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16.360 Lecture 3

I.  I. r. B =. 2 r. . . . . B. H =. . I. =. 2 r. 16.360 Lecture 3. Last lecture:. Magnetic field by constant current.  =  r  0 ,.  r: relative magnetic permeability.  r =1 for most materials. 16.360 Lecture 3. Last lecture:. Traveling wave.

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16.360 Lecture 3

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  1. I I r B = 2r     B H =  I = 2r 16.360 Lecture 3 Last lecture: • Magnetic field by constant current  = r 0, r: relative magnetic permeability r =1 for most materials

  2. 16.360 Lecture 3 Last lecture: • Traveling wave y(x,t) = Acos(2t/T-2x/), y(x,t) = Acos(x,t), (x,t) = 2t/T-2x/,

  3. 16.360 Lecture 3 Last lecture: • Traveling wave y(x,t) = Acos(2t/T+2x/), Velocity = 0.6/0.6T = /T Phase velocity: Vp = dx/dt = - /T

  4. Vs(t) VC(t) i (t) i(t)dt/C, VC(t) = 16.360 Lecture 3 • Phasor VR(t) Vs(t) = V0Sin(t+0), VR(t) = i(t)R, Vs(t) = VR(t) +VC(t), V0Sin(t+0) = i(t)dt/C + i(t)R, Integral equation, Using phasor to solve integral and differential equations

  5. jt ) Z(t) = Re( Z e Z is time independent function of Z(t), i.e. phasor jt jt e e = Re(V ), j(0 - /2) , V = V0 e 16.360 Lecture 3 • Phasor Vs(t) = V0Sin(t+0) j(0 - /2) ) = Re(V0 e

  6. jt ) i(t) = Re( I e 1 jt )/C + = Re(I i(t)dt= Re( I e )dt j jt jt jt e e e jt ), Re( I e Re(V ) 16.360 Lecture 3 • Phasor 1 ), = Re(I j time domain equation: V0Sin(t+0) = i(t)dt/C + i(t)R, phasor domain equation:

  7. jt 1 = I ) i(t) = Re( I e jC V j(0 - /2) j(0 - /2) V0 e V0 e 16.360 Lecture 3 • Phasor domain V + I R , I = R + 1/(jC) = , R + 1/(jC) Back to time domain: jt e = Re ( ) R + 1/(jC)

  8. 16.360 Lecture 3 • An Example : VR(t) Vs(t) = V0Sin(t+0), VR(t) = i(t)R, Vs(t) VL(t) i (t) Ldi(t)/dt, VL(t) = Vs(t) = VR(t) +VL(t), V0Sin(t+0) = Ldi(t)/dt + i(t)R, differential equation, Using phasor to solve the differential equation.

  9. jt ) i(t) = Re( I e jt jt jt e e e jt ), Re( I e Re(V ) 16.360 Lecture 3 • Phasor jt di(t)/dt= Re(d I e )/dt j = Re(I ), time domain equation: V0Sin(t+0) = Ldi(t)/dt + i(t)R, phasor domain equation: j )L + = Re(I

  10. jt ) i(t) = Re( I e V j(0 - /2) j(0 - /2) V0 e V0 e 16.360 Lecture 3 • Phasor domain jL + I R V = I , I = R + (jL) = , R + jL) Back to time domain: jt e = Re ( ) R + (jL)

  11. 16.360 Lecture 3 • Steps of transferring integral or differential equations to linear • equations using phasor. • Express time-dependent variables as phsaor. • Rewrite integral or differential equations in phasor domain. • Solve phasor domain equations • Change phasors variable to their time domain value

  12. 16.360 Lecture 3 • Electromagnetic spectrum. Recall relation: f = v. • Some important wavelength ranges: • Fiber optical communication:  = 1.3 – 1.5m. • Free space communication: ~ 700nm – 980nm. • TV broadcasting and cellular phone: 300MHz – 3GHz. • Radar and remote sensing: 30GHz – 300GHz

  13. 16.360 Lecture 3 • Transmission lines • Transmission line parameters, equations • Wave propagations • Lossless line, standing wave and reflection coefficient • Input impedence • Special cases of lossless line • Power flow • Smith chart • Impedence matching • Transients on transmission lines

  14. 16.360 Lecture 3 • Today • Transmission line parameters, equations B A VBB’(t) Vg(t) VAA’(t) L A’ B’ VAA’(t) = Vg(t) = V0cos(t), Low frequency circuits: VBB’(t) = VAA’(t) Approximate result VBB’(t) = VAA’(t-td) = VAA’(t-L/c) = V0cos((t-L/c)),

  15. B A VBB’(t) Vg(t) VAA’(t) L A’ B’ 16.360 Lecture 3 • Transmission line parameters, equations Recall: =c, and  = 2 VBB’(t) = VAA’(t-td) = VAA’(t-L/c) = V0cos((t-L/c)) = V0cos(t- 2L/), If >>L, VBB’(t)  V0cos(t) = VAA’(t), If <= L, VBB’(t) VAA’(t), the circuit theory has to be replaced.

  16. 16.360 Lecture 3 • Next lecture • Types of transmission lines • Lumped-element model • Transmission line equations • Wave propagation

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