Electromagnetic Theory¶

$\Omega$

$\Omega = 2\pi[1-cos(\theta)]$ for a cone whose cross-section subtends the angle $2\theta$

$\Omega = 4\pi$ for full sphere
$\mu$ and $\epsilon$

$\mu_o = 4\pi \times 10^{-7} \frac{H}{m}$

$\epsilon _o = 8.854187817 × 10⁻¹² \frac{F}{m}$

$B= \mu H$

$D = \epsilon E$
gauss law
- Integral form of Gauss law
  
  \[\displaystyle\unicode{x222F} D\cdot dS = \psi _{e} \varpropto Q\]
- Gauss Divergence theorem
  
  \[\displaystyle\unicode{x222F} D\cdot dS = \iiint \nabla \cdot D dV\]
- Point form of Gauss Law
  
  \[\rho _{v} = \nabla \cdot \vec{D}\]
ampere’s law
- Integral form of Ampere’s law
  
  \[\displaystyle\oint \vec{H}\cdot \vec{dl} = I\]
- Ampere Circulation
  
  \[\displaystyle\oint H\cdot dl = \iint (\nabla \times H) dS\]
- Point form of Ampere Law
  
  $\vec{J} = \nabla \times \vec{H}$
- modified ampere’s law
  
  \[\displaystyle\oint H\cdot dl = I_c + I_d\]
  
  $\nabla\times H = J_c+J_d$

rate of change of any scalar with del is gradient

what is condition for path independent line integral

$\nabla \times A=0$

solenoidal → $\nabla . \vec F =0$

irrotational → $\nabla \times \vec F =0$

curl of gradient

is always zero and its line integral is path independent
current density
- surface current
  
  $K=\dfrac{I}{width}$$\dfrac{ampere}{m}$
- volume current
  
  $J=\dfrac{I}{S}$ $\dfrac{ampere}{m^2}$
right hand (index-middle-thumb) sequence of unit vectors
- $a_x \bot a_y \bot a_z$
- $a_{\rho}\bot a_{\phi} \bot a_{z}$
- $a_r \bot a_\theta \bot a_\phi$
scaling factors

$\begin{array}{c:c} \color{orange} \text{parameters} & \color{orange} \text {scaling\space factors} \\ \hdashline \begin{array}{c:c:c} x & y & z \\ \hdashline \rho & \phi & z \\ \hdashline r & \theta & \phi \\ \hdashline u & v & w \end{array} & \begin{array}{c:c:c} 1 & 1 & 1 \\ \hdashline 1 & \rho & 1 \\ \hdashline 1 & r & r\cdot sin(\theta) \\ \hdashline h_1 & h_2 & h_3 \end{array} \end{array}$

$\vec{dl} = (h_1du)a_u+(h_2dv)a_v+(h_3dw)a_w$

$\boxed{\vec{dS} = (h_2h_3dvdw)a_u+(h_3h_1dwdu)a_v+(h_1h_2dudv)a_w}$

$dV = h_1h_2h_3dudvdw$
divergence and curl using scaling factors

$\nabla\cdot A \neq A\cdot\nabla$
- $\nabla$
  
  $=\dfrac{1}{h_1}\dfrac{\partial}{\partial u}a_u+\dfrac{1}{h_2}\dfrac{\partial}{\partial v}a_v+\dfrac{1}{h_3}\dfrac{\partial}{\partial w}a_w$
- $\nabla\cdot\vec{A}$ if $\vec{A}=A_ua_u+A_va_v+A_wa_w$
  
  $=\dfrac{1}{h_1h_2h_3}\Big\lbrack\dfrac{\partial (h_2h_3A_u)}{\partial u}+\dfrac{\partial(h_3h_1A_v)}{\partial v}+\dfrac{\partial(h_1h_2A_w)}{\partial w}\Big\rbrack$
- $\nabla\times\vec{A}$ if $\vec{A}=A_ua_u+A_va_v+A_wa_w$
  
  $=\dfrac{1}{h_1h_2h_3}\begin{vmatrix} h_1a_u & h_2a_v & h_3a_w \\ \frac{\partial}{\partial u} & \frac{\partial}{\partial v} & \frac{\partial}{\partial w}\\ h_1A_u & h_2A_v & h_3A_w \end{vmatrix}$
- $\nabla ^2 G$
  
  $\color{skyblue}\nabla ^2 = \nabla \cdot ( \nabla G)$
  
  $\nabla ^2 G =\dfrac{1}{h_1h_2h_3}\Big\lbrack\dfrac{\partial }{\partial u}\Big(\dfrac{h_2h_3}{h_1}\dfrac{\partial G}{\partial u}\Big)+\dfrac{\partial }{\partial v}\Big(\dfrac{h_1h_3}{h_2}\dfrac{\partial G}{\partial v}\Big)+\dfrac{\partial }{\partial w}\Big(\dfrac{h_1h_2}{h_3}\dfrac{\partial G}{\partial w}\Big)\Big\rbrack$
$\displaystyle\int_{\theta=0}^{\theta=\pi} \int_{\phi=0}^{\phi=2\pi} sin(\theta)d\theta d\phi$

$=4\pi$
$\displaystyle \int_{0}^{\frac{\pi}{2}} sin^3(\theta)d\theta$

$=\dfrac{2}{3}$
position vectors
- cartesian
  
  $\vec{r}=x a_x +y a_y +z a_z$
- cylindrical
  
  $\vec{r}=\rho a_\rho$
  
  $\neq \rho a_\rho + \phi a_\phi +z a_z$
- spherical
  
  $\vec{r}=r a_r$
  
  $\neq r a_\rho + \theta a_\theta +\phi a_\phi$
gauss law application
- point charge
  
  $D(r)_r = \dfrac{Q}{4\pi r^2} a_r$
- hollow sphere - surface
  
  $D(r) = \rho_s \frac {R^2}{r^2}$ , $r>R$
- infinite line
  
  $D=\dfrac{\rho_L}{2\pi\rho}a_\rho$
- hollow cylinder
  
  $D=\dfrac{\rho_s}{\rho}R$ , $\rho > R$
- solid cylinder
  
  $D=\frac{\rho_v\rho}{2}$ , $\rho<R$
  
  $D=\frac{\rho_vR^2}{2\rho}$ , $\rho>R$
- infinite sheet
  
  $D=\frac{\rho_s}{2}(\pm a_N)$
ampere law application
- infinite wire
  
  $H=\frac{I}{2\pi\rho}a_\phi$
  
  $a_\phi=I\times \rho$
- biot-savart law
  
  $\boxed{dH = \dfrac{Idl \times a_r}{4\pi r^2} = \dfrac{KdS \times a_r}{4\pi r^2} = \dfrac{JdV \times a_r}{4\pi r^2} }$
- hollow cylinder
  
  $H=\dfrac{K}{\rho}R$ , $\rho > R$
- solid cylinder
  
  $H=\frac{J\rho}{2}$ , $\rho<R$
  
  $H=\frac{JR^2}{2\rho}$ , $\rho>R$
- infinite sheet
  
  $H=\dfrac{K\times a_N}{2}$
un-symmetric charge

$\displaystyle\int dD = \int \dfrac{\rho_L dl}{4\pi r^2}a_r = \iint \frac{\rho_S dS}{4\pi r^2}a_r = \iiint \frac{\rho_V dV}{4\pi r^2}a_r$
un-symmetric current
- finite length wire
  
  $H = \dfrac{I}{4\pi \rho}[sin(\alpha_1)-sin(\alpha_2)]a_\phi$
  
  ($\pm\alpha_1$ and $\pm\alpha_2$ are measured from horizontal level )
dipole

$torque = \vec R \times \vec F$
- electric
  
  $V=\dfrac{Q \cdot d\cdot cos(\theta)}{4\pi \epsilon r^2}$
  
  can calculate E by $E = -\nabla V$
- magnetic
  
  $torque = \vec M \times \vec B$
  
  $\vec M = \displaystyle \iint I d\vec s$
laplace and poisson

$\nabla^2V = \dfrac{-\rho_v}{\epsilon}$ (poisson)

$\nabla ^2V = 0$ (laplace)
vector potential

$\vec A = \dfrac{W}{I dl_1}=\displaystyle { \int \vec B\times \vec {dl_2} \space} \frac{weber}{m} = \frac{\mu I dl_3}{4\pi r}$

$\vec B = \nabla \times \vec A$
boundary conditions of fields
- electric field
  
  $D_{n_2} = D_{n_1}+\rho_s$
  
  $E_{t_1}=E_{t_2}$
- magnetic field
  
  $B_{n_1}=B_{n_2}$
  
  $H_{t_2} = H_{t_1}+\vec K\times a_N$
maxwell’s equations

\[\begin{split}\displaystyle\begin{array}{c:c} \color{orange} \text{integral} & \color{orange} \text {point} \\ \hdashline \unicode{x222F} D\cdot dS = Q & \nabla \cdot \vec{D}=\rho _{v} \\ \oint E\cdot dl = 0 & \nabla \times \vec{E}=0 \\\unicode{x222F} B\cdot dS = 0 & \nabla \cdot \vec{B}=0 \\ \oint H\cdot dl = I & \nabla \times \vec{H}=0 \end{array}\end{split}\]

$j = e^{j90^{\circ}}$ (delay)

continuity equation

$\nabla\cdot J = \dfrac{\partial \rho_v}{\partial t}$
conduction and displacement current

$J_c = \sigma E$

$J_d = \dfrac{\partial D}{\partial t} =\dfrac{\partial \epsilon E}{\partial t}$
faraday law, lenz law and lorentz law

$\displaystyle\boxed{\nabla\times E = - \dfrac{\partial B}{\partial t}}$

$\boxed{\displaystyle\nabla\times H = \dfrac{\partial D}{\partial t} + J_C}$

$\displaystyle\oint E \cdot dl= - \iint\dfrac{\partial B}{\partial t}\cdot dS$

$\displaystyle\oint H \cdot dl= \iint\dfrac{\partial D}{\partial t}\cdot dS + I_C$
helmholtz

$\nabla ^2 E = \gamma^2 E$

$\nabla ^2 H = \gamma^2 H$
$\gamma$

$\boxed{{{\gamma }} = \alpha + j \beta =\sqrt {j\omega \mu \left( {\sigma + j\omega \epsilon} \right)}}$

$e^{-\gamma z} = e^{-\alpha z} e^{-j\beta z}$ (amplitude and phase variations)

skin depth, $\delta$ = $\dfrac{1}{\alpha} =\dfrac{1}{\sigma R_S }$ —→ $|E| = E_o e^{-1}$

loss tangent $=|\dfrac{J_c}{J_d}| = \dfrac{\sigma}{\omega \epsilon}$

$\alpha = \omega \sqrt{\dfrac{\mu\epsilon}{2}(\sqrt{1+(\frac{\sigma}{\omega \epsilon} )^2}-1)}$

$\beta = \omega \sqrt{\dfrac{\mu\epsilon}{2}(\sqrt{1+(\frac{\sigma}{\omega \epsilon} )^2}+1)}$
$\eta$ (intrinsic impedance)

$\boxed{\eta = R + j X = \dfrac{E\angle \theta_1}{H\angle \theta_2} = \sqrt{\dfrac{j\omega\mu}{\sigma+j\omega\epsilon}}=\sqrt{\dfrac{\mu}{\epsilon }}}$

$\eta_o = \sqrt{\dfrac{\mu_o}{\epsilon _o}}= 120\pi = 377\Omega$
$v_p$ (phase velocity) and $v_g$ (group velocity)

$v_p=\dfrac{\omega}{\beta}= \dfrac{1}{\sqrt{\mu\epsilon}}$

$\theta = \omega t = \beta z$

$v_g =\dfrac{d \omega}{d \overline \beta}$

$v_g v_p =c^2$
poynting vector

$P_{avg} = \dfrac{1}{2}|E\times H|=E_{rms} \times H_{rms}$

$\displaystyle power= \iint P_{avg}\cdot dS$

ohmic power dissipated per unit volume : $J\cdot E = \sigma E^2 \dfrac{watts}{m}$
reflections and transmission
- normal incidence
  
  $\dfrac{E^t_o}{E^i_o} = \tau$
  
  $\boxed{\Gamma_E =\dfrac{E^r_o}{E^i_o} = \dfrac{\eta_2-\eta_1}{\eta_2+\eta_1} }$
  
  $\boxed{\Gamma_H =\dfrac{H^r_o}{H^i_o} =\dfrac{\eta_1-\eta_2}{\eta_1+\eta_2} = -\Gamma_E }$
  - boundary conditions
    - $E_{tang1}=E_{tang2}$
      
      $\boxed{1+\Gamma = \tau}$
    - $H_{tang1}=H_{tang2}$
      
      $\boxed{1-\Gamma = \dfrac{\eta_1}{\eta_2}\tau}$
- inclined incidence
  
  $\theta _i=\theta _t$
  - snell’s law
    
    $\dfrac{sin(\theta_i)}{sin(\theta_t)}=\sqrt{\dfrac{\epsilon_2}{\epsilon_1}}$
  - brewster’s angle
    
    $tan(\theta_i) =\sqrt{\dfrac{\epsilon_2}{\epsilon_1}}$
  $\boxed{\Gamma _S = \dfrac{\dfrac{\eta_2}{cos(\theta_t)}-\dfrac{\eta_1}{cos(\theta_i)}}{\dfrac{\eta_2}{cos(\theta_t)}+\dfrac{\eta_1}{cos(\theta_i)}}}$
  
  $\boxed{\Gamma _P = \dfrac{\eta_2cos(\theta_t)-\eta_1{cos(\theta_i)}}{\eta_2{cos(\theta_t)}+\eta_1{cos(\theta_i)}}}$
transmission line

$\gamma=\sqrt{(R+j\omega L)(G+j\omega C)}$

$Z_o = \sqrt{\dfrac{(R+j\omega L)}{(G+j\omega C)}}$

$Z_{series} = R+j\omega L$

$Y_{shunt} = G+j\omega C$

$\boxed{\Gamma = \dfrac{Z_L - Z_o}{Z_L + Z_o}}$

$\boxed{\text{SWR} = \dfrac{1 + |\Gamma|}{1 - |\Gamma|}} = \dfrac{|V_\text{max}|}{|V_\text{min}|}= \dfrac{|I_\text{max}|}{|I_\text{min}|}$
- standing waves
  
  $2\beta x_{\text{max}} = 2n\pi +\theta$
  
  $2\beta x_{\text{min}} = (2n+1)\pi +\theta$
- lossless → $\alpha =0$
  
  $\boxed{R=G=0}$
- distortionless → $\omega \propto \beta$
  
  $V(t) \leftrightarrow V(x) \implies \omega t = \beta x$
  
  $\boxed{\dfrac{L}{R} = \dfrac{C}{G}}$
- distortions
  - delay distortion
    
    $\beta\propto \omega$
  - frequency distortion
    
    $\alpha \propto \omega$
- $Z_{in}$
  
  $Z_{in} = R_o(\dfrac{Z_Lcos(\beta x)+jR_o sin(\beta x)}{R_ocos(\beta x)+jZ_L sin(\beta x)})$
  
  $Z_{in} = \begin{cases} R_o\dfrac{Z_L +j R_o}{R_o +j Z_L} & l=\dfrac{\lambda}{8} \\ \dfrac{R_o^2}{Z_L} & l=\dfrac{\lambda}{4} \\ Z_L & l=\dfrac{\lambda}{2} \end{cases}$
- capacitance and inductance
  
  $c=\dfrac{2\pi\epsilon l}{ln(\frac{b}{a})}$ for co-axial cable
  
  $c=\dfrac{\pi\epsilon l}{ln(\frac{D}{r})}$
  
  $\dfrac{L}{l}\cdot\dfrac{C}{l} = \mu\epsilon$
- smith chart
antenna

$|E_\theta| = \dfrac{I_m}{2}\dfrac{dl}{\lambda}\dfrac{\sin \theta}{r}\eta$

$P_{avg} = \frac{1}{2}E_\theta H_\phi$
$$P_{total} = \unicode{x222F} P_{avg} \cdot ds$$

$\boxed{W_r =I_{rms}^2 R_r= I_{rms}^2 80\pi^2 \Big ( \dfrac{dl}{\lambda} \Big)^2}$ ( isotropic antenna )

uniform current : $dl<<\lambda$
- linear current : $l \rightarrow \frac{\lambda}{10} -\frac{\lambda}{20}$
  
  $I_{avg} = \frac{I_o+0}{2}$ → $\frac{I_{rms}^2}{4}$
- harmonic current : $l \rightarrow \frac{\lambda}{2} , \frac{\lambda}{4},\frac{\lambda}{8}$
  
  $I_{avg} = \dfrac{\int_0^{\pi}I_m\sin(t)dt}{\pi}=\dfrac{2I_m}{\pi}$ → $\dfrac{4I_{rms}^2}{\pi^2}$
  
  $\frac{\lambda}{2}$ → $R_r = 80\pi^2\dfrac{4}{\pi^2}\Big(\dfrac{\frac{\lambda}{2}}{\lambda}\Big)^2 =73 \Omega$
  
  $\frac{\lambda}{4}$ → $R_r =73\Big(\dfrac{\frac{\lambda}{4}}{\frac{\lambda}{2}}\Big)= 36.5 \Omega$
  
  $\frac{\lambda}{8}$ → $R_r = 36.5\Big(\dfrac{\frac{\lambda}{8}}{\frac{\lambda}{4}}\Big)=18.25 \Omega$
  
  $R_r \not\propto \frac{1}{\lambda^2}$
- fraunhofer region
  
  for practical antenna of ‘D’ dimension radiation fields are considered strong beyond
  
  $r> \dfrac{2D^2}{\lambda}$
- radiation intensity
  
  $\dfrac{dW_r}{dS} = P(r,\theta,\phi)$
  
  $\dfrac{dW_r}{d\Omega} = U(\theta,\phi)$
  
  $U(\theta,\phi)= P(r,\theta,\phi) r^2$
  
  $dS \rightarrow r^2 \sin \theta d\theta d\phi$
  
  $d\Omega \rightarrow \sin \theta d\theta d\phi$
  
  $dS = r^2 d\Omega$
- radiation pattern
  - half power beam width (HPBW)
    
    $\theta _{HPP_1} -\theta _{HPP_2}$ via $\theta _{max}$
    
    $\displaystyle \theta_{HPBW} =\int_{\theta_1}^{\theta_2}\sin \theta d\theta$
    
    $\phi_{HPBW}=\phi _{HPP_2} -\phi _{HPP_1}$
    
    beam solid angle : $\displaystyle \Omega _A = \theta_A \times \phi _A =\int_{\theta_1}^{\theta_2}\sin \theta d\theta \times \phi_A$
  - beam width first null (BWFN)
    
    $\theta _{NP_1} -\theta _{NP_2}$
  $HPBW \approx \dfrac{BWFN}{2}$ for complex pattern
- directive gain, $G_D$
  
  $U_{avg} = \dfrac{W_r}{4\pi}$
  
  $\displaystyle G_D(\theta ,\phi) = \dfrac{U(\theta ,\phi)}{U_{avg}}=\dfrac{U(\theta ,\phi)}{\Bigg(\dfrac{\int_{\theta=0}^{\pi}\int_{\phi=0}^{2\pi}U(\theta ,\phi) d\Omega}{4\pi}\Bigg)}$
  
  $G_D = \dfrac{4\pi}{\Omega_A}$
  - maximum gain
    
    hertzian dipole : maximum gain = 1.5
    
    half wave dipole : maximum gain = 1.63
- antenna array
  
  $E_{total} = E_o + E_o e^{j\psi}=2E_o \cos(\frac{\psi}{2})$
  
  $\psi =\alpha +\beta d\cos\theta$
  
  end fire : $\psi =0$, $\alpha = \pm \beta d$, $\theta_{max} =0^{\circ} , 180^{\circ}$
  
  broadside array : $\psi =0$, $\alpha =0$, $\theta_{max} =90^{\circ}$
  - N-element-uniform-isotropic sources
    
    $E_T = E_o \dfrac{\sin(\frac{N\psi}{2})}{\sin(\frac{\psi}{2})}$
    
    $\psi =0\rightarrow NE_o$
- multiplication of pattern
  
  \[\text{final pattern = unit pattern $\times$ array factor } = U(\theta,\phi) \cos (\frac{\psi}{2})\]
- friis formula
  
  capture area: $G_D= \dfrac{4\pi}{\lambda^2}A_e$
  
  $\boxed{W_r=\dfrac{W_tG_tG_r}{(\frac{4\pi d}{\lambda})^2}}$
  
  $\Big(\dfrac{4\pi d}{\lambda}\Big)^2 = L_s$ loss due to radial scattering or spreading
- polarization loss factor
  
  $PLF = |\cos \theta |^2 =|\hat{E_T}\cdot \hat{E_R}|^2$
polarization
- linear
  - single component of E
  - 2 components of E (both in phase)
- circular
  - 2 components having equal amplitude and out of phase by $90 ^{\circ}$
- elliptical
  - 2 components of E
    - unequal amplitude with any phase other than $0^{\circ} / 180 ^{\circ}$
    - equal amplitude with any phase other than $0^{\circ}/ 90^{\circ} / 180 ^{\circ}$
- sense of rotation
  
  propagation > thumb
  
  field rotation > fingers
  
  which ever hand will follow the above will be rotation
waveguide
- parallel plane
  
  $E_{tang} =0$ $E_{normal} =E_{max}$
  
  $H_{normal} =0$ $H_{tang} = H_{max}$
  
  $\boxed{\beta_x a = m\pi}$
  
  $\boxed{\beta_x^2+\overline \beta^2 = \beta ^2 = \omega^2 \mu\epsilon}$
  
  cut off → $\overline \beta =0$
  
  $\overline \beta = \beta \cos (\theta)$ $\beta_c=\beta _x= \beta \sin (\theta)$
  
  $\overline \lambda = \dfrac{\lambda}{\cos(\theta)}$
- rectangular
  
  $\omega_c = \Bigg(\sqrt{\Big(\dfrac{m\pi}{a}\Big)^2 +\Big(\dfrac{n\pi}{b}\Big)^2} \Bigg) c$
  
  $TE_{mn}$ /$TM_{mn}$ → m horizontal axis feeds and n vertical axis feeds
  
  non-existing modes in TM → evanescent modes → m = 0 / n =0
  
  dominant mode → least cut off
  
  degenerate modes → same cut off
  
  $\eta _{TE} \text{ or } \eta _{TM} = \dfrac{E_{transverse}}{H_{transverse}}$
  
  $\eta _{TE} = \dfrac{120\pi}{\sqrt{1-(\frac{f_c}{f})^2}}$
  
  $\eta _{TM} = 120\pi\sqrt{1-(\frac{f_c}{f})^2}$
- circular
  
  $E(\rho =a) _{tang} =0$
  
  $H(\rho =a)_{normal} =0$
  - TE
    
    $J'_n(\beta_\rho \rho) =0$ → $\beta_\rho \rho = X'_{np}$ $(H_{tang} = H_{max})$
    
    $\boxed{\omega_c = \dfrac{X_{np}' c}{ a}}$
  - TM
    
    $J_n(\beta_\rho \rho) =0$ → $\beta_\rho \rho = X_{np}$ $(E_{tang} = 0)$
    
    $\boxed{\omega_c = \dfrac{X_{np} c}{ a}}$
  - $X_{np}$ and $X'_{np}$
    
    $`p^{th}$ root of $n^{th}$ order`
    
    n = 0, 1, 2 …
    
    p = 1, 2, 3 …
    
    n → number of feeds in $\phi$
    
    p → number of feeds in $\rho$
    
    n=0 → $\phi=0^{\circ} \text{ or } 180^{\circ}$
    
    $TE_{n0}$ and $TM_{n0}$ → evanescent modes
    
    $X_{11} ' = 1.84$ $(TE_{11} )$ → dominant mode
    
    $X_{01} =2.40$ $(TM_{01} )$→least $f_c$TM
    
    $X'_{21} =3.05$ $(TE_{21} )$
    
    $X'_{01} =X_{11} = 3.83$ $(TM_{11} \text{ and } TE_{01})$→first degenerate modes
    
    for any p : $X'_{0p} =X_{1p}$ $(TM_{1p} \text{ and } TE_{0p})$→degenerate modes
- TE and TM
  - RWG
    - TM waves → $H_z =0$
      
      $E_z = E_o \sin \Big(\dfrac{m\pi}{a}x\Big)\sin \Big(\dfrac{n\pi}{b}y\Big) e^{-j\overline \beta z} e^{j\omega t} a_z$
      
      $E_x = C_1\dfrac{\partial E_z}{\partial x}$
      
      $E_y = C_2\dfrac{\partial E_z}{\partial y}$
      
      $H_x = C_3 E_y$
      
      $H_y = C_4 E_x$
    - TE waves → $E_z =0$
      
      $H_z = H_o \cos \Big(\dfrac{m\pi}{a}x\Big)\cos \Big(\dfrac{n\pi}{b}y\Big) e^{-j\overline \beta z} e^{j\omega t} a_z$
      
      $H_x = C_1\dfrac{\partial H_z}{\partial x}$
      
      $H_y = C_2\dfrac{\partial H_z}{\partial y}$
      
      $E_x = C_3 H_y$
      
      $E_y = C_4 H_x$
  - CWG
    - TM waves → $H_z =0$
      
      $E_z = E_o J_n(\beta_{\rho}\rho)\cos (n\phi) e^{-j\overline \beta z} e^{j\omega t} a_z$
      
      $E_\rho = C_1\dfrac{\partial E_z}{\partial \rho}$
      
      $E_\phi = C_2\dfrac{\partial E_z}{\partial \phi }$
      
      $H_\phi = C_3 E_\rho$
      
      $H_\rho = C_4 E_\phi$
    - TE waves → $E_z =0$
      
      $H_z = H_o J_n(\beta'_{\rho}\rho)\cos (n\phi) e^{-j\overline \beta z} e^{j\omega t} a_z$
      
      $H_\rho = C_1\dfrac{\partial H_z}{\partial \rho}$
      
      $H_\phi = C_2\dfrac{\partial H_z}{\partial \phi }$
      
      $E_\phi = C_3 H_\rho$
      
      $E_\rho = C_4 H_\phi$

J-Book

Electromagnetic Theory¶