MICROWAVE COMMUNICATIONS AND RADAR Generic Architecture: Signal Amplificatin Guide Antenna Prcessing Micrwave r ptical Signal Prcessing Detectin Guide Antenna tuning, resnance waveguides transitins cupling matching micrwave integrated circuits Cmmunicatins, bi-static radar separately lcated systems Radar, lidar, data recrding c-lcated systems Passive sensing uses receiver side nly Systems fail at the weakest link, therefre understand all parts L16-1
MICROWAVE CIRCUITS Printed Circuits Exhibit R,L,C Behavir: y Equivalent TEM line circuit: x i() i( ) L L L C C C - - v(t,) v(t, ) e { jωt Let v(,t) = R V()e : } d v(t,) - i(t,) TEM E ˆ = H ˆ = 0 Difference Equatins v( ) v() =L di() dt i( ) i() =C dv() dt Limit as 0: dv d =Ldidt di d =Cdv dt dv() d =ω j L I() Wave Equatin di() d =ω j C V() d V() d ω LCV() = 0 L16-
TEM SINUSOIDAL STEADY STATE EQUATIONS Wave Equatin: Vltage Slutin: Test slutin: Passes test iff: Current I(): Since: Therefre d V()/d ω LCV() = 0 V() = V e -jk V - e jk [(-jk) V e -jk (jk) V - e jk ] ω LC[V e -jk V - e jk ] = 0 k = ω LC V()/ = -jωl I() I() = (1/jωL)jk(V e -jk V - e jk ) = Y (V e -jk V - e jk ) [Characteristic admittance Y = k/ωl = ω(lc) 0.5 /ωl = (C/L) 0.5 = 1/ ] Transmissin Line Equatins: V() = V e -jk V - e jk I() = Y (V e -jk V - e jk ) L16-3
Impedance: COMPLEX LINE IMPEDANCE () I() - V() () = V()/I() = R() jx() Resistance Reactance () Equivalent circuit - V - V - ( () = V() I() = V e jk V e jk ) 1Γ() V e jk V e jk = 1Γ() Cmplex Reflectin Cefficient Γ(): Γ() = V e jk V e jk = Γ L e jk where Γ L = Γ ( = 0) = V V Examples: Γ = 0 () = Γ = 1 = Γ = -1 = 0 L16-4
GENERAL EXPRESSIONS FOR () Γ Cmplex Reflectin Cefficient Γ(): Since () = [1 Γ() [1 Γ () = () Therefre: Γ() = [() [ () as a Functin f L,, k, and : Substituting: Γ L () = [ L ] [ L ] ] ] n ] () ] = [ n () 1 ] [ n () 1] V e jk V e jk Int: () = V() I() = Yields: = V e jk V e jk (e jk Γ L e jk ( ) e jk Γ L e jk ) () = ( L )e jk ( L )e jk ( L )e jk ( L )e jk csk j sink L = jl sink csk Therefre: j tank () = L j L tank L16-5
EXAMPLES OF () TRANSFORMATIONS Transfrmatin Equatin: j tank () = L j L tank Example Open Circuit, L = : = -^ ( ^) = j ctk^ = j k ^ fr k^ << 1 ( ) 0.5 0.5 =jlc ω (LC) ^ = 1jωC^ = 0 C (capacitr) = 0 when = -λ/4, -3λ/4, Im{} = when = 0, -λ/, In general: -j < (-^) < j -λ 3λ λ λ 4 (ANY capacitance r inductance at a SINGLE frequency) 0 L16-6
MORE EXAMPLES OF () TRANSFORMATIONS Example Inductive Lad, L = jωlω fr = -λ/4: λ Recall: Since: Therefre: Nte: L j tank () =? j L tank λ/4 0 k = k^ = π λ λ 4 = π, and tan( k ^ ) = ( ) ( ) () = L = (L C ( ) jω L ) = 1 ( jωcl L) = 1 j C ) 0) () = 1 j ωl if ^ = λ, λ,...(tan (π λ λ = Example Transfrmatin f Surce Impedances: L ω s j tank^ s Th = js tank^ V s V A - - ^ 0 V A = V S A /( S A ) where A = j ct k^ = V (e jk ^ e -jk ^ ) = V cs k^ Therefre: V Th = V = V S A /[( S A )cs k^ ] V Th A Th L16-7
ALTERNATE APPROACH TO FINDING () I() V(),c Cmplex Reflectin Cefficient Γ(): () 0 L V e jk V e jk V e jk V e jk 1) () = V() I() = = 1Γ( ) 1Γ( ) = () ) Γ () = V e V e = Γ L e = Γ () where Γ L = Γ ( = 0) = V V 3) jk jk jk [ ] [ ] Γ = L L L Γ-Plane Slutin Methd: L Γ L Γ() () (3) () (1) Recall: n = / n = 0 Γ = n = j increasing ^ n = 1 Im{Γ} Γ = 1 n = -j Γ = 1 n = Re{Γ} tward generatr L16-8
Gamma Plane: Re{ n } = 0 GAMMA PLANE SMITH CHART R e { n } = 1 Γ = j ^ I() V() -,c () 0 L Γ = -1 Γ = 1 ^ Γ = 0 Im{ n } = 0 n = n = (1 Γ)/ (1 - Γ) Tward generatr (larger ^) Im{ n } = -j (- directin; λ/ full rtatin) Γ = -j Smith Chart: Thus: e -jk^ ges clckwise as ^ j Γ() = Γ L e jk = Γ L e jk^ ωt e jωt 0 1 L Ln Γ L Γ() n () () ωt = 0 ω Recall: e j t = cs ωt jsin ωt L16-9