Real part and Imaginary Part of the Propagation constant γ in Electromagnetics Ang Man Shun 2012-9-26 Reference David Griffiths Introduction to Electrodynamics David M. Pozar Microwave Engineering 1 Review of some related mathematics 1.1 Square root of z = a + jb The square root of a complex number will appear in propagation constant, here is the general form : r r p r + a r − a γ = a + jb = + j Sgn (b) 2 2 p where r = a2 + b2 (r ) ( r ) r + a r − a Thus , the real part of the complex number γ is , and the imaginary part is Sgn (b) 2 2 Proof. The First Proof, using Rectangular Cooredinate ( p a = x2 − y2 (∗) a + jb = x + jy () a + jb = x2 − y2 + j2xy () b = 2xy (∗∗) Consider a2 + b2 ( ) a2 + b2 = x4 + y4 − 2x2y2 + 4x2y2 = x4 + y4 + 2x2y2 = x2 + y2 2 p ) x2 + y2 = ± a2 + b2 = ±r (∗ ∗ ∗) Since x ; y 2 R p x2 + y2 = + a2 + b2 = +r p (− a2 + b2 is rejected ) Consider (∗) + (∗ ∗ ∗) s p p a + a2 + b2 2x2 = a + a2 + b2 ) x = ± 2 Consider (∗ ∗ ∗) − (∗) s p p −a + a2 + b2 2y2 = −a + a2 + b2 ) y = ± 2 From equation (∗∗) , ( x ; y same sign if b > 0 x ; y different sign if b < 0 1 Thus 8 "r p r p # > 2 2 − 2 2 > a + a + b a + a + b > ± + j if b > 0 <> 2 2 γ = x + jy = > "r p r p # > > a + a2 + b2 −a + a2 + b2 :> ± − j if b < 0 2 2 p Using r = a2 + b2 and sgn(b), "r r # r + a r − a γ = ± + j Sgn (b) 2 2 ± sign is related to the direction / determined by direction so finally r r r + a r − a γ = + j Sgn (b) 2 2 The second proof, using polar coordinate [ ] b p z = a + jb = r exp j tan−1 r = a2 + b2 a [ ] [ ] [ ] p p j b p 1 b p 1 b z = r exp tan−1 = r cos tan−1 + j r sin tan−1 2 a 2 a 2 a ( ) b The sign of b is important , since tan−1 give out a angle, and sin (±θ) = ± sin θ : 8 [ a ] [ ] > p 1 p 1 <> r cos tan−1 b + j r sin tan−1 b b > 0 p 2 a 2 a z = [ ] [ ] > p 1 p 1 :> r cos tan−1 b − j r sin tan−1 b b < 0 2 a 2 a Using Sgn notation on b [ ] [ ] p p 1 b p 1 b z = r cos tan−1 + jSgn(b) r sin tan−1 2 a 2 a Turn sin into cos [ ] s [ ] p 1 b p 1 b = r cos tan−1 + jSgn(b) r 1 − cos2 tan−1 2 a 2 a r cos 2A + 1 Recall, the Double Angle Formula : cos 2A = 2 cos2 A − 1 () cos A = 2 2 v v u ( ) u ( ) u b u b ucos tan−1 + 1 u cos tan−1 + 1 p t a p t a = r + jSgn(b) r 1 − 2 2 v v u ( ) u ( ) u b u b ucos tan−1 + 1 ucos tan−1 − 1 p t a p t a = r + jSgn(b) r 2 2 b a a For tan θ = , cos θ = p = 2 2 a v a + b r v u u u a u a − r r p t + 1 p t 1 a + r a − r = r r + jSgn(b) r r = + jSgn(b) 2 2 2 2 ) r r p r + a r − a γ = a + jb = + j Sgn (b) 2 2 1.2 Curl of Curl r × r × A = r (r · A) − r2A Proof. (This is a proof by brute force, just expand everything from definition) 0 1 x^ y^ z^ *( ) ( ) ( )+ B @ @ @ C @A @A @A @A @A @A r × A = det B C = Z − Y ; X − Z ; Y − X @ @x @y @z A @y @z @z @x @x @y AX AY AZ 0 1 x^ y^ z^ B @ @ @ C r × r × A = det B C @ @x @y @z A r × r × r × ( A)X ( A)Y ( A)Z * ( ) ( ) ( ) + @ (r × A) @ (r × A) @ (r × A) @ (r × A) @ (r × A) @ (r × A) = Z − Y ; X − Z ; Y − X @y @z @z @x @x @y * ( ) ( ) ( ) + @2A @2A @2A @2A @2A @2A @2A @2A @2A @2A @2A @2A = Y − X + Z − X ; Z − Y − Y + X ; X − Z − Z + Y @y@x @y2 @x@z @z2 @z@y @z2 @x2 @x@y @x@z @x2 @y2 @y@z @A @A @A r · A = X + Y + Z @x @y @z * ( ) ( ) ( )+ @ @A @A @A @ @A @A @A @ @A @A @A r (r · A) = X + Y + Z ; X + Y + Z ; X + Y + Z @x @x @y @z @y @x @y @z @z @x @y @z 3 *( ) ( ) ( )+ @2A @2A @2A @2A @2A @2A @2A @2A @2A r (r · A) = X + Y + Z ; X + Y + Z ; X + Y + Z @x2 @x@y @x@z @y@x @y2 @x@z @z@x @z@y @z2 * + ( ) ( ) ( ) 2 2 2 2 r A = r AX ; r AY ; r AZ *( ) ( ) ( )+ @2A @2A @2A @2A @2A @2A @2A @2A @2A = X + X + X ; Y + Y + Y ; Z + Z + Z @x2 @y2 @z2 @x2 @y2 @z2 @x2 @y2 @z2 ) r (r · A) − r2A = *( ) ( ) ( )+ @2A @2A @2A @2A @2A @2A @2A @2A @2A @2A @2A @2A Y + Z − X − X ; X + Z − Y − Y ; X + Y − Z − Z @x@y @x@z @y2 @z2 @y@x @x@z @x2 @z2 @z@x @z@y @x2 @y2 They are equal, thus r × r × A = r (r · A) − r2A 2 The propagation constant in EM wave 2.1 The Wave Equation The Phasor form Maxwell’s Eqautions in source free region 8 > @H < r × E = −µ Faraday’s Law @t > @E : r × H = σE + ϵ Ampère’s circuital law @t The vector identity : curl of curl r × r × A = r (r · A) − r2A Take the curl of both equation 8 ( ) > @E <> r (r · E) − r2E = −µ σE + ϵ ( ) @t( ) > @H @ @H :> r (r · H) − r2H = σ −µ + ϵ −µ @t @t @t With the help of Gauss’s Law (No source) r · E = r · H = 0 The 2 Maxwell’s Equation is now then 8 > @E @2E < r2E = µσ + µϵ @t @t2 Helmholtz Equation > @H @2H : r2H = µσ + µϵ @t @t2 4 Use phasor ( r2E = +j!µσE − !2µϵE = j!µ (σ + j!ϵ) E r2H = +j!µσH − !2µϵH = j!µ (σ + j!ϵ) H p Let γ = j!µ (σ + j!ϵ) ( r2E − γ2E = 0 P hasor Helmholtz Equations r2H − γ2H = 0 2.2 The propagation constant p γ = j!µ (σ + j!ϵ) = α + jβ Recall, r r p r + a r − a γ = a + jb = + jsgn (b) 2 2 Now p p γ = j!µ (σ + j!ϵ) = −!2µϵ + jσ!µ 8 > 2 > a = −! µϵ > > b = σ!µ 2 R+ ) sgn (b) = +1 8 r > r ( ) > p p 2 > r + a > 2 σ > α = Re (γ) = <> r = a2 + b2 = !µ σ2 + !2ϵ2 = ! µϵ 1 + <> " !ϵ # 2 r ( ) r ( ) ) > σ 2 σ 2 > r > 2 − 2 2 − > > r + a = ! µϵ 1 + ! µϵ = ! µϵ 1 + 1 :> r − a > !ϵ " !ϵ # β = Im (γ) = sgn (b) > r ( ) r ( ) 2 > 2 2 > 2 σ 2 2 σ :> r − a = ! µϵ 1 + + ! µϵ = ! µϵ 1 + + 1 !ϵ !ϵ i.e. v " # u r ( ) uµϵ σ 2 α = !t 1 + − 1 2 !ϵ p γ = j!µ (σ + j!ϵ) = α + jβ v " # u r ( ) uµϵ σ 2 β = !t 1 + + 1 2 !ϵ 5 3 The propagation constant in Transmission Line Model 3.1 The Telegrapher Equation & Wave Equation from KCL, KVL The Transmission Line model KVL : @i (z; t) v (z + ∆z; t) − v(z; t) @i (z; t) v(z; t)−R∆z·i (z; t)−L∆z· −v (z + ∆z; t) = 0 ) = −R·i (z; t)−L @t ∆z @t @v (z + ∆z; t) KCL : i(z; t)−G∆z·v (z + ∆z; t)−C∆z· −i (z + ∆z; t) = 0 @t i (z + ∆z; t) − i (z; t) @v (z + ∆z; t) ) = −Gv (z; t)−C ∆z @t 8 > v (z + ∆z; t) − v(z; t) @v(z; t) @i (z; t) > lim = = −Ri (z; t) − L < ∆z!0 ∆z @z @t Shrink segment ∆z ! 0 > :> i (z + ∆z; t) − i (z; t) @v (z; t) lim = −Gv (z; t) − C ∆z!0 ∆z @t 8 > dV (z) − <> = (R + j!L) I(z) Using phasor dz T elegrapher Equation > :> dI(z) = − (G + j!C) V (z) dz d (T elegrapher Equation) : dz 8 8 8 > > 2 > 2 > 2 > d V (z) > d V (z) dI(z) >d V (z) > − γ2V (z) = 0 > = − (R + j!L) > = (R + j!L)(G + j!C) V (z) > dz2 < dz2 dz < dz2 < ! ! > > > d2I(z) >d2I(z) dV (z) > d2I(z) > − γ2I(z) = 0 > = − (G + j!C) > = (G + j!C)(R + j!L) I(z) > 2 : 2 : 2 :> pdz dz dz dz γ= (R + j!L)(G + j!C) 3.2 The propagation constant γ p γ = (R + j!L)(G + j!C) = α + jβ Recall again, r r p r + a r − a γ = a + jb = + jsgn (b) 2 2 6 Now p p γ = (R + j!L)(G + j!C) = (RG − !2LC) + j (!LG + !RC) a = RG − !2LC b = !LG + !RC 2 R+ ) sgn (b) = +1 p q p ) r = a2 + b2 = (RG − !2LC)2 + (!LG + !RC)2 = R2G2 + !4L2C2 + !2L2G2 + !2R2C2 p p = G2 (R2 + !2L2) + C2!2 (!2L2 + R2) = (G2 + C2!2)(R2 + !2L2) 8 8 r > p > r + a > 2 > α = Re (γ) = <> r + a = (G2 + C2!2)(R2 + !2L2) + RG − ! LC <> 2 ) > p > r > r − a = (G2 + C2!2)(R2 + !2L2) − RG + !2LC > r − a : : β = Im (γ) = sgn (b) 2 ) s p RG − !2LC + (G2 + C2!2)(R2 + !2L2) α = p 2 γ = (R + j!L)(G + j!C) = α + jβ s p −RG + !2LC + (G2 + C2!2)(R2 + !2L2) β = 2 − − END 7.