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1 MIT OpenCureWare /ESD.03J Eletrmagneti and ppliatin, Fall 005 Pleae ue the fllwing itatin frmat: Marku Zahn, Erih Ippen, and David Staelin, 6.03/ESD.03J Eletrmagneti and ppliatin, Fall 005. (Maahuett Intitute f Tehnlgy: MIT OpenCureWare). (aeed MM DD, YYYY). Liene: Creative Cmmn ttributin- Nnmmerial-Share like. Nte: Pleae ue the atual date yu aeed thi material in yur itatin. Fr mre infrmatin abut iting thee material r ur Term f Ue, viit:

2 6.03 Frmula Sheet attahed. Prblem Maahuett Intitute f Tehnlgy Department f Eletrial Engineering and Cmputer Siene 6.03 Eletrmagneti and ppliatin Qui, Nvember 7, () = 00 ( π 0 t) vt _ R = 50Ω it () λ = meter Z 0 = 00Ω jx λ = 0.4 meter 4 Z 0 = 50Ω ZL ( j) = 50 Ω tranmiin line ytem inrprate tw tranmiin line with harateriti impedane f Z0 = 00Ω and Z0 = 50Ω a illutrated abve. vltage ure i applied at the left end, 8 λ vt ( ) = 00( π 0 t). t thi frequeny, line ha length f = meter and line ha length f λ = 0.4 meter, where λ 4 and λ are the wavelength alng eah repetive tranmiin line. The tw tranmiin line are nneted by a erie reatane jx and the end f line i laded by impedane ZL = 50( j) Ω. The vltage ure i nneted t line thrugh a ure reitane R = 50Ω. a) What are the peed and f eletrmagneti wave n eah line? X be hen that the ure urrent i ( t ) I ( 8 0 π 0 t ) b) It i deired that phae with the vltage ure. What i X? = i in ) Fr the value f X in part (b), what i the peak amplitude I 0 f the ure urrent i( t )? Nte that the value f X itelf i nt needed t anwer thi quetin r part ( d ).

3 Prblem parallel plate waveguide i t be deigned that nly TEM mde an prpagate in the frequeny range 0 < f < GH. The dieletri between the plate ha a relative dieletri ntant f ε r = 9 and a magneti permeability f free pae μ 0. a) What i the maximum allwed paing d max between the parallel plate waveguide plate? b) If the plate paing i. m, and f = 0 GH, what TEn and TMn mde will prpagate?

4 Prblem 3 R = 00Ω Swith pen at t = 0 00 Vlt _ Z0 = 00 Ω, T = Z = 300Ω L 0 tranmiin line f length, harateriti impedane Z 0 = 00Ω, and ne-way time f flight T = i nneted at = 0 t a 00 vlt DC battery thrugh a erie ure reitane R = 00Ω and a with. The = end i laded by a 300Ω reitr. a) The with at the = 0 end ha been led fr a very lng time that the ytem i in the DC teady tate. What are the value f the pitive and negative traveling wave vltage amplitudev t and V t? ( ) ( ) Part b, n the next page, t be handed in with yur exam. Put yur name at the tp f the next page.

5 Name: b) With the ytem in the DC teady tate, the with i uddenly pened at time t = 0. i) Plt the pitive and negative traveling wave vltage amplitude, V t) and V ( t), a a funtin f at time t = T. ( V ( T /) V ( T /) ii) Plt the tranmiin line vltage (, t) (, T ) v t = v a a funtin f at time t = T Pleae tear ut thi page and hand in with yur exam. Dn t frget t put yur name at the tp f thi page.

6 6.03 Qui Frmula Sheet Nvember 7, 005 Carteian Crdinate (x,y,): Ψ = x ˆ y ˆ ˆ x y x y i = x y y x y = xˆ ˆ ˆ y y x x x y Ψ= x y Cylindrial rdinate (r,φ,): Ψ = rˆ ˆ φ ˆ r r φ ( rr ) φ i = r r r φ rˆ rφˆ ˆ φ r ( r ˆ φ ) = r r ˆ ˆ d r φ = et r φ φ r r r φ r r rφ ( r ) Ψ= r r r r φ Spherial rdinate (r,,φ): Ψ = r ˆ ˆ ˆ φ r r rin φ ( r ) ( in ) r φ i = r r rin rin φ ( in φ) r ( r ) ( r ˆ ˆ ) r φ ˆ φ r = r in φ r in φ r r r r rˆ rˆ rin φˆ = det r φ r in r rin r ( r ) ( in ) Ψ= r r r r in r in φ φ Gau Divergene Therem: V ig dv = G inˆ da Stke Therem: ( G ) inˆ da = Gid C Vetr lgebra: = xˆ x yˆ y ˆ B= xbx yby B ( ) = 0 ( ) = ( )

7 Bai Equatin fr Eletrmagneti and ppliatin Fundamental E// E// = 0 f = q( E v μ H)[ N] H// H// = J nˆ E = B t B B = 0 d E d = B da dt D D = ρ H = J D t 0 = if σ = d H d = J da D da dt Eletrmagneti Wave D =ρ D da = ρ με t E = 0 [Wave Eqn.] dv ( ) V ( ) B= 0 B da = 0 k E = 0, E = E e jk ir J = ρ t k = ω(με) 0.5 = ω/ = π/λ E = eletri field (Vm - ) k x k y k = k = ω με H = magneti field (m - ) v p = ω/k, v g = ( k/ ω) - D = eletri diplaement (Cm - ) r = i B = magneti flux denity (T) int in i = ki kt = ni nt Tela (T) = Weber m - = 0,000 gau in ( n n ) = t i ρ = harge denity (Cm -3 ) = ( ε ε ) 0.5 J = urrent denity (m - ) tan fr TM B t i > = σ = ndutivity (Siemen m - ) k = k' jk'' J = urfae urrent denity (m - ) Γ = T x jk Et EiTe α ρ = urfae harge denity (Cm - ) TTE = ( η [ i t ηt i] ) ε Fm - TTM = ( η [ t t ηi i] ) μ = 4π 0-7 Hm - Tranmiin Line = (ε μ ) m - Time Dmain e = C v(,t)/ = -L i(,t)/ t η 377 hm = (μ /ε ) 0.5 i(,t)/ = -C v(,t)/ t με t E = 0 [Wave Eqn.] v/ = LC v/ t ( ) jωt E y (,t) = E (-t) E - (t) = R e{e ()e } H x (,t) = η - [E (-t)-e - (t)] [r(ωt-k) r (t-/)] ( ) ( ) ( ε μ ) y v(,t) = V (t /) V - (t /) i(,t) = Y [V (t /) V - (t /)] E H da d dt E H dv = (LC) -0.5 = (με) -0.5 V = J dv (Pynting Therem) Z = Y - = (L/C) 0.5 V E Media and Bundarie Γ L = V - /V = (R L Z )/(R L Z ) D =ε EP Frequeny Dmain D =ρf, τ=εσ (d /d ω LC)V() = 0 ε E =ρ ρ p V() = V e -jk V - e jk f P = ρ p, J =σe I() = Y [V e -jk V - e jk ] B=μ H =μ ( ) H M k = π/λ = ω/ = ω(με) ε=ε ωp ω, ω = ( Ne mε ) (Plama) Z() = V() I() = Z Z n () ( ) p ε eff =ε( jσ ωε ) Z n () = [ Γ() ] [ Γ ()] = Rn jxn kin depth δ = (/ωμσ) 0.5 jk [m] Γ () = ( V V ) e = [ Z n() ] [ Z n() ] Z() = Z ( ZL jz tan k) ( Z jz tan k) L VSWR = V V max min ˆn