Voltage Induced in a Rotating Loop

Transcription

1 Voltage Induced in a Rotating Loop Assumptions: Air gap flux density is radial. The flux density is uniform under magnet poles and vanishes midpoint between poles (Neutral plane). As the rotor moves at velocity vin a magnetic field B, the voltage induced in each segment is given by equation 1. Segment ab ind ( v B) e l vbl positive into page under pole face e ba 0 beyond the pole edge Figure 7-1 A simple rotating loop between curved pole faces. (a) erspective view; (b) field lines; (c) loop view; (d) front view Segment bc. e cb 0 Since v x B is into (or out of) page and thus perpendicular to l. 3. Segment cd. The total induced voltage on the loop is given by e 2vB under the pole faces e + e + e + e l 0 beyond th epole edges ind ba ad dc cb vbl positive out of page under pole face e dc 0 beyond the pole edges 4. Segment da e ad 0 3 Figure 7-3 The output voltage of the loop 4 1

2 The tangential velocity v of the loop can be expressed as v r ω m Getting DC Voltage Out of Rotating Loop: Commutator Hence, 2 0 Where, ( l) p φ π r B A B 5 6 The Induced Torque in the Rotating Loop Suppose a battery is connected to the loop The force on a current-carrying conductor placed in a magnetic field is given by 1. Segment ab. F i l ( x B) F Bl i tangent to direction of motion ab The torque on the rotor caused by the force is o rf sinθ r(ilb)sin rilb ab 90 CCW (Theta is angle between r and F) 2. Segment bc. Figure 7-6 (a) A DC motor with commutator. (b) Derivation of an equation for induced torque. 7 F bc 0 since l is parallel to B 8 2

3 3. Segment cd Fcd i ( l x B) ilb tangent to direction of motion The torque on the rotor caused by the force is 4. Segment da. F da 0 since is parallel to B l 2 rilb under the pole face ind ab bc cd da 0 beyond the pole edge Since ( l) p φ π r B A B o rf sinθ ri l B sin90 ri l B CCW cd 2 φ i under the pole face ind π 0 beyond the pole edges 9 10 Given: r 0.5 m R 0.3 Ω V B 120 v l 1.0 m B 0.25 T What happens when the switch is closed? (1) Current flows in loop , 0 3

4 What happens when the switch is closed? (2) The current produces a torque Combining the forces on each segment What happens when the switch is closed? (3) The rotor begins to turn, an induced voltage develops ( ) Combining the voltages on each segment Nm What happens when the switch is closed? (4) Both the current iand torque Twill fall as increases since and steady state will be reached when, 0, 0 The steady state velocity will be / What happens when a 10 Nm load torque is applied? The load torque will cause speed to fall. Then as 2 falls, the current increases since 2. and steady state will be reached when The steady state velocity will be 40 A /

5 How much power is supplied to the shaft? Tω (10)(432) 4320 w How much power is supplied by the battery? V B i (120)(40) 4800 w How much power is lost in the resistor? i 2 R (1600)(0.3) 480 w Suppose a torque of 7.5 Nm is applied in the direction of rotation. What is the new steady-state speed? The steady state velocity will be 30 A out of machine! Since.. the speed increase will cause > / Induced Voltage Equations in DC Machine Z Z E e vb A l a a v rω Z E φω Where: 2π a Kis a machine s constant Z is total number of conductors is number of pole ais the number of current paths 19 m Zrω Bl m E A a Flux per pole BA B Therefore, φ Kφω A m m 2π rl Induced Torque Equations in DC Machine cond ri cond Bl IA i cond a rbli A Z ind a φ 2π rl BA B Therefore, Z φi KφI 2π a ind A A 20 5

6 LAB 5 RIOR REARATION: Complete the following at a time determined by the laboratory instructor. 1.Show that the mechanical power output in watts of a motor can be found from the equation mech n( rpm) T ( Nm) 2 π ( rad / revolution) n T Nm n T ( watts) n( rpm) T ( Nm) ( ) ( Watts) (sec/ min) 9.55 sec 9.55 LAB 5 RIOR REARATION: 2. A DC motor turns at a speed of 1460 rpm and produces an output torque of 3.0 Nm. The DC voltage applied to the motor is 100V and a current of 5.1A flows through the motor. a. What is the efficiency of the motor? η out in mech e 1460( rpm) 3.0Nm / % 100V 5.1A 510 b. How much are the power losses in the motor? (loss) in out W Introduction to Rotating Machines Introduction to Rotating Machines 6