Principles of High-efficiency Electric Flight
J. Philip Barnes* 11 Jan 2016
* Technical Fellow, Pelican Aero Group
www.HowFliesTheAlbatross.com
1
Abstract
Principles of High-efficiency Electric Flight
J. Philip Barnes, Technical Fellow, Pelican Aero Group
To maximize the energy efficiency of aircraft electric propulsion with propeller(s)
and “permanent-magnet-type” electrical machines, we first introduce a
simplified and practical circuit model of the battery, speed control, and motorgenerator with emphasis on a “fixed-torque-loss” model of the motor-generator,
validated by test data. We show that for a given propeller torque and rotational
speed, maximum system efficiency requires specific values of the motor EMF
constant and battery EMF, both of which are readily solved with the proposed
method. To accommodate wide-ranging scales from model aircraft to inhabited
aircraft, we normalize and approximate the fixed torque loss as on the order of
1% of motor stall torque. We then apply the method to show, again validated
with test data, that the fixed torque loss dominates the losses associated with
the well-known pulse-width modulation (PWM) method of speed control, and
that PWM losses at cruise may exceed 30%, excluding the “chopping losses” of
energy dissipation in speed-control flyback diodes. This then leads to the
suggested application of ground electric-vehicle regenerative-braking
technology in the form of a DC boost converter which, by boosting either battery
voltage with motoring or generator voltage with regeneration, reduces powerconditioning losses to about 3% across the board. Finally, we show that a multibladed, high-pitch propeller (or “windprop” with regeneration) exhibits superior
peak efficiency and much-reduced RPM for a given thrust and diameter.
www.HowFliesTheAlbatross.com
2
Presentation Contents
• Equivalent circuit & system efficiency
• Fundamental, and previously-unpublished
•
•
•
•
•
•
Component efficiency: test data
Non-dimensional torque & current
Solve for battery EMF & motor constant
Losses with Pulse-width-mod. (PWM)
DC boost converter: efficient alternative
Advantages of the multi-blade propeller
www.HowFliesTheAlbatross.com
3
Equivalent circuit with fixed-torque loss model
i
1. Definitions:
Torque, τ ; rotation speed, ω
Fixed torque loss, λ
EMF, ε ; Current, i
EMF ratio, ν ≡ εmg/εb= keω/εb
EMF constant, ke = εmg/ω = (τ+λ)/i
System resistance, Rs
Non-dim. torque loss, ψ ≡ λRs/(keεb)
~ 0.005 full-scale, ~ 0.02 model scale
εmg = keω
ω
Motoring
εb
τ = ke i - λ
i
εb
Rs
Rs
Generating
εmg = keω
2. Circuit model equations:
Non-dim current, iRs/εb = 1-ν
System motoring eff., ηs = τω/(εbi)
ω
τ = ke i +λ
Model accommodates motor, gen, or motor-gen
Neglecting losses, motor efficiency = EMF ratio
Neglecting losses, gen. efficiency = 1 / EMF ratio
Model predicts M-G performance at any Voltage
Fixed torque loss (λ) ≈ 0.5% of stall torque, kεb/Rs
Torque loss & system resistance degrade efficiency
Next chart: Const.-torque-loss model matches data
Phil Barnes Sept 2015
Combine circuit model EQs:
motoring: ν < 1 - λRs/(keεb):

η s = ν 1 −

λ Rs  
ψ 
 = ν 1 −

ke ε b (1 −ν ) 
 (1 −ν ) 
Generating: ν > 1,
ηs ≡ εbi /(τω):

λ Rs  
ψ 
1 / η s = ν 1 +
 = ν 1 +

 (ν − 1) 
 ke ε b (ν − 1) 
www.HowFliesTheAlbatross.com
4
Fixed-torque-loss motor-generator model matches data
Starter/motor-generator system efficiency (η)
Brushed-DC or Brushless with inverter/rectifier Sys.
"4-const. EqDC" model, sys. resistance (R) & fixed torque loss (λ)
εb = battery EMF, k = EMF constant, τ = torque, ω = rotation speed
1
System efficiency,
τω/(εbi)
or
εbi/(τω)
0.9
0.8
0.7
0.6
MOTORING
VisForVoltage.org
1-HP Scott motor
εb=24V / 15,000 RPM
ke = 0.070 N-m/A
Rs = 0.054 Ohm
ψ ≈ 0.0065
generator & battery
ideal motor system
0.5
ideal generator sys.
test_data
0.4

ψ 
1 / η s = ν 1 +

 (ν − 1) 
0.3

η s = ν 1 −
0.2

ψ

(1 −ν ) 
const. torque-loss model,
generating
const. torque-loss model
motoring
0.1
Phil Barnes Sept 2015
GENERATING
LMCLTD.net
εb=48V / 3,600 RPM
ke = 0.16 N-m/A
Rs = 0.041 Ohm
ψ ≈ 0.010
motor and battery
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
νν≡≡EMF
ratio ε=mgke/ε
ω/ε
=ωspeed
ratio, ratio,
ω / (εωb/k
EMF ratio,
/εb = speed
/ (ε
b =b k
e)b /k)
www.HowFliesTheAlbatross.com
5
Non-dimensional speed, torque, & current
Torque group,
τ Rs/(keεb) or
Current group,
i Rs/εb
MOTORING
VisForVoltage.org
1-HP Scott motor
εb=24V / 15,000 RPM
ke = 0.070 N-m/A
Rs = 0.054 Ohm
ψ ≈ 0.0065
GENERATING
LMCLTD.net
εb=48V / 3,600 RPM
ke = 0.16 N-m/A
Rs = 0.041 Ohm
ψ ≈ 0.010
Lines/curves: model
symbols: test data
Non-dimensional rotation speed: ν = ω / (εb /ke)
Non-dimensional current: i Rs /εb = 1-ν
Non-dim. torque: τ Rs / (ke εb ) = 1 - ν - ψ
Torque & current change sign, generator mode
ν ≡ EMF ratio = keω/εb = speed ratio, ω / (εb/ke)
www.HowFliesTheAlbatross.com
6
Max efficiency: Solving for battery EMF & motor constant
Objective: solve for key parameters yielding max motoring efficiency
i.e. speed ratio, ν ≡ ke ω/εb ≈ 0.9
Given: prop. torque (τ) & rotation speed (ω) {separate tech. paper!}
Assume: Unity duty cycle (δ) if power conditioning implements PWM
1) Current:
2) Power:
i Rs = εb (1-ν)
τ ω = ηs εb i
Combine, equate batt. EMF, εb:
τ ω (1 −ν )
i=
Rs η s
Combine, equate current, i:
Rs τ ω
εb =
η s (1 −ν )
Req’d motor EMF constant:
ke = ν ε b / ω
www.HowFliesTheAlbatross.com
Sys. resistance, Rs = Rb+Rm
Rb = f(.., εb) ; Rm = f(.., ke)
Thus, iterative sol’n req’d.
See next two slides
Voltage constant:
“Kv” RPM/Volt
= 60 / (2 π ke)
7
Motor resistance varies with design and scale
Trend, Motor Resistance, Rm Vs. Kv
Brushless Outrunners, modelmotors.cz
Grouped by (DL)0.5 ~ mm
1000
24_35
31_36
41_43
53_58
61_63
71_71
curve fit
Rm ~ mOhm
100
Indoor
Full scale
Giant scale
Kv ~ RPM/Volt
10
10
100
1000
10000
“Order of magnitude” correlation
Design features are proprietary
www.HowFliesTheAlbatross.com
8
Algorithm, solution for battery EMF & motor constant
‘ Visual Basic pseudo code, solution for battery EMF & motor EMF constant, given
‘ prop. torque, speed, batt. cell properties, & specific torque loss est.
‘
‘ inputs:
EMFb_V
' 1st guess for required battery EMF
RPM
' rotational speed, RPM
tau_Nm
' torque, Nm
EMFc_V
' battery cell EMF, Volts
Rc_mO
' battery cell resistance, mOhm
Np
' number of parallel strings of cells
nu
' speed ratio = EMF ratio = Ke*omega/EMFb
psi
' specific torque loss, Lambda*Rs/(Ke*EMFb)
'
om_rads = RPM * 2 * 3.14159 / 60
'
For i = 1 To 5
' predict-correct iteration
Ns = EMFb_V / EMFc_V
' number of series cells
Rb_mO = Rc_mO * Ns / Np
' battery resistance
Ke_NmA = nu * EMFb_V / om_rads
' motor EMF constant, N-m/Amp = Volts/(rad/s)
Kv_RPMV = 9.55 / Ke_NmA
' motor voltage constant, RPM/Volt
log10Kv = Log(Kv_RPMV) / Log(10)
log10Rm = 0.4869 * log10Kv + 0.5154
Rm_mO = 10 ^ log10Rm
‘ approx. motor resistance, mOhm
Rs_mO = Rb_mO + Rm_mO
' system resistance, mOhm
Lam_Nm = psi * Ke_NmA * EMFb_V / (Rs_mO / 1000)
' fixed torque loss, Nm
etas = nu * (1 - psi / (1 - nu))
' system efficiency
i_A = tau_Nm * om_rads / (etas * EMFb_V)
' current, Amps
' update current and battery EMF with 0.5 damping factor:
i_A = 0.5 * (i_A + Sqr(tau_Nm * om_rads * (1 - nu) / ((Rs_mO / 1000) * etas)))
EMFb_V = 0.5 * (EMFb_V + Sqr((Rs_mO / 1000) * tau_Nm * om_rads / (etas * (1 - nu))))
Next i
See next chart →
www.HowFliesTheAlbatross.com
9
Example, solution for battery EMF & motor constant
Make first guess (EMFb) and monitor iteration
guess_V EMFb_V
20
20.0
RPM
19.6
12500
19.3
tau_Nm
19.1
0.0520
18.9
18.8
RUN
EMFc_V
3.70
Rc_mO
110
Rb_mO Ke_Vrads Kv_RPMV Rm_mO
297
0.0138
694
79
291
0.0134
710
80
286
0.0132
721
81
283
0.0131
729
81
281
0.0130
734
81
280
0.0130
737
82
Np
2
nu
0.900
psi
0.0220
Rm_mo
0
Rs_mO
377
371
367
364
363
362
o'ride
if > 0
i_A
4.8
5.0
5.0
5.1
5.1
5.1
Ns
5.4
5.3
5.2
5.2
5.1
5.1
eWatts Lam_Nm
97
0.0161
97
0.0156
97
0.0153
97
0.0151
97
0.0149
97
0.0148
etas
0.702
0.702
0.702
0.702
0.702
0.702
sWatts
68
68
68
68
68
68
etam
0.757
0.758
0.759
0.759
0.760
0.760
etab
0.928
0.926
0.925
0.924
0.924
0.924
Mini UAV results:
0.100 kWe class
5S2P LiPo battery
www.HowFliesTheAlbatross.com
10
Equivalent “no-load current” (for reference only)
•
•
•
Readers are familiar with “fixed-no-load current” method
“Fixed-torque-loss” method recommended & applied herein
Compare below to “fixed-no-load-current” method* for ref.
• In practice, “fixed” no-load current increases with voltage*
• Method herein: No-load current is proportional to voltage
* AIAA 2010-483, Figs. 20-21
ψ
1
Torque group,
τ Rm / (keεb)
0
= 1−ν−ψ
0
Current group,
i Rm / εb = 1−ν
No-load current
group, io Rm / εb
ν ≡ keω/εb
1
“Equivalent” no-load
current, io = ψ εb / Rm
www.HowFliesTheAlbatross.com
11
Pulse-width Modulation (PWM) duty cycle(δ) & efficiency(η)
on
PWM is applied
to commutation
Intent: preserve full-power
efficiency at reduced power
off | τ | | | δτ
•
•
•
•
•
•
Assume: Max efficiency at δ=1 (climb) ; δ<1 (cruise)
Assume: Fixed torque loss persists during “PWM off”
Neglect: Transistor switching loss relative to torque loss
Define: Fixed torque loss, λ ; PWM cycle period, τ
Define: Specific torque loss, ψ ≡ λ Rs /(keεb ) order ~ 0.01
Recall: Speed ratio, ν ≡ keω /εb ; current, i Rs / εb = 1-ν
Efficiency = output / input = (input – losses) / input
= 1 – (energy loss / energy input)
= 1 – λ ω (1-δ) τ / (εb i δ τ) = 1 – λ ω (1-δ) / (δ εb i )
= 1 – (ψ ke εb /Rs) ω (1-δ) / [ δ εb (1-ν) εb / Rs) ]
= 1 – ψ ke ω (1-δ) / [ δ (1-ν) εb ) ]
PWM efficiency, η ≈ 1 – ψ (ν/δ) (1-δ) / (1-ν) See next chart →
www.HowFliesTheAlbatross.com
12
PWM efficiency: theory, test data, & empirical curve fit
Efficiency of Pulse-width Modulation
Comparison of Models Vs. Test Data
1.0
0.9
Climb
PWM
Efficiency,
ηPWM
0.8
0.7
1) Theory
2) Test Data
0.6
3) Empirical fit
0.5
Cruise
0.4
1) ν = 0.9, ψ = 0.01
2) AIAA 2010-483
3) η = δ 0.41 - 0.25 δ
0.3
0.2
0.1
PWM Duty Cycle, δ
0.0
0.0
•
•
•
•
•
•
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Theory matches the test data down to 40% duty cycle
PWM: ~30% loss when power is reduced to cruise
Chopping loss was ignored for theoretical prediction
Model-scale: MOSFET, low (i&V), ~05% chopping loss
Full-scale: iGBT, high (i&V), chop. loss approaches 10%
Any scale: Fixed torque loss dominates PWM losses
www.HowFliesTheAlbatross.com
13
DC boost architecture – increased efficiency & added regen.
233 Vdc in
Regen
Motor
εB
L
PWM
iGBT
C
M-G
5
10
15
20 kW
"Evaluation of 2004 Toyota Prius,"
Oakridge National Lab, U.S. Dept. of Energy
•
•
•
•
•
•
•
DC boost architecture enables high-efficiency bi-directional power
Age-old regen problem at reduced RPM: motor-gen EMF < battery
Solution: DC boost conv. (DCBC) boosts either Voltage, up to ~ 2.5x
2x voltage → ½ current (ok ; issue is low voltage, not low current)
Enables shorter battery “totem pole” & efficient regen. capability
Low-Voltage PWM duty cycle at IGBT gate sets DCBC Voltage gain
High elec. flight efficiency, with or without interest in regeneration
www.HowFliesTheAlbatross.com
14
Battery current with DC boost converter
•
•
•
•
•
•
•
Objective: Get battery current with DC boost architecture
With the DCBC, current “gain” is inverse of Voltage gain (G)
Boost either batt. voltage to motor or M-G voltage to regen.
Say “half resistance (Rh)” resides up & downstream of DCBC
System resistance Rs = Rb + Rm = 2Rh ; Approx. Rh ≈ Rb ≈ Rm
Solve for Voltage at node “a” to get battery current by mode
Efficiency has trends shown earlier, but Vs. νe ≡ Gmkeω/(Gbεb)
G ≡ DCBC voltage gain
Rh
ib
εb
a
Gb
ib = [εb Gb2- Gb keω] / [Rh (1+Gb2)] motoring
ib = [keωGm - εb] / [Rh (1+Gm2)]
Rh
Rh
ib /Gb εm=keω
τω
Motoring
ib
εb
www.HowFliesTheAlbatross.com
a
Gm
regeneration
Rh
G m ib
εm=keω
τω
Generating
15
Sample propeller geometry & lifting-line aero analysis
•
•
•
•
•
•
8-blade, high-pitch propeller at cruise
25 horseshoe vortices per blade
Method validated with test data*
BEM (dash curves) added for ref. *
EXCEL computational platform
Sym. sections ; 86% prop. efficiency
www.HowFliesTheAlbatross.com
* SAE 1999-01-1581
Lifting-line method
* Blade-element method
(BEM) over predicts the
thrust, torque, and
wake-induced velocities
16
Windprop aero: comparison of two blades Vs. eight
Same diameter & climb thrust
Symmetrical blade sections
Incompressible flow assumed
•
•
•
1
Efficiency
0.9
0.8
0.7
Turbine
η=τω/(fv)
Propeller
η = f v / (τ ω)
0.6
0.5
0.4
0.3
0.2
0.1
8-blades
Low-RPM
βtip = 30o
0
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
0.9
0.8
Climb
0.7
8-blade rotor has 8 wakes, but at higher pitch,
moving farther downstream than for 2 blades.
Note: available test data holds common pitch,
showing an expected penalty of 8 blades Vs. 2
0.6
0.5
0.4
Cruise
0.3
0.2
Force coefficient,
F = f / (q π R2)
0.1
0
2-blades
High-RPM
βtip = 13.6o
Regen
-0.1
-0.2
-0.3
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
Windprop speed ratio, σ ≡ V / (ω R tan βtip )
www.HowFliesTheAlbatross.com
1.4
1.5
1.6
17
Summary – Principles of High-efficiency Electric Flight
• New way of modeling & understanding
permanent-magnet motor generators
• Solved for battery EMF and motor-gen
EMF constant for maximum efficiency
• Fixed torque loss persists in PWM “off,”
with ~30% cruise efficiency penalty of
PWM, not including the chopping loss
• DC boost converter: efficient alternative
and enabler of efficient regeneration
• Multi-blade, high-pitch prop./windprop
has low RPM/noise, highest efficiency
“Electric flight is coming soon to an airport near you”
www.HowFliesTheAlbatross.com
18
About the Author
Phil Barnes has a Master’s Degree in
Aero Engineering from Cal Poly Pomona
and BSME from the University of
Arizona. He is a 35-year veteran of air
vehicle, propulsion, and subsystems
performance analysis at Northrop
Grumman. Phil authored a “landmark”
study of dynamic soaring, and he is
pioneering the science of regenerative
electric flight. Author of numerous SAE,
AIAA, and other technical papers, he is
often invited to present travel-paid
lectures at various universities. The
charter of his free website (see footer)
is to apply “green aero engineering” to
prevent or delay extinction of the
wandering albatross.
www.HowFliesTheAlbatross.com
19