Lecture 12. Time Varying Electromagnetic Fields

Transcription

1 Lecture. Time Varying Electromagnetic Fields For static electric and magnetic fields: D = ρ () E = 0...( ) D= εe B = 0...( 3) H = J H = B µ...( 4 ) For a conducting medium J =σ E From Faraday s observations, postulate was modified to B E =...( a ) t From Lenz s Law Maxwell s formulation B E ds = E dl = ds t ( ) S C S Lecture - Time varying fields - F. Rahman

2 The closed path C encompasses the area S E is no longer a gradient of a scalar such as V For a stationary magnetic circuit d dλ E dl = v = B ds = C S where λ is the flux linkage through S bounded by C. B Three possibilities for non zero ds S t.. B varies with time and the flux linkage of a stationary circuit changes with time; e.g., transformers. B is stationary and a moving circuit links the field, e.g., motors, generators, loudspeakers 3. B varies with time and a moving circuit links the flux, e.g., motors, generators, rotating transformers. Lecture - Time varying fields - F. Rahman

3 . Time varying field and a stationary coil A sinusoidal time varying field B (or φ) is set up in the core by the primary winding on the left side of the core. Assuming that B or φ are sinusoidal functions of time, φ = φm cos π ft v dλ = = Ndφ V rms N V V core. = φm π fn sinπ ft rms π fnφ 4.44fNφ m = = m Volts = 4.44fNB S where S c is the cross-section area of the or rms m c Note that the induced voltage v is 90 out of phase with the flux linkage λ. Voltage φ,v,t BH Curve i,t M agnetising current The magnetizing current can be very large if the core undergoes saturation. Lecture - Time varying fields -3 F. Rahman

4 An application: The Transformer I I N N V V Load If leakage flux is neglected, the same flux (mutual) φ links both coils. Thus dλ Ndφ From E dl = v C = = and dλ Ndφ E dl = v C = = Thus, v V N = = v V N From H dl = N I L NI For an infinitely permeable core (µ = ), H c required to establish the field in the core is negligible. Hence Thus I I L H dl = NI NI = 0 = N N Due to non ideal core, an exciting or magnetizing current I m, in addition to I, must flow. Lecture - Time varying fields -4 F. Rahman

5 An AC generator Stator Coil S r N φ = 0 Rotor Coil The rotor electromagnet has steady flux (non varying with time) around the two poles. Its distribution is co-sinusoidal with φ. The poles rotate at an angular speed ω rad/sec, where ω = πf rad/sec, and f is the frequency of rotation in revolutions/sec (Hz). Let B = a B cosφ r m The flux linkage with the N turn stator coil is λ = lrnbm cos ωt t where φ = ω + φ and φ o is assumed to be zero. 0 o dφ v = N = Nlrω Bm sinωt Note that at t = 0, the rotor magnet is horizontal, the flux linkage λ of the stator coil is maximum but its dλ/ is zero and hence the induced voltage of the stator coil is zero. When the rotor is vertical, the stator coil flux linkage is zero, dλ/ is maximum and the induced voltage in the stator coil is maximum. Lecture - Time varying fields -5 F. Rahman

6 . A moving conductor in a static magnetic field A moving conductor in a static B field experiences dφ/ and an electric field is induced. The free electrons in the conductor experience a force F = q( u B) and drift. The separation of charges produce an electric field. E = F / q V/m 3 Sliding contact v B l u 4 u ( ) V = u B dl = Blu Volts V is the induced motional voltage Note: Right-Hand Rule applies. No induced voltage in conductors 3 and 4. Lecture - Time varying fields -6 F. Rahman

7 3. A moving circuit in a time varying B field F = q( E + u B) N F E' = = E+ u B V/m q ( ) C C C E' dl = E dl + u B dl B = ds + u B dl S t C or v ( ) Transformer voltage d v= B ds = S dλ Motional voltage Lecture - Time varying fields -7 F. Rahman

8 z x w 3 α a n B y 4 h (a) Transformer voltage Let B= aybo sinω ot and the coil is initially stationary at an angle α with the y-axis. λ = Φ = S B ds= aybo sin ωot anhw since N =. = Bhwsinω tcosα o dλ v = = o ( Bhw ) ( t) ω cos ω cos α V o o o For α = 90, when the coil lies flat, the induced voltage of the coil, v = 0. Lecture - Time varying fields -8 F. Rahman

9 (b) The coil rotates at an angular velocity ω m where The motional voltage v= u B dl C ( ) w an m aybosin ot axdx ω ω = ( ) ( ) 3 w an m aybosin ot axdx 4 ω ω + = hwω mbosinωot sinα t α = ω t + φ 0 m o ( ) ( ) Note: when the coil is vertical, ie., α = 0, the induced voltage of the coil, v = 0. Total induced voltage For the special case when ω m = ω o and φ o = 0, the total induced voltage is given by ( ) ( ) ( ) ( ) v = hwω B cos ω t cos ω t t m o o o + hwω B sin ω t sin ω t m o o o ( ω ) = hwω B cos t V m o o For other angular speeds, what is the total induced voltage? Lecture - Time varying fields -9 F. Rahman