1. www.HowFliesTheAlbatross.com Principles of High-efficiency Electric Flight J. Philip Barnes* 15 Jan 2016 * Technical Fellow, Pelican Aero Group 1 DRAFT

2. www.HowFliesTheAlbatross.com 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 motor- generator 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 power- conditioning losses to about 3% across the board. Finally, we show that a multi- bladed, high-pitch propeller (or “windprop” with regeneration) exhibits superior peak efficiency and much-reduced RPM for a given thrust and diameter. 2

3. www.HowFliesTheAlbatross.com Presentation Contents Equivalent circuit & system efficiency Fundamental, and previously-unpublished Component efficiency: test data Non-dimensional torque & current Solve for battery EMF & motor constant Losses with Pulse-width-mod.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 3

4. www.HowFliesTheAlbatross.com i  k e i -  bb RsRs  mg = k e  Motoring i  k e i +  bb RsRs  mg = k e  Generating Phil Barnes Sept 2015 Equivalent circuit with fixed-torque loss model 1.Definitions: Torque,  ; rotation speed,  Fixed torque loss, EMF,  ; Current, i EMF ratio,  ≡  mg /  b = k e  /  b EMF constant, k e =  mg /  = (  )/i System resistance, R s Non-dim. torque loss,  ≡ R s /(k e  b ) ~ 0.005 full-scale, ~ 0.02 model scale 2. Circuit model equations: Non-dim current, iR s /  b = 1- System motoring eff.,  s  =  /(  b i) Combine circuit model EQs: 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 ( ) ≈ 0.5% of stall torque, k  b /R s Torque loss & system resistance degrade efficiency Next chart: Const.-torque-loss model matches data 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 ( ) ≈ 0.5% of stall torque, k  b /R s Torque loss & system resistance degrade efficiency Next chart: Const.-torque-loss model matches data Generating:  > 1,  s  ≡  b i /(  ): 4 motoring:  < 1 - R s /(k e  b ):

5. www.HowFliesTheAlbatross.com  ≈ 0.0065 GENERATING LMCLTD.net  b =48V / 3,600 RPM k e = 0.16 N-m/A R s = 0.041 Ohm MOTORING VisForVoltage.org 1-HP Scott motor  b =24V / 15,000 RPM k e = 0.070 N-m/A R s = 0.054 Ohm  ≈ 0.010 Phil Barnes Sept 2015 Fixed-torque-loss motor-generator model matches data Ideal motoring  Ideal generating  const. torque-loss model motoring const. torque-loss model, generating  ≡ EMF ratio = k e  /  b = speed ratio,  / (  b / k e ) 5

6. www.HowFliesTheAlbatross.com Non-dimensional speed, torque, & current Non-dimensional rotation speed: =  / (  b /k e ) Non-dimensional current: i R s /  b = 1- Non-dim. torque:  R s / (k e  b ) = 1 -  -  Torque & current change sign, generator mode  ≈ 0.0065 GENERATING LMCLTD.net  b =48V / 3,600 RPM k e = 0.16 N-m/A R s = 0.041 Ohm MOTORING VisForVoltage.org 1-HP Scott motor  b =24V / 15,000 RPM k e = 0.070 N-m/A R s = 0.054 Ohm  ≈ 0.010 Current group, i R s /  b Torque group,  R s /( k e  b ) Motor eff.,  /(  b i) Gen eff.,  b i/(  ) Lines/curves: model symbols: test data Lines/curves: model symbols: test data Torque group,  R s /( k e  b ) or Current group, i R s /  b  ≡ EMF ratio = k e  /  b = speed ratio,  / (  b / k e ) 6

7. www.HowFliesTheAlbatross.com Max efficiency: Solving for battery EMF & motor constant Objective: solve for key parameters yielding max motoring efficiency i.e. speed ratio,  ≡ k e  /  b ≈ 0.9 Given: prop. torque (  ) & rotation speed (  ) {separate tech. paper!} Assume: Unity duty cycle (  ) if power conditioning implements PWM 1) Current: i R s =  b (1- ) 2) Power:  =  s  b i Combine, equate batt. EMF,  b : Combine, equate current, i: Req’d motor EMF constant: Voltage constant: “K v ” RPM/Volt = 60 / (2  k e ) Voltage constant: “K v ” RPM/Volt = 60 / (2  k e ) 7 Sys. resistance, R s = R b +R m R b = f(..,  b ) ; R m = f(.., k e ) Thus, iterative sol’n req’d. See next two slides Sys. resistance, R s = R b +R m R b = f(..,  b ) ; R m = f(.., k e ) Thus, iterative sol’n req’d. See next two slides

8. www.HowFliesTheAlbatross.com 8 Motor resistance varies with design and scale “Order of magnitude” correlation Design features are proprietary Indoor Full scale Giant scale

9. www.HowFliesTheAlbatross.com 9 ‘ 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 ‘ 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 Algorithm, solution for battery EMF & motor constant See next chart →

10. www.HowFliesTheAlbatross.com 10 Example, solution for battery EMF & motor constant Make first guess (EMFb) and monitor iteration Mini UAV results: 0.100 kWe class 5S2P LiPo battery Mini UAV results: 0.100 kWe class 5S2P LiPo battery

11. www.HowFliesTheAlbatross.com 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 Current group, i R m /  b = 1  1010 0 1  ≡ k e  /  b No-load current group, i o R m /  b Torque group,  R m / ( k e  b ) = 1   “Equivalent” no-load current, i o =  b / R m “Equivalent” no-load current, i o =  b / R m * AIAA 2010-483, Figs. 20-21 11

12. www.HowFliesTheAlbatross.com 12 Pulse-width Modulation (PWM) duty cycle(  ) & efficiency(  ) Assume: Max efficiency at  =1 (climb) ;  <1 (cruise) Assume: Fixed torque loss persists during “PWM off” Neglect: Transistor switching loss relative to torque loss Define: Fixed torque loss,  PWM cycle period,  Define: Specific torque loss,  ≡  R s /(k e  b ) order ~ 0.01 Recall: Speed ratio, ≡ k e  /  b ; current, i R s /  b = 1- Assume: Max efficiency at  =1 (climb) ;  <1 (cruise) Assume: Fixed torque loss persists during “PWM off” Neglect: Transistor switching loss relative to torque loss Define: Fixed torque loss,  PWM cycle period,  Define: Specific torque loss,  ≡  R s /(k e  b ) order ~ 0.01 Recall: Speed ratio, ≡ k e  /  b ; current, i R s /  b = 1-        on off PWM is applied to commutation Intent: preserve full-power efficiency at reduced power Efficiency = output / input = (input – losses) / input = 1 – (energy loss / energy input) = 1 –  (1-  )  / (  b i   ) = 1 –  (1-  ) / (  b i ) = 1 – (  k e  b /R s )  (1-  ) / [  b (1- )  b / R s ) ] = 1 –  k e  (1-  ) / [  (1- )  b ) ] Efficiency = output / input = (input – losses) / input = 1 – (energy loss / energy input) = 1 –  (1-  )  / (  b i   ) = 1 –  (1-  ) / (  b i ) = 1 – (  k e  b /R s )  (1-  ) / [  b (1- )  b / R s ) ] = 1 –  k e  (1-  ) / [  (1- )  b ) ] PWM efficiency,  ≈ 1 –  (  ) (1-  ) / (1- ) See next chart →

13. www.HowFliesTheAlbatross.com 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 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 PWM efficiency: theory, test data, & empirical curve fit Cruise Climb 13

14. www.HowFliesTheAlbatross.com 14 DC boost architecture – increased efficiency & added regen. "Evaluation of 2004 Toyota Prius," Oakridge National Lab, U.S. Dept. of Energy 233 Vdc in 5 10 15 20 kW Regen M-G Motor PWM iGBT C L BB 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 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

15. www.HowFliesTheAlbatross.com 15 Battery current with DC boost converter i b / G b  GbGb RhRh  m =k e  bb RhRh ibib a Motoring G m i b  GmGm RhRh  m =k e  bb RhRh ibib a Generating i b = [  b G b 2 - G b k e  ] / [R h (1+G b 2 )] motoring i b = [k e  G m -  b ] / [R h (1+G m 2 )] regeneration G ≡ DCBC voltage gain 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 (R h )” resides up & downstream of DCBC System resistance R s = R b + R m = 2R h ; Approx. R h ≈ R b ≈ R m Solve for Voltage at node “a” to get battery current by mode Efficiency has trends shown earlier, but Vs. e ≡ G m k e  /(G b  b ) 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 (R h )” resides up & downstream of DCBC System resistance R s = R b + R m = 2R h ; Approx. R h ≈ R b ≈ R m Solve for Voltage at node “a” to get battery current by mode Efficiency has trends shown earlier, but Vs. e ≡ G m k e  /(G b  b )

16. www.HowFliesTheAlbatross.com 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 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 * SAE 1999-01-1581 Lifting-line method * Blade-element method (BEM) over predicts the thrust, torque, and wake-induced velocities 16

17. www.HowFliesTheAlbatross.com Windprop aero: comparison of two blades Vs. eight 8-blades Low-RPM  tip = 30 o 8-blades Low-RPM  tip = 30 o 2-blades High-RPM  tip = 13.6 o 2-blades High-RPM  tip = 13.6 o 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 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 Same diameter & climb thrust Symmetrical blade sections Incompressible flow assumed Same diameter & climb thrust Symmetrical blade sections Incompressible flow assumed Climb Cruise Regen 17

18. www.HowFliesTheAlbatross.com 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 18 “Electric flight is coming soon to an airport near you”

19. www.HowFliesTheAlbatross.com 19 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. About the Author