6.061 / Introduction to Electric Power Systems

Transcription

1 MIT OpenCourseWare / 6.69 Introduction to Electric Power Systems Spring 7 For information about citing these materials or our Terms of Use, visit:

2 Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.6/6.69 Introduction to Power Systems Problem Set Solutions May 6, 7 Problem :. If the motor is producing kw at radians/second, torque is T = = 4N-m. The motor constant is then: T N-m G = = =.4Ωsec I 6 A Now, back voltage would be E b = GΩI =.4 = V, and if terminal voltage is 6 V, the motor resistance must be RI = 6 = V or R =.Ω. The torque/speed relationship is: GV T = R + GΩ This is evaluated in the attached script and is shown in Figure. 6 Series Motor for Problem. Current, A x Torque, N m Speed, radians/sec Figure : Torque vs. Speed Characteristic: Series Connected Motor. For the vehicle application, (a) The equivalent wheel radius is m/sec R eq = =.8m Rad/sec (b) If the car weighs, kg and is accelerating at m/s, the accelerating force is F A = N.

3 (c) Drag force is ( ) u F D = m/s (d) Then the drive force required is: ( ) G F A + F D F A + F D = I Or I = R G eq Velocity and required force are shown in Figure. Speed is simply u = At, so then we can find voltage at the terminals by: V = RI + Figure 3 shows required motor current, which for the resistive controller is also drive current. The middle chart in this figure is voltage drop across the controller. Power required from the source is just P s = V s I. G R eq ui R eq Acceleration Transient Required force Velocity, m/s x Figure : Velocity and drive force In the next part we replace the resistive controller by an ideal buck converter (chopper). The duty cycle of the buck converter is d = and of course, once that is found source current is: V V s I s = di Current in the motor and required power are shown in Figure 4. Of course if the buck converter (chopper) is ideal, power into the motor and power from the source are the same.

4 7 Resistive Controller Source Power, W Controller V Current, A x Figure 3: Motor Current, Resistance Voltage Drop and Source Power: Resistive Controller 3. For Simulation of this situation is a fairly straightfoward exercise. Perhaps the most ambitious assumption here is the the car does not blow a circuit breaker. The scripts required to do this simulation are appended and the result is shown in Figure. Problem Since the two motors achieve base power at different speeds they have different low speed torque limits. For Motor A that low speed torque limit is, / = 4 N, for Motor B it is /6. = 8 N. Then the motor coefficients are G A I f = 4/9 4.3 and G B I f = 8/ The same force requirements as were calculated for the series connected motor hold and current is just I a = F A + F D GI f R eq Field current is constant: I f =, /6 = A. With a resistive controller source power is simply P s = V s (I a + I f ) The comparison is shown in Figure 6. The limited jerk rate acceleration example assumes a jerk rate of.m/s 3. This is about / g per second and is actually pretty gentle. During this limited period velocity is u = t After an initial transient of seconds we reach the stated acceleration rate of a = m/s (about / g) and u = m/s. The end of the acceleration is a mirror image. The resulting acceleration and velocity profiles are shown in Figure 7. All of the other details of this calculation are simple extensions of those done earlier. As this is a separately excited machine, armature current is directly proportional to required force. Acceleration and drag force are shown in Figure 8. Voltage components: back voltage E b = GΩI f, resistive drop and total terminal voltage and curents (armature current is proportional to force and source current at the input to the armature chopper) are shown in Figure 9. 3

5 6 Chopper Controller Source Power, W Current, A x Figure 4: Source Current and Source Power: Ideal Chopper Controller Finally, power is shown in Figure : mechanical power is force times velocity (dotted) and electrical power is source voltage times current (solid). 4

6 4 Uncontrolled acceleration Current, A Speed, m/s Figure : Simulation of uncontrolled acceleration Motor Comparison 4 Currents Motor A Motor B x 3 Power. Motor A Motor B Accel Figure 6: Motor Comparison

7 Jerk Limited Acceleration.8 Total Newtons Velocity Acc Figure 7: Limited Jerk Rate Acceleration x 4 Forces Acceleration Drag x Figure 8: Force addition 6

8 6 Chopper Controlled Voltage and Current Voltages 4 3 Internal Resistive Terminal Amperes 3 Armature Total Figure 9: Voltages and Currents x 3 Chopper Controlled Input and Mechanical Power. Mechanical Output Electrical Input Watts Figure : Mechanical and Electrical Power 7

9 % 6.6/6.96 Spring 7 Problem Set % Problem R =.; G =.4; V = 6; om = ::; I = V./ (R + G.* om); T = G.* I.^; % motor series resistance, ohms % motor constant, ohm-sec % terminal voltage % speed range % current drawn by series motor % and resulting torque figure() subplot() plot(om, I) title( Series Motor for Problem. ) ylabel( Current, A ) subplot() plot(om, T) ylabel( Torque, N-m ) xlabel( Speed, radians/sec ) % Part M = ; D = ; Req =.8; a = ; t = :.:; u = a.* t; Fa = M*a; Fd = D.* (u./ ).^; Ft = Fa+Fd; Ia = sqrt(ft./ (G/Req)); Psr = V.* Ia; Eb = (G/Req).* Ia.* u; Er = R.* Ia; Erc = V - Eb - Er; Vt = Eb + Er; vr = Vt./ V; Is = Ia.* vr; Psc = V.* Is; % we are gonna accelerate this thing % drag force ( kw at m/s) at m/s % equivalent wheel radius % required acceleration % time for acceleration % so this is speed % acceleration force % drag force % required current % power from the source: Resistive controller % back voltage % resistive drop in motor % drop in resistive controller % chopper output voltage % chopper duty cycle % current from source % power from source: chopper controller figure() subplot 8

10 plot(t, u) title( Acceleration Transient ) ylabel( Velocity, m/s ) subplot plot(t, Ft) ylabel( Required force ) xlabel( ) figure(3) subplot 3 plot(t, Ia) title( Resistive Controller ) ylabel( Current, A ) subplot 3 plot(t, Erc) ylabel( Controller V ) subplot 33 plot(t, Psr) ylabel( Source Power, W ) xlabel( ) figure(4) subplot plot(t, Is) title( Chopper Controller ) ylabel( Current, A ) subplot plot(t, Psc) ylabel( Source Power, W ) xlabel( ) % now to simulate the uncontrolled acceleration example ts = :.:; S = [ ] ; [tsim, S] = ode3( tc, ts, S); Isim = S(:,); Usim = S(:,); figure() subplot plot(tsim, Isim) title( Uncontrolled acceleration ) ylabel( Current, A ) subplot plot(tsim, Usim) 9

11 ylabel( Speed, m/s ) xlabel( ) % Problem GA = /(*9); GB = /(6.*9); If = /6; t = :.:; u = a.* t; Fa = M*a; Fd = D.* (u./ ).^; Ft = Fa+Fd; R_A = (V -GA*)/9; fprintf( Motor_A R = %g\n, R_A); % acceleration force % drag force IA = Ft./ GA; IB = Ft./ GB; I_a = IA + If; I_b = IB + If; P_a = V.* I_a; P_b = V.* I_b; % required armature current % total current: add back field % power required figure(6) subplot() plot(t, I_a, -, t, I_b, -- ) legend( Motor A, Motor B ) ylabel( Currents ) title( Motor Comparison ) subplot() plot(t, P_a, -, t, P_b, -- ) ylabel( Power ) xlabel( Accel ) legend( Motor A, Motor B ) % Jerk rate limit J =.; a = ; dt =.; T = ; T = ; T3 = ; t = :dt:t;

12 a = J.* t; u =.*J.* t.^; U = u(length(t)); t = T+dt:dt:T; a = a.* ones(size(t)); u = U + a.* (t-t); U = u(length(u)); t3 = T+dt:dt:T3; a3 = a - J.* (t3 - T); u3 = U + a.* (t3-t) -.*J.* (t3 - T).^; t = [t t t3]; a = [a a a3]; u = [u u u3]; figure(7) subplot plot(t, a) title( Jerk Limited Acceleration ) ylabel( Acc ) subplot plot(t, u) ylabel( Velocity ) xlabel( ) Fa = M.* a; Fd = D.* (u./ ).^; Ft = Fa + Fd; Ia = Ft./ (GA/Req); V_r = R_A.* Ia; E_a = (GA/Req).* u; Vt = E_a + V_r; vr = Vt/V; Is = vr.* Ia + If; Ps = V.* Is; Pm = u.* Ft; % required current % internal resistance drop % back voltage % chopper ratio % source current figure(9) subplot plot(t, Fa, t, Fd, -- ) title( Forces ) ylabel( Newtons ) legend( Acceleration, Drag ) subplot plot(t, Ft) ylabel( Total )

13 xlabel( ) figure() subplot plot(t, E_a, --, t, V_r, -., t, Vt) title( Chopper Controlled Voltage and Current ) ylabel( Voltages ) legend( Internal, Resistive, Terminal ) subplot plot(t, Ia, --, t, Is) legend( Armature, Total ) ylabel( Amperes ) xlabel( ) figure(8) plot(t, Pm, --, t, Ps) title( Chopper Controlled Input and Mechanical Power ) ylabel( Watts ); xlabel( ) legend( Mechanical Output, Electrical Input )