
proportionality constant given by (k = e/w = t/i). Both torque and current change sign with
generator operation, but (e) retains the same sign.
For motor operation, the EMF (e) is less than the terminal voltage (E). For generator
operation (e>E). If we imagine the terminal connected to a battery of EMF (eb), we can
define an "EMF ratio," equivalent to a "speed ratio," which we shall designate (k), defined
by k  e/eb = kw/eb. Unity speed ratio (k=1) then marks the boundary between motor and
generator operation.
11
Above we show the electrical power system consisting of a battery with EMF (eb) and
internal resistance (Rb) , electronic speed control with both pulse-width-modulation (PWM)
and inverter functions, and motor generator with internal resistance (Rmg) and torque loss
increment (Dt<0). When PWM is progressively "throttled down," the controller continues
to sustain switching losses, but motor-generator power vanishes, whereby controller
efficiency approaches zero, approximately and empirically according to the one-fourth
power of its duty cycle (d) representing the fraction of "on" time.
With or without PWM, the inverter feature is necessary if a brushless motor-gen is used.
Such inversion would convert DC power to 3-phase AC power or vice versa. A brushless
machine can rotate much faster than a brushed machine, but must be electronically
commutated by the controller.
With a "first-principle" model of the system, we can readily grasp the fundamentals of
system efficiency for each mode. Ignoring both the torque loss and electronic controller
loss, we obtain the system efficiencies shown above for motoring and regeneration modes.
First, system motoring efficiency is found linearly proportional to rotational speed (w),
reaching unity when the motor-gen EMF matches the battery EMF (i.e., emg = kw = eb).
Second, system regen efficiency, taken downstream of battery resistance loss, is found
inversely proportional to rotational speed. Thus, as generator speed increases without
limit, regen efficiency falls toward zero as (emg) rises such that resistance voltage drop
eventually far overshadows the battery EMF. The next slide will compare component test
data to our "first-principle" model.
12
The chart above applies and illustrates the "first principle" model, first by showing the
upper limits of system motoring and regen efficiencies (dashed blue curves), together with
component efficiency test data (solid red curves), and second by characterizing the torque
and current as dimensionless groups (see solid straight lines).
At a speed ratio below unity, we have motoring operation. Beyond unity speed ratio we
have regeneration. Near unity speed ratio, the first-principle model fails because there, for
a real machine, the torque loss remains as motoring torque or generated current vanish.
Thus for a real machine, the efficiency follows the trends of the test data shown. For the
most efficient machines, the efficiency peaks will penetrate more deeply into the "corners"
of the theoretical limits. But with PWM, the efficiency peaks will move away from the
corners, particularly at low throttle. At any rate, for both motoring and regeneration, the
test data approaches asymptotically the trends predicted by the first-principle model.
Two DC machines, with some similarities but also with many differences, were used for the
above plot (data for a given machine, both motoring and generating, is scarce). However, a
wide range of DC machines (with permanent magnets or otherwise fixed magnetic field
strength) can be expected to tightly follow the trends seen above.
13
As a "low-tech," but perhaps useful alternative to PWM, here we offer a regenerative DC
propulsion circuit with a unique "battery shuffler" which ensures that each battery of a
series spends equal time at each "totem pole" position (without changing the physical
position of the battery in the aircraft). The envisioned rotary switch for the battery shuffler
would be periodically rotated manually or automatically, back and forth to share the
workload on each battery in the circuit.
The motor-generator would then connect to one of several voltage nodes, depending on
operating regime. For example, takeoff and climb would use voltage node (A), whereas max
regen might use node (E). Node (E) might also be used for ground windmilling where
windprop rotational speed is quite low. In flight, nodes in the middle (B, C, D) would
accommodate efficient motoring or regeneration. To minimize line lengths and losses, the
motor-gen would connect closely to the preferred node of the battery shuffler. If necessary
to limit current, a lower voltage node would be selected for motoring, or a higher voltage
node selected for regeneration.
This arrangement has the advantage that regenerative electrical power enters only the
active batteries, thus reducing the battery resistance loss during regeneration. An
"inverter-only" speed control would be used if the brushless configuration were chosen for
the motor-gen.
14
15
Our rationale for the design of “RegenoSoar” begins with our intent to minimize in-flight aerodynamic interference
between the windprops and other aerodynamic surfaces, while also providing self-contained and robust ground
handling by the pilot alone. Thus, the counter-rotating windprops, which allow steering on the ground, are kept
aerodynamically clear of the wing and empennage via twin pod installations.
The windprops are arranged in a pusher configuration, whereby the sudden rotational flow imparted by the
blades (the upstream flow has no rotation) cannot impinge on the leading edges of downstream lifting surfaces
which otherwise would suffer interference and induced drag penalties. If necessary, pod-boom trailing-edge
blowing may mitigate any adverse affects of the pod-boom wake on windprop operation.
Windprop noise is dramatically reduced via low rotational speed afforded by multiple blades operating at high
pitch. The windprop has the smallest diameter meeting requirements for climb thrust and cruise/regen efficiency.
The speed control and motor-generator units, housed and air-cooled in the pods, are relatively close to the
fuselage-enclosed energy storage unit, mitigating line losses and effects on c.g. position.
The system enjoys the simplicity of fixed geometry for the windprops and their installation. Retraction or folding
mechanisms are not required, and as illustrated later herein, the windprops simply “pinwheel,” with minimal drag
penalty, when neither the propeller nor turbine mode is used. A study of a “constant-speed” windprop (actuated
blades) yielded 40% greater max-capacity regen power, but did not offer gains in efficiency for either operational
mode. Uniform fixed pitch was therefore retained for our study herein.
Finally, the wing design incorporates downward-pointing winglets with integrated tip wheels, the latter required
regardless of wingtip configuration. The winglets, which develop aerodynamic thrust in flight, are somewhat
elevated above the ground via wingtip dihedral. Such clearance is enhanced as the wing flexes upward under
steady lift load. Above a threshold ground-roll speed during takeoff and landing, the empennage and tail wheels
will lift off from the ground.
16
The 3D geometry of “RegenoSoar” is fully characterized with equations. The fuselage,
wings, empennage, and windprop blades are modeled as “distorted cylinders.” Canopybody, wing-body, and windprop blade-spinner intersections are iteratively determined. We
show here a wireframe model consisting of a fuselage “prime meridian and equator,”
together with section cuts of the fuselage, wing, empennage, and windprop blades, as well
as “perimeters” for the wing, empennage, and blades.
An earlier paper by the author introduces methods of mathematically characterizing
streamlined shapes. Such characterization reduces drag, promotes sharing of consistent
geometry for inter-disciplinary analysis, and takes advantage of today’s precision
manufacturing technologies. Interested readers may consult the paper 961317 “Math
Modeling of Airfoil Geometry,” available at SAE.org. An analysis of winglet aerodynamic
thrust can be found in the author’s paper 975559, “Semi-empirical Vortex Step Method for
the Lift and Induced Drag of 2D and 3D Wings.”
17
Above we plot the drag polars of both the wing airfoil and total vehicle. Our “thrustdrag accounting” for the regen defines drag to represent the “clean” configuration
(windprop system removed), but holding total system weight. All windprop forces,
including drag increments associated with windprop system addition, are treated as
thrust, with installation penalty modeled as a non-dimensional drag increment
(DD/D). We assume cruising at max-L/D airspeed and also assume thermaling, with
or without regeneration, at min-sink airspeed.
18
To relate the normal load factor (nn) to sink rate and airspeed, we first recognize that
the lift coefficient (cL) includes the load factor as shown in the formula at the upper
right. The drag polar then provides the drag coefficient, and the ratio of drag-to-lift
(D/L) is then equal to the ratio of drag-to-lift coefficients (cD/cL).
Now we can calculate the still-air clean sink rate, [nn(D/L)v], the latter including the
load factor. For example, the aircraft in max L/D glide (1.0-g) sinks at 0.75 m/s at 85
km/h airspeed. However, the aircraft turning at 1.4-g sinks at 1.25-m/s at 100 km/r
airspeed. The left-hand tip of each curve represents operation at max lift coefficient,
and the maximum of each curve represents minimum-sink operation.
Finally, we note that the graph above shows the “clean sink rate.” When the windprop
system is added, operating in the turbine mode, the regen aircraft will fall more
quickly. We will calculate the still-air sink rate with regeneration later herein.
20
To evaluate regen performance, we must know the climb rate (or sink rate) of the
maneuvering aircraft, taken relative to the local airmass. In particular, we are interested in the
effects of g-load, or normal load factor (nn), lift-to-drag ratio (L/D), and thrust-to-drag ratio
(T/D). Our diagram and analysis together describe the effects of the forces acting on the
aircraft climbing at a flight path angle (g) and banked at the angle (f). The lift vector (L), normal
to the airspeed vector (v), has the value (nnw), where (w) designates weight. Note that flight
path angle (g), taken relative to the local airmass, will be negative if the aircraft is sinking in
relation to the surrounding airmass.
After “normalizing” the various forces in terms of dimensionless ratios, we find that the steadystate climb rate (dz/dt), whether in still air or as seen by a balloon-based observer rising with
the thermal, is given by the product of an “aerodynamic group” [nn(D/L)v] and a “propulsive
group” [(T/D)-1]. Indeed, the aerodynamic group is the sink rate in still air with the propulsion
system “aerodynamically removed.” For a sailplane or frigate bird (T/D=0), still-air climb rate is
of course negative.
For either the sailplane or “clean” regen, sink or climb performance is degraded as load factor
(nn) is increased, with (L/D) evaluated at the lift coefficient under load. Thus, turning “twice
increases” the drag penalty, and this calls for high aspect ratio (learning from the albatross
and frigate bird!) to mitigate this effect.
For the regen, climb rate depends on the “clean sink rate” for the chosen airspeed, and the
propulsive group. The latter will be positive for climb, zero for cruise (dz/dt=0), and negative
during regen. As expected, the regen sinks faster when the windprop operates as a turbine. In
the glide between thermals, the windprop pinwheels with a small drag penalty (T/D<0).
21
A key product of our study is a fundamental “Regenerative Flight Equation” (RFE)
relating the total climb rate to the updraft and total sink rate. Interested readers can
consult the SAE technical paper “Flight Without Fuel,” for its derivation. Whereas
the updraft provides the specific power into the system, "total sink" represents the
specific power lost to aerodynamic drag and windprop operation.
The RFE is generally applicable to either a soaring frigate bird (where T/D=0) or a
regen in any operating mode. The “exchange ratio” (e), determined by operating
mode and not applicable to a glider, is set to zero if the regen is pinwheeling,
whereby the system “exchanges” no shaft power, and whereby the term (T/D, about
-0.10) represents pinwheeling thrust (negative) as a fraction of aircraft drag.
Otherwise, the exchange ratio is set to turbine system efficiency or the inverse of
propeller system efficiency, whichever is applicable. Recall that thrust is negative in
the turbine mode.
22
The left-hand contour plots above represent ground-observed climb rate, versus load factor
and elevation, for the regen in the thermal operating with either max-efficiency (upper) or
max-capacity (lower plot) regeneration. The aircraft can thermal over a wide range of
normal load factors, with associated bank angles and turn radii (also as described earlier),
but a specific "load-factor-altitude trajectory" yields the fastest climb, should rapid climb
with regeneration be of interest. For both left-hand plots, the white contour represents
equilibrium regeneration, whereby the regen falls through the air at the same rate as the
air rises. Such equilibrium can be sustained either "very low" or "very high" in the thermal.
In the region between the upper and lower limits of equilibrium regeneration, the aircraft
climbs in the thermal, even with maximum regeneration. One scenario for rapid-regen
climb might be to make the most of a thermal which will soon vanish.
Perhaps a more likely objective would be maximum total energy gain in the thermal. This is
shown in the right-hand contour plots, again for maximum-efficiency regen or maximumcapacity regen. The optimum energy trajectories is indicated by the dashed curve, showing
tight turns (~ 1.45-g) near the base of the thermal and wide turns (~ 1.15-g) near the top.
Whatever trajectory is chosen for energy gain (right-hand plots) will have a corresponding
trajectory for ground-relative climb rate (left-hand plots). Working "left-to-right," we notice
that equilibrium regeneration (per the white contours) does not yield attractive rates of
change of total energy. Thus the best strategy for regeneration when thermaling is to climb
in the thermal. Nevertheless, equilibrium regeneration proves useful in wave or ridge lift.
23
Finally, we apply the Regenerative Flight Equation (and related formulas) to compute
the performance of the regen at key flight conditions. The table shows the various rates
(dz_/dt) with applicable sign conventions. Table entries at lower left show how the
propeller climb mode exercises the system capacity.
Notice that thrust/drag ratio (T/D) is 6.33 in climb, but -1.01 for max-capacity regen as the
aircraft turns at 1.3-g with the windprop spinning at only 324 RPM. For this example, the
max-capacity regen condition can be interpreted as having the drag doubled by windprop
operation.
After takeoff, the aircraft climbs in still air at (dz/dt = 4.00 m/s) as total specific energy
(kinetic, potential, & stored) decreases (dzt/dt = - 5.40 m/s). Once the regen is well into the
thermal and regenerating, say at max capacity, a balloon-based observer rising with the
updraft at 3.72 m/s sees the aircraft falling (dz/dt = - 2.06 m/s). At the same time, a
ground-based observer sees the aircraft climbing (dzo/dt = 1.66 m/s).
Although we include max-capacity regen here for study, only max-efficiency regen has
competitive flight performance. In particular, while battery energy storage rate (last row) is
highest with max-capacity regeneration, the total energy rate (dzt/dt) increases most
rapidly with max-efficiency regen. Nevertheless, max-capacity regeneration proves useful
in many scenarios, including strong wave lift, slope lift, and final descent. Indeed, if the
last encountered updraft is near the airport, landing on a full charge could become routine.
An optional solar panel, and/or ground windmilling (with a safety perimeter) on windy
days, could overcome the loss of charge between flights.
24
25
The chart above summarizes the new ideas, methods, and formulas which we developed to
support our multi-disciplinary study of regenerative electric-powered flight. We hope to
have shown that adding the regeneration feature to an electric-powered aircraft is feasible
and attractive, making our flight longer, farther, safer, and greener. Therefore, let’s do our
best to emulate the regen optimized over the last fifty million years ~ the Frigate Bird.
26
27