Propagation Constant Γ in Electromagnetics Ang Man Shun 2012-9-26 Reference David Griﬃths Introduction to Electrodynamics David M

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 + a r − a γ = a + jb = + j Sgn (b) 2 2 √ where r = a2 + b2 (√ ) ( √ ) 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 { √ 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 √ ∴ x2 + y2 = ± a2 + b2 = ±r (∗ ∗ ∗) Since x , y ∈ R √ x2 + y2 = + a2 + b2 = +r √ (− a2 + b2 is rejected ) Consider (∗) + (∗ ∗ ∗) √ √ √ a + a2 + b2 2x2 = a + a2 + b2 ⇒ x = ± 2 Consider (∗ ∗ ∗) − (∗) √ √ √ −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  [√ √ √ √ ]  2 2 − 2 2  a + a + b a + a + b  ± + j if b > 0  2 2 γ = x + jy =  [√ √ √ √ ]   a + a2 + b2 −a + a2 + b2  ± − j if b < 0 2 2 √ Using r = a2 + b2 and sgn(b), [√ √ ] r + a r − a γ = ± + j Sgn (b) 2 2 ± sign is related to the direction / determined by direction so ﬁnally √ √ r + a r − a γ = + j Sgn (b) 2 2 The second proof, using polar coordinate [ ] b √ z = a + jb = r exp j tan−1 r = a2 + b2 a [ ] [ ] [ ] √ √ j b √ 1 b √ 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 :  [ a ] [ ]  √ 1 √ 1  r cos tan−1 b + j r sin tan−1 b b > 0 √ 2 a 2 a z = [ ] [ ]  √ 1 √ 1  r cos tan−1 b − j r sin tan−1 b b < 0 2 a 2 a Using Sgn notation on b [ ] [ ] √ √ 1 b √ 1 b z = r cos tan−1 + jSgn(b) r sin tan−1 2 a 2 a Turn sin into cos [ ] √ [ ] √ 1 b √ 1 b = r cos tan−1 + jSgn(b) r 1 − cos2 tan−1 2 a 2 a √ 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 √ t a √ 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 √ t a √ t a = r + jSgn(b) r 2 2 b a a For tan θ = , cos θ = √ = 2 2 a v a + b r v u u u a u a − √ √ √ t + 1 √ t 1 a + r a − r = r r + jSgn(b) r r = + jSgn(b) 2 2 2 2 ∴ √ √ √ r + a r − a γ = a + jb = + j Sgn (b) 2 2

1.2 Curl of Curl ∇ × ∇ × A = ∇ (∇ · A) − ∇2A Proof. (This is a proof by brute force, just expand everything from deﬁnition)

  xˆ yˆ zˆ ⟨( ) ( ) ( )⟩  ∂ ∂ ∂  ∂A ∂A ∂A ∂A ∂A ∂A ∇ × A = det   = Z − Y , X − Z , Y − X  ∂x ∂y ∂z  ∂y ∂z ∂z ∂x ∂x ∂y AX AY AZ   xˆ yˆ zˆ  ∂ ∂ ∂  ∇ × ∇ × A = det    ∂x ∂y ∂z  ∇ × ∇ × ∇ × ( A)X ( A)Y ( A)Z ⟨ ( ) ( ) ( ) ⟩ ∂ (∇ × A) ∂ (∇ × A) ∂ (∇ × A) ∂ (∇ × A) ∂ (∇ × A) ∂ (∇ × 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 ∇ · A = X + Y + Z ∂x ∂y ∂z

⟨ ( ) ( ) ( )⟩ ∂ ∂A ∂A ∂A ∂ ∂A ∂A ∂A ∂ ∂A ∂A ∂A ∇ (∇ · 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 ∇ (∇ · 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 ∇ A = ∇ AX , ∇ AY , ∇ 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

∴ ∇ (∇ · A) − ∇2A =

⟨( ) ( ) ( )⟩ ∂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 ∇ × ∇ × A = ∇ (∇ · A) − ∇2A

2 The propagation constant in EM wave

2.1 The Wave Equation The Phasor form Maxwell’s Eqautions in source free region   ∂H  ∇ × E = −µ Faraday’s Law ∂t  ∂E  ∇ × H = σE + ϵ Ampère’s circuital law ∂t The vector identity : curl of curl

∇ × ∇ × A = ∇ (∇ · A) − ∇2A Take the curl of both equation  ( )  ∂E  ∇ (∇ · E) − ∇2E = −µ σE + ϵ ( ) ∂t( )  ∂H ∂ ∂H  ∇ (∇ · H) − ∇2H = σ −µ + ϵ −µ ∂t ∂t ∂t With the help of Gauss’s Law (No source)

∇ · E = ∇ · H = 0 The 2 Maxwell’s Equation is now then   ∂E ∂2E  ∇2E = µσ + µϵ ∂t ∂t2 Helmholtz Equation  ∂H ∂2H  ∇2H = µσ + µϵ ∂t ∂t2

4 Use phasor { ∇2E = +jωµσE − ω2µϵE = jωµ (σ + jωϵ) E ∇2H = +jωµσH − ω2µϵH = jωµ (σ + jωϵ) H √ Let γ = jωµ (σ + jωϵ) { ∇2E − γ2E = 0 P hasor Helmholtz Equations ∇2H − γ2H = 0

2.2 The propagation constant √ γ = jωµ (σ + jωϵ) = α + jβ Recall, √ √ √ r + a r − a γ = a + jb = + jsgn (b) 2 2 Now √ √ γ = jωµ (σ + jωϵ) = −ω2µϵ + jσωµ

  2  a = −ω µϵ   b = σωµ ∈ R+ ⇒ sgn (b) = +1  √  √ ( )  √ √ 2  r + a  2 σ  α = Re (γ) =  r = a2 + b2 = ωµ σ2 + ω2ϵ2 = ω µϵ 1 +  [ ωϵ ] 2 √ ( ) √ ( ) ⇒  σ 2 σ 2  √  2 − 2 2 −   r + a = ω µϵ 1 + ω µϵ = ω µϵ 1 + 1  r − a  ωϵ [ ωϵ ] β = Im (γ) = sgn (b)  √ ( ) √ ( ) 2  2 2  2 σ 2 2 σ  r − a = ω µϵ 1 + + ω µϵ = ω µϵ 1 + + 1 ωϵ ωϵ

i.e. v [ ] u √ ( ) uµϵ σ 2 α = ωt 1 + − 1 2 ωϵ √ γ = jωµ (σ + jωϵ) = α + jβ v [ ] u √ ( ) uµϵ σ 2 β = ωt 1 + + 1 2 ωϵ

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   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   dV (z) −  = (R + jωL) I(z) Using phasor dz T elegrapher Equation   dI(z) = − (G + jωC) V (z) dz

d (T elegrapher Equation) : dz      2  2  2  d V (z)  d V (z) dI(z) d V (z)  − γ2V (z) = 0  = − (R + jωL)  = (R + jωL)(G + jωC) V (z)  dz2  dz2 dz  dz2  → →    d2I(z) d2I(z) dV (z)  d2I(z)  − γ2I(z) = 0  = − (G + jωC)  = (G + jωC)(R + jωL) I(z)  2  2  2  √dz dz dz dz γ= (R + jωL)(G + jωC)

3.2 The propagation constant γ √ γ = (R + jωL)(G + jωC) = α + jβ Recall again, √ √ √ r + a r − a γ = a + jb = + jsgn (b) 2 2

6 Now √ √ γ = (R + jωL)(G + jωC) = (RG − ω2LC) + j (ωLG + ωRC)

a = RG − ω2LC b = ωLG + ωRC ∈ R+ ⇒ sgn (b) = +1

√ √ √ ∴ r = a2 + b2 = (RG − ω2LC)2 + (ωLG + ωRC)2 = R2G2 + ω4L2C2 + ω2L2G2 + ω2R2C2

√ √ = G2 (R2 + ω2L2) + C2ω2 (ω2L2 + R2) = (G2 + C2ω2)(R2 + ω2L2)

  √  √  r + a  2  α = Re (γ) =  r + a = (G2 + C2ω2)(R2 + ω2L2) + RG − ω LC  2 ⇒  √  √  r − a = (G2 + C2ω2)(R2 + ω2L2) − RG + ω2LC  r − a   β = Im (γ) = sgn (b) 2 ∴

√ √ RG − ω2LC + (G2 + C2ω2)(R2 + ω2L2) α = √ 2 γ = (R + jωL)(G + jωC) = α + jβ √ √ −RG + ω2LC + (G2 + C2ω2)(R2 + ω2L2) β = 2

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