DC Motor / Propeller Matching 3 Mar 05 Lab 5 Lecture Notes

Transcription

1 DC Motor / Propeller Matching 3 Mar 05 Lab 5 Lecture Notes Nomenclature prop thrust prop torque m motor torque P shaft shaft power P elec electrical power C thrust coefficient based on tip speed C P power coefficient based on tip speed λ advance ratio propeller efficiency motor efficiency ρ air density µ air viscosity Re Reynolds number flight velocity motor and propeller rotation rate v motor terminal voltage R motor resistance i motor current K v motor speed constant () o quantity at zero load Motor Parameters he behavior of an electric motor is fairly accurately described by the equivalent circuit model shown in Figure 1. he internal back-emf v m is proportional to the rotation rate via the motor speed constant K v. Applying the usual circuit equations, together with the conservation of energy, gives the following motor parameters as functions of the motor current i and motor terminal voltage v: m (i) = (i i o )/K v (1) (i, v) = (v ir)k v (2) P shaft (i, v) = m = (i i o )(v ir) (3) P elec (i, v) = v i (4) (i, v) = P shaft /P elec = (1 i o /i)(1 ir/v) (5) hese relations depend on the motor constants R, i o, K v, which can be measured via simple benchtop experiments. he equations here assume K v is in rad/s/olt, although it s usually given in RPM/olt. i m + R + υm = /Kv υ Figure 1: Equivalent circuit for a brushed DC electric motor. For the purpose of matching the motor to a load, such as a propeller, we first manipulate relation (2) into a function for the current, ( i(, v) = v ) 1 (6) K v R i 1

2 which is then substituted into all the other righthand sides to give the following functions of motor speed and voltage: m (, v) = [( v ) ] 1 1 K v R i o Kv (7) P shaft (, v) = [( v ) ] 1 K v R i o Kv (8) (, v) = [ 1 i o R v /K v ] v K v (9) hese functions are sketched in Figure 2 versus motor speed, for three applied voltages. m P shaft Figure 2: Motor output variables versus motor speed and applied voltage. Propeller Parameters A propeller is characterized by the thrust and power coefficients, which depend primarily on the advance ratio λ, the blade Reynolds number Re, and on the prop geometry. C = C (λ, Re, geometry) (10) C P = C P (λ, Re, geometry) (11) λ = R (12) Re = ρr c ave µ (13) 2

3 Once the propeller geometry is specified and the C, C P, functions generated by measurement or analysis, the dimensional thrust and torque can be computed for any other and by dimensionalizing the coefficients. λ(, ) = R (, ) = 1 2 ρ(r)2 πr 2 C (, ) = 1 2 ρ(r)2 πr 3 C P (14) = 1 2 ρ 2 πr 2 C (λ, Re) λ 2 (15) = 1 2 ρ 2 πr 3 C P(λ, Re) λ 2 (16) Figure 3 shows the resulting and variation for a propeller operating at two fixed values (presumably set by the aircraft trim), as functions of the rotational speed (presumably varied by the effective motor voltage). Strictly speaking, the correct Re which increases with should be used to evaluate the C and C P functions, although its effect is usually weak and a typical constant Re is often assumed. increasing = 5 C P C λ 4 λ 3 λ 2 λ 1 λ o λ = R = 500 o = = 800 o Figure 3: Propeller,, versus for two flight speeds =5, 8, generated using common C, C P, curves. Zero-thrust speeds o occur at the same / o R=λ o values. Motor/Prop Matching he equilibrium operating speed of the motor/prop combination occurs when the torques are equal: m (, v) = (, ) (17) 3

4 A common situation is the need to determine all the prop and motor operating parameters which result from a specified flight speed and applied motor voltage v. his is shown graphically in Figure 4, where the torque-matching condition (17) is applied to first determine the motor speed. All the other prop and motor parameters can then be determined from the propeller and motor characteristic curves. prop efficiency at specified prop thrust at specified motor torque at specified υ m prop torque at specified m = specified motor voltage Figure 4: Prop and motor parameters obtained from a specified flight speed and a specified motor voltage v. he motor speed is read from the intersection of of the prop and motor torque curves, and all the other prop and motor parameters follow. 4

5 Another situation is the need to determine the motor voltage which will give a required thrust at some specified flight speed (e.g. to sustain level flight). his procedure is shown graphically in Figure 5. Again, the torque-matching condition (17) is used, but this time to read off the required voltage, suitably interpolated from the computed m (, v) lines. prop efficiency at specified prop thrust at specified specified thrust prop torque at specified required voltage m = υ = 7.3 motor torque lines for different voltages m υ = 7.3 Figure 5: Prop and motor parameters obtained from a specified flight speed and a required thrust. he required motor voltage v is read from the intersection of the prop and motor torque curves, and all the other motor parameters follow. 5

6 An alternative to graphical interpolation is obtained if we take equation (7), which defines the m (, v) lines in Figure 5, and invert it, also replacing m with. v(, ) = (K v + i o ) R + K v (18) Equation (18) gives the motor voltage v in terms of the propeller speed and torque, both of which follow directly from the specified thrust. Well-Matched Systems A power supplier and a power sink (the motor and the propeller in this case), are said to have a good impedance match if they are both operating at close to their peak efficiency. his means that their efficiency curves must both have their peaks at roughly the same speed, and at the required thrust. Figure 6 shows good and poor match cases. A common approach to get a better impedance match in propeller systems is with gearing, which allows shifting the efficiency peak of the motor/gearbox system to a lower output-shaft, where the prop wants to operate. But the gearing has drawbacks in its own losses, plus more weight, more cost, and possibly less reliability. Hence, a careful tradeoff is called for. Well matched Poorly matched required thrust Figure 6: Well-matched and poorly matched motor and propeller pairs. 6