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Actuators & the Coil Gun Outline Review of Electrostatic Actuators -Energy method for Electrostatics Rail Gun -Energy method for Magnetostatics Coil Gun ∂λ fx = i ∂x 1 TRUE or FALSE? 1. Since W = ½ CV2, the force always acts to decrease capacitance to go to lower energy. 2. If the gap is decreased, the magnetic flux will increase. 3. The energy stored in the electric field is ½ εoE2 and has units of Joules. 2 Attractive Force Between Parallel Plates + + + + + ++ + + ++++ + -- --- - -- - - -- - - -- - - CAPACITOR PLATES ARE PULLED TOWARDS EACH OTHER (FORCE ACTS TO INCREASE C) 3 Linear Electrostatic Actuator + + + + + + + + + + + + + + + + + + ++ + + + + - - - - - - - - - - - - - - - - - - - - - - - - - - CONSTANT FORCE ALONG DIRECTION OF MOTION (FORCE ACTS TO INCREASE C) 4 Gap Closing Electrostatic Actuators o Q2 w f (y) = 2 C2 d Moderate force Weak force 5 MEMS Actuator Mirror Hinges Yoke Landing Tip 6 Rail Gun Force … Using Energy Conservation Perfectly conducting plates and slider K f x Depth w L(x) = μo Gx/w … one-turn solenoid i = Kw 1 ∂L 1 f = i2 = μo K 2 Gw 2 ∂x 2 7 G Rail Gun Force -Energy Conservation Again Perfectly conducting plates and slider H, B K v x W = 1 μo H 2 dV = 2 f Depth w 1 2 B /μo dV ; B = μo H 2 1 = μo K 2 xGw (Recall that H = K ) 2 1 … (Keep λ constant) = λ2 /(μo xG) 2 1 −∂W f= = μo K 2 Gw ∂x 2 8 G Coil Gun How can we use a solenoid to propel a bullet if the force is always into the solenoid? 9 Consider this energy model: R Back voltage v is due to motion dx dx dλ = vi = i Pm = fm u = fm dt dt dx dλ Then, force is fm = i dx simply: L + i - v= dx dλ dt dx Some caution here: this is the force due to the interaction with the coil -- It doesnt mean there may not be other forces! To find back voltage: run the system open circuited This will give the right back voltage; If you know how fast the magnet is going, you get flux change + v - 10 + - v= dx dλ dt dx Permanent Ws = i(λ)dλ = = fm dλ =i dx 11 λf 0 fx dx 0 λf ∂λ i dx ∂x Key Takeaways Energy method for calculating Forces calculated at constant flux linkage Coil Gun fx = i ∂λ ∂x Wmechanical = fm dx = M ghthrow Force acts to increase Capacitance o Q2 w f (y) = 2 C2 d Moderate force Weak force 12 MIT OpenCourseWare http://ocw.mit.edu 6.007 Electromagnetic Energy: From Motors to Lasers Spring 2011 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.